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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review is concerned with a beautiful elegant algebraic framework to deal with the \textsl{intersection theory of flag schemes}, based on the theory of \textsl{universal factorization and/or splitting algebras} of a set of polynomials with coefficients in some unital commutative ring. Intersection theory of flag schemes is the key ingredient to prove many formulas which are useful to compute fundamental classes of degeneracy loci of maps of vector bundles (e.g. Thom--Porteous' formula or its generalization à la Kempf-Laksov).
The exceptionally well written introduction illustrates aims and scopes of the paper and, at the same time, already provides a comprehensive description of the main results of Section 4, where the authors \textsl{``give the complete connection between the bivariant Chow group and Chow rings for flag schemes and Grassmannians on the one side, and splitting and factorization algebras on the other'')}.
Section 1 is devoted to recall a few basic notions concerning \textsl{splitting} and \textsl{factorization} algebras. The notion of partial flag schemes is fully worked out in Section 2, whose intersection theory is treated in Section 3, exploiting the notion of bivariant Chow ring. The generalized Schur determinant of a locally free sheaf with respect to a partial flag is the key formula of this section. Chow groups for flag schemes are studied in Section 4 and entirely described in terms of suitable splitting algebras, those associated, roughly speaking, to the Chern polynomial of a locally free sheaf on some regular scheme.
Corollary 4.5 turns out to be a transparent translation of the celebrated determinantal formula by Kempf and Laksov, which is stated again in Section 6 under the shape of a general Giambelli's-like formula. Another crucial notion emphasized by the authors is that of Gysin homomorphism which is related with the \textsl{divided differences operators}, studied in that famous paper by \textit{A. Lascoux} and \textit{M.-P. Schuetzenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)] where the double Schubert polynomials were introduced for the first time.
The appendix sheds additional light on many combinatorial properties of symmetric functions, like the nice comparison with MacDonald's book on symmetric functions, Example 20, p. 54--57, which are useful for the proof of the important polarity formula stated in section 1.20. The paper concludes itself with a rich reference list which includes all the previous works the authors did in relation with the subject. Indeed, the origin of this paper is the main result appeared in [\textit{D. Laksov} and \textit{A. Thorup}, Indiana Univ. Math. J. 56, No. 2, 825--845 (2007; Zbl 1121.14045)], which proves that the canonical symmetric structure of the tensor power of a polynomial ring gets rid of the formalism of Schubert calculus for Grassmann bundles.
The reviewer shares with the authors the opinion that the results proposed in this beautiful paper are definitely interesting for researchers in Algebra, Algebraic Geometry, Combinatorics and Representation Theory. splitting algebra; Schubert calculus; flag scheme; Grassmannian; Chow group; Giambelli's formula; polarity formula; Gysin manps; Schubert conditions; determinantal formulas; Schur determinants Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Polynomials over commutative rings, Galois theory and commutative ring extensions Splitting algebras 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 The aim of this paper is to prove a Chevalley restriction theorem and its double analogue for a cyclic quiver. When the quiver is of type \(\widetilde A_0\), we recover the results for \(\mathfrak{gl}_n\). The proof of our Chevalley restriction theorem is similar to the proof for \(\mathfrak{gl}_n\); however, the proof of the double analogue uses a theorem of \textit{W. Crawley-Boevey} on decomposition of quiver varieties [Compos. Math. 130, No. 2, 225-239 (2002; Zbl 1031.16013)]. The double analogue is the limiting case of an isomorphism between a Calogero-Moser space and the center of a symplectic reflection algebra proved by \textit{P. Etingof} and \textit{V. Ginzburg} [Invent. Math. 147, No. 2, 243-348 (2002; Zbl 1061.16032)]. It is also the associated graded version of a conjectural Harish-Chandra isomorphism for the cyclic quiver. Chevalley restriction theorem; cyclic quivers; decompositions of quiver varieties; Calogero-Moser spaces; symplectic reflection algebras Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Chevalley restriction theorem for the cyclic quiver. | 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 the \(K\text{-}k\)-Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. \textit{T. Lam} et al. [Compos. Math. 146, No. 4, 811--852 (2010; Zbl 1256.14056)] identified the \(K\text{-}k\)-Schur functions as Schubert representatives for \(K\)-homology of the affine Grassmannian for \(\mathrm{SL}_{k + 1}\). Our perspective reveals that the \(K\text{-}k\)-Schur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of \textit{S. Baldwin} and \textit{S. Kumar} [Represent. Theory 21, 35--60 (2017; Zbl 1390.19010)]. We further show that a slight adjustment of our formulation for \(K\)-\(k\)-Schur functions produces a second shift-invariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of \textit{T. Ikeda} et al. [Int. Math. Res. Not. 2020, No. 19, 6421--6462 (2020; Zbl 1479.14017)], we conjecture that this second basis gives the images of the Lenart-Maeno quantum Grothendieck polynomials under a \(K\)-theoretic analog of the Peterson isomorphism. symmetric functions; affine Grassmannian; \(K\text{-}k\)-Schur; Katalan function Symmetric functions and generalizations, Classical problems, Schubert calculus \(K\)-theoretic Catalan 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 Consider a matrix polynomial of the form
\[
W(z) = B^0 z^m + \cdots + B^m, \quad B^i \in \mathrm{Mat}_n(\mathbb{C}).
\]
Let \(C \to \mathbb{CP}^1\) be the associated spectral curve whose affine part is given by
\[
\{(z,w)\in \mathbb{C}^2 ~|~\det\left(w\cdot \mathbf{1}_n - W(z)\right) = 0\}.
\]
If \(W\) is generic, then \(C\) is smooth and irreducible with \(n\) pairwise distinct points \(P_1, \dots, P_n\) over \(\infty\). Note that the spectral curve \(C\) comes with a natural line bundle \(\mathcal{L}\) defined by the eigenvectors of \(W\). Finally, we denote by \(\theta\) the Riemann theta function of \(C\) associated with a fixed canonical basis of \(H_1(C,\mathbb{Z})\).
The present article studies \(N\)-differentials \(\nu_{N, \mathbb{Q}}\) on \(C\) for \(N\geq 2\) and \(N\)-tuples \(\mathbb{Q}\) of points on \(C\). The \(\nu_{N, \mathbb{Q}}\) are defined by the \(N\)-th logarithmic differential of \(\theta\) at the point in the Jacobian corresponding to \(\mathcal{L}\) (after tensoring with an appropriate divisor) and the \(N\)-tuples \(\mathbb{Q}\). The surprising main result of this articles expresses each \(\nu_{N, \mathbb{Q}}\) explicitly in terms of \(W\) and \(\mathbb{Q}\) only. If \(W\) has rational coefficients, i.e. \(B^i \in \mathrm{Mat}_n(\mathbb{Q})\), then it is further shown as a corollary that \(\nu_{N, \mathbb{Q}}\) has only rational coefficients when expanded around \(\infty\).
The key of proving these statements (for \(N \geq 3\); for \(N=2\) the methods are related but more direct) is the relationship to the \(n\)-wave (or AKNS-D) hierarchy defined by
\[
[L_{a, k}, L_{b,l}] = 0,\quad L_{a, k} = \frac{\partial}{\partial t^a_k} - U_{a, k}(\mathbf{t}; z).
\]
Here \(U_{a, k}(\mathbf{t}; z)\) are any \(n \times n\)-matrix-valued polynomials in \(z\) of degree \(k+1\) of a special form. Now given a spectral curve \(C\) as above and \(N\geq 0\), a solution to the \(n\)-wave hierarchy is constructed (Proposition 2.20) following Krichever's approach and using a vector-valued Baker-Akhiezer function. We note that the \(U_{a,k}\) of the hierarchy are in fact constructed from the spectral curve \(C\). A key result (Proposition 2.24) is that the tau-function attached to such a solution (reviewed in Appendix A) can be expressed as
\[
\tau(\mathbf{t}) = a(\mathbf{t})\cdot \theta(V(\mathbf{t}) - \mathbf{u}_0).
\]
Here \(\theta\) is the Riemann theta function of \(C\) as above and \(a(\mathbf{t})\), \(V(\mathbf{t})\) are certain scalar-/vector-valued functions.
From the previous formula it follows that the \(N\)-differentials \(\nu_{N, \mathbb{Q}}\) are equivalently defined as certain \(N\)-th logarithmic differentials of the tau-function \(\tau(\mathbf{t})\). Applying results on such differentials for more general solutions of the \(n\)-wave hierarchy (proven in Appendix A), it is possible to express them in terms of \(W(z)\) for the solutions attached to \(C\). Thereby the main theorem follows (see end of Section 2.3).
Despite being technical, this article is well written and organized. It contains several interesting results along the way. For example, results on the divisor of normalized eigenvectors of \(W\) (see Section 2) and its relation to the relative Jacobian \(J(C; P_1, \dots, P_n)\) (which can be considered as the Jacobi variety of the singular curve obtained from \(C\) by identifying the points \(P_i\) over \(\infty\)). Finally, explicit examples of the main results are provided as well as an appendix on tau-functions of solutions to the \(n\)-wave hierarchy. Riemann surfaces; Riemann theta function; tau-functions; integrable hierarchies Relationships between algebraic curves and integrable systems, Theta functions and abelian varieties, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Algebraic spectral curves over \(\mathbb{Q}\) and their tau-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 give Kazhdan-Lusztig polynomials \(P_\mu(\nu)\) for all vexillary permutations \(\mu\) (i.e. \(\mu\) does not contain any substring \(jikh\), \(i<j<h<k\)), and all \(\nu\). We use the embedding of the symmetric group into its enveloping lattice, and the characterization of a permutation by the set of bi-Grassmannian ones which are below it with respect to the Ehresmann-Bruhat order. Schubert varieties; Bruhat order; Kazhdan-Lusztig polynomials; vexillary permutations Alain Lascoux, Polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 6, 667 -- 670 (French, with English and French summaries). Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Kazhdan-Lusztig polynomials for vexillary 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 Matrix Schubert varieties are the closures of the orbits of \(B \times B\) acting on all \(n \times n\) matrices, where \(B\) is the group of invertible lower triangular matrices. Extending work of \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)], \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes, and derived a related primary decomposition in terms of reduced pipe dreams. These results lead to a geometric proof of the Billey-Jockusch-Stanley formula for a Schubert polynomial, among many other applications. We define skew-symmetric matrix Schubert varieties to be the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices. In analogy with Knutson and Miller's work, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams. We show that these initial ideals are likewise the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Our methods differ from \textit{A. Knutson} and \textit{E. Miller}'s [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] and can be used to give new proofs of some of their results, as we explain at the end of this article. Schubert varieties; Gröbner bases; Grothendieck polynomials; simplicial complexes Combinatorial aspects of algebraic geometry, Combinatorial aspects of simplicial complexes, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Permutations, words, matrices Gröbner geometry for skew-symmetric matrix 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 \(q_1, \dots, q_n\) be some variables and consider the ring \(K := \mathbb Z[q_1,\dots,q_n]/( \prod_{i=1}^n q_i)\). We show that there exists a \(K\)-bilinear product \(\star\) on \(H^*(F_n;\mathbb Z)\otimes K\) which is uniquely determined by some quantum cohomology like properties (most importantly, a degree two relation involving the generators and an analogue of the flatness of the Dubrovin connection). Then we prove that \(\star\) satisfies the Frobenius property with respect to the Poincaré pairing of \(H^*(F_n;\mathbb Z)\); this leads immediately to the orthogonality of the corresponding Schubert type polynomials. We also note that if we pick \(k\in\{1,\dots,n\}\) and we formally replace \(q_k\) by 0, the ring \((H^*(F_n;\mathbb Z)\otimes K,\star)\) becomes isomorphic to the usual small quantum cohomology ring of \(F_n\), by an isomorphism which is described precisely. flag manifolds; cohomology; quantum cohomology; periodic Toda lattice; Schubert polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Homology and cohomology of homogeneous spaces of Lie groups A quantum type deformation of the cohomology ring of 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 show that the dual character of the flagged Weyl module of any diagram is a positively weighted integer point transform of a generalized permutahedron. In particular, Schubert and key polynomials are positively weighted integer point transforms of generalized permutahedra. This implies several recent conjectures of \textit{C. Monical} et al. [Sel. Math., New Ser. 25, No. 5, Paper No. 66, 37 p. (2019; Zbl 1426.05175)]. Schubert polynomial; key polynomial; dual character of the flagged Weyl module; Newton polytope; integer point transform; generalized permutahedron Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Combinatorial aspects of matroids and geometric lattices, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Representations of finite symmetric groups, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Schubert polynomials as integer point transforms of generalized permutahedra | 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 As a basis in the space of symmetric functions the Schur functions enjoy a number of special properties such as Littlewood-Richardson and Young rules, Jacobi-Trudy and Cauchy identities. They also play a central role in the representation theory of the symmetric group and cohomology of complex Grassmanians. There exist \(q\)-deformed versions of the latter, namely Hecke algebras and quantum cohomology, respectively. In a previous paper with \textit{A. Lascoux} [Duke Math. J. 116, No.\,1, 103--146 (2003; Zbl 1020.05069)] the authors introduced a new basis of \(k\)-Schur functions that play a similar role in the deformed context and have similar algebraic properties. The original definition was purely combinatorial and in this paper algebraic and geometric applications are worked out.
First, it is proved that \(k\)-Littlewood-Richardson coefficients are the same as the structure constants in the representation ring of Hecke algebra \(H_\infty(q),\,q^n=1\) when \(k=n-1\). By a work of \textit{F. M. Goodman} and \textit{H. Wenzl} [Adv. Math. 82, No.\,2, 244--265 (1990; Zbl 0714.20004)] they coincide with the fusion coefficients for the Wess-Zumino-Witten CFT associated with the affine Lie algebra \(\widehat{\mathfrak{su}}(l)\). Furthermore, as shown by \textit{E. Witten} [Conf. Proc. Lect. Notes Geom. Topol. 4, 357--422 (1995; Zbl 0863.53054)] the fusion algebra of \(\widehat{\mathfrak{u}}(l)\) Wess-Zumino-Witten CFT is isomorphic to the small quantum cohomology of the Grassmanian \(\text{Gr}_{l,n}(\mathbb{C})\). Since \(\widehat{\mathfrak{u}}(l)=\widehat{\mathfrak{su}}(l)\times\widehat{\mathfrak{u}}(1)\) one expects a connection between the \(k\)-Schur functions and quantum cohomology as well and it is described in detail in the paper. In the main result the cohomology structure constants (\(3\)-point Gromov-Witten invariants of \(\text{Gr}_{l,n}(\mathbb{C})\)) are explicitly reduced to certain \(k\)-Littlewood-Richardson coefficients.
In the above applications only a subset of \(k\)-Littlewood-Richardson coefficients is involved. On the other hand, \textit{T. Lam} proved recently [J. Am. Math. Soc. 21, No.\,1, 259--281 (2008; Zbl 1149.05045)] that all \(k\)-Schur functions form the Schubert basis in the homology of the affine Grassmanian of \(GL_{k+1}(\mathbb{C})\). The authors introduce dual \(k\)-Schur functions and show that the expansion coefficients of their product are the same as the coproduct structure constants for the \(k\)-Schur functions. This implies that the dual Schurs form the Schubert basis in the integral cohomology of the affine Grassmanian and coincide with Lam's affine Schur functions. \(k\)-Schur functions; Littlewood-Richardson coefficients; Grassmanian; Schubert basis; quantum cohomology Lapointe, L.; Morse, J., Quantum cohomology and the \textit{k}-Schur basis, Trans. amer. math. soc., 360, 4, 2021-2040, (2008) Symmetric functions and generalizations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology and the \(k\)-Schur basis | 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 begins with a historical account of Schubert's approach to the problem of finding the number of quadrics tangent to a given set of linear subspaces of a projective space, and he gives a description of a naive parameter space for the general problem. The purpose of the paper is to describe Schubert's problem and its solution in detail, in a modern setting. He introduces the notion of complete quadrics, and the corresponding tangency conditions. Modern intersection theory is applied to the various parameter spaces, leading to the determination of the Schubert numbers. In addition, he derives a series of incidence formulas and he presents some new explicit formulas for the Schubert numbers. He stresses that the explicit formulas are only of theoretical interest, since Schubert's recursive procedure has been verified by several authors and in practice might be easier to use than formulas. number of quadrics; intersection theory; Schubert numbers Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Quadratic forms over general fields Parameter spaces for quadrics | 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 analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a \(\hbar\)-difference equation :
\[
\psi(x+\hbar)=\left(e^\hbar\frac{d}{dx}\right)\psi(x)=L(x;\hbar)\psi(x),\quad L(x;\hbar)\in \mathrm{GL}_2((\mathbb{C}(x))[\hbar]).
\]
In particular, the author extends the notion of determinantal formulas and topological type property proposed for formal WKB solutions of \(\hbar\)-differential systems to this setting. He applies his results to a specific \(\hbar\)-difference system associated to the quantum curve of the Gromov-Witten invariants of \(\mathbb{P}^1\) for which he proves that the correlation functions are reconstructed from the Eynard-Orantin differentials computed from the topological recursion applied to the spectral curve \(y=\coth^{-1}\frac{x}{2}\). Finally, identifying the large \(x\) expansion of the correlation functions, proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new generating series for Gromov-Witten invariants of \(\mathbb{P}^1\). In other words, the purpose of this paper is to prove on the simple example of the quantum curve arising in the enumeration of Gromov-Witten invariants of \(\mathbb{P}^1\) that the determinantal formulas and the topological type property may be used in the context of \(\hbar\)-difference systems rather than \(\hbar\)-differential systems. However, as presented in this paper, it seems that the construction might be adapted for more general situations. WKB expansion; topological recursion; Gromov-Witten invariants; difference systems; determinantal formulas; topological type property Marchal, O.: WKB solutions of difference equations and reconstruction by the topological recursion (2017). arXiv:1703.06152 [math-ph] Relationships between algebraic curves and integrable systems, Difference equations, scaling (\(q\)-differences), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds WKB solutions of difference equations and reconstruction by the topological recursion | 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 classical Schubert calculus describes the Chow ring of the Grassmannian (parametrizing all $d$-planes of a fixed $n$-space) with the help of its Schubert classes indexed by $\binom{n}{d}$ partitions contained in the rectangle $(n- d)^d$. The fundamental theorems of Schubert calculus include: the basis theorem, the duality theorem and the Giambelli formula expressing a general Schubert class in terms of special ones.
In the present paper, the authors study various push-forward formulas related to Grassmann bundles, Schubert bundles and Kempf-Laksov bundles. They use intersection theory and push-forward maps for the mentioned bundles. push-forward; Grassmann bundle; Schubert bundle; duality theorem; Schubert classes Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations Gysin maps, duality, and Schubert classes | 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 each infinite series of the classical Lie groups of type \(B\), \(C\) or \(D\), we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur \(Q\)- or \(P\)-functions defined earlier by Ivanov. Schubert polynomial; flag variety; factorial \(Q\)-function Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Double Schubert polynomials for the classical Lie 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 \(X=G/P\) be a complex projective homogeneous space endowed with the action of the maximal torus \(T\). The \(T\)-equivariant cohomology of \(X\) is generated by the equivariant Schubert cycles, and the corresponding structure constants are called the equivariant Littlewood-Richardson coefficients. These can be thought of as polynomials, after a suitable choice of generators for \(H^{\star}_{T}(\text{point})\), corresponding to the negative simple roots. Graham showed that these polynomials have nonnegative coefficients [\textit{W. Graham}, Duke Math. J. 109, No.~3, 599--614 (2001; Zbl 1069.14055)]. The paper under review proves the same statement in quantum cohomology. The author extends Graham's result, showing that the structure constants of the quantum equivariant cohomology of \(X\) are also polynomials of with non-negative coefficients. Schubert calculus; Littlewood-Richardson coefficients; quantum cohomology L. C. Mihalcea, ''Positivity in equivariant quantum Schubert calculus,'' Amer. J. Math., vol. 128, iss. 3, pp. 787-803, 2006. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Positivity in equivariant quantum 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 flagged Grothendieck polynomials defined as generating functions of flagged set-valued tableaux of \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] can be expressed by a Jacobi-Trudi-type determinant formula generalizing the work of \textit{T. Hudson} and \textit{T. Matsumura} [Eur. J. Comb. 70, 190--201 (2018; Zbl 1408.14030)]. We also introduce the flagged skew Grothendieck polynomials in these two expressions and show that they coincide. Grothendieck polynomials; flagged set-valued tableaux; vexillary permutations; Jacobi-Trudi formula Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Grothendieck groups, \(K\)-theory and commutative rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry Flagged Grothendieck 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 Schubert polynomials for the classical groups were defined by S. Billey and M. Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety over a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for Weyl groups of classical types. RC-graphs; Weyl groups Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations, Combinatorial aspects of representation theory Pipe dreams for Schubert polynomials of the classical 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 We have developed a patch implementing multivariate polynomial seen as a multi-base algebra. The patch is to be released into the software Sage and can already be found within the Sage-Combinat distribution. One can use our patch to define a polynomial in a set of indexed variables and expand it into a linear basis of the multivariate polynomials. So far, we have the Schubert polynomials, the Key polynomials of types \(A\), \(B\), \(C\), or \(D\), the Grothendieck polynomials and the non-symmetric Macdonald polynomials. One can also use a double set of variables and work with specific double-linear bases like the double Schubert polynomials or double Grothendieck polynomials. Our implementation is based on a definition of the basis using divided difference operators and one can also define new bases using these operators. Sage-Combinat distribution; Schubert polynomials; key polynomials; Grothendieck polynomials; non-symmetric Macdonald polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Multivariate polynomials in Sage | 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 proves a Chevalley formula for the equivariant quantum multiplication of two Schubert classes in a homogeneous variety \(X=G/P\). An efficient algorithm to compute 3 point, genus zero equivariant GW-invariants on \(X\) is also given. equivariant quantum cohomology; homogeneous space; Chevalley formula L. C. Mihalcea, ''On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms,'' Duke Math. J., vol. 140, iss. 2, pp. 321-350, 2007. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Equivariant algebraic topology of manifolds, Algebraic combinatorics On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms | 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 Positive presentations of Schubert cycles such as classical Schubert polynomials play a key role in the Schubert calculus. Ideas of toric geometry and theory of Newton (or moment) polytopes motivated search for positive presentations with a more convex geometric flavor. For instance, Schubert cycles on the complete flag variety for \(\mathrm{GL}_n\) were identified by various means with unions of faces of Gelfand-Zetlin polytopes.
In the present paper, the author develops an algorithm for representing Schubert cycles by faces of convex polytopes in the case of complete flag varieties for arbitrary reductive groups.
In this paper,first she defines geometric mitosis operations \(M_1,\ldots, M_r\) on faces of polytope \(P_{\lambda}\) as convex geometric counterparts of Demazure operators \(D_1, \ldots , D_r\).
The definition of mitosis operations is elementary, and its main ingredient is mitosis on parallelepipeds introduced in [\textit{V. A. Kirichenko} et al., Russ. Math. Surv. 67, No. 4, 685--719 (2012); translation from Usp. Mat. Nauk 67, No. 4, 89--128 (2012; Zbl 1258.14055), Section 6]. Mitosis assigns to every face a collection (possibly empty) of faces of dimension one greater. Mitosis on parallelepipeds can be viewed as a convex geometric realization of the mitosis of Knutson-Miller restricted to two consecutive rows of pipe dreams. Then she uses mitosis on parallelepipeds as a building block for mitosis on more general polytopes \(P_{\lambda}\) (called parapolytopes) that admit \(r\) different fibrations by parallelepipeds. The combinatorics of mitosis depends significantly on the combinatorics of \(P_{\lambda}\). For instance, for Gelfand-Zetlin polytopes she gets mitosis on usual pipe dreams, and for a polytope associated with the cone of adapted strings in type \(C\) she gets different combinatorial objects that we call skew pipe dreams. For \(\mathrm{Sp}_4\) and a Newton-Okounkov polytope of the symplectic flag variety, the algorithm yields a new combinatorial rule that extends to \(\mathrm{Sp}_{2n}\). Demazure operator; flag variety; Newton-Okunkov polytope; Schubert calculus Kiritchenko, V., Geometric mitosis, Math. Res. Lett., 23, 1069-1096, (2016) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Geometric mitosis | 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 determines explicitly the irreducible components of any Schubert variety \(\text{GL}_n(K)\) for any algebraically closed field \(K\) and describes the generic singularities along them. Similar results have been given by \textit{L. Manivel} [Int. Math. Res. Not. 2001, 849--871 (2001; Zbl 1023.14022)], \textit{S. Billey} and \textit{G. S. Warrington} [Trans. Am. Math. Soc. 355, 3915--3945 (2003; Zbl 1037.14020)], and \textit{C. Kassel, A. Lascoux} and \textit{C. Reutenauer} [J. Algebra 269, 74--108 (2003; Zbl 1032.14012)]. The methods of the present article are geometric and give a different and useful perspective of the field. Quasi-resolutions play an important role, and may be useful in computing Kazhdan-Lusztig polynomials for arbitrary polynomials. The proof builds heavily upon the results of \textit{A. Cortez} [Adv. Math. 178, 396--445 (2003; Zbl 1044.14026)]. Schubert varieties; generic singularities; quasi-resolutions; linear groups; singular loci Cortez, Aurélie, Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire, Adv. Math., 178, 2, 396-445, (2003), MR 1994224 Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Generic singularities and quasi-resolutions of Schubert varieties for the lineary group. | 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 present booklet is an expanded version of the author's article [Schubert calculus via Hasse--Schmidt derivations. To appear in Asian J. Math. (2005)] where he presents a new point of view on Schubert calculus on a Grassmann manifold. In this interpretation the Chow ring of the Grassmann manifold of \(k\)-dimensional subspaces on an \(n\)-space is the \(k\)'th exterior product of an \(n\)-dimensional vector space \(M\) with a fixed base \(e_1,\dots, e_n\), and the Chern classes are presented as certain differential operators on the exterior product. These operators the author calls Schubert derivations, and are obtained from the operator \(D_1:M\to M\) given by \(D_1(e_i)=e_{i+1}\) for \(i=1,\dots, n-1\) and \(D_1(e_n)=0\).
The treatment gives an easy and natural approach to Schubert calculus, that is well adopted to computations. In particular it gives a satisfactory explanation of the determinants appearing in Giambelli's formula.
It is refreshing and surprising that it is possible to take a new perspective on this classical part of geometry, particularly taken into account the massive amount of work in the field, coming from many different parts of mathematics like geometry, algebra and combinatorics.
Chapter by chapter the contents is as follows:
Chapter 1: The Schubert calculus of the author is indicated and related to Wronski determinants and the usual differential calculus.
Chapter 2: This is a short review of the necessary combinatorics and intersection theory.
Chapter 3: The Grassmann manifold and Schubert varieties are constructed and the basic results of Schubert calculus recalled.
Chapter 4: Here the new material is presented. The Schubert derivations are constructed and it is shown how Pieri's formula is a special case of the action of the Schubert derivations on the exterior product mentioned above. Giambelli's formula is discussed and proved, and the formal results of Schubert calculus are obtained. Chow ring; Schubert derivation; Grassmann manifold; Pieri's formula; Giambelli's formula Gatto, L.: Schubert calculus: an algebraic introduction, 25 colóquio brasileiro de matemática, (2005) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Schubert calculus: an algebraic introduction. Paper from the 25th Brazilian mathematics colloquium---colóquio Brasileiro de matemática, Rio de Janeiro, Brazil, July 24--29, 2005 | 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 result of D.~Peterson identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of \(\text{GL}_n\) is proven. The methods developed are used to give a new proof of a formula of C.~Vafa, K.~Intriligator, and A.~Bertram for the structure constants (Gromov-Witten invariants). Certain inequalities for Schur polynomials at roots of unity are proven. Gromov-Witten invariants; quantum cohomology rings; Grassmannians; reduced coordinate rings; Schur polynomials K. Rietsch, Quantum cohomology rings of Grassmannians and total positivity, Duke Math. J. 110 (2001), no. 3, 523--553. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions Quantum cohomology rings of Grassmannians and total positivity | 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 survey article based on ideas of Grothendieck which provide geometric and combinatorial ways of studying the Galois theory of algebraic numbers. We report on some recent results in the field, and point out the relationship with the theory of hypermaps introduced by \textit{R. Cori} in 1975.
A connected Riemann surface \(X\) is compact if and only if it is the Riemann surface of \(\Phi (x, y) =0\) for some irreducible polynomial \(\Phi\in \mathbb{C} [x, y]\); it is said to be defined over a subfield \(K\) of \(\mathbb{C}\) if \(\Phi\in K[x, y]\). Belyi's theorem states that \(X\) is defined over the field \(\overline {\mathbb{Q}}\) of algebraic numbers if and only if there is a Belyi function \(\beta: X\to \Sigma\), that is, a meromorphic function from \(X\) to the Riemann sphere \(\Sigma= P^1 (\mathbb{C})\) with all its critical values in \(\{0, 1, \infty\}\). Such a covering \(\beta\) is determined by three monodromy permutations \(g_0\), \(g_1\) and \(g_\infty\) which show how the sheets are permuted by continuation around 0, 1 and \(\infty\); these generate a transitive group, and satisfy \(g_0 g_1 g_\infty =1\). One can always choose \(\beta\) so that \(g^2_1 =1\), in which case these permutations correspond to a map (a 2-cell imbedding of a graph) on the surface \(X\); this gives the correspondence, first pointed out by Grothendieck, between algebraic curves over \(\overline {\mathbb{Q}}\) and maps, or dessins d'enfants as he called them.
A general Belyi function \(\beta\) (without the restriction on \(g_1\)) similarly corresponds to a hypermap (or hypergraph imbedding) on \(X\), and in this survey we show how these give a more natural combinatorial representation of Belyi functions. In either case, the natural action of the absolute Galois group \(\mathbb{G}= \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})\) of \(\overline {\mathbb{Q}}\) on Belyi pairs \((X, \beta)\) induces a faithful action of \(\mathbb{G}\) on maps or on hypermaps, thus giving an alternative insight into this important and complicated group. We explain how these topics are closely related to certain congruence subgroups of the modular group and to elliptic curves. Galois theory of algebraic numbers; hypermaps; Riemann surface; Belyi function; dessins d'enfants; Galois group; congruence subgroups; modular group; elliptic curves Jones, G. A.; Singerman, D., Belyǐ functions, hypermaps and Galois groups, Bull. Lond. Math. Soc., 28, 561-590, (1996) Riemann surfaces; Weierstrass points; gap sequences, Galois theory, Planar graphs; geometric and topological aspects of graph theory, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Coverings of curves, fundamental group, Structure of modular groups and generalizations; arithmetic groups, Geometric group theory, Compact Riemann surfaces and uniformization Belyĭ functions, hypermaps and Galois 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 Schur polynomials \(s_{\lambda }\) are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For \(\rho = (n, n-1, \dots , 1)\) a staircase shape and \(\mu \subseteq \rho\) a subpartition, the Stembridge equality states that \(s_{\rho /\mu } = s_{\rho /\mu^T}\). This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials \(G_{\lambda }\), and the dual stable Grothendieck polynomials \(g_{\lambda }\), developed by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)], \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)], are variants of the Schur polynomials and describe the \(K\)-theory of the Grassmannians. Using the Hopf algebra structure of the ring of symmetric functions and a generalized Littlewood-Richardson rule, we prove that \(G_{\rho /\mu } = G_{\rho /\mu^T}\) and \(g_{\rho /\mu } = g_{\rho /\mu^T}\), the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials. Stembridge equality; Grothendieck polynomial; Young tableau; Hopf algebra Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Connections of Hopf algebras with combinatorics The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck 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 This survey paper is devoted to the explicit computation of Thom polynomials of singularities [\textit{R.~Thom}, Ann. Inst. Fourier 6, 43--87 (1956; Zbl 0075.32104)] in the form of linear combinations of Schur functions. In particular, the authors recall the notion of Schur functions and Thom polynomials, sketch the proof of positivity of coefficients in the Schur function expansion of a Thom polynomial [\textit{P.~Pragacz} and \textit{A.~Weber}, Fundam. Math. 195, No. 1, 85--95 (2007; Zbl 1146.05049)], the Rimanyi method of computation of Thom polynomials [\textit{R. Rimanyi}, Invent. Math. 143, No. 3, 499--521 (2001; Zbl 0985.32012)], overview Thom polynomials of the singularities \(A_i\), \((xy, x^2, y^3)\) and \((xy, x^2+y^2)\), and give some new computational results for the next simplest \((2,0)\)-Thom-Boardman singularities \((xy, x^2+y^3)\) and \((xy, x^3, y^3)\). The survey is well written and can serve as a quick introduction to the subject and an overview of preceding works by the authors. Thom polynomial; Schur function; singularity; Grassmanian; cotangent map; degeneracy locus; resultant Global theory of complex singularities; cohomological properties, Singularities of differentiable mappings in differential topology, Global theory of singularities, Symmetric functions and generalizations, Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus On Schur function expansions of Thom 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 We use Young's raising operators to introduce and study \textit{double theta polynomials}, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur \(S\)-polynomials and \(Q\)-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations. Tamvakis, H.; Wilson, E., \textit{double theta polynomials and equivariant Giambelli formulas}, Math. Proc. Cambridge Philos. Soc., 160, 353-377, (2016) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Classical problems, Schubert calculus, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Double theta polynomials and equivariant Giambelli 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 In this paper, the author gives a construction of the \textit{elliptic double shuffle Lie algebra} \({{\mathfrak{ds}_\mathrm{ell}}}\) that generalizes the double shuffle Lie algebra \({{\mathfrak{ds}}}\) constructed by G.Racinet to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the Lie algebra \({{\mathfrak{ds}}}\) express two families of algebraic relations between multiple zeta values that conjecturally generate all relations. In analogy with this, elements of the elliptic double shuffle Lie algebra \({{\mathfrak{ds}_\mathrm{ell}}}\) are Lie polynomials whose behaviors conjecturally describe the (dual of the) set of algebraic relations between elliptic multiple zeta values (constructed explicitly in [\textit{P. Lochak} et al., Int. Math. Res. Not. 2021, No. 1, 698--756 (2021; Zbl 1486.11107)]). The major construction relies on the notion of \textit{moulds} due to J.Écalle whose basic properties are concisely summarized in Appendix of this article, where one finds that various combinatorial behaviors of Lie polynomials can be controlled in terms of certain symmetric properties of associated (sequence of) rational functions in \(\prod_r\mathbb{Q}(u_1,\dots,u_r)\). The counterparts of \({{\mathfrak{ds}}}\), \({{\mathfrak{ds}_\mathrm{ell}}}\) in the Grothendieck-Teichmüller theory are the Lie algebra \({{\mathfrak{grt}}}\) of Drinfeld and its elliptic version \({{\mathfrak{grt}_\mathrm{ell}}}\) by \textit{B. Enriquez} [Sel. Math., New Ser. 20, No. 2, 491--584 (2014; Zbl 1294.17012)] respectively, and a conjecturally isomorphic injection from \({{\mathfrak{grt}}}\) to \({{\mathfrak{ds}}}\) is established by \textit{H. Furusho} [Ann. Math. (2) 174, No. 1, 341--360 (2011; Zbl 1321.11088)]. It is shown that the tangential sections \({{\mathfrak{grt}}}\hookrightarrow{{\mathfrak{grt}_\mathrm{ell}}}\) (Enriquez) and \({{\mathfrak{ds}}}\hookrightarrow{{\mathfrak{ds}_\mathrm{ell}}}\) (Écalle) arising from the Tate elliptic curve compatibly terminate into the common target derivation algebra \(\mathrm{Der}\mathrm{Lie}[a,b]\), while constructing a compatible morphism between \({{\mathfrak{grt}_\mathrm{ell}}}\) and \({{\mathfrak{ds}_\mathrm{ell}}}\) is posed as an important open problem. double shuffle relations; multiple zeta values; elliptic associators Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Multiple Dirichlet series and zeta functions and multizeta values, Lie algebras and Lie superalgebras Elliptic double shuffle, Grothendieck-Teichmüller and mould 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 find presentations by generators and relations for the equivariant quantum cohomology rings of the maximal isotropic Grassmannians of types B, C and D, and we find polynomial representatives for the Schubert classes in these rings. These representatives are given in terms of the same Pfaffian formulas which appear in the theory of factorial \(P\)- and \(Q\)-Schur functions. After specializing to equivariant cohomology, we interpret the resulting presentations and Pfaffian formulas in terms of Chern classes of tautological bundles. T. Ikeda , L. C. Mihalcea and H. Naruse , 'Factorial \(P\) - and \(Q\) -Schur functions represent equivariant quantum Schubert classes', Osaka J. Math., to appear. Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Factorial \(P\)- and \(Q\)-Schur functions represent equivariant quantum Schubert classes | 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 introduce coplactic raising and lowering operators \(E^{\prime}_{i}, F^{\prime}_{i}, E_{i}\) and \(F_{i}\) on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of ``doubled crystal'' structure that recovers the combinatorics of type \(B\) Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are the (skew) Schur \(\mathcal Q\)-functions. We give local axioms for these crystals, which closely resemble the Stembridge axioms for type \(A\). Finally, we give a new criterion for such tableaux to be ballot. Schubert calculus; shifted tableaux; jeu de taquin; crystal base theory Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Exact enumeration problems, generating functions Shifted tableau crystals | 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 compact connected Lie group and let \(H\subset G\) be the centralizer of a one parameter subgroup in \(G\): the homogeneous space \(G/H\) is a flag variety. Let \(T\subset H\) be a maximal torus and \(W\) (resp \(W'\)) the Weyl group of \(G\) (resp of \(H\)).
The Grothendieck cohomology of coherent sheaves on \(G/H\) is canonically isomorphic to the \(K\)-theory of complex vector bundles on \(G/H\). The group \(K(G/H)\) is a free \(\mathbb{Z}\)-module with rank equal to the quotient of the order of \(W\) by the order of \(W'\). There are two distinguished additive bases of \(K(G/H)\): the first valid for the case \(H= T\), is indexed by elements from the Weyl group \(\{a_w\in K(G/T)\mid w\in W\}.\)
The second comes from identifying the set of left cosets of \(W'\) in \(W\) with
\[
\overline W= \{w\in W/l(w)\leq l(w_1)\text{ for all }w_1\in W'\}
\]
and taking a canonical partition of the flag variety \(G/H\) into Schubert subvarieties
\[
G/H= \bigcup_{w\in\overline W} X_w(H),\quad\dim X_w(H)= 2l(w).
\]
Then the coherent sheaves \(\Omega_w(H)\in K(G/T)\) form a basis for the \(\mathbb{Z}\)-module \(K(G/H)\). Therefore, in order to give a complete description of the ring \(K(G/T)\) (resp \(K(G/H)\)), one has to specify the structure constants \(C^w_{u,v}\in\mathbb{Z}\) (resp. \(K^w_{u,v}\in\mathbb{Z}\)) where \(u,v,w\in W\) (resp. \(\in\overline W\)), which express the product of the basis elements \(a_u\cdot a_v= \sum C^w_{u,v}\) in the first basis or
\[
\Omega_u(H)\cdot\Omega_v(H)= \sum K^w_{u,v}(H) \Omega_w(H)
\]
in the second basis.
In this paper the author proves a formula that expresses the constants \(C^w_{u,v}\) (resp \(K^w_{u,v}\)) in terms of the Cartan numbers of \(G\). These formulae are computable, in the sense that an existing algorithm for multiplying Schubert classes, can be extended to implement the \(C^w_{u,v}\) (resp. the \(K^w_{u,v}\)). Duan, Haibao: Multiplicative rule in the Grothendieck cohomology of a flag variety. (2004) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Multiplicative rule in the Grothendieck cohomology of a flag 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 We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric functions, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtaining a combinatorial chain construction of Schubert polynomials. pieri formula; Bruhat order; Schubert polynomial; Stanley symmetric function; flag manifold; jeu de taquin; weak order \beginbarticle \bauthor\binitsN. \bsnmBergeron and \bauthor\binitsF. \bsnmSottile, \batitleSkew Schubert functions and the Pieri formula for flag manifolds, \bjtitleTrans. Amer. Math. Soc. \bvolume354 (\byear2002), no. \bissue2, page 651-\blpage673 \bcomment(electronic). \endbarticle \OrigBibText ----, Skew Schubert functions and the Pieri formula for flag manifolds , Trans. Amer. Math. Soc. 354 (2002), no. 2, 651-673 (electronic). \endOrigBibText \bptokstructpyb \endbibitem Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorics of partially ordered sets, Enumerative problems (combinatorial problems) in algebraic geometry Skew Schubert functions and the Pieri formula for 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 The author gives a very precise and explicit description of the quiver category of perverse sheaves constructible with respect to the Schubert stratification on a Hermitian symmetric space \(X\) of type A (a Grassmannian) and of type B (an isotropic Grassmannian). He uses microlocal techniques and the action of the Borel group on the conormal variety to the Schubert stratification of \(X\). He also discusses why his methods fail for a general flag variety \(G/B\). microlocal geometry; constructible sheaf; quiver category; perverse sheaves; Hermitian symmetric space; Grassmannian; isotropic Grassmannian; conormal variety; flag variety T. Braden, ''Perverse sheaves on Grassmannians,'' Canad. J. Math., vol. 54, iss. 3, pp. 493-532, 2002. Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Sheaves of differential operators and their modules, \(D\)-modules, Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds Perverse sheaves on 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 The aim of this series of (three) papers is to prove rigorously some enumerative formulae of Schubert for triple contacts of plane curves which move in suitable families. Two smooth plane curves have at least triple contact at a point P iff they have the same tangent line at P and the same second-order data at P. Here the problem is to give a conceptual approach for the second-order data. If C is a smooth plane curve and \(x,y,z\in C\) are three points approaching a given point P, consider the family \(\Sigma(x,y,z)\) of all concis through \(x,y,z.\) Consider \(P=P^ 2\times P^ 2\times P^ 2\) and \v{P}\(=\check P^ 2\times \check P^ 2\times \check P^ 2\) and define \(W=\{(x,y,z;L,M,N)\in P\times \check P|\) \(x,y\in L,\) \(x,z\in M\) and \(y,z\in N\},\) Then one has a rational map \(W\to G(2,5)\) defined by \((x,y,z;L,M,N)\to \Sigma(x,y,z)\) (one thinks of G(2,5) as the parameter variety for 2-dimensional linear families of conics). Let \(W^*\subset W\times G(2,5)\) be the closure of the graph of the above rational map. The variety \(W^*\), called the model of Schubert triangles, is the interesting object to be studied for the problem the authors have in mind. To this end one also considers the blow-up \(\bar W\) of W along the closed subvariety \(X=\{(x,x,x;L,L,L)|\) \(x\in P^ 2,L\in \check P^ 2,x\in L\}\) (X is just the singular locus of W and \(\bar W\) is smooth), and the full-diagonal blow-up B of P. Then one has a commutative diagram
\[
\begin{tikzcd}\bar {W} \ar[r,"p"]\ar[d,"p" '] & W^\ast\ar[d,"qw"] \\ B \ar[r,"p_W" '] & W \end{tikzcd}
\]
in which \(\bar W\) is identified with the blow-up of B along \(X_ B=p_ W^{-1}(X)\), and also with the blow-up of \(W^*\) along \(X^*=q_ W^{-1}(X)\). Both blow-ups have the same exceptional locus \(\bar X\) of \(\bar W.\) In this first part of the paper the authors begin the program aiming to determine the rational equivalence ring \(A^{\bullet}(W^*)\), by establishing some basic properties of B and \(W^*\) and by computing \(Pic(W^*)\). The variety \(X^*\) is called the variety of the second-order data of \(P^ 2.\)
[See also the following two reviews.] enumerative formulae; triple contacts of plane curves; Schubert triangles; variety of the second-order data Roberts-Speiser , '' Enumerative Geometry of Triangles, I '' Comm. in Alg. 12(10) 1213-1255 (1984). Enumerative problems (combinatorial problems) in algebraic geometry, (Equivariant) Chow groups and rings; motives Enumerative geometry of triangles. 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 Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [\textit{A. Knutson} and \textit{E. Miller}, Gröbner geometry of Schubert polynomials, Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)]. The original proof of the duality was roundabout, relying on the algebra of certain monomial ideals and a recursive characterization of reduced pipe dreams. This paper provides a direct combinatorial proof. Jia, N.; Miller, E.: Duality of antidiagonals and pipe dreams, Sém. lothar. Combin. 58 (2008) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds Duality of antidiagonals and pipe dreams | 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 \(H_n=\text{Hilb}^n(\mathbb{C}^2)\) be the Hilbert scheme which parametrizes the subschemes \(S\) of length \(n\) of \(\mathbb{C}^2\). To each such subscheme \(S\) corresponds a unordered \(n\)-tuple with possible repetitions \(\sigma(S)=[[P_1,...,P_n]]\) of points of \(\mathbb{C}^2\). There exists an algebraic variety \(X_n\) (called the isospectral Hilbert scheme) which is finite over \(H_n\) and which consists of all ordered \(n\)-tuples \((P_1,...,P_n)\in(\mathbb{C}^2)^n\) whose underlying unordered \(n\)-tuple is \(\sigma(S)\). The main aim of the paper under review is to study the geometry of \(X_n\), which is more complicated than the geometry of \(H_n\). For instance, a classical result of J. Fogarty asserts that \(H_n\) is irreducible and non-singular. The main result of the paper under review asserts that \(X_n\) is normal and Gorenstein (in particular, Cohen-Macaulay). Earlier work of the author indicated that there is a far-reaching correspondence between the geometry and sheaf cohomology of \(H_n\) and \(X_n\) on the one hand, and the theory of Macdonald polynomials on the other hand. The link between Macdonald polynomials and Hilbert schemes comes from some recent work [see \textit{A. M. Garsia} and \textit{M. Haiman}, Proc. Nat. Acad. Sci. USA 90, No. 8, 3607-3610 (1993; Zbl 0831.05062)].
The main result proved in this paper is expected to be an important step toward the proof of the so-called \(n!\)-conjecture and Macdonald positivity conjecture. The main theorem is based on a technical result (theorem 4.1) which asserts that the coordinate ring of a certain type of subspace arrangement is a free module over the polynomial ring generated by some of the coordinates. Macdonald polynomials; isospectral Hilbert schemes; \(n!\)-conjecture; Macdonald positivity conjecture; subspace arrangement K.B. Alkalaev and V.A. Belavin, \textit{Conformal blocks of}\( {\mathcal{W}}_n \)\textit{Minimal Models and AGT correspondence}, arXiv:1404.7094 [INSPIRE]. Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Hilbert schemes, polygraphs and the Macdonald positivity 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 The maximal minors of a \(p\times (m+p)\)-matrix of univariate polynomials of degree \(n\) with indeterminate coefficients are themselves polynomials of degree \(np\). The sub-algebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree \(np\) in the Grassmannian of \(p\)-planes in \((m+p)\)-space. These subalgebra generators are shown to form a sagbi basis. The resulting flat deformation from the quantum Grassmannian to a toric variety gives a new ``Gröbner basis style'' proof of the Ravi-Rosenthal-Wang formulas in quantum Schubert calculus [\textit{M. S. Ravi}, \textit{J. Rosenthal} and \textit{X. Wang}, Math. Ann. 311, No.1, 11-26 (1998; Zbl 0902.14036)]. The coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, Cohen-Macaulay, and Koszul, and the ideal of quantum Plücker relations has a quadratic Gröbner basis. This holds more generally for skew quantum Schubert varieties. These results are well-known for the classical Schubert varieties \((n=0)\). We also show that the row-consecutive \((p\times p)\)-minors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations. quantum Grassmannian; sagbi basis; quantum Plücker relations; quadratic Gröbner basis; quantum Schubert varieties Sottile, Frank; Sturmfels, Bernd, A sagbi basis for the quantum Grassmannian, J. Pure Appl. Algebra, 158, 2-3, 347-366, (2001) Computational aspects of higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Rings with straightening laws, Hodge algebras, Determinantal varieties A sagbi basis for the quantum 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 The main result of the paper under review is: any three-point genus zero Gromov-Witten invariant on a Grassmannian \(X\) is equal to a classical intersection number on a homogeneous space \(Y\) of the same Lie type. The statement was proven when \(X\) is a type \(A\) Grassmannian, and, in types \(B\), \(C\), and \(D\), when \(X\) is the Lagrangian or orthogonal Grassmannian parametrizing maximal isotropic subspaces in a complex vector space equipped with a non-degenerate skew-symmetric or symmetric form. Their key identity for Gromov-Witten invariants is based on an explicit bijection between the set of rational maps counted by a Gromov-Witten invariant and the set of points in the intersection of three Schubert varieties in the homogeneneous space \(Y\). It should be noted that the proof of their result does not use moduli spaces of maps and only uses basic algebraic geometry. In types \(B\), \(C\), and \(D\), their result may be used to give new proofs of the structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians obtained by \textit{A. Kresch} and \textit{H. Tamvakis} in [J. Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070) and Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)]. Their methods can also be used to prove a quantum Pieri rule for the quantum cohomology of sub-maximal isotropic Grassmannians. Gromov-Witten invariants; Grassmannians; Flag varieties; Schubert varieties; quantum cohomology; Littlewood-Richardson rule A. Buch, A. Kresch, and H. Tamvakis. ''Gromov-Witten invariants on Grassmannians''. J. Amer. Math. Soc. 16 (2003), pp. 901--915.DOI. Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Gromov-Witten invariants on 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 Continuing their work from a previous paper [Adv. Math. 140, 1-43 (1998; Zbl 0951.14035)], the authors develop a calculus of divided differences for types \(B\) and \(D\) (i.e., for the orthogonal groups). (In the quoted paper, a corresponding calculus had been developed for type \(C\).) The main result of this paper describes the action of a certain divided difference operator on the product of a (variant of a) Schur \(P\)-polynomial times a (type \(A\)) Schubert polynomial, the result being either 0 or an explicitly described (variant of a) \(P\)-polynomial (up to a multiplicative constant). Applications of this result to type \(D\) Schubert polynomials are briefly discussed, and it is indicated that these results have also implications for the cohomological study of Schubert varieties for the orthogonal groups and the related degeneracy loci. divided differences; vertex operators; jeu de taquin; symmetric group; orthogonal groups; Schur \(P\)-polynomials; orthogonal Schubert polynomials Alain Lascoux and Piotr Pragacz, Orthogonal divided differences and Schubert polynomials, \?-functions, and vertex operators, Michigan Math. J. 48 (2000), 417 -- 441. Dedicated to William Fulton on the occasion of his 60th birthday. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Classical problems, Schubert calculus Orthogonal divided differences and Schubert polynomials, \(\widetilde P\)-functions, and vertex operators | 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 linear algebraic group over the complex numbers and let \(B\) be a Borel subgroup. The varieties \(X_ w = \overline {BwB/B}\), as \(w\) varies over the Weyl group \(W\), are called the Schubert varieties of the flag variety \(G/B\). The purpose of the present article is to give an elementary algebraic treatment of the cohomology algebra, over the rational numbers \(\mathbb{Q}\), of the Schubert varieties. A precise description of the cohomology algebra is given and an interesting precise connection with the theory of \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surv. 28, No. 3, 1-26 (1973); translation from Usp. Mat. Nauk 28, No. 3(171), 3-26 (1973; Zbl 0286.57025)] for the flag variety, is presented. It is also shown that the description of the cohomology algebra \(H^ \bullet (G/B, \mathbb{Q})\), as the coinvariant algebra of \(W\) extends to Schubert varieties. cohomology algebra; Schubert varieties of flag variety James B. Carrell, Some remarks on regular Weyl group orbits and the cohomology of Schubert varieties, Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 33 -- 41. Grassmannians, Schubert varieties, flag manifolds, Cohomology theory for linear algebraic groups, Classical real and complex (co)homology in algebraic geometry Some remarks on regular Weyl group orbits and the cohomology 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 [\textit{C. Monical} et al., Transform. Groups 26, No. 3, 1025--1075 (2021; Zbl 1472.05152)], we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard set-valued tableaux of any fixed rectangular shape. Here, we establish this conjecture by explicitly constructing the K-crystal operators. As a consequence, we establish the first combinatorial formula for Lascoux polynomials \(L_{w\lambda}\) when \(\lambda\) is a multiple of a fundamental weight as the sum over flagged set-valued tableaux. Using this result, we then prove corresponding cases of conjectures of \textit{C. Ross} and \textit{A. Yong} [Sémin. Lothar. Comb. 74, B74a, 11 p. (2015; Zbl 1328.05200)] and \textit{C. Monical} [ibid. 78B, 78B.35, 12 p. (2017; Zbl 1384.05160)] by constructing bijections with the respective combinatorial objects. Grothendieck polynomial; crystal; Lascoux polynomial; quantum group; set-valued tableau; Kohnert move; skyline tableau Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial identities, bijective combinatorics, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations K-theoretic crystals for set-valued tableaux of rectangular shapes | 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 expository talk discusses recent results by the author and others concerning the smoothness of a Schubert variety \(X(w)\) in a flag variety \(G/B\) where \(G\) is a semisimple algebraic group over \(\mathbb{C}\), and \(B\) is a Borel subgroup of \(G\). Here, the two notions of smoothness are involved: smoothness in the sense of algebraic geometry and rational smoothness in the sense that local Poincaré duality holds. The two notions coincide when \(G= SL_n (\mathbb{C})\) [\textit{V. V. Deodhar}, Commun. Algebra 13, 1379-1388 (1985; Zbl 0579.14046)], and for all simply laced \(G\), as announced by the author [in: Algebraic groups and their generalizations: classical methods, Proc. Symp. Pure Appl. Math. 56, Part I, 53-61 (1994; Zbl 0818.14020)] and in the present paper.
One of the main themes of the paper is the action of the maximal torus \(T\subset B\) of \(G\) on the Schubert variety \(X(w)\), and the topological and combinatorial information on rational smoothness given by \(T\)-invariant curves in \(X(w)\). The author's point of view is shortly expressed in his words: ``Schubert varieties are trying to be smooth''. smoothness of a Schubert variety; flag variety J. B. Carrell, ''On the smooth points of a Schubert variety,'' in Representations of Groups, Allison, B. N. and Cliff, G. H., Eds., Providence, RI: Amer. Math. Soc., 1995, pp. 15-33. Grassmannians, Schubert varieties, flag manifolds, Reflection groups, reflection geometries On the smooth points 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 We obtain new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of 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-Melou and S.~Butler} [Ann. Comb. 11, 335--354 (2007; Zbl 1141.05011)], and \textit{A.~Woo and A.~Yong} [Adv. Math. 207, 205--220 (2006; Zbl 1112.14058)]. We also show how mesh patterns, introduced by \textit{P.~Bränden and A.~Claesson} [Electron. J. Comb. 18, No. 2, Research Paper P5, 14 p. (2011; Zbl 1220.05003)], subsume many other types of patterns and define an extension of them called marked mesh patterns. We use these new patterns to further simplify the description of Gorenstein Schubert varieties and give a new description of Schubert varieties that are defined by inclusions, introduced by \textit{V.~Gasharov and V.~Reiner} [J. Lond. Math. Soc., II. Ser. 66, 550--562 (2002; Zbl 1064.14056)]. We also give a description of 123-hexagon avoiding permutations, introduced by \textit{S. C.~Billey} and \textit{G. S.~Warrington} [J. Algebr. Comb. 13, 111--136 (2001; Zbl 0979.05109)], Dumont permutations and cycles in terms of marked mesh patterns. permutation patterns; singularities of Schubert varieties; Gorenstein varieties; bivincular patterns; smoothness; factoriality; mesh patterns; marked mesh patterns Úlfarsson, H., A unification of permutation patterns related to Schubert varieties, Pure Mathematics and Applications, 22, 273-296, (2011) 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 For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra \(U_q^- (\mathfrak{g})\), where \(U_q^- (\mathfrak{g})\) is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig's category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra \(U_q^- (\mathfrak{g})\). quiver Grassmannian; dual canonical basis; Poincaré polynomial; quantum group Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations On the cohomology of quiver Grassmannians for acyclic quivers | 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 well written survey on the path model in the representation theory of complex semisimple algebraic groups, and its geometric applications to Schubert varieties and related varieties. The authors construc canonical bases of irreducible representations by a purely algebraic approach, based on the quantum Frobenius map for quantum groups at roots of unity thus completing the program of standard monomial theory, initiated by the first author and \textit{C. S. Seshadri} [see, e.g., Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras 279-322 (1991; Zbl 0785.14028)]. Then applications to the geometry of Schubert varieties, their normality, defining equations and singular locus are given. The authors also treat ladder determinantal varieties and quiver varieties, shown to be normal and Cohen-Macaulay by embedding them into Schubert varieties as open subsets. Finally, they give an extension of standard monomial theory to Bott-Samelson varieties and configuration varieties. survey; path model; representation theory; complex semisimple algebraic groups; ladder determinantal varieties; canonical bases; Schubert varieties; quiver varieties; Bott-Samelson varieties Venkatramani Lakshmibai, Peter Littelmann, and Peter Magyar, Standard monomial theory and applications, Representation theories and algebraic geometry (Montreal, PQ, 1997) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 319 -- 364. Notes by Rupert W. T. Yu. Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Homogeneous spaces and generalizations Standard monomial theory and applications | 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 paper is about some pieces of algebra concerning the \textsl{quantum cohomology of the Lagrangian Grassmannian}, as the title announces. It is a very important subject, of very much current interest, and most of its beauty is due to the fact that its investigation requires a skillful tasty blend of algebra, geometry, algebraic geometry, combinatorics and some splash of theoretical mechanics flavor coming from equipping even dimensional complex vector spaces with a symplectic form. Key words for this paper are: \textsl{Grassmannians} (from geometry), \textsl{posets and dosets} (from combinatorics), \textsl{straightening law} (from algebra), \textsl{Drinfeld's Grassmannians} (from quantum cohomology of Lagrangian Grassmannians).
The main result, invoked in the introduction as Theorem 1.1., states that \textsl{the coordinate ring of any Schubert subvariety of the Drinfel'd Lagrangian Grassmannian is an algebra with straightening law on a doset}: this extends previous analogous achievements by Sottile and Sturmfels, regarding ordinary grassmannians.
Here are a few words to give the reader the feeling of what the above main result is about. What a Grassmannian \(G(k,n)\) is, the variety parameterizing \(k\)-dimensional vector subspaces of the \(n\)-dimensional complex space \(\mathbb{C}^n\), everybody knows. Equipping \(\mathbb{C}^{2n}\) with a maximal rank skew symmetric bilinear form \(\omega\), briefly said a \textsl{symplectic form}, one is led to distinguish \textsl{isotropic}, \textsl{coisotropic} or \textsl{lagrangian} subspaces (both \textsl{isotropic} and \textsl{coisotropic}), and \(LG(n)\) is the subvariety of \(G(n, 2n)\) parameterizing \(\omega\)-lagrangian subspaces, which are necessarily \(n\)-dimensional. The \textsl{Drinfeld's Grassmannian} is a suitable compactification of the space of holomorphic map from the Riemann sphere \(\mathbb{C}\mathbb{P}^1\) to \(LG(n)\). Suitable Schubert subvarieties of the Drinfeld's Grassmannian can be defined. The quantum cohomology of the Lagrangian grassmannian is the intersection theory of the Drinfeld's Grassmannian. Thus, the main theorem of the paper under review has to do with the algebraic properties of the coordinate rings of Schubert subvarieties of the Drinfeld's Grassmannian, a result which can be easily spelled using a language borrowed from combinatorics. The term \textsl{poset} is just a shortcut standing for \textsl{partially ordered set}. For instance, if \([n]\) is the set of all first positive integers, the set of all increasingly ordered subsets of \(k\) elements is a \textsl{poset}. If \(P\) is a poset, let \(\Delta_P\) be the diagonal of \(P\times P\) and \(O_P:=\{(\alpha,\beta)\in P\times P\,|\, \alpha\leq \beta\}\) be the graph of the order relation defined in \(P\). A \textsl{doset} is a subset of \(P\times P\), which contains \(\Delta_P\) and is contained in \(O_P\), such that if \(\alpha\leq \beta\leq \gamma\) and \((\alpha,\gamma)\in D\), then \((\alpha,\beta)\) and \((\beta, \gamma)\) belongs to \(D\) as well. The set \(P\) is said to be the \textsl{underlying poset} of the doset \(D\).
We shall not recall here what an \textsl{algebra with straightening law on a doset} is, a definition due to De Concini and Lakshmibai, and which is very clearly explained in the Definition 3.3 of the paper. However we must say that such an algebraic structure has to do with a special case of Hodge algebra which manifests itself within the coordinate ring of the projective embedding of a homogeneous space \(G/P\) (a generalized flag variety), and it turns out that the coordinate ring of the projective embedding is an \textsl{algebra with straightening law}, a fact noticed by Sottile and Sturmfels.
The paper is divided into five sections. The introduction is followed by a section of preliminaries which is somewhat extended to Section 3, where the algebras with straightening laws are detailedly discussed, with many explicit computations. The intersection theory of Drinfeld's Grassmannian is analyzed in Section 4, while Section 5 goes back again on the straightening law, specifically for the coordinate rings of Schubert varieties of the Drinfeld's Grassmannian. The list of the references ends a paper which has evidently been written with the attempt of not loosing the non specialistic reader. The latter, however, could get some initial worries because in the preliminary section 2A, the notation \({[n]\choose k}\), used to define the set \({[n]\choose k}_d\), is seemingly introduced with no explanation. Its meaning, however, is explained in Section 2B, a few lines before Proposition 2.2, and then it suffices, for the interested reader, to give again a quick further reading of section 2A to get newly rid of what is going on. algebra with straightening law; quasimap; Lagrangian grassmannian; quantum cohomology; Drinfeld's Grassmannian Rings with straightening laws, Hodge algebras, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus Quasimaps, straightening laws, and quantum cohomology for 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 We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call ``excited Young diagrams'', and the second one is written in terms of factorial Schur \(Q\)- or \(P\)-functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety. torus-equivariant cohomology ring of isotropic Grassmannians T. Ikeda, H. Naruse, Excited Young diagrams and equivariant Schubert calculus. \textit{Trans. Amer. Math. Soc}. \textbf{361}(2009), 5193-5221. MR2515809 (2010i:05351) Zbl 1229.05287 Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Excited Young diagrams and 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 In 2002, \textit{M. Kapovich, B. Leeb} and \textit{J. Millson} studied certain structure constants for two related rings [The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra, preprint, \texttt{http://arxiv.org/abs/math.RT/0210256}, Mem. Am. Math. Soc. (to appear)]: the spherical Hecke algebra of a split connected reductive group over a local non-Archimedean field, and the representation ring of the Langlands dual group. The structure constants of the former are defined relative to characteristic functions of double cosets, and the latter relative to highest weight representations. They proved that the nonvanishing of one of the latter structure constants always implies the nonvanishing of the corresponding former one. For \(\text{GL}_n\), the reverse implication also holds, and is due to P. Hall. Both proofs are combinatorial in nature. In this paper, we provide geometric proofs for both results, using affine Grassmannians. We also provide some additional results concerning minuscule coweights and the equidimensionality of the fibers of certain Bott-Samelson resolutions of affine Schubert varieties for \(\text{GL}_n\). Haines, T.: Structure constants for Hecke and representation rings. Int. Math. Res. Not. 39, 2103--2119 (2003) math.RT/0304176 Representations of Lie and linear algebraic groups over local fields, Linear algebraic groups over local fields and their integers, Group schemes Structure constants for Hecke and representation rings | 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 survey the recent study of involution Schubert polynomials and a modest generalization that we call degenerate involution Schubert polynomials. We cite several conditions when (degenerate) involution Schubert polynomials have simple factorization formulae. Such polynomials can be computed by traversing through chains in certain weak order posets, and we provide explicit descriptions of such chains in weak order for involutions and degenerate involutions. As an application, we give several examples of how certain multiplicity-free sums of Schubert polynomials factor completely into very simple linear factors. Schubert polynomial; involution; degenerate involution; weak order; symmetric subgroup Classical problems, Schubert calculus Schubert polynomial analogues for degenerate involutions | 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 action of a differential operator on Schubert polynomials. Using this action, we first give a short new proof of an identity of \textit{I. Macdonald} [Notes on Schubert polynomials. Montréal: Publications du LACIM, Université du Québec (1991)]. We then prove a determinant conjecture of \textit{R. Stanley} [``Some Schubert shenanigans'', Preprint, \url{arXiv:1704.00851}]. This conjecture implies the (strong) Sperner property for the weak order on the symmetric group, a property recently established by \textit{C. Gaetz} and \textit{Y. Gao} [Proc. Am. Math. Soc. 148, No. 1, 1--7 (2020; Zbl 07144479)]. Sperner property; weak order; Schubert polynomial; Macdonald identity Combinatorial aspects of representation theory, Symmetric functions and generalizations, Group actions on combinatorial structures, Combinatorics of partially ordered sets, Determinants, permanents, traces, other special matrix functions, Grassmannians, Schubert varieties, flag manifolds Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley | 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 complex semisimple, simply-connected algebraic group and let \(P\) be a parabolic subgroup. Consider the flag variety \(X:= G/P\) and the Schubert subvarieties \(X_w:= \overline{BwP/P}\subset G/P\) for any \(w\in W/W_P\), where \(W\) is the Weyl group of \(G\), \(W_P\) is the Weyl group of \(P\) and \(B\) is a Borel subgroup of \(G\) contained in \(P\). Then, any subvariety \(V\) of \(X\) is rationally equivalent to a linear combination of Schubert cycles \([X_w]\) with uniquely determined nonnegative integral coefficients. Then, Brion calls \(V\) multiplicity free if these coefficients are 0 or 1. Examples of multiplicity free \(V\) include the Schubert varieties \(X_w\) themselves, \(G\)-stable (irreducible) subvarieties of \(X\times X\) (under the diagonal action of \(G\)), irreducible hyperplane sections of \(X\) in its smallest projective embedding and the irreducible hyperplane sections of Schubert varieties in Grassmannians embedded by the Plücker embedding.
The main theorem of the paper under review asserts that any multiplicity-free subvariety \(V\subset X\) is normal and Cohen-Macaulay. Further, \(V\) admits a flag degeneration inside \(X\) to a reduced Cohen-Macaulay union of Schubert varieties. Hence, for any globally generated line bundle \(G\) on \(X\), the restriction map \(H^0(X,{\mathcal L})\to H^0(V,{\mathcal L})\) is surjective and \(H^i(V,{\mathcal L})= 0\) for all \(i\geq 1\). If \(L\) is ample, then \(H^i(V, {\mathcal L}^{-1})= 0\) for any \(i<\dim V\). Thus, \(V\) is arithmetically normal and Cohen-Macaulay in the projective embedding given by any ample \({\mathcal L}\). Schubert varieties; Cohen-Macaulay; arithmetically normal; projective embedding Brion, M.: Multiplicity-free subvarieties of flag varieties. Commutative algebra (Grenoble/Lyon, 2001), 13-23, Contemp. Math., \textbf{331}, Amer. Math. Soc., Providence, RI, 2003 Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Multiplicity-free subvarieties of flag 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 survey article the authors collect many of the most important results on singular loci of Schubert varieties. The article can be used as a handbook by geometers and combinatorists. It covers the topics:
1. Generalities of \(G/B\) and \(G/Q\).
2. Schubert varieties in \(SL(n)/B\).
3. Tangent spaces and smoothness.
4. Rational smoothness.
5. Determination of the singular locus of \(X(w)\).
6. Descriptions of \(T(w, \tau)\).
7. Computationally efficient criteria for smoothness and rational smoothness.
8. Irreducible components of the singular locus of a Schubert variety.
9. Groups of rank 2.
10. Factoring the Poincaré polynomial of a Schubert variety.
11. Counting smooth Schubert varieties.
The treatment is short and technical. For a more complete and leisurely presentation the interested reader should consult the book ``Singular loci of Schubert varieties'' [Prog. Math. 182 (2000; Zbl 0959.14032)] by \textit{S. Billey} and \textit{V. Lakshmibai}. singular locus; Schubert varieties; rational smoothness; Chevalley-Bruhat order; Plücker coordinates; Kazhdan-Lusztig polynomials; quotient Sara Billey and V. Lakshmibai, On the singular locus of a Schubert variety, J. Ramanujan Math. Soc. 15 (2000), no. 3, 155 -- 223. Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry On the 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 The concatenation product of permutations enjoys many nice properties with respect to the Schubert calculus; that is, from a combinatorial point of view, with respect to the Lascoux-Schützenberger calculus of Schubert polynomials. We give explicit formulas for the product of the Schubert cycles (resp. polynomials) which are associated to the corresponding permutations with general Schubert cycles (resp. polynomials). Those formulas complete the partial known results about the combinatorics of intersection products on flag manifolds (Monk's formula, generalized Pieri formula of Lascoux and Schützenberger, some properties of vexillary permutations). Monk formula; Lascoux-Schützenberger calculus of Schubert polynomials; Schubert cycles; permutations; flag manifolds; Pieri formula F. Patras, Le calcul de Schubert des permutations décomposables , Sém. Lothar. Combin. 35 (1995), Art. B35f, approx. 10 pp. (electronic), http://cartan.u-strasbg.fr:80/~slc/wpapers/s35patras.html. Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices The Schubert calculus of decomposable permutations | 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 article under review provides a proof of a theorem in H. Schubert's classical text [Calcül der abzählenden Geometrie. Leipzig: Teubner (1879; JFM 11.0460.01)] on the enumerative relations among six special lines associated to a given planar cuspidal cubic curve and a given point in the plane. A cuspidal cubic curve in the projective complex plane has three special points: its cusp point, its inflection point, and the intersection point of its inflection tangent line with the tangent line at its cusp. Given such a curve and a general point in the plane, there is six special lines passing through the point, namely the three lines which connect the given point with the three special points plus the three lines which are tangent to the curve and pass through the given point. Schubert's theorem states that, fixing any four of these six special lines, there are only finitely many possibilities for the other two. This theorem also gives the exact numbers of possibilities, depending on which four of the six lines have been fixed. To rephrase the theorem, one can consider the map which, for a fixed point in the plane, sends a planar cuspidal cubic curve to the six special lines associated to the curve and the fixed point. Its image lives in a product of six projective lines, each of these six factors consisting of all lines in the plane that pass through the fixed point. Schubert's theorem says that this image is four-dimensional and it states its multidegree. As Schubert did not provide a full formal proof of his theorem, the author of the article at hand shows that Schubert's theorem is correct. The proof combines theoretical insights with computations and makes heavy use of a Maple implementation of the Ritt-Wu method. In addition, the author proves that the singular locus of the four-dimensional image of the map described above is an irreducible surface. cuspidal cubic curves; Schubert calculus; multidegree; Maple computations; singular locus; Ritt-Wu method Classical problems, Schubert calculus, Special algebraic curves and curves of low genus, Computational aspects of algebraic curves Hilbert problem 15 and Ritt-Wu method. 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 This paper is concerned with the quantum \(K\)-theory of cominuscule homogeneous spaces. To help the reader, who is not specialist in the subject, to catch the flavor of the paper under review, some basic vocabulary is needed. Recall that if \(G\) is an algebraic group, an algebraic subgroup \(P\) of \(G\) is said to be \textsl{parabolic} if the quotient \(G/P\) is a complete algebraic variety. Any parabolic subgroup contains a Borel subgroup \(B\) which contains a maximal torus \(T\) of \(G\). The pair \((G,T)\) defines \textsl{weights} (which are characters of \(T\) satisfying certain non triviality conditions) and a \textsl{root system}. A fundamental weight \(\omega\) is said to be \textsl{minuscule} if and only if \(|<\omega, \alpha>|\leq 0\), for each positive root \(\alpha\), where \(<,>\) is the pairing induced by the natural duality between the characters and the co-characters of \(T\). A fundamental weigth \(\omega\) is said to be \textsl{co-minuscule} if and only if \(<\omega, \alpha^\vee>=1\), where \(\alpha\) denotes the highest root. To each such weight a parabolic subgroup \(P_\omega\) of \(G\) can be associated, and the corresponding quotient \(G/P_\omega\) is said to be a (co)minuscule homogeneous space.
Example of cominuscules homogeneous varieties are type A Grassmannians, Lagrangian grassmannians \(LG(n,2n)\) and, more exotically, the two exceptional homogeneous spaces: the Cayley Plane and the Freudenthal variety.
The paper under review deals with the finiteness of the cominuscule quantum \(K\)-theory. The product of two Schubert classes in the quantum \(K\)-theory of a homogeneous space \(X=G/P\) is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on \(X\). The remarkable result proven by the authors is that if \(X\) is cominuscule then the power series expansion of the product has only finitely many non zero terms (Theorem 1 stated in the Introduction and proven in Section 5). The proof consists in a fine analysis of the geometry of the Schubert varieties of cominuscule homogeneous spaces. They all have at most rational singularities. Furthermore boundary Gromov-Witten varietes defined by two Schubert varieties are either empty or unirational. General details on the nature of the theorems and their proofs are provided in the comprehensive introduction to the paper. In order to prove their main results the authors are led to use an adaptation of a result by \textit{M. Brion} [J. Algebra 258, No. 1, 137--159 (2002; Zbl 1052.14054)] about a Kleiman-Bertini's like theorem regarding rational singularities (Theorem 2.5), which is interesting in its own. Section 3 is devoted to the geometry of the Gromov-Witten varieties. Section 4 is very important as it supplies the list of the Gromov-Witten varieties of cominuscule spaces. Indeed, this is what the authors need to reach the climax of the paper in the last section, where the main theorem about the finiteness of the expansion of the product in quantum \(K\)-theory, stated in the introduction, is finally proven. quantum \(K\)-theory; Gromov-Witten varieties; rational singularities; rational connectedness; quantum Schubert calculus; cominuscule homogeneous spaces Buch, A. S.; Chaput, P. E.; Mihalcea, L. C.; Perrin, N., Finiteness of cominuscule quantum \textit{K}-theory, Ann. Sci. Éc. Norm. Supér. (4), 46, 3, 477-494, (2013) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Finiteness of cominuscule quantum \(K\)-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 Using a combinatorial approach that avoids geometry, this paper studies the structure of \(K_T(G/B)\), the \(T-\) equivariant \(K\)-theory of the generalized flag variety \(G/B\). This ring has a natural basis \(\{[\mathcal O_{X_w}]| w\in W\}\) (the double Grothendieck polynomials), where \(\mathcal O_{X_w}\) is the structure sheaf of the Schubert variety \(X_w\). For rank two cases we compute the corresponding structure constants of the ring \(K_T(G/B)\) and, based on this data, make a positivity conjecture for general \(G\) which generalizes the theorems of \textit{M. Brion} [J. Algebra 258, No.1, 137--159 (2002; Zbl 1052.14054)] (for \(K(G/B)\) and
\textit{W. Graham} [Duke Math. J. 102, 599--614 (2001; Zbl 1069.14055)] (for \(H_T^*(G/B)\)). Let \([X^{\lambda}]\in K_T(G/B)\) be the class of the homogeneous line bundle on \(G/B\) corresponding to the character of \(T\) indexed by \(\lambda\). For general \(G\) we prove ``Pieri-Chevalley formulas'' for the products \([X^{\lambda}][\mathcal O_{X_w}]\), \([X^{-\lambda}][\mathcal O_{X_w}]\),\([X^{w_0\lambda}][\mathcal O_{X_w}]\) and \([\mathcal O_{X_{w_0s_i}}][\mathcal O_{X_w}]\), where \(\lambda\) is dominant. By using the Chern character and comparing lowest degree terms the products which are computed in this paper also give results for the Grothendieck polynomials, double Schubert polynomials, and ordinary Schubert polynomials in, respectively \(K(G/B)\), \(H_T^*(G/B)\) and \(H^*(G/B)\). flag variety; \(K\)-theory; affine Hecke algebras; Schubert varieties DOI: 10.1016/j.ejc.2003.10.012 Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Equivariant \(K\)-theory, Linear algebraic groups over arbitrary fields Affine Hecke algebras 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 The authors prove a new formulation of the \(K\)-theoretic Pieri rule regarding multiplication of stable Grothendieck polynomials using iterated residues. They also use theirs method to establish straightening laws to transform Grothendieck polynomials corresponding to general integer sequences to linear combinations of those corresponding to partitions.
The introduction is very well written and they lay down the notations in a very clear matter. The proofs involves Young tableau, residue of meromorphic functions, among other notions and techniques. Although the computations and proofs deal with a considerable amount of indexes and have a naturally convoluted writing due to the subject the authors make a very clear exposition and as a result the work is very pleasant to read. Grothendieck polynomial; \(K\)-theory; Pieri rule; iterated residue Classical problems, Schubert calculus, Applications of methods of algebraic \(K\)-theory in algebraic geometry \(K\)-theoretic Pieri rule via iterated residues | 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 \(( \mathfrak{g},\mathsf{g})\) be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with \(\mathsf{g}\) being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories \(\mathscr{C}_{\mathfrak{g}}\) and \(\mathscr{C}_{\mathsf{g}}\) of finite-dimensional representations over the quantum loop algebras of \(\mathfrak{g}\) and \(\mathsf{g} \), respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan-Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced \(\mathfrak{g} \). In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we prove the analog of Kazhdan-Lusztig conjecture (formulated in [\textit{D. Hernandez}, Adv. Math. 187, No. 1, 1--52 (2004; Zbl 1098.17009)]) for simple modules in remarkable monoidal subcategories of \(\mathscr{C}_{\mathfrak{g}}\) for any non-simply-laced \(\mathfrak{g} \), and for any simple finite-dimensional modules in \(\mathscr{C}_{\mathfrak{g}}\) for \(\mathfrak{g}\) of type \(\text{B}_n \). In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of \(T\)-systems, and also we generalize the isomorphisms of [\textit{D. Hernandez} and \textit{B. Leclerc}, J. Reine Angew. Math. 701, 77--126 (2015; Zbl 1315.17011); \textit{D. Hernandez} and \textit{H. Oya}, Adv. Math. 347, 192--272 (2019; Zbl 1448.17019)] to all \(\mathfrak{g}\) in a unified way, that is, isomorphisms between subalgebras of the quantum group of \(\mathsf{g}\) and subalgebras of the quantum Grothendieck ring of \(\mathscr{C}_{\mathfrak{g}} \). Finite-dimensional groups and algebras motivated by physics and their representations, Relationship to Lie algebras and finite simple groups, Simple, semisimple, reductive (super)algebras, Grothendieck groups, \(K\)-theory, etc., Affine algebraic groups, hyperalgebra constructions, Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Ring-theoretic aspects of quantum groups Isomorphisms among quantum Grothendieck rings and propagation of positivity | 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 book gathers research papers and surveys on the latest advances in Schubert Calculus, presented at the International Festival in Schubert Calculus, held in Guangzhou, China on November 6--10, 2017. With roots in enumerative geometry and Hilbert's 15th problem, modern Schubert Calculus studies classical and quantum intersection rings on spaces with symmetries, such as flag manifolds. The presence of symmetries leads to particularly rich structures, and it connects Schubert Calculus to many branches of mathematics, including algebraic geometry, combinatorics, representation theory, and theoretical physics. For instance, the study of the quantum cohomology ring of a Grassmann manifold combines all these areas in an organic way.
The book is useful for researchers and graduate students interested in Schubert Calculus, and more generally in the study of flag manifolds in relation to algebraic geometry, combinatorics, representation theory and mathematical physics.
The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry, Semisimple Lie groups and their representations, Proceedings of conferences of miscellaneous specific interest Schubert calculus and its applications in combinatorics and representation theory. Selected papers presented at the ``International Festival in Schubert Calculus'', Guangzhou, China, November 6--10, 2017 | 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 Affine Schubert calculus is a subject that ties together combinatorics, algebraic geometry and representation theory. Its modern development is motivated by the relation between k-Schur functions and the (co)homology of the affine Grassmannian of \(SL(n)\). The \(k\)-Schur functions were introduced by Lapointe, Lascoux, Morse in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory.
In this paper, the author prove the affine Pieri rule for the cohomology of the affine flag variety conjectured in [\textit{T. Lam} et al., Math. Z. 264, No. 4, 765--811 (2010; Zbl 1230.05279)]. He introduces the cap operators acting on the affine nilCoxeter ring \(A_0\) by investigating the work of Kostant and Kumar and show that the cap operators for Pieri elements are the same as the Pieri operators sing strong strips. The affine Pieri rule gives us a geometric interpretation of the skew strong Schur functions as an affine Grassmannian part of the cap product of the Schubert classes in (co)homology of the affine flag variety. Then he describes these two operators. affine flag variety; k-Schur function; nilCoxeter algebra; Pieri rule; strong Schur function Lee, Seung Jin, Pieri rule for the affine flag variety, Adv. Math., 304, 266-284, (2017) Homogeneous spaces and generalizations, Classical problems, Schubert calculus Pieri rule for the affine flag 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 A flagged Weyl module is a representation of the group of invertible upper triangular \(n\times n\) matrices, associated with a diagram \(D\) with \(n\) columns. Schubert polynomials and key polynomials are special cases of the dual chararacters of these modules. The principal specialization of Schubert polynomials has long been of interest. The authors prove a lower bound on the principal specialization of the dual characters of flagged Weyl modules; when specialized to Schubert polynomials, this gives a new proof of a conjecture of Stanley originally proved in [\textit{A. E. Weigandt}, Algebr. Comb. 1, No. 4, 415--423 (2018; Zbl 1397.05205)]. They also characterize the diagrams which give equality. In the case of equality, all nonzero coefficients of the character are 1. There is a conjectural characterizaton of all cases in which the nonzero coefficients are all 1, which is known for Schubert and key polynomials. There is also a conjecture for when the known upper bound gives equality. flagged Weyl module; Schubert polynomials; key polynomials; Schur functions; principal specialization Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus Principal specialization of dual characters of flagged Weyl modules | 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 introduce genomic tableaux, with applications to Schubert calculus. We report a combinatorial rule for structure coefficients in the torus-equivariant \(K\)-theory of Grassmannians for the basis of Schubert structure sheaves. This rule is positive in the sense of \textit{D. Anderson} et al. [J. Eur. Math. Soc. (JEMS) 13, No. 1, 57--84 (2011; Zbl 1213.19003)]. We thereby deduce an earlier conjecture of \textit{H. Thomas} and \textit{A. Yong} [``Equivariant Schubert calculus and jeu de taquin
'', Ann. Inst. Fourier (to appear)] for the coefficients. Moreover, our rule specializes to give a new Schubert calculus rule in the (non-equivariant) \(K\)-theory of Grassmannians. From this perspective, we also obtain a new rule for \(K\)-theoretic Schubert structure constants of maximal orthogonal Grassmannians, and give conjectural bounds on such constants for Lagrangian Grassmannians. Schubert calculus; equivariant \(K\)-theory; Grassmannians; genomic tableaux Pechenik, O., Yong, A.: Genomic tableaux and combinatorial \(K\)-theory. Discrete Math. Theor. Comput. Sci. Proc. \textbf{FPSAC'15}, 37-48 (2015) Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Genomic tableaux and combinatorial \(K\)-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 For positive integers \(k<n\), the Schubert classes \(\sigma_\lambda\) (parametrized by partitions \(\lambda\subset(n-k)^{k}\)) form a basis for the integral cohomology of the Grassmannian \(G(k,n)\). For \(\lambda, \mu\subset(n-k)^{k}\), let \(\#(\sigma_\lambda\cdot\sigma_\mu)_{G(k,n)}\) denote the intersection number of two Schubert classes in \(G(k,n)\). The First (resp. Second) reduction formula expresses \(\#(\sigma_\lambda\cdot\sigma_\mu\cdot\sigma_{\nu^c})_{G(k,n)}\) in terms of the intersection number of three Schubert classes in \(G(k-1,n-1)\) obtained from \(\lambda, \mu, \nu^c\) by deleting a row (resp. column) from each. (\(\nu^c\) denotes the complement partition of \(\nu\).) In combinatorial terms, \(\#(\sigma_\lambda\cdot\sigma_\mu\cdot\sigma_{\nu^c})_{G(k,n)}\) corresponds to the Littlewood-Richardson coefficient \(c_{\lambda,\mu}^\nu\), which is the number of LR-tableaux of shape \(\nu/\lambda\) and content \(\mu\), so the reduction formulae can be expressed in terms of LR-tableaux.
The authors have previously [J. Comb. Theory, Ser. A 114, No.\,7, 1199-1219 (2007; Zbl 1124.05092)] given an explicit bijective proof of the First reduction formula, but their proof of the second formula was via a bijection between LR-tableaux of conjugate shapes. In the current article, they give an explicit bijection for the Second reduction formula. The key result is the algorithm in Definition 3.3, which gives a map \(\Phi\) from LR-tableaux of shape \(\nu/\lambda\) and content \(\mu\) to LR-tableaux of shape \((\nu\ominus(k-\gamma))/(\lambda\ominus\alpha)\) and content \(\mu\ominus\beta\). (\(\lambda\ominus\alpha\) is the partition obtained by deleting the \(\alpha\)'th column of \(\lambda\).) The inverse map \(\Psi\) is described in Sect. 4, which gives the proofs that \(\Phi\) is a well-defined bijection between LR-tableaux of the requisite shapes and contents. The authors conclude with remarks about the connections between the two reduction formulae and factorization of LR-polynomials, and about finding bijective proofs for other combinatorial models of LR-coefficients, e.g. hives, puzzles, checker boards and Mondrian tableaux. LR-coefficient; LR-tableau; Reduction formula; Grassmannian; Schubert class Cho S., Jung E.-K., Moon D.: A bijective proof of the second reduction formula for Littlewood-Richardson coefficients. Bull. Korean Math. Soc. 45(3), 485--494 (2008) Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds A bijective proof of the second reduction formula for Littlewood-Richardson coefficients | 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 An expanded version of the article can be found in the booklet of \textit{L. Gatto} [``Schubert calculus: an algebraic introduction''. Publicações Matemáticas do IMPA (2005; Zbl 1082.14054)]. We cite from the review of that book:\dots he presents a new point of view on Schubert calculus on a Grassmann manifold. In this interpretation the Chow ring of the Grassmann manifold of \(k\)-dimensional subspaces on an \(n\)-space is the \(k\)th exterior product of an \(n\)-dimensional vector space \(M\) with a fixed base \(e_1,\dots,e_n\), and the Chern classes are presented as certain-differential operators on the exterior product. These operators the author calls Schubert derivations, and are obtained from the operator \(D_1 :M\to M\) given by \(D_1(e_i) = e_{i+1}\) for \(i = 1,\dots,n-1\) and \(D_1(e_n) = 0\).
The treatment gives an easy and natural approach to Schubert calculus, that is well adopted to computations. In particular it gives a satisfactory explanation of the determinants appearing in Giambelli's formula.
It is refreshing and surprising that it is possible to take a new perspective on this classical part of geometry, particularly taken into account the massive amount of work in the field, coming from many different parts of mathematics like geometry, algebra and combinatorics. Chow ring; Schubert derivation; Schubert varieties; Pieri's formula; Giambelli's formula Gatto, L.: Schubert calculus via Hasse-Schmidt derivations. Asian J. Math. 3, 315-322 (2005) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Schubert calculus via Hasse-Schmidt derivations | 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 [Adv. Math. 217, No. 6, 2401--2442 (2008; Zbl 1222.14107)], the second author defined a Landau-Ginzburg model for homogeneous spaces \(G/P\). In this paper, we reformulate this LG model in the case of the odd-dimensional quadric \(X=Q_{2m-1}\). Namely we introduce a regular function \(\mathcal{W}_{\mathrm{can}}\) on a variety \(\check{X}_{\mathrm{can}}\times\mathbb{C}^\ast\), where \(\check{X}_{\mathrm{can}}\) is the complement of a particular anticanonical divisor in the projective space \(\mathbb{C}\mathbb{P}^{2m-1}=\mathbb{P}(H^\ast(X,\mathbb{C})^\ast)\). Firstly we prove that the Jacobi ring associated to \(\mathcal{W}_{\mathrm{can}}\) is isomorphic to the quantum cohomology ring of the quadric, and that this isomorphism is compatible with the identification of homogeneous coordinates on \(\check{X}_{\mathrm{can}}\subset \mathbb{C}\mathbb{P}^{2m-1}\) with elements of \(H^\ast(X,\mathbb{C})\). Secondly we find a very natural Laurent polynomial formula for \(\mathcal{W}_{\mathrm{can}}\) by restricting it to a `Lusztig torus' in \(\check{X}_{\mathrm{can}}\). Thirdly we show that the Dubrovin connection on \(H^\ast(X,\mathbb{C}[q])\) embeds into the Gauss-Manin system associated to \(\mathcal{W}_{\mathrm{can}}\) and deduce a flat section formula in terms of oscillating integrals. Finally, we compare \((\check{X}_{\mathrm{can}},\mathcal{W}_{\mathrm{can}})\) with previous Landau-Ginzburg models defined for odd quadrics. Namely, we prove that it is a partial compactification of \textit{A. B. Givental}'s original LG model [Int. Math. Res. Not. 1996, No. 13, 613--663 (1996; Zbl 0881.55006)]. We show that our LG model is isomorphic to the Lie-theoretic LG model from [loc. cit.]. Moreover it is birationally equivalent to an LG model introduced by \textit{V. Gorbounov} and \textit{M. Smirnov} [Glasg. Math. J. 57, No. 3, 481--507 (2015; Zbl 1341.14023)], and it is algebraically isomorphic to Gorbounov and Smirnov's mirror for \(Q_3\), implying a tameness property in that case. mirror symmetry; quadrics; Lie theory; Gromov-Witten theory; quantum cohomology; Landau-Ginzburg model; Gauss-Manin system Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homogeneous spaces and generalizations, Mirror symmetry (algebro-geometric aspects), Homology and cohomology of homogeneous spaces of Lie groups A comparison of Landau-Ginzburg models for odd dimensional quadrics | 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 Motivated by a question in Schubert calculus, we study the interplay of quasisymmetric polynomials with the divided symmetrization operator, which was introduced by \textit{A. Postnikov} [Int. Math. Res. Not. 2009, No. 6, 1026--1106 (2009; Zbl 1162.52007)] in the context of volume polynomials of permutahedra. Divided symmetrization is a linear form which acts on the space of polynomials in \(n\) indeterminates of degree \(n-1\). We first show that divided symmetrization applied to a quasisymmetric polynomial in \(m\) indeterminates can be easily determined. Several examples with a strong combinatorial flavor are given. Then, we prove that the divided symmetrization of any polynomial can be naturally computed with respect to a direct sum decomposition due to \textit{J. C. Aval} et al. [Adv. Math. 181, No. 2, 353--367 (2004; Zbl 1031.05127)], involving the ideal generated by positive degree quasisymmetric polynomials in \(n\) indeterminates. divided symmetrization; quasisymmetric function; symmetric function Symmetric functions and generalizations, Permutations, words, matrices, Factorials, binomial coefficients, combinatorial functions, Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), \(n\)-dimensional polytopes Divided symmetrization and quasisymmetric 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 It is well known that a quantum loop algebra was defined as a generalization of the Drinfeld's new realization of the quantum affine algebra to the loop algebra of any Kac-Moody algebra. It has been shown by Schiffmann that the Hall algebra of the category of coherent sheaves on a weighted projective line is closely related to the quantum loop algebra of some Kac-Moody algebra with a star-shaped Dykin diagram. In this paper the authors study Drinfeld's presentation of the quantum loop algebra in the double Hall algebra setting and find out a collection of generators of the double composition algebra and verify that they satisfy all the Drinfeld relations. quantum loop algebra; Drinfeld's presentation; Hall algebra; weighted projective line; coherent sheaf Dou, R; Jiang, Y; Xiao, J, The Hall algebra approach to drinfeld's presentation of quantum loop algebras, Adv. Math., 231, 2593-2625, (2012) Vector bundles on curves and their moduli, Quantum groups (quantized enveloping algebras) and related deformations, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) The Hall algebra approach to Drinfeld's presentation of quantum loop 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 There are many variants of Gromov-Witten theory and many remarkable formulas relating the various invariants. This was discovered back when \textit{E. Witten} [in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 243--310 (1991; Zbl 0757.53049)] described 2D quantum gravity and his conjecture was proved by \textit{M. Kontsevich} [Commun. Math. Phys. 147, No. 1, 1--23 (1992; Zbl 0756.35081)] using the matrix Airy function. Various Gromov-Witten invariants may be combined together into generating functions that satisfy various systems of PDE such as the KdV hierarchy. One recent progress is the development of the double ramification hierarchy in the paper [Commun. Math. Phys. 336, No. 3, 1085--1107 (2015; Zbl 1329.14103)] by \textit{A. Buryak}. In fact that paper provides nice exposition on the background for the paper under review. The double ramification hierarchy is based on covers of \(P^1\) branched over zero and infinity with given partitions.
The paper under review develops several new recursion relations for the double ramification hierarchy. The new recursion relations recover the full hierarchy from just one of the Hamiltonians. The paper ends with a proof of the Miura equivalence between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for the Gromov-Witten theory of the complex projective line. Gromov-Witten; double ramification; hierarchy Buryak, A.; Rossi, P., Recursion relations for double ramification hierarchies, Commun.Math. Phys., 342, 533-568, (2016) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Hodge theory in global analysis Recursion relations for double ramification hierarchies | 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 give a characterization of arithmetically Gorenstein Schubert varieties in a minuscule generalized flag variety \(G/P\) over an algebraic closed field. Recall that \(G/P\) is minuscule if \(P\) is a maximal parabolic subgroup associated to a minuscule (fundamental) weights. If \(G=\mathrm{SL}_n\), then \(G/P\) is just a Grassmannian.
The minuscule weights plus zero are a set of ``minimal'' representatives for the cosets of the root lattice in the weight lattice.
All Schubert varieties (in a flag variety) are Cohen-Macaulay, but not all of them are smooth. The smooth ones are completely classified. The Gorenstein property is a geometric property between Cohen-Macaulayness and the smoothness property. The problem of classifying the Gorenstein Schubert varieties is an open problem. The minuscule varieties have a canonical closed embedding in a projective space (associated to the ample generator of the Picard group). A Schubert variety is arithmetically Gorenstein (with respect to this embedding) if the affine cone over it is Gorenstein; this property implies the Gorenstein property.
In [Adv. Math. 207, No. 1, 205--220 (2006; Zbl 1112.14058)], \textit{A. Woo} and \textit{A. Yong} have given a characterization of Gorenstein Schubert varieties in \(\mathrm{SL}_n/B\). In this case the Weyl group is isomorphic to \(S_n\) and the characterization can be given in terms of the Young diagram associated to the Schubert variety. This implies a combinatorial characterization of the Gorenstein Schubert varieties in the Grassmannian. The authors of this work give stronger results, namely, a characterization of the arithmetic Gorenstein property (for the Plücker embedding).
The first main result of this work is a generalization of this combinatorial characterization in the case of the orthogonal Grassmannian where \(G=\mathrm{SO}_n\) and \(P\) is associated to one of the right end roots. The Schubert varieties can be again be represented by a ``generalized'' Young diagram and the combinatorial condition is the same.
The idea of the proof is the following: the homogeneous coordinate ring of a Schubert variety \(X\) (with respect to the canonical embedding of \(G/P\)) is a Cohen-Macaulay and graded Hodge algebra with a set of generators indexed by the Bruhat poset of Schubert subvarieties of \(X\). This facts together with a result of Stanley give a characterization of the Gorenstein property for this algebra.
When \(P=P_1\) and \(G\) is \(\mathrm{SO}_{2m}\) or \(\mathrm{Sp}_{2m}\), the authors prove that all Schubert varieties are arithmetically Gorenstein. In particular, \(\mathrm{Sp}_{2m}/P_1\) is a projective space and the Schubert subvarieties are linear subspaces, in particular smooth.
The authors give also a list of the arithmetically Gorenstein Schubert varieties in the exceptional cases. Finally, they described the arithmetically Gorenstein Schubert varieties of the generalized flag varieties of \(\mathrm{SL}_n\), even if they are non minuscule. Schubert varieties; minuscule; Gorenstein Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Arithmetically Gorenstein Schubert varieties in a minuscule \(G/P\) | 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{A. Chiodo} and \textit{Y. Ruan} [Adv. Math. 227, No. 6, 2157--2188 (2011; Zbl 1245.14038)], using the transposition methods of \textit{P. Berglund} and \textit{T. Hübsch} [Nucl. Phys., B 393, No. 1--2, 377--391 (1993; Zbl 1245.14039)], provided a way of obtaining pairs of Calabi-Yau manifolds whose Hodge diamonds are symmetric. The paper under review is devoted to applying this method to a class of \(K3\) surfaces with a non-symplectic involution and as a result producing pairs of lattice mirror \(K3\) surfaces. To be more precise, let \(W\) be a \(K3\) surface defined in a weighted projective space by a specific non-degenerate Delsarte type polynomial whose matrix of exponents \(A\) is invertible over \(\mathbb{Q}\). Let \(G\) be a subgroup of the diagonal symmetries of \(W\) and \(J\) be the monodromy group of the affine Milnor fiber. Suppose that \(J\subset G\) is of determinant 1. Denote by \(W^T\) the hypersurface associated to the transpose \(A^T\), and let \(\widetilde{G}=G/J\) and \(\widetilde{G^T}=G^T/J^T\). The main result of the paper under review proves that the two orbifolds \([W/\widetilde{G}]\) and \([W^T/\widetilde{G^T}]\) belong to the lattice mirror families. These mirror orbifolds are not necessarily Gorenstein. This is the main difference with Batyrev mirror symmetry. \(K3\) surfaces; mirror symmetry; non-symplectic involutions Artebani, M.; Boissière, S.; Sarti, A., The berglund-Hübsch-chiodo-ruan mirror symmetry for K3 surfaces, Journal de Mathématiques Pures et Appliquées, 102, 758-781, (2014) \(K3\) surfaces and Enriques surfaces, Mirror symmetry (algebro-geometric aspects), Automorphisms of surfaces and higher-dimensional varieties, Families, moduli, classification: algebraic theory The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for \(K3\) 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 Let \(\mathcal{S}\) be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of \(\mathcal{S}\) by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities. Schubert varieties; intersection cohomology Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Topological properties in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Global theory of complex singularities; cohomological properties, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Topological properties of mappings on manifolds Polynomial identities related to special 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 The Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson's map is only a surjection, and one needs to quotient by a suitable ideal on the affine side to map isomorphically onto the quantum cohomology. We provide a detailed exposition of this parabolic Peterson isomorphism in the case of the Grassmannian of \(m\)-planes in complex \(n\)-space, including an explicit recipe for doing quantum Schubert calculus in terms of the appropriate subset of non-commutative \(k\)-Schur functions. As an application, we recast Postnikov's affine approach to the quantum cohomology of the Grassmannian as a consequence of parabolic Peterson by showing that the affine nilTemperley-Lieb algebra arises naturally when forming the requisite quotient of the homology of the affine Grassmannian. Peterson isomorphism; quantum cohomology; non-commutative \(k\)-Schur functions; Grassmannian; affine nilTemperley-Lieb algebra; Schubert calculus Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Reflection and Coxeter groups (group-theoretic aspects), Classical problems, Schubert calculus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Applying parabolic Peterson: affine algebras and the quantum cohomology of the 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 The quantum cohomology ring \(H^*_q(G(k,n), \mathbb{C})\) is introduced, \(G(k,n)\) being the Grassmannian of \(k\)-plane in \(\mathbb{C}^n\), using intersection data on the moduli space of maps from \(\mathbb{P}^1\) to \(G(k,n)\). The quantum Giambelli and quantum Pieri formulae are derived. These are similar to the classical Giambelli and Pieri formulae. Two generalizations are discussed, namely, the generalization using intersection data on the moduli space of maps from positive genus curves to the Grassmannian, and the form of the quantum cohomology ring for the full flag variety. quantum Giambelli formulae; quantum cohomology; quantum cohomology ring; moduli space of maps; quantum Pieri formulae Bertram, A.: Computing Schubert's calculus with Severi residues. (1996) Grassmannians, Schubert varieties, flag manifolds, Quantization in field theory; cohomological methods, Enumerative problems (combinatorial problems) in algebraic geometry Computing Schubert's calculus with Severi residues: An introduction to 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 Let \(G\) be a complex simply connected semisimple Lie group, let \(B\) be a Borel subgroup and let \(X=G/B\) be the associated flag manifold. Inside \(X\) is a certain affine subvariety \(Q\), determined by the choice of a principal nilpotent element \(f\) of \(\text{Lie}(G)\), whose closure is denoted \(P_f\) and whose affine coordinate ring, denoted \(A(Q)\), is a polynomial ring. The intersection of \(P_f\) and the Schubert cell defined by \(B\) is denoted \(R\). Furthermore, there is an isomorphism of affine varieties which identifies \(R\) and a certain subset \(Y_0\) of \(\text{Lie}(G)\); under this isomorphism \(Q\cap R\) gets identified with the Toda leaf \(Y^*_0\). The author describes the affine algebras \(A(Q)\), \(A(Y_0)\), \(A(R)\), and \(A(Q\cap R)\) in terms of polynomial generators and relations.
As the author explains in the first sections of the paper, this work was inspired by some conjectures in the case \(G=SL_n(\mathbb{C})\) where the quantum cohomology algebra \(CH(X,\mathbb{C})\) was conjectured and then proven to have a certain description in terms of generators and relations. The author shows this algebra is isomorphic to \(A(Y_0)\) in the \(SL_n\) case.
There is much more in the paper than this brief review can summarize. Although it relies extensively on previous work of the author and others, the paper features examples and exposition making it largely self-contained. flag manifold; affine algebras Kostant, Bertram, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \textit{\({\rho}\)}, Selecta Math. (N.S.), 2, 1, 43-91, (1996) Grassmannians, Schubert varieties, flag manifolds, Applications of linear algebraic groups to the sciences Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \(\rho\) | 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 results of the paper contribute to the combinatorial understanding of degeneracy loci formulas. \par For a sequence of vector bundle maps $E_1\to E_2 \to \ldots \to E_{n-1} \to H_{n-1} \to \ldots \to H_2\to H_1$, and a permutation in $S_n$ one can associate a degeneracy locus, a subvariety in the base, consisting of points over which the induced maps $E_\bullet \to H_\bullet$ are of certain ranks. The fundamental cohomology class represented by this locus, in terms of characteristic classes of the bundles, is called double Schubert polynomial, and their $K$-theory analogue (the class of the structure sheaf) is called double Grothendieck polynomial. Both of these sets of polynomials are defined recursively. \par Combinatorial formulas are often sought for these classes, sometimes for just special classes of permutations. These formulas also depend on the basic classes the formulas are expressed in terms of. \par In the present paper the authors prove determinant formulas for the $K$-theory classes of the structure sheaves of the degeneracy loci, in case the permutation is $2143$-avoiding (vexillary), in terms of Grothendieck polynomials of one-row partitions. The proof depends on the careful study of a resolution of the loci. In the last section the formula is generalized to algebraic cobordism. vexillary permutations; Lascoux-Schützenberger's double Grothendieck polynomials; degeneracy loci Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, \(K\)-theory in geometry Vexillary degeneracy loci classes in \(K\)-theory and algebraic cobordism | 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 how the threshold level of affine fusion, the fusion of Wess-Zumino-Witten (WZW) conformal field theories, fits into the Schubert calculus introduced by Gepner. The Pieri rule can be modified in a simple way to include the threshold level, so that calculations may be done for all (non-negative integer) levels at once. With the usual Giambelli formula, the modified Pieri formula deforms the tensor product coefficients (and the fusion coefficients) into what we call threshold polynomials. We compare them with the \(q\)-deformed tensor product coefficients and fusion coefficients that are related to \(q\)-deformed weight multiplicities. We also discuss the meaning of the threshold level in the context of paths on graphs. Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Classical problems, Schubert calculus Schubert calculus and threshold polynomials of affine fusion | 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 \({\mathbb{W}}_{{\mathbb{Q}}}=\Pr oj({\mathbb{Q}}[g_ 2,g_ 3,X,Y,Z]/(\hom ogeneous\quad ideal\quad generated\quad by\quad -Y^ 2Z+4X^ 3-g_ 2XZ^ 2-g_ 3Z^ 3)).\) This is said to be the Weierstrass family over the field \({\mathbb{Q}}\). Then the first homology with compact supports of the Weierstrass family is computed explicitly, i.e., it is generated by \(\{C^{-k}dX\wedge dY\}_{k\geq 1}\) and \(\{XC^{-k}dX\wedge dY\}_{k\geq 1}\) over the ring \({\mathbb{Q}}[g_ 2,g_ 3]\), where C is a polynomial \(Y^ 2-4X^ 3+g_ 2X+g_ 3.\) When one tensors the homology of the Weierstrass family with \(\Delta^{-1}{\mathbb{Q}}[g_ 2,g_ 3]\), being localized at the discriminant \(\Delta =g^ 3_ 2-27g^ 2_ 3\), over \({\mathbb{Q}}[g_ 2,g_ 3]\), the first homology is generated by \(C^{- 1}dx\wedge dY\) and \(XC^{-1}dX\wedge dY\). One also obtains the first homologies with compact supports of singular fibres over \(\wp =(g_ 2=g_ 3=0)\) and \(\wp =(g_ 2=3, g_ 3=1)\) as corollaries. elliptic curve; Weierstrass family; first homology with compact supports G. C. Kato, On the generators of the first homology with compact supports of the Weierstrass family in characteristic zero, A.M.S. Transactions, to appear. Classical real and complex (co)homology in algebraic geometry, Special algebraic curves and curves of low genus, \(p\)-adic cohomology, crystalline cohomology, Elliptic curves, Homology and cohomology theories in algebraic topology On the generators of the first homology with compact supports of the Weierstrass family in characteristic zero | 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 a formula for the structure constants of multiplication of equivariant Schubert classes in both equivariant cohomology and equivariant \(K\)-theory of Kac-Moody flag varieties \(G/B\). We introduce new operators whose coefficients compute these (in a manifestly polynomial, but not positive, way), resulting in a formula much like and generalizing the positive Andersen-Jantzen-Soergel/Billey and Graham/Willems formulæ for the restriction of classes to fixed points.
Our proof involves Bott-Samelson manifolds, and in particular, the \((K)\)-cohomology basis dual to the \((K)\)-homology basis consisting of classes of sub-Bott-Samelson manifolds. Schubert calculus; equivariant cohomology; Bott-Samelson manifolds Grassmannians, Schubert varieties, flag manifolds, Equivariant homology and cohomology in algebraic topology, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Classical problems, Schubert calculus Schubert structure operators and \(K^\ast_T (G/B)\) | 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 The authors study the intersection cohomology of Schubert varieties. Let \(\mathsf{k}\) be a field of characteristic \(p\). Let \(G\) be a connected reductive algebraic group over \(\mathbb{C}\), let \(B\subset G\) be a Borel subgroup of \(G\) and let \((W, S)\) be the corresponding Weyl group and its simple reflections. Let \(G/B\) be the flag variety with its classical topology and consider the intersection cohomology sheaves \(\mathbf{IC}(w)\) which belong to the bounded derived category of sheaves of \(\mathsf{k}\)-vector spaces on \(G/B\) constructible along \(B\)-orbits.
Let \(\mathcal{H}\) be the Hecke algebra of \((W, S)\) over \(\mathbb{Z}[v, v^{-1}]\) and let \(\{H_w | w\in W\}\) be the Kazhdan-Lusztig basis of \(\mathcal{H}\). If \(\mathsf{k}\) is of characteristic zero, a theorem of Kazhdan and Lusztig says that \(\mathrm{ch}(\mathbf{IC}(w)) = H_w\), where \(\mathrm{ch}(\mathbf{IC}(w))\) is the character sheaf of \(\mathbf{IC}(w)\). Thus the Poincaré polynomials of the stalks of the intersection cohomology sheaves are given by Kazhdan-Lusztig polynomials. It then follows that the same is true in almost all characteristics, however for any given characteristic almost nothing is known.
It has been known since the original papers of Kazhdan and Lusztig that in non-simply laced cases the intersection cohomology complexes may have a different character in characteristic 2. In 2002 Braden discovered examples of Schubert varieties in simply laced types \(A_7\) and \(D_4\) where the character of the intersection cohomology sheaf in characteristic 2 is different to all other characteristics.
The authors define a new basis \(\{B_w | w\in W\}\) using the information encoded in the \( W\)-graph and say that \(x\in W\) is separated if \(H_x=B_x\). The first main result states that \(ch(\mathbf{IC}(w)) = H_w\) for any field \(\mathsf{k}\) if \(w\) is separated.
The determination of the characters of \(\mathbf{IC}(w)\) is closely related to the decomposition theorem. Given a simple reflection \(s\in S\) let \(P_s\) be the corresponding standard minimal parabolic subgroup and consider the associated quotient map \(\pi_s\) over \(G/P_s\). If \(\mathsf{k}\) is of characteristic zero, the decomposition theorem implies that \(\pi_{s*}(\mathbf{IC}(w))\) is a direct sum of shifts of intersection cohomology sheaves. The second main result states some conditions, using the previous definition of separateness, so that the same holds in positive characteristic.
Unfortunately, the calculation of the sets \(\sigma(W)\) of separated elements can be very complicated. The authors use Fokko du Cloux's program Coxeter to prove that \(\sigma(W)=W\) if \(G\) is of type \(A_n\) for \(n<7\). By results of Soergel, this implies a part of Lusztig's conjecture for \(SL(n)\) with \(n<8\). The authors say also that their techniques are not as affective for the other groups.
Finally, in the appendix Braden shows that, both in type \(D_4\) and \(A_7\), \(ch(\mathbf{IC}(w)) \neq H_w\) if \(w\) is a minimal element in \(W \;\sigma(W)\) and \(\mathsf{k}\) has characteristic 2.
The main theoretical tool in the demonstration of the main results is the existence and uniqueness of parity sheaves. Williamson, Geordie; Braden, Tom, Modular intersection cohomology complexes on flag varieties, Math. Z., 0025-5874, 272, 3-4, 697\textendash 727 pp., (2012) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Hecke algebras and their representations Modular intersection cohomology complexes on flag 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 authors give a construction of a class of GKZ equations arising from the equivariant Gromov-Witten theory of projective spaces via the topological recursion, by using the idea of quantum curves which relates the topological recursion correlators to the WKB formal solution of a certain Schrödinger-type differential or difference equation.
The authors focus on the GKZ equations arising from the equivariant Gromov-Witten theory with the target space \(X\) which is either the projective spaces \(\mathbb{CP}^{N-1}_{\boldsymbol{w}}\) or smooth Fano complete intersections \(X_{\boldsymbol{w},\boldsymbol{\lambda}}\) of \(n\) (\(< N\)) degree 1 hypersufaces inside \(\mathbb{CP}^{N-1}\), specified by the equivariant parameters \({\boldsymbol{w}}\in\mathbb{C}^N\) and \(\boldsymbol{\lambda}\in\mathbb{C}^n\) with respect to the torus actions on the target space.
The authors observe that the GKZ equations are of the form \(\widehat{A}(\widehat{x},\widehat{y})J_X(x)=0\), where \(J(x)\) denotes the Givental's \(J\)-function, and \(\widehat{x}\) and \(\widehat{y}\) act on it by \(\widehat{x}J_X(x)=xJ_X(x)\) and \(\widehat{y}J_X(x)=\hbar x\frac{d}{dx}J_X(x)\). On the B-model side (i.e., the Landau-Ginzburg model side) of the mirror symmetry, this GKZ equation is also regarded as the equation satisfied by the ``equivariant oscillatory integral \(\mathcal{I}_X (x)\).''
Since the GKZ equation is a Schrödinger-type equation, one can take the classical limit \(A_X (x, y) \in \mathbb{C}[x, y]\) of the differential operator \(\widehat{A}(\widehat{x},\widehat{y})\). The authors call the algebraic curve \(A_X (x, y) = 0,\) the GKZ curve and show that all the GKZ curves for \(X = \mathbb{CP}^{N-1}_{\boldsymbol{w}}\) and \(X_{\boldsymbol{w},\boldsymbol{\lambda}}\) are of genus 0. Therefore there exist a pair of rational functions \(x = x(z)\) and \(y = y(z)\) which parametrize it. Thus, one may regard the GKZ curve as a spectral curve \((\mathbb{CP}^1, x(z), y(z))\) for the topological recursion.
The authors show that the GKZ curves satisfy the ``admissibility condition'' proposed in [\textit{V. Bouchard} and \textit{B. Eynard}, J. Éc. Polytech., Math. 4, 845--908 (2017; Zbl 1426.14009)], indicating that there exist a Schrödinger-type differential equation, with the GKZ curve being its classical limit, which annihilates the wave function \(\psi(D)\) defined by a certain WKB-type formal series where \(D\) denotes an integration divisor on \(\mathbb{CP}^1\).
The authors formulate their main result as a theorem stating that for the GKZ curve corresponding to \(\mathbb{CP}^{N-1}_{\boldsymbol{w}}\) or \(X_{\boldsymbol{w},\boldsymbol{\lambda}}\), the corresponding wave function \(\psi (D)\) satisfies the GKZ equation which is satisfied by the \(J\)-function \(J_X\) and the mirror equivariant oscillatory integral \(\mathcal{I}_X\).
This theorem claims that the GKZ equations are reconstructible from the GKZ curve through the topological recursion. More precisely, there exists an appropriate choice of the integration divisor \(D\) which realizes the GKZ equation as a quantum curve.
This theorem also implies that the WKB solution has an alternative meaning in the B-model description; it agrees with the saddle point expansion of the oscillatory integral \(\mathcal{I}_X\) up to some normalization factor. For certain oscillatory integrals the authors specify the factor and obtain a full-order coincidence of the asymptotic expansion of \(\mathcal{I}_X\) and the wave function.
The authors discuss the string dualities behind their proposal, especially by considering the \(J\)-function as the vortex partition function and brane partition function in the open topological A-model and B-model. The authors also compute the total Stokes matrices for the quantum curve arising from equivariant \(\mathbb{CP}^1\) by using the exact WKB method. They also examine a wall-crossing formula and equivariant version of the Dubrovin's conjecture [\textit{B. Dubrovin}, Doc. Math. Extra Vol., 315--326 (1998; Zbl 0916.32018)] in this particular case. GKZ equations; Gromov-Witten theory of projective spaces; topological recursion; mirror symmetry Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory Reconstructing GKZ via topological recursion | 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 conormal bundle of a Schubert variety \(S_I\) in the cotangent bundle \(T^\ast\mathrm{Gr}\) of the Grassmannian \(\mathrm{Gr}\) of \(k\)-planes in \(\mathbb C^n\). This conormal bundle has a fundamental class \(\kappa_I\) in the equivariant cohomology \(H^\ast_{\mathbb T}(T^\ast\mathrm{Gr})\). Here \(\mathbb T=(\mathbb C^\ast)^n\times\mathbb C^\ast\). The torus \(\mathbb C^\ast)^n\) acts on \(T^\ast\mathrm{Gr}\) in the standard way and the last factor \(\mathbb C^\ast\) acts by multiplication on fibers of the bundle. We express this fundamental class as a sum \(Y_I\) of the Yangian \(Y(\mathfrak{gl}_2)\) weight functions \((W_J)_J\). We describe a relation of \(Y_I\) with the double Schur polynomial \([S_I]\). A modified version of the \(\kappa_I\) classes, named \(\kappa'_I\), satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold \(T^\ast\mathrm{Gr}\). This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on \(\mathrm{Gr}\). The classes \((\kappa'_I)_I\) form a basis in the suitably localized equivariant cohomology \(H^\ast_{\mathbb T}(T^\ast\mathrm{Gr})\). This basis depends on the choice of the coordinate flag in \(\mathbb C^n\). We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix. Rimányi, R.; Tarasov, V.; Varchenko, A., \textit{cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions}, Math. Z., 277, 1085-1104, (2014) Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Lagrangian submanifolds; Maslov index, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Equivariant homology and cohomology in algebraic topology Cohomology classes of conormal bundles of Schubert varieties and Yangian weight 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 \(G\) be a complex, connected, reductive Lie group, and \(P\) a parabolic subgroup. The cohomology of the generalized flag variety \(G/P\) has a geometric basis, represented by so-called Schubert varieties. It is a longstanding goal in combinatorial algebraic geometry to find visibly positive combinatorial rules that calculate the structure constants of the cohomology ring with respect to this basis. The paper under review provides such a concise, root system uniform rule for the structure constants, in selected cases of \(G\) and \(P\) (called minuscule, and cominuscule). The main result is the extension of the well-known \textsl{jeu de taquin} formulation of these structure constants in the Grassmannian case. This result seems to be the first uniform generalization of the classical Littlewood-Richardson rule that involves both classical and exceptional Lie types. Although the proof of the main result is not type free (in particular, does not provide a new proof in the classical cases), the new viewpoint of the paper is an important contribution to the connections between Schubert calculus and representation theory. Schubert calculus; Littlewood-Richardson rule; minuscule; cominuscule; jeu de taquin Hugh Thomas & Alexander Yong, ``A combinatorial rule for (co)minuscule Schubert calculus'', Adv. Math.222 (2009) no. 2, p. 596-620 Enumerative problems (combinatorial problems) in algebraic geometry A combinatorial rule for (co)minuscule 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 \((W, S)\) be a finite Coxeter system with root system \(R\) and with set of positive roots \(R^+\). For \(\alpha \in R\), \(v, w \in W\), we denote by \(\partial_\alpha\), \(\partial_w\) and \(\partial_{w / v}\) the divided difference operators and skew divided difference operators acting on the coinvariant algebra of \(W\). Generalizing the work of \textit{R. I. Liu} [J. Algebr. Comb. 42, No. 3, 861--874 (2015; Zbl 1326.05173)], we prove that \(\partial_{w / v}\) can be written as a polynomial with nonnegative coefficients in \(\partial_\alpha\) where \(\alpha \in R^+\). In fact, we prove the stronger and analogous statement in the Nichols-Woronowicz algebra model for Schubert calculus on \(W\) after \textit{Y. Bazlov} [J. Algebra 297, No. 2, 372--399 (2006; Zbl 1101.16027)]. We draw consequences of this theorem on saturated chains in the Bruhat order, and partially treat the question when \(\partial_{w / v}\) can be written as a monomial in \(\partial_\alpha\) where \(\alpha \in R^+\). In an appendix, we study related combinatorics on shuffle elements and Bruhat intervals of length two. braided differential calculus; Nichols-Woronowicz algebra model for Schubert calculus on finite Coxeter groups; positivity; (skew) divided difference operators; saturated chains in the Bruhat order Reflection and Coxeter groups (group-theoretic aspects), Classical problems, Schubert calculus, Combinatorial aspects of groups and algebras Skew divided difference operators in the Nichols algebra associated to a finite Coxeter group | 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 fine paper delivers a combinatorial formula for the characters of the homogeneous components of the coinvariant algebra---\(\mathbb{Q}[x_1,\dots, x_n]\) factored out by those polynomials invariant under the action of the symmetric group with no constant term.
It begins with a clear and concise exposition of all the ingredients needed including Schubert polynomials, Monk's formula, and Kazhdan-Lusztig cells. From here the derivation of the action of the symmetric group on Schubert polynomials and some related inner product results lead to a swift proof of the aforementioned character formula, which is also shown to be equivalent to the decomposition of homogeneous components of the coinvariant algebra into irreducible representations.
Finally we are given a taster for two subsequent works ``Deformation of the coinvariant algebra'' and ``Major index of shuffles and restriction of representations'', the former of which yields a \(q\)-analogue of the character formula in this paper, the latter an algebraic interpretation of the set of all permutations of length \(k\) in the symmetric group. coinvariant algebra; Kazhdan-Lusztig cells; Schubert polynomials; character formula; irreducible representations; symmetric group Y. Roichman, Schubert polynomials, Kazhdan--Lusztig basis and characters, (with an, appendix: On characters of Weyl groups, co-authored with, R. M. Adin, and, A. Postnikov, ), Discrete Math, to appear. Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Symmetric groups Schubert polynomials, Kazhdan-Lusztig basis and characters | 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 symmetrizable Kac-Moody algebra, and \(U_q ({\mathfrak g})\) the quantized enveloping algebra of \({\mathfrak g}\). Let \(\lambda\) be a dominant, integral weight, and \(V(\lambda)\) the corresponding simple \(U_q ({\mathfrak g})\)-module (\(q\) being generic). Let \(V_A (\lambda)\) be the canonical \(A\)-form of \(V(\lambda)\) where \(A= \mathbb{Z} [q,q^{-1} ]\). Let \(W\) be the Weyl group of \({\mathfrak g}\). For \(w\in W\), let \(V_{A,w}\) be the (quantum) Demazure submodule of \(V_A (\lambda)\). We construct an \(A\)-basis for \(V_A (\lambda)\) compatible with \(\{V_{a,w}\), \(w\in W\}\), consisting of \(\{De\}\), where \(e\) is the highest weight vector in \(V(\lambda)\), and \(D\) is either 1 or of the form \(F_{\beta_r}^{ (n_r)} \dots F_{\beta_1}^{ (n_1)}\), \(\beta_i\) simple, \(n_i>0\) (for some suitable \(n_i\)'s), and \(s_{\beta_r} \dots s_{\beta_1}\) is reduced. We also show that for \(w\in W\), the transition matrix from our basis for \(V_{A,w}\) to Kashiwara's global basis is upper triangular with diagonal entries equal to 1 (for a suitable indexing). We also give an explicit expression for the crystal base \(B(\lambda)\). Given \(w\in W\), and \(\alpha\) a simple root such that \(w< s_\alpha w\) (\(= \tau\), say), we exhibit a unique ``Demazure'' \(U_q (sl_2)\) structure on \(V_\tau/ V_w\). symmetrizable Kac-Moody algebra; quantized enveloping algebra; dominant, integral weight; Weyl group; Demazure submodule; crystal base V. Lakshmibai, Bases for quantum Demazure modules, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 199 -- 216. Quantum groups (quantized enveloping algebras) and related deformations, Representation theory for linear algebraic groups, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds Bases for quantum Demazure modules | 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 Given a vector bundle \(V\) of rank \(n\) on a variety \(X\), together with two complete flags of subbundles, there is a degeneracy locus \(X_w\subset X\) for each \(w\) in the symmetric group \(S_n\). With suitable genericity hypotheses, the class of \(X_w\) in the Chow group of \(X\) is given by a double Schubert polynomial in the first Chern classes of the quotient line bundles of the flags [\textit{W. Fulton}, Duke Math. J. 65, No. 3, 381-420 (1992; Zbl 0788.14044)]. In this note we give similar formulas for corresponding loci when \(V\) has an orthogonal or symplectic structure and the flags are isotropic; there is one such locus \(X_w\) for each \(w\) in the corresponding Weyl group. determinantal formulas; symplectic degeneracy loci; intersection of fibers; isotropic flags; complete flags; Weyl group Fulton, W, \textit{determinantal formulas for orthogonal and symplectic degeneracy} loci, J. Diff. Geom., 43, 276-290, (1996) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Determinantal varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) Determinantal formulas for orthogonal and symplectic degeneracy loci | 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 develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: One is derived from a Gröbner basis for the Plücker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed. numerical homotopy algorithms; systems of polynomial equations; Schubert calculus; SAGBI basis Huber, B; Sottile, F; Sturmfels, B, Numerical Schubert calculus, J. Symb. Comp., 26, 767-788, (1998) Enumerative problems (combinatorial problems) in algebraic geometry, Configurations and arrangements of linear subspaces, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Computational aspects of higher-dimensional varieties, Symbolic computation and algebraic computation Numerical 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 book aims to describe the beautiful connection between Schubert varieties and their Standard Monomial Theory (SMT) on the one hand and Classical Invariant Theory (CIT) on the other. The roots of SMT are to be found in the work of Hodge, who described nice bases for the homogeneous coordinate ring of Schubert varieties of the Grassmannian in the Plücker embedding (over a field of characteristic zero). Grassmannians being precisely the homogeneous spaces that arise as quotients of special linear groups by maximal parabolic subgroups, it is natural to try to generalize Hodge's approach to projective embeddings of other spaces \(G/Q\), where \(G\) is a semisimple algebraic group and \(Q\) a parabolic subgroup. In the early '70s Seshadri initiated this generalization and called it SMT.
CIT concerns diagonal actions of classical groups on direct sums of the tautological representations and their duals. A description of the algebra of invariants for these actions comprises of two theorems. The First Fundamental Theorem specifies a finite set of generators for the algebra of invariants, and the Second Fundamental Theorem provides a finite set of generators of the ideal of relations among the algebra generators. A central role here plays the article [\textit{C.~De Concini} and \textit{C.~Procesi}, ``A characteristic free approach to Invariant Theory'', Adv. Math. 21, 330--354 (1976; Zbl 0347.20025)], where the First Fundamental Theorem for all classical groups and the Second Fundamental Theorem for general, orthogonal and symplectic linear groups were obtained (the case when the characteristic of the ground field is zero goes back to H.~Weyl). The idea to use SMT in the proof of the First and the Second Fundamental Theorems appeared in [\textit{V.~Lakshmibai} and \textit{C. S.~Seshadri}, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1--54 (1978; Zbl 0447.14011)] and turned out to be very fruitful. More precisely, one should realize the subalgebra of the algebra of invariants generated by ``basic'' invariants (which will in fact coincide with the algebra of invariants) as the algebra of regular functions on an affine variety related to a Schubert variety. Then there is a morphism from the spectrum of the algebra of invariants to this affine variety. Using Zariski's Main Theorem, one shows that this is an isomorphism. A difficult part of this program is to prove that our affine variety is normal. Normality follows from normality of Schubert varieties, and that is a consequence of SMT. Nowadays this approach is realized in complete generality, and the book under review provides an excellent account of there results.
The authors tried to make the presentation self-contained keeping in mind the needs of prospective graduate students and young researchers. After a detailed introduction, generalities on algebraic varieties and algebraic groups are given. Next chapters are devoted to classical, symplectic and orthogonal Grassmannians, determinantal varieties, Geometric Invariant Theory (GIT), basic results of SMT and their interrelations with CIT. The proof of the main theorem of SMT is given in an appendix. The authors also included some important applications of SMT: to the determination of singular loci of Schubert varieties, to the study of some affine varieties related to Schubert varieties --- ladder determinantal varieties, quiver varieties, varieties of complexes, etc. --- and to toric degenerations of Schubert varieties.
The book may be recommended as a nice introduction to SMT and related active research areas. It may be used for a year long course on Invariant Theory and Schubert varieties. Classical Invariant Theory; Grassmannians; Schubert varieties; homogeneous coordinates Lakshmibai, V.\!; Raghavan, K.\,N.\!, Standard monomial theory, Encyclopaedia of Mathematical Sciences (Invariant Theory and Alg. Transform. Groups VIII) 137, (2008), Springer-Verlag, Berlin Grassmannians, Schubert varieties, flag manifolds, Actions of groups on commutative rings; invariant theory, Rings with straightening laws, Hodge algebras, Determinantal varieties, Classical groups (algebro-geometric aspects) Standard monomial theory. Invariant theoretic approach | 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 study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials \(f_{i,j}\) were introduced by \textit{T. Abe} et al. [``Hessenberg varieties and hyperplane arrangements'', Preprint, \url{arXiv:1611.00269}]. We show that every polynomial \(f_{i,j}\) is an alternating sum of certain Schubert polynomials. flag varieties; Hessenberg varieties; Schubert polynomials Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds The cohomology rings of regular nilpotent Hessenberg varieties and 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 \(L_w\) be the Levi part of the stabilizer \(Q_w\) in \(GL_N\) (for left multiplication) of a Schubert variety \(X(w)\) in the Grassmannian \(G_{d,N}\). For the natural action of \(L_w\) on \(\mathbb {C}[X(w)]\), the homogeneous coordinate ring of \(X(w)\) (for the Plücker embedding), we give a combinatorial description of the decomposition of \(\mathbb {C}[X(w)]\) into irreducible \(L_w\)-modules; in fact, our description holds more generally for the action of the Levi part \(L\) of any parabolic subgroup \(Q\) that is contained in \(Q_w\). This decomposition is then used to show that all smooth Schubert varieties, all determinantal Schubert varieties, and all Schubert varieties in \(G_{2,N}\) are spherical \(L_w\)-varieties. Schubert varieties; Levi subgroup; representation theory; coordinate rings; spherical varieties; algebraic groups Grassmannians, Schubert varieties, flag manifolds, Representations of finite groups of Lie type, Representation theory for linear algebraic groups, Compactifications; symmetric and spherical varieties, Combinatorial aspects of representation theory Levi subgroup actions on Schubert varieties, induced decompositions of their coordinate rings, and sphericity consequences | 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 apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-\(A\) affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a \(k\)-Schur function, consequently proving that a subclass of three-point Gromov-Witten invariants of complete flag varieties for \(\mathbb C^n\) enumerate the highest weight elements under these operators. Included in this class are the Schubert structure constants in the (quantum) product of a Schubert polynomial with a Schur function \(s_\lambda\) for all \(|\lambda^\vee| < n\). Another by-product gives a highest weight formulation for various fusion coefficients of the Verlinde algebra and for the Schubert decomposition of certain positroid classes. Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Crystal approach to affine 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 In this survey article, we review past results (obtained by Hirzebruch, Libgober-Wood, Salamon, Gritsenko, and Guan) on Hodge and Betti numbers of Kähler manifolds, and more specifically of hyper-Kähler manifolds, culminating in the bounds obtained by Guan in 2001 on the Betti numbers of hyper-Kähler fourfolds. Let \(X\) be a compact Kähler manifold of dimension \(m\). One consequence of the Hirzebruch-Riemann-Roch theorem is that the coefficients of the \(\chi_y\)-genus polynomial
\[
p_X (y):= \sum_{p, q=0}^m (-1)^q h^{p,q}(X)y^p \in\mathbb{Z}[y]
\]
are (explicit) universal polynomials in the Chern numbers of \(X\). In 1990, Libgober-Wood determined the first three terms of the Taylor expansion of this polynomial about \(y=-1\) and deduced that the Chern number \(\int_X c_1 (X)c_{m-1}(X)\) can be expressed in terms of the coefficients of the polynomial \(p_X (y)\) (Proposition 2.1). When \(X\) is a hyper-Kähler manifold of dimension \(m=2n\), this Chern number vanishes. The Hodge diamond of \(X\) also has extra symmetries which allowed Salamon to translate the resulting identity into a linear relation between the Betti numbers of \(X\) (Corollary 2.5). When \(X\) has dimension 4, Salamon's identity gives a relation between \(b_2 (X),b_3 (X),\) and \(b_4 (X)\). Using a result of Verbitsky's on the injectivity of the cup-product map that produces an inequality between \(b_2 (X)\) and \(b_4 (X)\), it is easy to conclude \(b_2 (X)\leq 23\). Guan established in 2001 more restrictions on the Betti numbers (Theorem 3.6). Holomorphic symplectic varieties, hyper-Kähler varieties, Algebraic topology on manifolds and differential topology, Global differential geometry of Hermitian and Kählerian manifolds, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Research exposition (monographs, survey articles) pertaining to differential geometry On the Hodge and Betti numbers of hyper-Kähler 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 For a quasi-homogeneous polynomial \(W\) with weights \((w_1, \ldots , w_N)\) and degree \(d\) (satisfying certain conditions), let \(X_W\) be the hypersurface in the weighted projective space \(\mathbb{P}(w_1, \ldots , w_N)\) defined by the vanishing of \(W\). Then, the Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence asserts the equivalence of two cohomological field theories (CohFTs): the Fan-Jarvis-Ruan-Witten (FJRW) theory of \(W\), and the Gromov-Witten (GW) theory of \(X_W\). The FJRW theory of \(W\) is defined via intersection numbers of the moduli spaces \(\mathcal{R}^d_{\vec k}, \epsilon \vec{l}\), parametrising \(\vec{l}\)-twisted \(d\)-spin structures on \((d,\epsilon)\)-stable curves. These intersection numbers are defined by integrating \(\psi\)-classes over the Witten class of \(\mathcal{R}^d_{\vec k}\).
It is expected that the LG/CY correspondence could be proved using the theory of \textit{gauged linear sigma model} (GLSM) of Witten (for a mathematically rigorous definition of the GLSM see [\textit{H. Fan} et al., Geom. Topol. 22, No. 1, 235--303 (2018; Zbl 1388.14041)]. In the case of hypersurfaces, the GLSM is a one-dimensional family of CohFTs parametrised by the rational numbers different from zero. The CohFTs lying over \(\mathbb{Q}_{>0}\) (the so-called geometric phase) are related to certain versions of quasi-maps, while those over \(\mathbb{Q}_{<0}\) (the Landau-Ginzburg phase) correspond to the FJRW theory of \(W\).
The main results of the the paper under review concerns the CohFTs lying over \(\mathbb{Q}_{<0}\). For a Fermat polynomial \(W\), the authors define the \textit{genus zero descendant potential} \(\mathcal{F}^\epsilon_W\), for any \(\epsilon \in \mathbb{Q}_{>0}\) (the authors normilize the GLSMs in such a way to work with positive \(\epsilon\)). This is a generating function for the intersection numbers of the moduli spaces \(\mathcal{R}^d_{\vec k}\) above. For \(\epsilon >1\) (denoted \(\epsilon = \infty\)) one recovers the narrow FJRW descendant potential \(\mathcal{F}^\infty\). Following Givental, the authors define a certain infinite-dimensional vector space \(\mathcal{H}\), and the graph of the differential of \(\mathcal{F}^\infty\) is the formal germ of a Lagrangian cone \(\mathcal{L} \subset \mathcal{H}\), whose geometry reflects the properties of \(\mathcal{F}^\infty\). The derivatives of \(\mathcal{F}^\epsilon_W\) with respect to \(t\) yields the so-called large \(\mathcal{J}^\epsilon (t,u,-z)\)-functions. The main theorem of the paper (Theorem 1.11) says that, for any \(\epsilon >0\), \(\mathcal{J}^\epsilon (t,u,-z)\) is an \(\mathcal{H} (u)\)-valued point of \(\mathcal{L}\). This means that \(\mathcal{J}^\epsilon (t,u,-z)\) is a formal series of a certain form.
In Section 3 the authors derive several important consequences of the previous main theorem. The first one is a formula that relates \(\mathcal{J}^{\epsilon_1}\) with \(\mathcal{J}^{\epsilon_2}\), for different \(\epsilon_1, \epsilon_2 >0\). Furthermore, for \(\epsilon \to 0\) they obtain a new geometric interpretation of the LG mirror theorem. Fan-Jarvis-Ruan-Witten theory; Landau-Ginzburg/Calabi-Yau correspondence; mirror symmetry Ross, D.; Ruan, Y., Wall-crossing in genus zero Landau-Ginzburg theory, (2015) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Mirror symmetry (algebro-geometric aspects), String and superstring theories; other extended objects (e.g., branes) in quantum field theory Wall-crossing in genus zero Landau-Ginzburg 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 It is known that multiplication of Schubert classes in the (small) quantum cohomology ring of a Grassmannian can be carried out using the rim hook algorithm (cf. [\textit{A. Bertram} et al., J. Algebra 219, No. 2, 728--746 (1999; Zbl 0936.05086)]). In this paper, the authors give an equivariant generalization of the rim hook algorithm which computes the product of equivariant canonical Schubert classes in the equivariant quantum cohomology of a Grassmannian. To be more precise, let \(T^n\) be an \(n\)-dimensional torus acting naturally on \(\mathrm{Gr}(k, n)\). The authors define a group homomorphism
\[
\varphi: H^*_{T^{2n-1}}(\mathrm{Gr}(k, 2n-1))\to QH_{T^n}(\mathrm{Gr}(k, n))
\]
which sends the \(i\)-th equivariant variable \(t_i\) to \(t_{i(\operatorname{mod}n)}\), and implements the classical rim hook algorithm to equivariant canonical Schubert classes. The equivariant Rim hook algorithm says that in order to compute the equivariant quantum product \(\sigma_\lambda\star\sigma_\mu\in QH_{T^n}^*(\mathrm{Gr}(k, n))\), where \(\lambda\) and \(\mu\) are Young diagrams within the \(k\times(n-k)\) rectangle, one can first compute within \(\mathrm{Gr}(k, 2n-1)\) the equivariant product
\[
\sigma_\lambda\cdot\sigma_\mu=\sum_{\nu\subseteq k\times(2n-1-k)}c_{\lambda\mu}^\nu\sigma_\nu\in H_{T^{2n-1}}^*(\mathrm{Gr}(k, 2n-1))
\]
which, for example, can be done using the Knutson-Tao puzzle rule (cf. [\textit{A. Knutson} and \textit{T. Tao}, Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)]), and then apply the map \(\varphi\):
\[
\sigma_\lambda\star\sigma_\mu=\sum_{\nu\subseteq k\times(2n-1-k)}\varphi(c_{\lambda\mu}^\nu)\varphi(\sigma_\nu).
\]
The authors prove the equivariant rim hook algorithm by showing that the product it defines satisfies two conditions which characterize equivariant quantum product, namely, associativity and the equivariant quantum Chevalley-Monk formula. In proving associativity, they introduce what they call the abacus model for Young diagrams in order to understand the action of \(\varphi\) on the equivariant variables. The authors also outline future directions relating the equivariant rim hook algorithm and cyclic factorial Schur polynomials they develop in this paper to the work on the connections between quantum and affine Schubert calculus by Peterson and Postnikov. Schubert calculus; equivariant quantum cohomology; factorial Schur polynomial; abacus diagram; core partition Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Equivariant homology and cohomology in algebraic topology, Symmetric functions and generalizations Equivariant quantum cohomology of the Grassmannian via the rim hook 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 In this expository paper, the author discusses some results, both old and new, on the cohomology of flag varieties, Schubert subvarieties and Springer fibres. He begins by describing the technique for calculating cohomology from local data at the zeros of a holomorphic vector field. This is then applied to flag varieties and the subvarieties mentioned above; calculations are possible in the usual finite-dimensional case and also in the Kac-Moody setting. The paper finishes with a presentation of some of D. Peterson's recent generalisation of the Schubert calculus. cohomology of flag varieties; Schubert subvarieties; Springer fibres J. Carrell. Vector fields, flag manifolds and Schubert calculus. Proc. 1989 Hyderabad Conference on Algebraic Groups and Applications, 23--55. Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Vanishing theorems, Vanishing theorems in algebraic geometry Vector fields, flag varieties 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 The author surveys a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for the Macdonald symmetric functions, and the ``\(n!\)'' and ``\((n+1)^{n-1}\)'' conjectures which relate Macdonald polynomials to the characters of doubly graded \(S_n\)-modules.
In 1987 Macdonald unified the theory of Hall-Littlewood symmetric functions with that of spherical functions on symmetric spaces, introducing a class of symmetric functions, now known as Macdonald polynomials, with coefficients depending on two parameters \(q\) and \(t\). There are bivariate analogues of the Kostka-Foulkes polynomials, and Macdonald conjectured that these more general Kostka-Foulkes polynomials should have positive integer coefficients. In 1993 Garsia and the author introduced some bigraded \(S_n\)-module and conjectured that its dimension is \(n!\) and this is known as the \(n!\) conjecture. It is known that the \(n!\) conjecture implies the Macdonald positivity conjecture. The spaces figuring in the \(n!\) conjecture are quotients of the ring of coinvariants for the diagonal action of \(S_n\) on \({\mathbb C}^n\oplus{\mathbb C}^n\), and it was natural to investigate the characters of the full coinvariant ring. The space of coinvariants has dimension \((n+1)^{n-1}\). This involved \(q\)-analogues of this number and the Catalan numbers \(C_n\) in the data. A menagerie of things studied earlier by combinatorialists for their own sake turned up unexpectedly in this new context. Further, Procesi suggested that the diagonal coinvariants might be interpreted as sections of a vector bundle on the Hilbert scheme \(H_n\) of points in the plane. Understanding this geometric context has led the author to the proofs of the \(n!\) and \((n+1)^{n-1}\) conjectures. The full explanation depends on properties of the Hilbert scheme which were not known before, and had to be established from scratch in order to complete the picture. One might say, that the main results are not the \(n!\) and \((n+1)^{n-1}\) theorems, but new theorems in algebraic geometry.
In order to make the exposition self-contained, the author includes background from combinatorics, theory of symmetric functions, representation theory and geometry. At the end of the paper he discusses future directions, new conjectures, and related work of other mathematicians. Macdonald polynomials; symmetric polynomials; partitions; Hilbert scheme; \(n!\) conjecture Mark Haiman, Combinatorics, symmetric functions, and Hilbert schemes, Current developments in mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 39 -- 111. Symmetric functions and generalizations, Parametrization (Chow and Hilbert schemes) Combinatorics, symmetric functions, and Hilbert schemes | 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 goal of this paper is at least two-fold. First we attempt go give a survey of some recent applications of the theory of symmetric polynomials and divided differences to intersection theory. Secondly, taking this opportunity, we complement the story by either presenting some new proofs of older results or providing some new results which arose as by-products of the author's work in this domain during last years. Being in the past a good part of the classical algebraic knowledge (related for instance to the theory of algebraic equations and elimination theory), the theory of symmetric functions is rediscovered and developed nowadays. Here, we discuss only some geometric applications of symmetric polynomials which are related to the present interest of the author. In particular, the theory of polynomials universally supported on degeneracy loci is surveyed in Section 1.
Divided differences appeared already in the interpolation formulas of I. Newton. Their appearance in intersection theory is about twenty years old starting with the papers [Russ. Math. Surv. 28, No. 3, 1-26 (1973; Zbl 0286.57025)] of \textit{I. N. Bernstein}, \textit{I. M. Gelfand} and \textit{S. I. Gelfand} and [Invent. Math. 21, 287-301 (1973; Zbl 0269.22010) and Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)] of \textit{M. Demazure}. A recent work [Duke Math. J. 65, No. 3, 381-420 (1992; Zbl 0788.14044)] of \textit{W. Fulton} has illuminated the importance of divided differences to flag degeneracy loci. This was possible thanks to the algebraic theory of Schubert polynomials developed recently by A. Lascoux and M.-P. Schützenberger.
The geometrical objects we study are: (ample) vector bundles, degeneracy loci of vector bundle homomorphisms, flag varieties, Grassmannians including isotropic Grassmannians, i.e. the parameter spaces for isotropic subspaces of a given vector space endowed with an antisymmetric or symmetric form, Schubert varieties and the parameter spaces of complete quadrics.
The algebro-combinatorial tools we use are: Schur polynomials including supersymmetric and \(Q\)-polynomials, binomial determinants and Pfaffians, divided differences, Schubert polynomials of Lascoux and Schützenberger, reduced decompositions in the Weyl groups and Young-Ferrers' diagrams.
The content of the article is as follows: (1) Polynomials universally supported on degeneracy loci, (2) Some explicit formulas for Chern and Segre classes of tensor bundles with applications to enumerative geometry, (3) Flag degeneracy loci and divided differences, (4) Gysin maps and divided differences, (5) Fundamental classes, diagonals and Gysin maps, (6) Intersection rings of spaces \(G/P\), divided differences and formulas for isotropic degeneracy loci, (7) Numerically positive polynomials for ample vector bundles with applications to Schur polynomials of Schur bundles and a vanishing theorem.
Apart of surveyed results, the paper contains also some new ones. Perhaps the most valuable contribution, contained in Section 5, is provided by a method of computing the fundamental class of a subscheme using the class of the diagonal of the ambient scheme. The class of the diagonal can be determined with the help of Gysin maps (see Section 5). This method has been applied successfully in [the author and \textit{J. Ratajski}, Formulas for Lagrangian and orthogonal degeneracy loci, to appear in Compos. Math.] and seems to be useful also in other settings. Other results that appear to be new are contained in Proposition 1.3(ii), Proposition 2.1 and Corollary 7.2. Moreover, the paper is accompanied by a series of appendices which contain an original material but of more technical nature than the main text of the paper. Some proofs in the Appendices use an operator approach and the operators involved are mostly divided differences. This point of view leads to more natural proofs of many results than the ones known before, and we hope to develop it in our forthcoming book (in collaboration with Laksov, Lascoux, and Thorup). symmetric polynomials; divided differences; intersection theory; symmetric functions; polynomials universally supported on degeneracy loci; flag degeneracy loci; flag varieties; Grassmannians; Schubert varieties; Schur polynomials; \(Q\)-polynomials; determinants; Pfaffians; Weyl groups; Young-Ferrers' diagrams; Segre classes; tensor bundles; Gysin maps; vector bundles; Schur bundles; vanishing theorem P. Pragacz, ''Symmetric polynomials and divided differences in formulas of intersection theory,'' in Parameter Spaces, Warsaw: Polish Acad. Sci., 1996, vol. 36, pp. 125-177. Symmetric functions and generalizations, (Equivariant) Chow groups and rings; motives, Algebraic cycles Symmetric polynomials and divided differences in formulas of intersection 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 describe the \( \widetilde G\times \widetilde G\)-equivariant \( K\)-ring of \( X\), where \( \widetilde G\) is a \textit{factorial} covering of a connected complex reductive algebraic group \( G\), and \( X\) is a regular compactification of \( G\). Furthermore, using the description of \( K_{\widetilde G\times\widetilde G}(X)\), we describe the ordinary \( K\)-ring \( K(X)\) as a free module (whose rank is equal to the cardinality of the Weyl group) over the \( K\)-ring of a toric bundle over \( G/B\) whose fibre is equal to the toric variety \( \overline{T}^{+}\) associated with a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see [1]). We also give an explicit presentation of \( K_{\widetilde G\times\widetilde G}(X)\) and \( K(X)\) as algebras over \( K_{\widetilde G\times\widetilde G}(\overline{G_{\operatorname{ad}}})\) and \( K(\overline{G_{\operatorname{ad}}})\) respectively, where \( \overline{G_{\operatorname{ad}}}\) is the wonderful compactification of the adjoint semisimple group \( G_{\operatorname{ad}}\). In the case when \( X\) is a regular compactification of \( G_{\operatorname{ad}}\), we give a geometric interpretation of these presentations in terms of the equivariant and ordinary Grothendieck rings of a canonical toric bundle over \( \overline{G_{\operatorname{ad}}}\). equivariant \(K\)-theory; regular compactification; wonderful compactification; toric bundle Equivariant \(K\)-theory, Toric varieties, Newton polyhedra, Okounkov bodies, Compactifications; symmetric and spherical varieties, Group varieties Equivariant \( K\)-theory of regular compactifications: further developments | 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 orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group \(S_n\). Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote as \({\hat{\mathfrak S}}_y\) (to be called \textit{involution Schubert polynomials}) and \(\hat{\mathfrak S}^{\mathtt{FPF}}_y\) (to be called \textit{fixed-point-free involution Schubert polynomials}). Our main results are explicit formulas decomposing the product of \({\hat{\mathfrak S}}_y\) (respectively, \(\hat{\mathfrak S}^{\mathtt{FPF}}_y\)) with any \(y\)-invariant linear polynomial as a linear combination of other involution Schubert polynomials. These identities serve as analogues of Lascoux and Schützenberger's transition formula for Schubert polynomials, and lead to a self-contained algebraic proof of the nontrivial equivalence of several definitions of \({\hat{\mathfrak S}}_y\) and \( \hat{\mathfrak S}^{\mathtt{FPF}}_y\) appearing in the literature. Our formulas also imply combinatorial identities about \textit{involution words}, certain variations of reduced words for involutions in \(S_n\). We construct operators on involution words based on the Little map to prove these identities bijectively. The proofs of our main theorems depend on some new technical results, extending work of Incitti, about covering relations in the Bruhat order of \(S_n\) restricted to involutions. Symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Reflection and Coxeter groups (group-theoretic aspects), Compactifications; symmetric and spherical varieties Transition formulas for involution 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 By a diagram the authors mean a finite collection of cells in \({\mathbb Z}\times {\mathbb Z}\). They consider balanced labellings of diagrams. Special cases of these objects are the standard Young tableaux and the balanced tableaux introduced by \textit{P. Edelman} and \textit{C. Greene} [Adv. Math. 63, 42-99 (1987; Zbl 0616.05005)]. It turns out that for certain diagrams associated with permutations of the symmetric group \(\Sigma_n\), the set of balanced labellings has a remarkable rich structure. Balanced labellings of permutation diagrams yield the symmetric functions introduced by \textit{R. P. Stanley} [Eur. J. Comb. 5, 359-372 (1984; Zbl 0587.20002)] in the same way as the Schur functions can be constructed from column-strict tableaux. Balanced labelled diagrams can be also viewed as encodings of reduced decompositions of permutations. Imposing flag conditions, the authors obtain the Schubert polynomials of Lascoux and Schützenberger. Finally the authors construct an explicit basis for the Schubert module introduced in 1986 by W. Kraskiewicz and P. Pragacz (i.e. the representation of the upper triangular group with formal character equal to the corresponding Schubert polynomial). diagram; tableaux; Subert polynomials; symmetric functions; Schubert module S. Fomin, C. Greene, V. Reiner, and M. Shimozono, ''Balanced labellings and Schubert polynomials,'' European J. Combin. 18 (1997), no. 4, 373--389. the electronic journal of combinatorics 25(3) (2018), #P3.4622 Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Balanced labellings and 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 author extends the approach to the construction of parametric families of \(r\)-functions as generating functions for weighted Hurwitz numbers which are initiated by him and Guay-Paquet and were extended to other cases by him and others. The general method is developed and used to derive an infinite parametric family of \(2D\) Toda \(r\)-functions of hypergeometric type depending also on an additional pair \((q,t)\) of quantum deformation parameters entering in the definition of the scalar product. For specific choices of the parameters defining the weight generating function the author gives specialized versions of the quantum weighted Hurwitz numbers. By making other specializations involving particular values for the pair \((q,t)\) or their limits reduce the Macdonald or Schur or Hall-Littlewood or Jack polynomials. hypergeometric functions; Macdonald polynomials; generating functions; enumerative combinatorics Harnad, J., Quantum Hurwitz numbers and Macdonald polynomials, J. Math. Phys., 57, 113505, (2016) Bessel and Airy functions, cylinder functions, \({}_0F_1\), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Coverings of curves, fundamental group, Exact enumeration problems, generating functions Quantum Hurwitz numbers and Macdonald polynomials | 0 |
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