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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(A\) be a matrix whose entries are algebraic functions defined on a reduced quasi-projective algebraic set \(X, e.g\). multivariate polynomials defined on \(X:= \mathbb C N\) . The sets \({\mathcal S}_k(A)\), consisting of \(x\epsilon X\) where the rank of the matrix function \(A(x)\) is at most \(k\), arise in a variety of contexts: for example, in the description of both the singular locus of an algebraic set and its fine structure; in the description of the degeneracy locus of maps between algebraic sets; and in the computation of the irreducible decomposition of the support of coherent algebraic sheaves, \(e.g\). supports of finite modules over polynomial rings. In this article we present a numerical algorithm to compute the sets \(\mathcal S_k(A)\) efficiently. rank deficiency; matrix of polynomials; homotopy continuation; irreducible components; numerical algebraic geometry; polynomial system; Grassmannians D.J. Bates, J.D. Hauenstein, C. Peterson, and A.J. Sommese, \textit{Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials}, in Approximate Commutative Algebra, Texts Monogr. Symbol. Comput., Springer, Vienna, 2009, pp. 55--77. Computational aspects of higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds, Computational aspects and applications of commutative rings, Matrices over function rings in one or more variables, Numerical computation of solutions to systems of equations Numerical decomposition of the rank-deficiency set of a matrix of multivariate 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 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 Let X be the fixed point set in the flag variety of a unipotent element u in \(GL_ n\). The irreducible components of X are in bijective correspondence with standard Young tableaux. The intersection properties of these components have connections with the theory of Kazhdan-Lusztig polynomials. The author determines the intersection codimensions in the case that u has two Jordan blocks. He uses the methods of \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Astérisque 87-88, 249-266 (1981; Zbl 0504.20007)]. fixed point set; flag variety; unipotent element; irreducible components; Young tableaux; Kazhdan-Lusztig polynomials; intersection codimensions; Jordan blocks J.Wolper, Some intersection properties of the fibres of Springer's resolution, Proc. Am. Math. Soc. 91 (1984), 182--188. Representation theory for linear algebraic groups, Classical groups (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Global theory and resolution of singularities (algebro-geometric aspects) Some intersection properties of the fibres of Springer's resolution | 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 varieties are Borel orbits in homogeneous spaces. The study of their geometric properties is old and well developed. Richardson varieties are defined as intersections of Schubert varieties and `opposite' Schubert varieties, they are also key objects in geometry. The authors prove a very general statement claiming that essentially all questions concerning singularities of Richardson varieties reduce to corresponding questions about Schubert varieties. The properties/quantities reduced from Schubert varieties to Richardson varieties include smoothness and the property `Cohen-Macaulay with rational singularities', as well as multiplicity and H-polynomial. The authors give two versions of their elementary proof, one in the language of algebraic groups and one that avoids it. Richardson variety; Schubert variety; singularity A. Knutson, A. Woo, and A. Yong, Singularities of Richardson varieties, \textit{Math. Res. Lett.}, 20 (2013), no. 2, 391--400.Zbl 1298.14053 MR 3151655 Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Singularities of Richardson 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 Given a Schubert class on \(\mathrm{Gr}(k,V)\) where \(V\) is a symplectic vector space of dimension \(2n\), we consider its restriction to the symplectic Grassmannian \(\mathrm{SpGr}(k,V)\) of isotropic subspaces. \textit{P. Pragacz} [J. Algebra 226, No. 1, 639--648 (2000; Zbl 0945.05065)] gave tableau formulae for positively computing the expansion of these \(H^\ast(\mathrm{Gr}(k,V))\) classes into Schubert classes of the target when \(k=n\), which corresponds to expanding Schur polynomials into Q-Schur polynomials. \textit{I. Coşkun} [J. Comb. Theory, Ser. A 125, 47--97 (2014; Zbl 1296.14037)] described an algorithm for their expansion when \(k\le n\). We give a puzzle-based formula for these expansions, while extending them to equivariant cohomology. We make use of a new observation that usual Grassmannian puzzle pieces are already enough to do some 2-step Schubert calculus, and apply techniques from quantum integrable systems (``scattering diagrams''). Schubert calculus; puzzles; Grassmannian; symplectic Grassmannian Grassmannians, Schubert varieties, flag manifolds Restricting Schubert classes to symplectic Grassmannians using self-dual puzzles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show how to compute the structure constants for cohomological multiplication of Schubert classes by exploiting the action of the Weyl group
and that of BGG-operators, on the cohomology ring of a flag variety. We illustrate this method with simple proofs of the Chevalley and Pieri formulas. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry Multiplying 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 the entire collection see Zbl 0509.00015.]
Consider the Grassmann variety G(k,n). For each non-increasing sequence of non-negative integers \(a=(a_ 1,...,a_ k)\) with \(n-k\geq a_ 1\), let \(\sigma_ a\) denote the homology class of a corresponding Schubert cycle. The intersection cycle of \(\sigma_ a\) and \(\sigma_ b\) can be written \(\sigma_ a\cdot \sigma_ b=\sum_{c}\delta(a,b;c)\sigma_ c\), where \(\delta\) (a,b;c) denotes the intersection number of \(\sigma_ a\cdot \sigma_ b\) with \(\sigma_{\tilde c}\) and where \(\tilde c=(n-k- c_ k,...,n-k-c_ 1)\). No general formula is known for this intersection number. The purpose of the present paper is to give a recursive formula for \(\delta\) (a,b;c) involving the intersection numbers \(\delta\) (a,b;d) for all \(d<c\) (in a suitable odering). The proof of the formula involves representation theory for the unitary group U(k). Schubert calculus; Grassmann variety; Schubert cycle; representation of unitary group; intersection numbers Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Representation theory for linear algebraic groups A note on 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 provide bijections between the cluster variables (and clusters) in two families of cluster algebras which have received considerable attention. These cluster algebras are the ones associated with certain Grassmannians of \(k\)-planes, and those associated with certain spaces of decorated \(\mathrm{SL}_k\)-local systems in the disk in the work of \textit{V. Fock} and \textit{A. Goncharov} [Publ. Math., Inst. Hautes Étud. Sci. 103, 1--211 (2006; Zbl 1099.14025)]. When \(k\) is 3, this bijection can be described explicitly using the combinatorics of Kuperberg's basis of non-elliptic webs. Using our bijection and symmetries of these cluster algebras, we provide evidence for conjectures of \textit{S. Fomin} and \textit{P. Pylyavskyy} [``Webs on surfaces, rings of invariants, and cluster algebras'', Preprint, \url{arXiv:1308.1718}] concerning cluster variables in Grassmannians of 3-planes. We also prove their conjecture that there are infinitely many indecomposable nonarborizable webs in the Grassmannian of 3-planes in 9-dimensional space. cluster algebra; web; quasi-isomorphism; Grassmannian Combinatorial aspects of algebraic geometry, Cluster algebras, Grassmannians, Schubert varieties, flag manifolds Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract) | 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 \(T\) be a maximal torus of \(G\), \(B\) a Borel subgroup of \(G\) containing \(T\) and \(W\) the Weyl group of \(G\). Let \(F = G/B\) be the full flag variety and let \(X_w\subset F\) be the Schubert variety, for any \(w\in W\). In the paper under review, the authors prove that if every irreducible component of \(G\) is of type \(B_n\) or \(C_n\), and \(w_1, w_2\) are two distinct involutions in \(W\), then the tangent cones at the base point \(eB\in X_{w_i}\) to the corresponding Schubert subvarieties \(X_{w_1} , X_{w_2}\) in \(F\) do not coincide as subschemes of the tangent space \(T_{eB}(F)\). Similarly, they also show that if every irreducible component of \(G\) is of type \(A_n\) or \(C_n\), then the reduced tangent cones to \(X_{w_1} , X_{w_2}\) do not coincide as subvarieties of \(T_{eB}(F)\). Their first result is proved by using (what they call) Kostant-Kumar polynomials (so was the following result by Eliseev and Ignatyev). Their second result is proved by using a connection between the tangent cones of \(X_w\) and the geometry of coadjoint orbits of \(B\). In a previous work [J. Math. Sci., New York 199, No. 3, 289--301 (2014); translation from Zap. Nauchn. Semin. POMI 414, 82--105 (2013; Zbl 1312.14116)], \textit{D. Yu. Eliseev} and \textit{M. V. Ignatyev} had proved that \(X_{w_1} , X_{w_2}\) (for distinct involutions \(w_1, w_2\)) do not coincide as subschemes of the tangent space \(T_{eB}(F)\) when the irreducible components of \(G\) are of type \(A_n\), \(F_4\) and \(G_2\) only (partially confirming an earlier conjecture by Panov in 2011 for any \(G\)). flag variety; Schubert variety; tangent cone; reduced tangent cone; involution in the Weyl group; Kostant-Kumar polynomial Bochkarev, M; Ignatyev, M; Shevchenko, A, Tangent cones to Schubert varieties in types \(A_n\), \(B_n\) and \(C_n\), J. Algebra, 465, 259-286, (2016) Grassmannians, Schubert varieties, flag manifolds, Root systems, Classical groups (algebro-geometric aspects) Tangent cones to Schubert varieties in types \(A_{n}\), \(B_{n}\) and \(C_{n}\) | 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 very thorough (over 100 page) manuscript, the authors provide a careful description of the moduli space of matroids. Motivated by the notion from algebraic geometry of using Grassmannians as a functor from schemes to sets, the authors build a full subcategory of ordered blueprints. Idylls; ordered blueprints; moduli spaces; tract; hyperfield Combinatorial aspects of matroids and geometric lattices, Grassmannians, Schubert varieties, flag manifolds The moduli space of matroids | 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 determine the Bruhat cells in \(G/B_+\) which are reached by exp(tX), when X is a Jacobi element of a semisimple complex Lie algebra G. They relate this result to the singularities in the complex domain of the solutions of the Toda lattices studied by \textit{B. Kostant} [Adv. Math. 34, 195-338 (1979; Zbl 0433.22008)]. flag manifold; Toda lattice; Bruhat cells; semisimple complex Lie algebra Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Simple, semisimple, reductive (super)algebras, Grassmannians, Schubert varieties, flag manifolds Variétés de drapeaux et réseaux de Toda. (Flag manifolds and Toda lattices) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [``The class of the affine line is a zero divisor in the Grothendieck ring'', Preprint, \url{arXiv:1412.6194}], \textit{L. A. Borisov} has shown that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. We improve the final formula by removing a factor. Martin, N., \textit{the class of the affine line is a zero divisor in the Grothendieck ring: an improvement}, C. R. Math. Acad. Sci. Paris, 354, 936-939, (2016) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Varieties and morphisms, Grassmannians, Schubert varieties, flag manifolds The class of the affine line is a zero divisor in the Grothendieck ring: an improvement | 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 Review of [Zbl 1310.13002]. External book reviews, Research exposition (monographs, survey articles) pertaining to commutative algebra, Cluster algebras, Combinatorial aspects of commutative algebra, Grassmannians, Schubert varieties, flag manifolds, Poisson algebras, Root systems, \(n\)-dimensional polytopes Book review of: R. J. Marsh, Lecture notes on cluster 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 Let \(\mathbb R_{\mathrm{triv}}\) and \(\mathbb R_{\mathrm{sgn}}\) be the trivial and nontrivial irreducible representations of the cyclic group \(C_2\) of order \(2\) and \(\mathbb R^{p,q} = (\mathbb R_{\mathrm{triv}})^{p-q} \oplus (\mathbb R_{\mathrm{sgn}})^q\). The action of \(C_2\) on \(\mathbb R^{p,q}\) induces one on \(Gr_k(\mathbb R^{p,q})\), the Grassmannian of \(k\) dimensional subspaces of \(\mathbb R^{p,q}\). The author computes the \(RO(C_2)\)-graded Bredon cohomology of \(Gr_k(\mathbb R^{n,1})\) and \(Gr_2(\mathbb R^{n,2})\) with coefficients in a constant \(\mathbb Z_2\)-valued Mackey functor as a module over the cohomology of a point. equivariant topology; Grassmannian manifold; Bredon cohomology Homology with local coefficients, equivariant cohomology, Equivariant algebraic topology of manifolds, Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) \(RO(C_2)\)-graded cohomology of equivariant Grassmannian 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 Let \(V\) be an \(N\)-dimensional complex vector space equipped with a symmetric or skew-symmetric bilinear form \(\omega\), which can be either trivial or non-degenerate. The Grassmannians \(I G_{\omega}(m, N)\) of classical Lie type parameterize \(m\)-dimensional isotropic vector subspaces of \(V\). The cohomology ring of an isotropic Grassmannian \(X =I G_{\omega}(m, N)\), or more generally of a homogeneous variety, has an additive basis of Schubert classes represented by Schubert subvarieties \(X_{\lambda}\). One of the central problems of Schubert calculus is to find a manifestly positive formula for the structure constants of the cup product of two Schubert cohomology classes, or equivalently, for the triple intersection numbers of three Schubert subvarieties in general position. Such a positive formula, called a Littlewood-Richardson rule, has deep connections to various subjects, including geometry, combinatorics and representation theory.
An isotropic Grassmannian \(X\) can be written as a quotient of a classical complex simple Lie group \(G\) by a maximal parabolic subgroup \(P\) (with two notable exceptions of Lie type \(D_n\)). Fix a choice of maximal complex torus \(T\) and a Borel subgroup \(B\) with \(T \subset B \subset P\). The Schubert varieties \(X_{\lambda}\) are closures of \(B\)-orbits, and hence are \(T\)-stable. They give a basis \([X_{\lambda}]^T\) for the \(T\)-equivariant cohomology \({H^*}_T (X)\) as a \({H^*}_T (pt)\)-module. The structure coefficients \({N^{\nu}}_{\lambda,\mu}\) in the equivariant product,
\[
[X_{\lambda}]^T \cdot [X_{\mu}]^T=\sum\limits_{\nu}{N^{\nu}}_{\lambda,\mu} [X_{\nu}]^T,
\]
are homogeneous polynomials which satisfy a positivity condition conjectured by \textit{D. Peterson} [Lectures on quantum cohomology of \(G/B\), MIT (1996)] and proved by \textit{W. Graham} [Duke Math. J. 109, No. 3, 599--614 (2001; Zbl 1069.14055)]. In particular, they are Graham-positive, meaning they are polynomials in the negative simple roots, with non-negative integer coefficients. These equivariant structure coefficients carry much more information than the triple intersection numbers of Schubert varieties, and are more challenging to study.
In the present paper, the authors give for the first time an equivariant Pieri rule for Grassmannians of Lie types \(B, C\), and \(D\), as well as a new proof of the Pieri rule in type \(A\). Such a rule concerns products with the special Schubert classes \([X_{p}]^T\) , which are related to the equivariant Chern classes of the tautological quotient bundle, and generate the \(T\) -equivariant cohomology ring. Using geometric methods, they give a manifestly positive formula for the structure coefficients \({N^{\mu}}_{\lambda,p}\) of the equivariant multiplication \([X_{\lambda}]^T \cdot [X_{p}]^T\). isotropic Grassmannian; Schubert calculus; Littlewood-Richardson rule; \(T\)-equivariant cohomology; Graham-positive; equivariant Pieri rule Li, C.; Ravikumar, V.: Equivariant Pieri rules for isotropic grassmannians Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Equivariant homology and cohomology in algebraic topology Equivariant Pieri rules for isotropic Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quaternionic Grassmannian \(\operatorname{H Gr}(r,n)\) is the affine open subscheme of the usual Grassmannian parametrizing those \(2r\)-dimensional subspaces of a \(2n\)-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular, we have \(\operatorname{HP}^n = \operatorname{H Gr}(1,n+1)\). For a symplectically oriented cohomology theory \(A\), including oriented theories but also the Hermitian \(\operatorname{K} \)-theory, Witt groups, and algebraic symplectic cobordism, we have \(A(\operatorname{HP}^n) = A(\operatorname{pt})[p]/(p^{n+1})\). Borel classes for symplectic bundles are introduced in the paper. They satisfy the splitting principle and the Cartan sum formula, and they are used to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes.
The cell structure of the \(\operatorname{H Gr}(r,n)\) exists in cohomology, but it is difficult to see more than part of it geometrically. An exception is \(\operatorname{HP}^n\) where the cell of codimension \(2i\) is a quasi-affine quotient of \(\mathbb{A}^{4n-2i+1}\) by a nonlinear action of \(\mathbb{G}_a\). symplectically oriented cohomology theory; Hermitian \(K\)-theory; Witt groups; algebraic symplectic cobordism; cell structure splitting principle Grassmannians, Schubert varieties, flag manifolds Quaternionic Grassmannians and Borel classes in algebraic geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple simply-laced algebraic group of adjoint type over the field \(C\) of complex numbers, \(B\) be a Borel subgroup of \(G\) containing a maximal torus \(T\) of \(G\). In this article, we show that \(\omega_\alpha\) is a minuscule fundamental weight if and only if for any parabolic subgroup \(Q\) containing \(B\) properly, there is no Schubert variety \(X_Q(w)\) in \(G/Q\) such that the minimal parabolic subgroup \(P_\alpha\) of \(G\) is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of \(X_Q(w)\). minuscule weights; co-minuscule roots; Schubert varieties; automorphism groups Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Minimal parabolic subgroups and automorphism groups 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 Computer Vision, from a particular scene consisting of images of projected objects, the reconstruction problem asks to recover the position of such objects as well as the projections. In this process, we can find in the projective space of the objects, sets of points for which the projective reconstruction fails; theses configurations of points constitute varieties called the critical loci.
In this paper, the authors work with projections both of points from \(\mathbb{P}^4\) to \(\mathbb{P}^2\) and of lines from \(\mathbb{P}^3\) to \(\mathbb{P}^2\). Since three views is the minimum number needed to reconstruct a scene, they consider three projections.
In the case of projections of points, the approach used here is to compute the equations of the critical locus by means of the Grassmann tensors introduced in [\textit{R. I. Hartley} and \textit{F. Schaffalitzky}, Lect. Notes Comput. Sci. 3021, 363--375 (2004; Zbl 1098.68775)]. The authors prove that, in the general case, the ideal of this locus defines either a Bordiga surface (the image of the embedding in \(\mathbb{P}^4\) of the blow-up of \(\mathbb{P}^2\) at \(10\) general points by means of the complete linear system of the quartics through these \(10\) points) or a scheme in the same irreducible component of the associated Hilbert scheme. They also prove that every Bordiga surface is the critical locus for the reconstruction for suitable projections. The Bordiga surface is also related to another classical problem in Computer Vision that is the reconstruction problem of a scene in \(\mathbb{P}^3\) consisting of a set of lines (a subset of the Grassmannian \(\mathbb{G}(1, 3)\)).
In the case of projections of lines, the authors, as they say in the abstract, compute the ideal that defines the critical locus. This is ``the union of \(3\) \(\alpha\)-planes and a line congruence of bi-degree \((3, 6)\) and sectional genus \(5\) in the Grassmannian \(\mathbb{G}(1, 3) \subset \mathbb{P}^5\)'', moreover, it is biregular to a Bordiga surface [\textit{A. Verra}, Manuscr. Math. 62, No. 4, 417--435 (1988; Zbl 0673.14026)]''. This fact is used to give a bridge between the two reconstruction problems and an algorithm to compute the projection matrices is also described. Bordiga surface; line congruences in Grassmannians; projective reconstruction in computer vision; multiview geometry; critical configurations or loci Special surfaces, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry The Bordiga surface as critical locus for 3-view reconstructions | 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 \(U_q\) denote the quantum group for \(sl_3\) over \(\mathbb{Q}(q)\) and let \(U_A\) be the \(A=\mathbb{Z}[q,q^{-1}]\)-form of \(U_q\) defined via quantum divided powers. If \(V\) is a finite-dimensional simple module for \(U_q\) with highest weight vector \(v\), consider the \(A\)-form \(V_A=U_Av\). The author proves that \(V_A\) has an \(A\)-basis which is compatible with all quantum Demazure submodules. She also checks that the transition matrix from this basis to Lusztig's canonical basis for \(V_A\) is upper triangular (with respect to an appropriate ordering).
The formulations of these results generalize naturally to other semisimple Lie algebras. The author conjectures that they do in fact hold in general, and she has proved her conjecture in [\textit{V. Lakshmibai}, Proc. Symp. Pure Math. 56, Pt. 2, 149-168 (1994; Zbl 0848.17020)]. quantum Demazure modules; crystal basis; transition matrix Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Bases for quantum Demazure modules. 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 Double Kostka polynomials \(K_{\lambda,\mu}(t)\) are polynomials in \(t\), indexed by double partitions \({\lambda,\mu}\). As in the ordinary case, \(K_{\lambda,\mu}(t)\) is defined in terms of Schur functions \(s_\lambda(x)\) and Hall-Littlewood functions \(P_\mu(x;t)\). In this paper, we study combinatorial properties of \(K_{\lambda,\mu}(t)\) and \(P_\mu(x;t)\). In particular, we show that the Lascoux-Schützenberger type formula holds for \(K_{\lambda,\mu}(t)\) in the case where \(\mu = (-,\mu^{\prime\prime})\). Moreover, we show that the Hall bimodule \(\mathscr{M}\) introduced by \textit{M. Finkelberg} et al. [Sel. Math., New Ser. 14, No. 3--4, 607--628 (2009; Zbl 1215.20041)] is isomorphic to the ring of symmetric functions (with two types of variables) and the natural basis \(\mathfrak{u}_\lambda\) of \(\mathscr{M}\) is sent to \(P_\lambda(x;t)\) (up to scalar) under this isomorphism. This gives an alternate approach for their result. Schur functions; Hall-Littlewood functions S. Liu and T. Shoji, Double Kostka polynomials and Hall bimodule, \doihref10.3836/tjm/1475723088Tokyo J. Math., 39 (2017), 743--776. Symmetric functions and generalizations, Combinatorial aspects of representation theory, Representation theory for linear algebraic groups, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Double Kostka polynomials and Hall bimodule | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let G be a reductive algebraic group defined over an algebraically closed field. Moreover let Q be a parabolic subgroup. At least three proofs of the normality of the Schubert subvarieties in G/Q are known [\textit{H. H. Andersen}, Invent. Math. 79, 611-618 (1985; Zbl 0591.14036), \textit{S. Ramanan} and \textit{A. Ramanathan}, Invent. Math. 79, 217-224 (1985; Zbl 0553.14023), \textit{C. S. Seshadri}, Proc. Bombay Colloquium on vector bundles 1984)]. In the present article it is shown that the normality is an easy consequence of the fact that the Schubert varieties are Frobenius split proved by \textit{V. B. Mehta} and \textit{A. Ramanathan} in Ann. Math., II. Ser. 122, 27-40 (1986; Zbl 0601.14043)] and the following lemma:
Let \(f: Y\to X\) be a proper surjective morphism of irreductible varieties in characteristic p. Suppose that Y is normal, the fibres of f are connected and that X is Frobenius split. Then X is normal. normality of the Schubert subvarieties in homogeneous space; Schubert varieties are Frobenius split V. B. Mehta and V. Srinivas, Normality of Schubert varieties, Amer. J. Math. 109 (1987), 987--989. JSTOR: Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Normality 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 Let \(M(l\times m)\) be the affine space of matrices over a field with \(l\) rows and \(m\) columns. Denote by \(V\) the algebraic set consisting of those pairs \((A_1, A_2)\) in \(M(n_2\times n_1) \times M(n_3\times n_2)\) such that \(A_2A_1=0\). Then each irreducible component of \(V\) is isomorphic to the opposite cell in a Schubert variety \(SL(n_1+n_2+n_3)/Q\) where \(Q\) is a parabolic group. In this article the singular locus of each irreducible component of \(V\) is determined. A conjecture of Lakshmibai and Sandhya on how to write the singular locus of the associated Schubert variety \(X(\nu) =\overline{B\nu B} \pmod B\) of \(C\) as a union of varieties \(X(\lambda)\) is also proved. singular loci; varieties of complexes; Schubert varieties V. Lakshmibai. Singular loci of varieties of complexes. \textit{J. Pure Appl. Algebra} 152 (2000), 217--230. Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields Singular loci of varieties of complexes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected, simply-connected, semisimple algebraic group defined over an algebraically closed field \(k\) of characteristic 0, \(B\subset G\) a fixed Borel subgroup, \(T\subset B\) a maximal torus and \(W = N_G(T)/T\) the Weyl group of \(G\). Consider the complete flag variety \(G/B\). The fixed points of the action from the left of \(T\) on \(G/B\) are \(e_w := wB\), \(w\in W\). The Schubert variety \(X_w\) is defined as the Zariski closure of \(Be_w\) in \(G/B\). If \(B\) acts from the right (resp., left) on a variety \(X\) (resp., \(Y\)) then \(B\) acts from the right on the product \(X\times Y\) by : \((x,y)b := (xb,b^{-1}y)\) and one defines \(X{\times}^BY\) as \(X\times Y/B\). If \(V\) is a rational representation of \(B\) one gets a vector bundle \({\mathcal L}(V):=G{\times}^BV\) over \(G/B\). In particular, if \(\lambda \in X^{\ast}(B)\simeq X^{\ast}(T)\) is a character and if \(k_{\lambda}\) is the representation associated to \(\lambda \) then \({\mathcal L}(\lambda ):={\mathcal L}(k_{\lambda})\) is a line bundle on \(G/B\).
In this paper, the author gives explicitly the vanishing and non-vanishing of the cohomology modules \(\text{H}^i(X_w,{\mathcal L}(\lambda ))\) for \(w\in W\) and ``most'' \(\lambda \in X^{\ast}(T)\) when \(G\) has rank 2 (i.e., it is of type \(A_2\), \(B_2\), or \(G_2\)). For his computations, the author uses the Bott-Samelson-Demazure-Hansen desingularizations of \(X_w\), the fact that the cohomology of \({\mathcal L}(\lambda )\) on \(X_w\) is the same as the cohomology of its pull-back on the BSDH desingularisation, the realization of the BSDH desingularisations as towers of \({\mathbb P}^1\)-bundles starting with \({\mathbb P}^1\) and the Leray spectral sequence.
The author's explicit tables reveal nice patterns. For example, the author conjectures that, for \(G\) of arbitrary rank, if \(\text{H}^m(X_w,{\mathcal L}(\lambda ))\neq 0\) and \(\text{H}^n(X_w,{\mathcal L}(\lambda ))\neq 0\) for some \(m\leq n\) then \(\text{H}^i(X_w,{\mathcal L}(\lambda ))\neq 0\) for all \(i\) with \(m\leq i\leq n\). Schubert variety; cohomology of line bundles; semisimple algebraic group; root system Paramasamy, K.: Cohomology of line bundles on Schubert varieties: the rank two case, Proc. indian acad. Sci. 114, No. 4, 345-363 (2004) Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Classical groups (algebro-geometric aspects) Cohomology of line bundles on Schubert varieties: The rank two case | 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 flag domain is an open real group orbit in a complex flag manifold. It has been shown that a flag domain is either pseudoconvex or pseudoconcave. Moreover, generically 1-connected flag domains are pseudoconcave. In this study, for flag domains contained in irreducible Hermitian symmetric spaces of type \(AIII\) or \(CI\), we determine which pseudoconcave flag domain is generically 1-connected. flag domain; Hermitian symmetric space; Weyl group Grassmannians, Schubert varieties, flag manifolds, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) Generic 1-connectivity of flag domains in Hermitian symmetric spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G=C_p\) be a cyclic group of prime order \(p\). This paper calculates the \(RO(G)\)-graded cohomology of a free \(G\)-CW complex \(X\) with constant \(\mathbb Z/p\) coefficients from the ordinary mod \(p\) cohomology of its orbit space \(X/G\). If \(X\) is connected as a space, then its \(RO(G)\)-graded cohomology is calculated as a module over the cohomology of a point.
Applications include a new proof of the topological Tverberg conjecture in the prime case and for finite dimensional \(X\), the introduction of a numerical index that is identified with the Fadell-Husseini index of \(X\). Finally, a counter-example due to Clover May shows that the freeness theorem for \(C_2\)-representation complexes of [\textit{W. C. Kronholm}, Topology Appl. 157, No. 5, 902--915 (2010; Zbl 1194.55011)] does not extend directly to odd primes \(p\), but an analogue of the freeness theorem is proved for \(C_p\)-representation complexes that do not have cells in consecutive dimensions. Bredon cohomology; Mackey functor; Tverberg theorem; equivariant cohomology Equivariant homology and cohomology in algebraic topology, Equivariant homotopy theory in algebraic topology, Grassmannians, Schubert varieties, flag manifolds, Finite transformation groups Bredon cohomology of finite dimensional \(C_p\)-spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main aim of the present article is to define and study polynomials that the authors propose as type \(B\), \(C\) and \(D\) double Schubert polynomials. For the general linear group the corresponding objects are the double Schubert polynomials of Lascoux and Schützenberger. These type \(A\) polynomials possess a series of remarkable properties and the authors propose a theory with as many of the analogous properties as possible. They succeed in obtaining several properties which are desirable both from the geometric and combinatorial points of view.
When restricted to maximal Grassmannian elements of the Weyl group, the single versions of the polynomials are the \(\widetilde P\)- and \(\widetilde Q\)-polynomials of \textit{P. Pragacz} and \textit{J. Ratajski} [J. Reine Angew. Math. 476, 143--189 (1996; Zbl 0847.14029)]. The latter polynomials play, in some sense, the role in types \(B\), \(C\) and \(D\) analogous to that of Schur's \(S\)-functions in type \(A\). The utility of the \(\widetilde P\)- and \(\widetilde Q\)-polynomials in the description of Schubert calculus and degeneracy loci was studied by \textit{P. Pragacz} and \textit{J. Ratajski} [Compos. Math. 107, No. 1, 11--87 (1997; Zbl 0916.14026)], and according to the authors [see Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083) and J. Reine Angew. Math. 516, 207--223 (1999; Zbl 0934.14018)] the multiplication of \(\widetilde Q\)-polynomials describes both the arithmetic and quantum Schubert calculus on the Lagrangian Grassmannian. Thus the double Schubert polynomials in the present article are closely related to natural families of representing polynomials.
In many cases the authors obtain an analogue of the determinantal formula for Schubert cycles in Grassmannians and they answer a question of \textit{W. Fulton} and \textit{P. Pragacz} [Schubert varieties and degeneracy loci. Lect. Notes Math. 1689 (1998; Zbl 0913.14016)]. The formulas generalize those obtained by Pragacz and Ratajski [loc. cit.].
The main ingredients in the proofs are the geometric work of \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044) and Isr. Math. Conf. Proc. 9, 241--262 (1996; Zbl 0862.14032)] and \textit{W. Graham} [J. Differ. Geom. 45, 471--487 (1997; Zbl 0935.14015)] and the algebraic tools developed by \textit{A. Lascoux} and \textit{P. Pragacz} [Adv. Math. 140, No. 1, 1--43 (1998; Zbl 0951.14035) and Mich. Math. J. 48, Spec. Vol., 417--441 (2000; Zbl 1003.05106)]. determinantal formula; Weyl groups; Grassmannians; Lagrangian; Schubert cycles; Chern classes A. Kresch and H. Tamvakis, ''Double Schubert Polynomials and Degeneracy Loci for the Classical Groups,'' Ann. Inst. Fourier 52(6), 1681--1727 (2002). Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Double Schubert polynomials and degeneracy loci for 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 introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [\textit{S. Fomin} and the author, in: Brylinski, Jean-Luc (ed.) et al., Advances in geometry. Boston, MA: Birkhäuser. Prog. Math. 172, 147--182 (1999; Zbl 0940.05070); \textit{A. Postnikov}, in: Brylinski, Jean-Luc (ed.) et al., Advances in geometry. Boston, MA: Birkhäuser. Prog. Math. 172, 371--383 (1999; Zbl 0944.14019)]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [\textit{E. Mukhin, V. Tarasov} and \textit{A. Varchenko}, ``Bethe Subalgebras of the Group Algebra of the Symmetric Group'', Preprint, \url{arXiv:1004.4248}].
Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra in a connection with the values of the \(\beta\)-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan-Robbins polytope. Dunkl and Gaudin operators at critical level; Catalan numbers, Schroder numbers; Schubert polynomials; Grothendieck polynomials A. N. Kirillov, \textit{On Some Combinatorial and Algebraic Properties of Dunkl Elements}, RIMS preprint, 2012. Polynomial rings and ideals; rings of integer-valued polynomials, Grassmannians, Schubert varieties, flag manifolds, Quadratic and Koszul algebras, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) On some algebraic and combinatorial properties of Dunkl elements | 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 three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron. total positivity; Grassmannian; unipotent group Combinatorial aspects of simplicial complexes, Grassmannians, Schubert varieties, flag manifolds, Positive matrices and their generalizations; cones of matrices, Polytopes and polyhedra The totally nonnegative Grassmannian is a ball | 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 positivity property for the cup product in the \(T\)-equivariant cohomology of the flag variety. This was conjectured by D. Peterson and has as a consequence a conjecture of \textit{S. Billey} [Duke Math. J. 96, No.1, 205--224 (1999; Zbl 0980.22018)]. The result for the flag variety follows from a more general result about algebraic varieties with an action of a solvable linear algebraic group such that the unipotent radical acts with finitely many orbits. The methods are those used by \textit{S. Kumar} and \textit{M. Nori} [Int. Math. Res. Not. 1998, No.14, 757--763 (1998; Zbl 1014.17023)]. William Graham, ``Positivity in equivariant Schubert calculus'', Duke Math. J.109 (2001) no. 3, p. 599-614 Homogeneous spaces and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations Positivity in 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 This paper deals with a construction of solutions of KdV-type equations through infinite dimensional Grassmannian constructions initiated by M. Sato. \textit{M. Sato} and \textit{Y. Sato} [Nonlinear partial differential equations in applied science, Proc. U.S.-Jap. Semin., Tokyo 1982, North-Holland Math. Stud. 81, 259--271 (1983; Zbl 0528.58020)] gave a method to construct general solutions of KdV equation systematically in 1979. It was soon developed by \textit{E. Date, M. Jimbo, M. Kashiwara} and \textit{T. Miwa} [Publ. Res. Inst. Math. Sci. 18, 1111--1119 (1982; Zbl 0571.35101)] in more general situations. The main aims of this paper are to determine what class of solutions is obtained by this method, to illustrate in detail how the geometry of the Grassmannian is reflected in properties of the solutions fit into the picture. Moreover they try to explain the geometric meaning of the ''\(\tau\)-function''. The authors describe the ''Sato''-theory from their view point by using loop groups in order to give a clear and self contained account of the theory, but no new type of solutions has been obtained in their framework. Sato-theory; KdV-type equations; Grassmannian constructions; \(\tau\)-function; loop groups Segal, Graeme and Wilson, George, Loop groups and equations of {K}d{V} type, Institut des Hautes Études Scientifiques. Publications Mathématiques, 61, 5-65, (1985) KdV equations (Korteweg-de Vries equations), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Grassmannians, Schubert varieties, flag manifolds Loop groups and equations of KdV type | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the first part of this paper [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] the authors studied the cone \(\text{BDRY}(n)\) which is the set of triples of weakly decreasing \(n\)-tuples \((\lambda,\mu,\nu)\in ({\mathbb R}^n)^3\) satisfying the three conditions (1) regarding \(\lambda,\mu,\nu\) as spectra of \(n\times n\) Hermitian matrices, there exist three Hermitian matrices with those spectra whose sum is the zero matrix; (2) regarding \(\lambda,\mu,\nu\) as dominant weights of \(\text{GL}_n({\mathbb C})\), the tensor product \(V_{\lambda}\otimes V_{\mu}\otimes V_{\nu}\) of the corresponding irreducible modules contains a \(\text{GL}_n({\mathbb C})\)-invariant vector; (3) regarding \(\lambda,\mu,\nu\) as possible boundary data on a honeycomb, there exist ways to complete it to a honeycomb.
These conditions were proved to be equivalent. A sufficient list of inequalities for this cone was given due to the efforts of several authors: Klyachko, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. In the present, second, part of the paper the authors introduce new combinatorial objects called puzzles, which are certain kinds of diagrams in the triangular lattice in the plane, composed from unit equilateral triangles and unit rhombi, with edges labeled by 0 and 1. Puzzles are used to compute Grassmannian Schubert calculus, and have much interest in their own right. In particular, the authors get new, puzzle-theoretic, proofs of results of Horn and the above-mentioned authors.
The authors also characterize ``rigid'' puzzles and use them to prove a conjecture of Fulton which states that if the irreducible module \(V_\nu\) appears exactly once in \(V_\lambda \otimes V_\mu\), then for all \(N\in{\mathbb N}\), \(V_{N\lambda}\) appears exactly once in \(V_{N\lambda}\otimes V_{N\mu}\). honeycombs; symmetric functions; Littlewood-Richardson rule; puzzles; Hermitian matrix; eigenvalue problems; Schubert calculus; Grassmannian A. Knutson, T. Tao and C. Woodward, The honeycomb model of GLn tensor products II: Puzzles determine facets of the Littlewood-Richardson cone. \textit{Journal of the American Mathematical Society }17 (2004), 19--48. arXiv:math/0107011.Zbl 1043.05111 MR 2015329 Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Inequalities involving eigenvalues and eigenvectors, Representation theory for linear algebraic groups, Special polytopes (linear programming, centrally symmetric, etc.), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Combinatorial aspects of representation theory The honeycomb model of \(\text{GL}_n({\mathbb C})\) tensor products. II: Puzzles determine facets of the Little\-wood-Richardson cone | 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 \(\sigma\) -models with symmetric target spaces are classically integrable. With the example of the model with target space the flag manifold \(\frac{\operatorname{U}(3)}{\operatorname{U}(1)^3}\) - a non-symmetric space - we show that the introduction of torsion allows to cast the equations of motion in the form of a zero-curvature condition for a one-parameter family of connections, which can be a sign of integrability of the theory. We also elaborate on geometric aspects of the proposed model. Bykov, D., Integrable properties of \(\sigma\)-models with non-symmetric target spaces, Nucl. Phys. B, 894, 254-267, (2015) Model quantum field theories, Groups and algebras in quantum theory and relations with integrable systems, Grassmannians, Schubert varieties, flag manifolds Integrable properties of \(\sigma\) -models with non-symmetric target spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors generalize the notion of ladder determinantal varieties, which was introduced by Abhyankar, by allowing ideals of minors of different size of a matrix of indeterminates. Then they explore the relation between these mixed ladder determinantal varieties and Schubert varieties. Next they show that, up to product by affine spaces, each of these varieties is a basic open set in a classical ladder determinantal variety and that it contains as a basic open set another classical ladder determinantal variety. mixed ladder determinantal varieties; Schubert varieties Gonciulea, N.; Miller, C., Mixed ladder determinantal varieties, \textit{J. Algebra}, 231, 1, 104-137, (2000) Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds Mixed ladder determinantal 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 classical Borel-Weil-Bott theorem describes the cohomology of line bundles over flag varieties. In this paper this theorem is generalized for the case of \textit{wonderful varieties of minimal rank}.
Wonderful varieties were introduced by [\textit{C. De Concini} and \textit{C. Procesi}, Complete symmetric varieties. Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 966, 1--44 (1983; Zbl 0581.14041)]; later \textit{D. Luna} [Transform. Groups 1, No. 3, 249--258 (1996; Zbl 0912.14017)] showed that they all are spherical. These are embeddings of a \(G\)-homogeneous space \(G/H\), where \(G\) is a reductive algebraic group, into a smooth complete \(G\)-variety \(X\) satisfying a number of good properties on the boundary \(X-(G/H)\) (in particular, the boundary should have codimension 1). The degree \(r\) of the boundary divisor is called the \textit{rank} of \(X\). For example, wonderful varieties of rank 0 are exactly the partial flag varieties \(G/P\). There is always an inequality
\[
r\geq \mathrm{rk} G -\mathrm{rk}H,
\]
where rank of an algebraic group is the dimension of its maximal torus. A wonderful variety is said to be \textit{of minimal rank} if this inequality turns into an equality.
In this paper A. Tchoudjem computes the cohomology groups of invertible sheaves on such varieties. The answer is formulated in terms of spherical data for the variety \(X\).
The technique used for this computation is as follows: the author considers a Bialynicki-Birula decomposition of \(X\), then for a given invertible sheaf computes the groups of cohomology with support given by the cells of this decomposition, and finally passes from the cohomology with support to the usual cohomology using the Grothendieck-Cousin complex. Borel-Weil-Bott theorem; cohomology with support; spherical varieties; wonderful varieties; flag varieties; Grothendieck-Cousin complex; cohomology of line bundles; Verma modules Tchoudjem, A., Cohomologie des fibrés en droites sur les variétés magnifiques de rang minimal, Bull. Soc. Math. France, 135, 2, 171-214, (2007) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Compactifications; symmetric and spherical varieties, Vanishing theorems in algebraic geometry, Other algebraic groups (geometric aspects), Grassmannians, Schubert varieties, flag manifolds Cohomology of line bundles over wonderful varieties of minimal rank | 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 rectangular elements in the symmetric group. In the framework of PBW degenerations, we show that in type A the degenerate Schubert variety associated with a rectangular element is indeed a Schubert variety in a partial flag variety of the same type with larger rank. Moreover, the degenerate Demazure module associated with a rectangular element is isomorphic to the Demazure module for this particular Schubert variety of larger rank. This generalises previous results by \textit{G. Cerulli Irelli} et al. for the PBW degenerate flag variety in [Pac. J. Math. 284, No. 2, 283--308 (2016; Zbl 1433.17008)]. Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Degenerate Schubert varieties in type A | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review studies projections of general nodal linear sections of certain Mukai varieties. \par The main result says that a general $\left\{\begin{matrix} \text{codimension } 5 \ \text{singular linear section} \\ \text{hyperplane section}\\ \text{hyperplane section} \end{matrix}\right\}$ $L$ of $\left\{\begin{matrix} M_7 \\ G(2,6)\\ G_2 \end{matrix}\right\}$ admits a single node. the projection of $L$ from the node is a proper $\left\{\begin{matrix} \text{intersection } G(2,5)\cap Q \text{ where } Q \text{ is a quadric in } \mathbb{P}^9\\ \text{codimension } 3 \text{ linear section of } OG(5,10)\\ \text{linear section of } LG(3,6) \end{matrix}\right\}$ containing a smooth $\left\{\begin{matrix} 4 \\ 6\\ 3 \end{matrix}\right\}$-dimensional quadric. Moreover, any generic section of the above form containing a smooth quadric arises in this way. Mukai varieties; linear sections Fano varieties, Families, moduli, classification: algebraic theory, Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry Projections of Mukai varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We apply the previous calculations of Chow-Witt rings of Grassmannians to develop an oriented analogue of the classical Schubert calculus. As a result, we get complete diagrammatic descriptions of the ring structure in Chow-Witt rings and twisted Witt groups. In the resulting arithmetic refinements of Schubert calculus, the multiplicity of a solution subspace is a quadratic form encoding additional orientation information. We also discuss a couple of applications, such as a Chow-Witt version of the signed count of balanced subspaces of \textit{L. M. Fehér} and \textit{K. Matszangosz} [Period. Math. Hung. 73, No. 2, 137--156 (2016; Zbl 1389.14026)]. Chow-Witt rings; characteristic classes; Grassmannians Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Motivic cohomology; motivic homotopy theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Oriented Schubert calculus in Chow-Witt rings of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a 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, the classes \([{\mathcal O}_{X_w}]\) of structure sheaves of \(X_w\) form a \(\mathbb{Z}\)-basis of the \(K\)-group \(K(X)\) of \(X\). Write, under the product in \(K(X)\), for any \(u,\,v\in W/W_P\):
\[
[{\mathcal O}_{X_u}]= \sum_{w\in W/W_P} c^w_{u,v}[{\mathcal O}_{X_w}],
\]
for some \(c^w_{u,v}\in \mathbb{Z}\).
Then, the main result of the paper under review asserts that
\[
c^w_{u,v}(-1)^{\text{codim\,}X_u+ \text{codim\,}X_v+ \text{codim\,}X_w}\geq 0.
\]
This was conjectured by \textit{A. Buch} [Acta Math. 189, 37--78 (2002; Zbl 1090.14015)] and proved by him for the Grassmannians.
The author, in fact, proves the following more general result asked by \textit{W. Graham} [Duke Math. J. 102, 599--614 (2001; Zbl 1069.14055)]: Let \(Y\subset X\) be a closed subvariety with rational singularities. Express
\[
[{\mathcal O}_Y]= \sum_{w\in W/W_P} c^w_Y[{\mathcal O}_{X_w}],\text{ for }c^w_Y\in \mathbb{Z}.
\]
Then, \((-1)^{\text{codim\,}X_w+ \text{codim\,}Y} c^w_Y\geq 0\). Schubert calculus \beginbarticle \bauthor\binitsM. \bsnmBrion, \batitlePositivity in the Grothendieck group of complex flag varieties, \bjtitleJ. Algebra \bvolume258 (\byear2002), no. \bissue1, page 137-\blpage159. \endbarticle \OrigBibText Michel Brion, Positivity in the Grothendieck group of complex flag varieties , J. Algebra 258 (2002), no. 1, 137-159. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Grothendieck groups, \(K\)-theory and commutative rings, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Positivity in the Grothendieck group of complex 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 Let G be a semi-simple algebraic group over an algebraically closed field of characteristic \(p>1.\) Let T be a maximal torus in G and B a Borel subgroup, \(B\supset T\). Let \(W\) (\(=N(T)/T,\) \(N(T)\) the normalizer of T in G) be the Weyl group of G. For \(w\in W\), let \(X(w)\) \((=\overline{BwB}(\mod B))\) be the Schubert variety in \(G/B\) associated to w. Using Frobenius-splitting Ramanathan proved that \(X(w)\) is Cohen-Macaulay. The authors give a short proof of Ramanthan's result, using two simple lemmas. Cohen-Macaulayness of Schubert variety; characteristic p; Frobenius-splitting Mehta, V.; Srinivas, V.: A note on Schubert varieties in G/B. Math. ann. 284, 1-5 (1989) Grassmannians, Schubert varieties, flag manifolds, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) A note on Schubert varieties in 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 We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [\textit{S. Fomin} and \textit{A. N. Kirillov}, Prog. Math. 172, 147--182 (1999; Zbl 0940.05070); \textit{A. Postnikov}, ibid. 172, 371--383 (1999; Zbl 0944.14019); \textit{A. N. Kirillov} and \textit{T. Maeno}, ``A note on quantum \(K\)-theory of flag varieties'', Preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group [\textit{E. Mukhin} et al., ``Bethe subalgebras of the group algebra of the symmetric group'', Preprint, \url{arXiv:1004.4248}].{
}Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra in a connection with the values of the {\(\beta\)}-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan-Robbins polytope. Dunkl operators at critical level; Gaudin operators at critical level; Catalan numbers; Schröder numbers; Schubert polynomials; Grothendieck polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds On some algebraic and combinatorial properties of Dunkl elements | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper takes into consideration that ``the saturation theorem of \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] concerns the nonvanishing of Littlewood-Richardson coefficients''. Further, ``in combination with work of \textit{A. A. Klyachko} [Sel. Math., New Ser. 4, No. 3, 419--445 (1998; Zbl 0915.14010)], it implies \textit{A. Horn}'s conjecture [Pac. J. Math. 12, 225--241 (1962; Zbl 0112.01501)] about eigenvalues of sums of Hermitian matrices''. Then ``the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians'' are illustrated. The main result of this paper deals with an extension of Schubert calculus presenting a Schubert calculus interpretation of Friedland's problem, via equivariant cohomology of Grassmannians, and it derives a saturation theorem for this setting. eigenvalue problem; equivariant cohomology; Schubert calculus; Littlewood-Richardson coefficients; Hermitian matrices; Grassmannians Inequalities involving eigenvalues and eigenvectors, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Equivariant algebraic topology of manifolds Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use a theorem of Chow [\textit{W.-L. Chow}, Ann. Math. (2) 50, 32--67 (1949; Zbl 0040.22901)] on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question of \textit{P. Beelen}, \textit{S. R. Ghorpade} and \textit{T. Høholdt} [Affine Grassmann codes, IEEE Trans. Inf. Theory 56, No. 7, 3166--3176 (2010; Zbl 1365.94579)] concerning the permutation automorphism groups of affine Grassmann codes. Finally, we prove an analogue of Chow's theorem for the case of Schubert divisors in Grassmannians and then use it to determine the automorphism group of linear codes associated to such Schubert divisors. In the course of this work, we also give an alternative short proof of MacWilliams theorem concerning the equivalence of linear codes and a characterization of maximal linear subspaces of Schubert divisors in Grassmannians. Grassmann variety; Schubert divisor; linear code; automorphism group; Grassmann code; affine Grassmann code Ghorpade S.R., Kaipa K.V.: Automorphism groups of Grassmann codes. Finite Fields Appl. \textbf{23}, 80-102 (2013). Linear codes (general theory), Geometric methods (including applications of algebraic geometry) applied to coding theory, Grassmannians, Schubert varieties, flag manifolds, Finite automorphism groups of algebraic, geometric, or combinatorial structures Automorphism groups of Grassmann codes | 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 Recall that a resolution \(p:\widetilde X \to X\) of an irreducible complex projective variety \(X\) is said to be small if, for all \(i>0\), \(\text{codim}_X \{x\in X : \dim p^{-1} (x)\geq i\}>2i\). Let \(G=SO(2n)\) or \(Sp(2n)\) and let \(P_n\) be the maximal parabolic subgroup obtained by deleting the right end root (following the Bourbaki convention). In an earlier work, ``Small resolutions of Schubert varieties in symplectic and orthogonal Grassmannians'' in Publ. Res. Inst. Math. Sci. 30, No. 3, 443-458 (1994), the authors exhibited `Bott-Samleson type' small resolutions \(p:\widetilde X(\lambda) \to X(\lambda)\) of certain Schubert varieties \(X(\lambda)\) in \(G/P_n\).
The authors, in the paper under review, give an inductive formula (following similar works of Zelevinskii in the case of Grassmannian Schubert varieties) to determine the Poincaré polynomials of the fibres of \(p\) over \(T\)-fixed points (where \(T\) is the maximal torus of \(G\) acting on \(G/P_n\) via the left multiplication). They use this result (and some results of Zelevinskii) to show that the Kazhdan-Lusztig (KL for short) polynomials \(P_{\theta, \lambda_0}\) (for the Weyl group associated to \(G)\), for certain pairs \(\theta\leq \lambda_0\), are equal to the KL-polynomials \(P_{\theta', \lambda_0'}'\) (where \(P'\) denotes the KL-polynomials for \(SL(M)\), for certain integer \(M\) and certain \(\theta'\leq \lambda_0'\) in the Weyl group of \(SL(M))\).
The authors also exhibit small resolutions for certain Schubert varieties in \(E_6/P_6\) and calculate the Poincaré polynomials of the fibres over \(T\)-fixed points for them (where \(P_6\) is again the maximal parabolic subgroup obtained by deleting the right end root). In addition, they explicitly determine the singular locus for most of the Schubert varieties in \(E_6/P_6\). Kazhdan-Lusztig polynomials; Schubert varieties; small resolutions Grassmannians, Schubert varieties, flag manifolds, Global theory and resolution of singularities (algebro-geometric aspects) Small resolutions of Schubert varieties and Kazhdan-Lusztig polynomials | 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 twisted Grassmann variety (a form of Grassmann variety), which is the variety representing the functor of right ideals of prescribed rank in a central simple algebra over a field, is represented by a linear section of a Grassmann variety (Theorem A). The Severi-Brauer schemes of some \(R\)-orders in the matrix ring \(M_4(R)\) of degree 4 over a regular local ring \(R\) are constructed (Theorem B). The variety of rank 4 right ideals of the associative \(k\)-algebra generated by \(x,y\) with the relations \(x^4=y^4=0\) and \(yx=\sqrt{-1}xy\) is described (Theorem C). orders; twisted Grassmann varieties; central simple algebras; Severi-Brauer schemes; regular local rings Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Grassmannians, Schubert varieties, flag manifolds, Brauer groups (algebraic aspects), Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) A realization of twisted Grassmann 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 \textit{W. Kucharz} [Boll. Unione Mat. Ital., VII. Ser., A8, No. 3, 345-352 (1994; Zbl 0829.14006)] established a criterion for a smooth map from a real algebraic manifold into a flag manifold to admit an approximation by a regular map. In the present paper, the authors generalize this to smooth maps from real algebraic manifolds into real algebraic manifolds that are fibered in flag manifolds over a real algebraic base. Examples of such manifolds are the blow-up of a real projective space along a linear subspace, or a \((1,1)\)-hypersurface in the product of two real projective spaces. For such a manifold \(Y\), the authors show that a smooth map \(f: X\to Y\) can be approximated by a regular map if and only if \(f^*\) maps the first cohomology group \(H^1(Y, \mathbb{Z}/2\mathbb{Z})\) of \(Y\) into the group \(H_{\text{alg}}^1(X, \mathbb{Z}/2\mathbb{Z})\) of algebraic cohomology classes of \(X\). real algebraic variety; flag manifolds over a real algebraic base; approximation by regular maps J. Bochnak and W. Kucharz, Smooth maps and real algebraic morphisms, Canad. Math. Bull. 42 (1999), 445-451. Zbl0955.14043 MR1727342 Topology of real algebraic varieties, Grassmannians, Schubert varieties, flag manifolds, Real algebraic sets, Rational and birational maps Smooth maps and real algebraic morphisms | 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 obtains explicit formulas for the spectrum of the Hodge Laplacian on 1--forms on complex Grassmanians \(\text{Gr}_ 2(\mathbb C^ {m+2})\) for \(m\geq 3\). He uses results of \textit{A. Ikeda} and \textit{Y. Taniguchi} [Osaka J. Math. 15, 515--546 (1978; Zbl 0392.53033)]. We mention that related results were obtained by \textit{F. El Chami} [Int. J. Pure Appl. Math. 12, No. 4, 395--418 (2004; Zbl 1121.58024)]. spectrum; Hodge Laplacian; complex Grassmannian General topics in linear spectral theory for PDEs, Differential complexes, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Spectrum of the Hodge Laplacian on complex Grassmannian \(\mathrm{Gr}_2(\mathbb C^{m+2})\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The so-called Laudal's lemma, after the improvement by Gruson and Peskine, states that, if a curve of degree \(d > s^ 2 + 1\) in \(\mathbb{P}^ 3\) has its general plane section contained in a curve of degree \(s\), then the curve itself is contained in a surface of degree \(d\). The general lifting problem can be stated as follows: Given a projective variety \(X\) of \(\mathbb{P}^ r\) of degree \(d\) and dimension \(n\) whose intersection with a general \(h\)-plane is contained in a variety of degree \(s\) of codimension \(t\) in the \(h\)-plane, find necessary conditions for \(X\) to be contained in a variety of degree \(s\) and codimension \(t\) in \(\mathbb{P}^ r\).
The authors give an answer in the case that \(t = \text{codim} (X) - 1\), i.e. when \(X\) is a hypersurface inside the variety of degree \(s\) we are looking for. More precisely, they give a condition of the type \(d > D (s,h,r,n)\), where \(D\) is a given explicit function. Their proof is based on the theory of focal loci. They give some examples and comment some generalizations to arbitrary codimension \(t\). In particular, their bounds are not the best possible, but asymptotically they are so. For example, in the case of curves in \(\mathbb{P}^ 3\) they obtain the original result of Laudal. general plane section; general lifting problem; projective variety Chiantini, L; Ciliberto, C, A few remarks on the lifting problem, Astérisque, 218, 95-109, (1993) Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Embeddings in algebraic geometry A few remarks on the lifting problem | 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 note, the real intersection cohomology \(IH^\bullet (X_\Delta)\) of a toric variety \(X_\Delta\) is described in a purely combinatorial way using methods of elementary commutative algebra only. We define, for arbitrary fans, the notion of a ``minimal extension sheaf'' \({\mathcal E}^\bullet\) on the fan \(\Delta\) as an axiomatic characterization of the equivariant intersection cohomology sheaf. This provides a purely algebraic interpretation of the \(h\)- and \(g\)-vector of an arbitrary polytope or fan under a natural vanishing condition. The results presented in this note originate from joint work with \textit{G. Barthel}, \textit{J. P. Brasselet} and \textit{L. Kaup} [Tohoku Math. J., II. Ser. 54, No.1, 1-41 (2002; Zbl 1055.14024)]. real intersection cohomology; toric variety; equivariant intersection cohomology K.-H. Fieseler, Towards a combinatorial intersection cohomology for fans, C. R. Acad. Sci. Paris, Sér. I Math. 330 (2000), 291--296. Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Towards a combinatorial intersection cohomology for fans | 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 the vector space \(V\) of \(r\times n\) matrices, acted upon by the group \(G=\mathrm{GL}_r(C) \times (C^{\times})^n\). Most points of \(V\) correspond to ordered \(n\)-tuples of points in the projective space \(P^{r-1}\). Such a tuple determines a matroid, a so-called representable matroid, via the dimensions of the spans of various subsets of the tuple. That matroid is invariant under the \(G\) action.
The paper under review is one of the many works that explore the relation between the matroid and certain aspects of the geometry of the orbit. In particular, consider the class of the orbit closure in the \(G\)-equivariant \(K\)-theory of the vector space \(V\). The authors conjecture (Conjecture 5.1) that the matroid determines this class.
Besides this conjecture the paper has two main results. In the first one the authors prove that certain coefficients of the \(K\)-class of the orbit closure are indeed determined by the matroid. In this result, the combinatorics of hook-shaped partitions/Schur functions play a role.
The other main result is the description of the ideal of the orbit closure using only the matroid -- up to radical. If the ideal was reduced this would prove the conjecture, and it indeed does prove it in special cases when the ideal is reduced. For example, rank 2 or corank 2 uniform matroids satisfy this property. matroid; determinantal variety; equivariant \(K\)-class Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Determinantal varieties, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Equivariant homology and cohomology in algebraic topology Matrix orbit closures | 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 every function on the unit sphere in \(\mathbb{R}^n\), which is invariant under rotations about some coordinate axis, is completely determined by a function of one variable. Similar results, when invariance of a function reduces dimension of its actual argument, hold for every compact symmetric space and can be obtained in the framework of Lie-theoretic consideration. In the present article, this phenomenon is given precise meaning for functions on the Grassmann manifold \(G_{n,i}\) of \(i\)-dimensional subspaces of \(\mathbb{R}^n\), which are invariant under orthogonal transformations preserving complementary coordinate subspaces of arbitrary fixed dimension. The corresponding integral formulas are obtained. Our method relies on bi-Stiefel decomposition and does not invoke Lie theory. compact symmetric space; complementary coordinate subspaces Ólafsson, G., Rubin, B.: Invariant functions on Grassmannians. In: Radon Transforms, Geometry and Wavelets. Cont. Math., vol. 464, pp. 201--211. AMS, Providence (2008) Harmonic analysis on homogeneous spaces, Analysis on real and complex Lie groups, Laplace transform, Grassmannians, Schubert varieties, flag manifolds Invariant functions 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 authors obtain an explicit presentation for the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation for the ordinary cobordism ring. It has another application for the equivariant Schubert calculus in cobordism theory. The authors also described the rational equivariant cobordism rings of wonderful symmetric varieties of minimal rank.
The equivariant cohomology and the equivariant Chow groups of these two classes of spherical varieties have been extensively studies before. In the setting of equivariant \(K\)-theory, the Borel presentation of complete flag variety is due to [\textit{B. Kostant} and \textit{S. Kumar}, J. Differ. Geom. 32, No. 2, 549--603 (1990; Zbl 0731.55005)]. The analogue results for the equivariant Chow ring and the singular cohomology ring are, respectively due to \textit{M. Brion} [Transform. Groups 2, 225--267 (1997; Zbl 0916.14003)] and \textit{T.-S. Holm} and \textit{R. Sjamaar} [Transform. Groups 13, No. 3-4, 585--615 (2008; Zbl 1221.55007)]. The ordinary cobordism rings of such varieties have been recently described by \textit{J. Hornbostel} and \textit{V. Kiritchenko} [J. Reine Angew. Math. 656, 59--85 (2011; Zbl 1226.14032)] and \textit{B. Calmes, V. Petrov} and \textit{K. Zainoulline} [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 3, 405--448 (2013; Zbl 1323.14026)]. cobordism; \(K\)-theory; flag varieties; symmetric varieties Kiritchenko, V; Krishna, A, Equivariant cobordism of flag varieties and of symmetric varieties, Transf. Groups, 18, 391-413, (2013) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Equivariant cobordism of flag varieties and of symmetric 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 present paper studies some components of the Springer fibers for \(\mathrm{GL}(n)\) which are associated to closed orbits of \(\mathrm{GL}(p) \times \mathrm{GL}(q)\) on the flag variety of \(\mathrm{GL}(n)\), where \(n =p+q\). Such components occur in any Springer fiber and, in contrast to the case of arbitrary components, they are always smooth. Building on the work of \textit{L.~Barchini} and \textit{R.~Zierau} [Represent. Theory 12, 403--434 (2008; Zbl 1186.22017)], the authors show that these components are iterated bundles and are stable for the action of a maximal torus \(H\) of \(\mathrm{GL}(n)\). Using this description of the components, the authors calculate the Betti numbers and they determine the \(H\)-fixed points and the weights of \(H\) acting on the tangent spaces at these points.
If \(\mathcal L\) is a line bundle on the flag variety of \(\mathrm{GL}(n)\) and if \(Z\) is one of the considered components of a Springer fiber regarded as a subvariety of the flag variety, using localization theorems in equivariant \(K\)-theory and Borel-Moore homology the authors obtain a character formula for the cohomology of \(\mathcal L_Z\) as a \(H\)-module, where \(\mathcal L_Z\) denotes the restriction of \(\mathcal L\) to \(Z\). If the weight corresponding to \(\mathcal L\) is sufficiently dominant, then it is shown that the higher cohomology groups of \(\mathcal L_Z\) vanish, therefore the character formula gives in this case the character of \(H^0(Z,\mathcal L_Z)\). This is of interest because of a result of \textit{J.~T.~Chang} [Trans. Am. Math. Soc. 341, No. 2, 603--622 (1994; Zbl 0817.22009)], which relates the dimension of \(H^0(Z,\mathcal L_Z)\) with the associated cycle of a discrete series representation of \(U(p,q)\). Applying localization theorems, they are also obtained formulas which express the classes defined by the considered components in equivariant cohomology and in \(K\)-theory in terms of Schubert bases. In the appendix, they are identified the tableaux corresponding to the considered components under the bijective correspondence between components of Springer fibers for \(\mathrm{GL}(n)\) and standard tableaux. Springer fibers; iterated bundles; flag varieties; nilpotent orbits Graham, W.; Zierau, R., Smooth components of Springer fibers, Ann. Inst. Fourier (Grenoble), 61, 5, 2139-2182, (2011) Grassmannians, Schubert varieties, flag manifolds, Classical groups (algebro-geometric aspects), Linear algebraic groups over the reals, the complexes, the quaternions, Semisimple Lie groups and their representations Smooth components of Springer fibers | 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 \(Gr_{d,n}\) denote the Grassmannian of \(d\)-dimensional vector subspaces of \(K^n\), \(K\) an algebraically closed field. If \(\alpha\), \(\gamma\) are \(d\)-element subsets of \(\{1,\dots,n\}\) and \((e_1,\dots,e_n)\) is the standard basis of \(K^n\), \(e_\alpha\) denotes the span of \(e_{\alpha_1},\dots,e_{\alpha_d}\). The Richardson variety \(X^\gamma_\alpha\) is then defined as the intersection of the Schubert variety \(X^\gamma\), the closure of the orbit of \(e_\alpha\) under the action of upper triangular matrices, and the opposite Schubert variety \(X_\alpha\), the orbit of \(e_\alpha\) under the action of lower triangular matrices. The point \(e_\beta\) belongs to \(X^\gamma_\alpha\) if and only if \(\alpha\leq \beta\leq\gamma\). In the article under review the author computes the degree of the tangent cone of \(X^\gamma_\alpha\) at \(e_\beta\), ie the multiplicity of \(X^\gamma_\alpha\) at \(e_\beta\). This follows as corollary from the explicit computation of a Gröbner basis \(G^\gamma_{\alpha,\beta}\) for a suitable monomial order of the ideal of \(Y^\gamma_{\alpha,\beta}=X^\gamma_\alpha\cap\mathcal O_\beta\), where \(\mathcal O_\beta\) is a special affine open subset of \(Gr_{d,n}\). The proof relies on a generalization of the Robinson-Schensted-Knuth correspondence, introduced by the author and called bounded RSK correspondence.
This paper extends previous results of \textit{V. Kodiyalam} and \textit{K. N. Raghavan} [J. Algebra 270, No. 1, 28--54 (2003; Zbl 1083.14056)] and \textit{V. Kreiman} and \textit{V. Lakshmibai} [in: Algebra, arithmetic and geometry with applications. Papers from Shreeram S. Ahhyankar's 70th birthday conference, Purdue University, West Lafayette, IN, USA, July 19--26, 2000. (Berlin): Springer. 553--563 (2003; Zbl 1092.14060)] on Schubert varieties. Schubert variety; Grassmannian; multiplicity; Robinson-Schensted-Knuth correspondence S. Upadhyay, \textit{Initial ideals of tangent cones to Richardson varieties in the orthogonal Grassmannian via a orthogonal-bounded-RSK-correspondence}, preprint, http://arxiv.org/abs/0909.1424. Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Computational aspects of higher-dimensional varieties, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study invariant generalized almost complex structures on real flag manifolds. In particular, they obtain the classification of real flag manifolds that admit generalized almost complex structures (Theorem 3.1). It is shown in Theorem 4.1 that no \(GM_2\)-maximal real flag manifolds admit integrable invariant generalized almost complex structures.
Section 5 is devoted to a concrete description of the generalized complex geometry on the maximal real flags of type \(B_2\), \(G_2\), \(A_3\), and \(D_l\) with \(l\geq 5\). For these cases, the authors study the action by invariant \(B\)-transformations on the space of invariant generalized almost complex structures. They prove the following: for every \(M\)-class root space, any generalized complex structure which is not of complex type is a \(B\)-transform of a structure of symplectic type; every element in the set of generalized complex structures of complex type is fixed for the action induced by \(B\)-transformations (Propositions 5.4 and 5.5). Moreover, the space of invariant generalized almost complex structures under invariant \(B\)-transformations is homotopy equivalent to a torus. The authors classify all invariant generalized almost Hermitian structures on them (Proposition 5.9). \(B\)-transformation; invariant generalized almost complex structure; integrability; generalized geometry; real flag manifold Generalized geometries (à la Hitchin), Grassmannians, Schubert varieties, flag manifolds, General geometric structures on manifolds (almost complex, almost product structures, etc.), Differential geometry of homogeneous manifolds Invariant generalized almost complex structures on real 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 For finite dimensional hermitean inner product spaces \(V\), over \(\ast\)-fields \(F\), and in the presence of orthogonal bases providing form elements in the prime subfield of \(F\), we show that quantifier-free definable relations in the subspace lattice \(\mathsf{L}(V)\), endowed with the involution induced by orthogonality, admit quantifier-free descriptions within \(F\), also in terms of Grassmann-Plücker coordinates. In the latter setting, homogeneous descriptions are obtained if one allows quantification type \(\Sigma _1\). In absence of involution, these results remain valid.
For Part I see [the authors, ibid. 79, No. 3, Paper No. 68, 26 p. (2018; Zbl 1472.03029)]. subspace lattice; involution; definable relations; constructible sets; Grassmann-Plücker coordinates Complemented modular lattices, continuous geometries, Interpolation, preservation, definability, Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Euclidean analytic geometry Definable relations in finite dimensional subspace lattices with involution. II. Quantifier-free and homogeneous descriptions | 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 PBW structures play a very important role in the Lie theory and in the theory of algebraic groups. The importance is due to the huge number of possible applications. The main goal of the workshop was to bring together experts and young researchers working in the certain areas in which PBW structures naturally appear. The interaction between the participants allowed to find new viewpoints on the classical mathematical structures and to launch the study of new directions in geometric, algebraic and combinatorial Lie theory. Collections of abstracts of lectures, Proceedings of conferences of miscellaneous specific interest, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Universal enveloping (super)algebras, Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Symmetric functions and generalizations, Proceedings, conferences, collections, etc. pertaining to nonassociative rings and algebras, Proceedings, conferences, collections, etc. pertaining to group theory Mini-workshop: PBW structures in representation theory. Abstracts from the mini-workshop held February 28 -- March 5, 2016 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a reductive algebraic group over \(\mathbb{C}\) and let \(N\) be a \(G\)-module. Choose a principal nilpotent \(X\) in \(\text{Lie\,}G\). For any subspace \(M\) of \(N\), the kernels of the powers of \(X\) define the Brylinski-Kostant filtration on \(M\). This filtration is related to a \(q\)-analog of weight multiplicity due to Lusztig.
The author generalizes this filtration to the case when \(X\) is a Richardson nilpotent of a parabolic and shows that this generalized filtration is related to ``parabolic'' versions of Lusztig's \(q\)-analog of weight multiplicity. For this he also needs to generalize results of Broer on cohomology vanishing of vector bundles on cotangent bundles of partial flag varieties. He concludes by computing some explicit examples. Brylinski-Kostant filtrations; cotangent bundle cohomology; flag varieties; Kazhdan-Lusztig polynomials; \(sl_2\)-triples; nilpotent elements Hague C.: Cohomology of flag varieties and the Brylinski-Kostant filtration. J. Algebra 321, 3790--3815 (2009) Cohomology theory for linear algebraic groups, Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds Cohomology of flag varieties and the Brylinski-Kostant filtration. | 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 the BGG category \({\mathcal O}\) associated to the general linear Lie algebra \({\mathfrak gl}_n\) and its principal block \({\mathcal O}_0\). Given a pair of (orthogonal) parabolic subalgebras of \({\mathfrak gl}_n\), the author associates a Serre quotient of a Serre subcategory of \({\mathcal O}_0\). In related work of the author [Sel. Math., New Ser. 22, No. 2, 669--734 (2016; Zbl 1407.17012)], these subquotients of \({\mathcal O}_0\) are used to categorify some representations of the quantized Lie superalgebra \({\mathfrak gl}(1|1)\). In the paper under review, these subquotient categories are shown to be equivalent to (graded) module categories of certain diagram algebras. This equivalence is then used to compute the endomorphism rings of two special functors.
The diagram algebras are explicitly constructed following the ideas used by \textit{J. Brundan} and \textit{C. Stroppel} [Mosc. Math. J. 11, No. 4, 685--722 (2011; Zbl 1275.17012)] to construct generalized Khovanov algebras. It is directly shown that these diagram algebras are graded cellular and properly stratified.
The approach used by Brundan and Stroppel to define the multiplication in the diagram algebras will not work in this setting. The author overcomes this obstruction by making use of morphisms between Soergel modules, which are modules for the complex polynomial ring \({\mathbb C}[x_1, \dots, x_n]\). Each module is associated to a permutation (of \(n\) objects), but they are not well understood in general. The computation of the relevant Soergel modules is aided by the fact that they are all cyclic, that is, the corresponding Schubert variety is rationally smooth in the full flag variety. A key portion of the paper (of independent interest) is a study of these particular Soergel modules. Each relevant Soergel module is shown to be isomorphic to \({\mathbb C}[x_1,\dots, x_n]/I\) for an explicit ideal \(I\) generated by symmetric polynomials. From this, the author obtains a description of the morphisms between the relevant Soergel modules.
The ideas in the paper are nicely developed with a thorough discussion of the tools and combinatorics being used, along with numerous examples. diagram algebra; symmetric polynomials; category \({\mathcal O}\); Soergel modules; Khovanov algebra; general linear Lie algebra; Serre category; categorification Sartori, A.: A diagram algebra for Soergel modules corresponding to smooth Schubert varieties. Trans. Amer. Math. Soc. (2013). 10.1090/tran/6346 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Symmetric functions and generalizations, Polynomial rings and ideals; rings of integer-valued polynomials, Grassmannians, Schubert varieties, flag manifolds, Graded rings and modules (associative rings and algebras) A diagram algebra for Soergel modules corresponding to smooth 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 Two-row Springer fibers are varieties of flags of vector spaces fixed by a nilpotent operator with two Jordan blocks. The authors give a topological description of the two-row Springer fiber over the real numbers by showing that its cohomology ring coincides with the oddification of the cohomology ring of the complex Springer fiber (Theorem 3.6, page 1425). They also realize a certain odd topological quantum field theory from pullbacks and exceptional pushforwards along inclusion and projection maps between hypertori (Theorem 4.20, page 1441). The authors conclude by giving an odd analog, via constructing the odd arc algebra as a convolution algebra over components of the real Springer fiber (Theorem 5.6, page 1447). two-row Springer fibers; real Springer fibers; odd arc algebras; oddified cohomology ring; odd topological quantum field theory Grassmannians, Schubert varieties, flag manifolds, Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.), Topology of real algebraic varieties, Graded rings and modules (associative rings and algebras), ``Super'' (or ``skew'') structure, Quantum groups (quantized function algebras) and their representations Real Springer fibers and odd arc 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 An integral basis of the cohomology of a flag manifold \(G/B\) is the Schubert classes \(\mathfrak S_w\) indexed by the elements \(w\) of the Weyl group of \(G\). Hence there are, for all pairs of elements \(u,v\) in the Weyl group, integers \(c_{uv}^w\) such that
\[
\mathfrak S_u\mathfrak S_v =\sum_wc_{uv}^w \mathfrak S_w,
\]
where the summation is over the elements in the Weyl group. A Pieri type formula is a formula that describes the structure constants \(c_{uv}^w\) when \(\mathfrak S _v\) is a special Schubert class pulled back from the projection \(G/B\to G/P\), where \(P\) is a maximal parabolic subgroup. When \(G=\text{Gl}_n(\mathbb{C})\) the classical Pieri formula gives such a description. For other \(G\) there are formulas by \textit{H. Hiller} and \textit{B. Boe} [Adv. Math. 62, 49-67 (1986; Zbl 0611.14036)] and by \textit{P. Pragacz} and \textit{J. Ratajski} [J. Reine Angew. Math. 476, 143-189 (1996; Zbl 0847.14029); C. R. Acad. Sci., Paris, Sér. I 317, 1035-1040 (1993; Zbl 0812.14034); Manuscr. Math. 79, 127-151 (1993; Zbl 0789.14041)]. When \(G=\text{Gl}_n(\mathbb{C})\) an interpretation of the structure constants \(c_{uv}^w\) in Pieri's formula, in terms of chains in the Bruhat order, was conjectured by \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, 257-269 (1993; Zbl 0803.05054)] and given algebraic, geometric and combinatorial proofs by \textit{A. Postnikov} [Prog. Math. 172, 371-383 (1999; Zbl 0944.14019)], \textit{F. Sottile} [Ann. Inst. Fourier 46, 89-110 (1996; Zbl 0837.14041)], and \textit{M. Kogan} and \textit{A. Kumar} [Proc. Am. Math. Soc. 130, 2525-2534 (2002; Zbl 1001.05121)], respectively. The main result of the present article are analogous Pieri type formulas when \(G\) is \(\text{Sp}_{2n}(\mathbb{C})\) and \(\text{SO}_{2n+1}(\mathbb{C})\). One of the techniques used is to explicitly determine triple intersections of Schubert varieties. \textit{F. Sottile} has used this technique with success earlier [see, e.g., Colloq. Math. 82, 49-63 (1999; Zbl 0977.14023)], and shows that the coefficients in the Pieri type formulas are the intersection number of a linear space with a collection of quadrics, and thus are either \(0\) or a power of \(2\). special Schubert classes; Schubert varieties; Bruhat order; Pieri type formulas; Weyl groups; parabolic groups; isotropic flag manifolds; cohomology \beginbarticle \bauthor\binitsN. \bsnmBergeron and \bauthor\binitsF. \bsnmSottile, \batitleA Pieri-type formula for isotropic flag manifolds, \bjtitleTrans. Amer. Math. Soc. \bvolume354 (\byear2002), no. \bissue7, page 2659-\blpage2705 \bcomment(electronic). \endbarticle \OrigBibText ----, A Pieri-type formula for isotropic flag manifolds , Trans. Amer. Math. Soc. 354 (2002), no. 7, 2659-2705 (electronic). \endOrigBibText \bptokstructpyb \endbibitem Classical groups (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Symmetric functions and generalizations, Combinatorics of partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] A Pieri-type formula for isotropic 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 homology of the space of Iwahori subalgebras containing a nilpotent element of a split semisimple Lie algebra over the field \(\mathbb{C}((\varepsilon))\) is isomorphic to the homology of the entire affine flag manifold. affine flag manifolds; homology; Iwahori subalgebras; nilpotent element; semisimple Lie algebra E. Sommers, Nilpotent Orbits and the Affine Flag Manifold, Ph.D. Thesis, M.I.T, 1997. Group structures and generalizations on infinite-dimensional manifolds, Homology and cohomology of Lie groups, Infinite-dimensional Lie groups and their Lie algebras: general properties, Classical real and complex (co)homology in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds The homology of the space of affine flags containing a nilpotent element | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0719.00022.]
Let \(F_ n\) denote the space of complete flags in \(k^ n\) (k\(\in \{{\mathbb{R}},{\mathbb{C}}\})\) and let \(c_{\sigma}\) be the cell of the standard Schubert cell decomposition \(Sch_ f\) of \(F_ n\), corresponding to a complete flag \(f\in F_ n\), whose cells are enumerated by permutations \(\sigma \in S_ n\). Define the link of \(c_{\sigma}\) to be the manifold \(A_{\sigma}=B\setminus Tn_ f\), where B is a sufficiently small (n(n- 1)/2)-dimensional (over k) ball centered at some point of \(c_{\sigma}\) and \(Tn_ f\) is the train of the flag f. Define the Euler characteristic of \(A_{\sigma}\) to be \(\chi_{\sigma}=\sum_{\ell}(-1)^{\ell}\dim (H^{\ell}(A_{\sigma})) \). Finally, define the number \(\chi_{\sigma}^{pq}=\sum_{\ell}(-1)^{\ell}\dim (Gr^ p_ FGr^ w_{p+q}H^{\ell}(A_{\sigma})) \) where \(Gr^ W\) (resp. \(Gr_ F)\) are the associated graded objects of the weight (resp. Hodge filtration). The authors show how to reduce the calculation of \(\chi_{\sigma}\) and \(\chi_{\sigma}^{pq}\) for \(F_ n\) to those for \(F_{n-1}\) and then do the calculation for low dimensions. They also find a relation between \(\chi_{\sigma}\) and \(\chi_{\sigma}^{pq}\). Euler characteristics for links of Schubert cells; Hodge filtrations; train of the flag; weight Shapiro, B. Z.; Vainshtein, A. D.: Euler characteristics for links of Schubert cells in the space of complete flags. Adv. sov. Math. 1, 273-286 (1990) Grassmannians, Schubert varieties, flag manifolds, Stratified sets Euler characteristics for links of Schubert cells in the space of complete flags | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{S. Franco} et al., [``Non-planar on-shell diagrams'', Preprint, \url{arXiv:1502.02034}] have proposed a boundary measurement map for a graph on any closed orientable surface with boundary. We consider this boundary measurement map which takes as input an edge weighted directed graph embedded on a surface and produces on element of a Grassmannian. Computing the boundary measurement requires a choice of fundamental domain. Here the boundary measurement map is shown to be independent of the choice of fundamental domain Also, a formula for the Plücker coordinates of the element of the Grassmannian in the image of the boundary measurement map is given. The formula expresses the Plücker coordinates as a rational function which can be combinatorially described in terms of paths and cycles in the directed graph. boundary measurement; plabic graphs; totally nonnegative Grassmannian; Plücker coordinates Grassmannians, Schubert varieties, flag manifolds, Planar graphs; geometric and topological aspects of graph theory Boundary measurement matrices for directed networks on 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 We describe the structure of the Chow-Arakelov ring of Grassmannians, and deduce from this the rationality of the height of their Plücker embedding. Schubert calculus; Chow-Arakelov ring; Grassmannians; height; Plücker embedding Grassmannians, Schubert varieties, flag manifolds, Arithmetic varieties and schemes; Arakelov theory; heights, (Equivariant) Chow groups and rings; motives, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry An arithmetic 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 \(\mathfrak g\) be a Kac-Moody algebra and \(Q\) be some orientation of the Dynkin diagram of \(\mathfrak g\). Let \(\widetilde Q\) be the double quiver of \(Q\). The main result of this paper asserts that a certain quiver Grassmannian for \(\widetilde Q\) is homeomorphic to a certain Lagrangian Nakajima quiver variety. The authors also refine this result by finding quiver Grassmannians that are homeomorphic to the Demazure quiver varieties, and others which are homeomorphic to the graded/cyclic quiver varieties. The Demazure quiver Grassmannian allows the authors to describe injective objects in the category of locally nilpotent modules over the preprojective algebra.
Motivated by an earlier version of the paper, \textit{I. Shipman} has recently proved [see Math. Res. Lett. 17, No. 5, 969-976 (2010; Zbl 1231.16010)] that Lusztig's bijection between the points of the Lagrangian Nakajima quiver variety and the points of a type of quiver Grassmannian inside a projective object is, in fact, an isomorphism of algebraic varieties. The present paper has an Appendix where the authors explain how this result allows one to conclude that the maps between quiver Grassmannians and Lagrangian Nakajima quiver varieties described in the paper under review are also isomorphisms of algebraic varieties. quivers; preprojective algebras; quiver Grassmannians; quiver varieties; Kac-Moody algebras; Demazure modules Savage, A; Tingley, P, Quiver Grassmannians, quiver varieties and the preprojective algebra, Pacific J. Math., 251, 393-429, (2011) Representations of quivers and partially ordered sets, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Group actions on affine varieties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Quiver Grassmannians, quiver varieties and the preprojective algebra. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Motivated by a recent conjecture of \textit{R. P. Stanley} [``Some Schubert shenanigans'', Preprint, \url{arXiv:1704.00851}] we offer a lower bound for the sum of the coefficients of a Schubert polynomial in terms of 132-pattern containment. Schubert polynomials; permutation patterns Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Schubert polynomials, 132-patterns, and Stanley's 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 Let \(Q_ 2=G_{4,2}({\mathbb{R}})\) be the non-degenerate quadric in \({\mathbb{P}}^ 3({\mathbb{C}})\) identified with the real Grassmann manifold of planes in \({\mathbb{R}}^ 4\). Let G denote the Grassmann bundle associated to the real tangent space \(TQ_ 2\) and let G' denote the sub-bundle of G corresponding fibre-wisely to the totally real planes in \(TQ_ 2\). If M is a totally real surface embedded into \(Q_ 2\), then its Gauss map \(t: M\to G\) factorizes through G'. Due to the topology of G', the manifold M carries by pull-back a form \(\alpha_ M\in H^ 1_{comp}(M,{\mathbb{Z}}_ 2)\) which reflects its position in G'. This cohomology class is analogous to the Maslov-Arnold index attached to Lagrangian submanifolds of \({\mathbb{R}}^{2n}.\)
The author presents an invariant construction of the class \(\alpha_ M\) in terms of the twistor space. As an application of these tools, the author studies the geometry of algebraic curves in \({\mathbb{P}}^ 3({\mathbb{C}})\), via the twistor map \({\mathbb{P}}^ 3({\mathbb{C}})\to S^ 4\). twistor transformation; Grassmann manifold; totally real surface Th. Fiedler, ''A characteristic class for totally real surfaces in the grassmanian of two-planes in four-space,''Ann. Global Anal. Geom.,4, 121--132 (1986). Twistor theory, double fibrations (complex-analytic aspects), Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) A characteristic class for totally real surfaces in the Grassmannian of two-planes in four-space | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper outlines the construction of the Langlands duality, i.e. the equivalent of the category of \(k\)-representations of the Langlands dual group \(^LG\) and the category of perverse sheaves with coefficients in \(\mathbb{K}\) on the affine Grassmannian associated in \(G\) for a complex algebraic reductive group \(G\). The result for \(\mathbb{K}\) the field of complex numbers is due to \textit{V. Ginzburg} (alg-geom/9511007). The authors generalize it to any commutative ring \(\mathbb{K}\), and claim new proofs for some crucial steps of the complex case, but the text only outlines the constructions and announces the results. Satake isomorphism; Langlands duality; perverse sheaves; affine Grassmannian I. Mirković and K. Vilonen, ''Perverse sheaves on affine Grassmannians and Langlands duality,'' Math. Res. Lett., vol. 7, iss. 1, pp. 13-24, 2000. Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Perverse sheaves on affine Grassmannians and Langlands duality | 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 formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of \(\operatorname{SL}_r\)-webs and is built upon the \(r\)-fold dimer model on the network. When \(r\) equals 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata. When \(r\) equals 2 or 3, it is a reformulation of the \(\operatorname{SL}_2\)- and \(\operatorname{SL}_3\)-web immanants defined by the second author. The basic result is that the higher-rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of \(\operatorname{SL}_r\)-webs, re-proving a result of \textit{S. Cautis} et al. [Math. Ann. 360, No. 1--2, 351--390 (2014; Zbl 1387.17027)]. We establish compatibility between our map and restriction to positroid strata and thus between webs and total positivity. dimer; web; boundary measurement; positroid; Grassmannian Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups, Planar graphs; geometric and topological aspects of graph theory From dimers to webs | 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 jeu-de-taquin-based Littlewood-Richardson rule of \textit{H. Thomas} and \textit{A. Yong} [Adv. Math. 222, No. 2, 596--620 (2009; Zbl 1208.14052)] for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, \textit{A. Skovsted Buch} and \textit{M. J. Samuel} [J. Reine Angew. Math. 719, 133--171 (2016; Zbl 1431.19001)] developed a combinatorial theory of ``unique rectification targets'' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to \(K\)-theory. Separately, \textit{P.-E. Chaput} and \textit{N. Perrin} [J. Lie Theory 22, No. 1, 17--80 (2012; Zbl 1244.14036)] used the combinatorics of Proctor's ``\(d\)-complete posets'' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of \(d\)-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain \(K\)-theoretic Schubert structure constants in the Kac-Moody setting. Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Equivariant \(K\)-theory Unique rectification in \(d\)-complete posets | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple algebraic group over an algebraically closed field of arbitrary characteristic, and \(B\subset G\) a Borel subgroup. Recall that for every element \(w\) of the Weyl group of \(G\), the Schubert variety \(X(w)\subset G/B\) is defined as the Zariski closure of the Bruhat cell \(BwB/B\subset G/B\). A connection between Schubert varieties and toric varieties has been studied extensively, in particular, flat toric degenerations of Schubert varieties were constructed in [\textit{P. Caldero}, Transform. Groups 7, No. 1, 51--60 (2002; Zbl 1050.14040)].
The present paper describes all Schubert varieties that are already toric. Namely, the author proves that \(X(w)\) is a toric variety if and only if \(w\) is a product of pairwise distinct simple reflections (in particular, the dimension of \(X(w)\) does not exceed the rank of \(G\)). Schubert variety; toric variety Karuppuchamy, P., On Schubert varieties, Commun. Algebra, 41, 4, 1365-1368, (2013) Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies On Schubert varieties | 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 normalized quantum K-theoretic vertex functions for cotangent bundles of partial flag varieties are the eigenfunctions of quantum trigonometric Ruijsenaars-Schneider (tRS) Hamiltonians. Using recently observed relations between quantum Knizhnik-Zamolodchikov (qKZ) equations and tRS integrable system we derive a nontrivial identity for vertex functions with relative insertions. Special quantum systems, such as solvable systems, Selfadjoint operator theory in quantum theory, including spectral analysis, \(n\)-vertex theorems via direct methods, Grassmannians, Schubert varieties, flag manifolds, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) qKZ/tRS duality via quantum K-theoretic counts | 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 \(G : = G (1,n,q)\) denote the Grassmannian of lines in \(PG(n,q)\), embedded as a point-set in \(PG(N,q)\) with \(N := \binom{n+1}{2}-1\). For \(n=2\) or 3 the characteristic function \(\chi(\overline G)\) of the complement of \(G\) is contained in the linear code generated by characteristic functions of complements of \(n\)-flats in \(PG(N,q)\). In this paper we prove this to be true for all cases \((n,q)\) with \(q=2\) and we conjecture this to be true for all remaining cases \((n,q)\). We show that the exact polynomial degree of \(\chi(\overline G)\) is \((q-1)(\binom{n}{2}-1+\delta)\) for \(\delta : = \delta(n,q) = 0\) or 1, and that the possibility \(\delta = 1\) is ruled out if the above conjecture is true. The result \(\deg(\chi(\overline G)) = \binom{n}{2}-1\) for the binary cases \((n,2)\) can be used to construct quantum codes by intersecting \(G\) with subspaces of dimension at least \(\binom{n}{2}\). Polynomial degree; Geometric codes; Quantum codes Glynn D.G., Maks J.G., Casse L.R.A.: The polynomial degree of the Grassmannian G(1, n, q)of lines in finite projective space PG(n, q). Des. Codes Cryptogr. 40, 335--341 (2006) Grassmannians, Schubert varieties, flag manifolds, Linear codes and caps in Galois spaces, Geometric methods (including applications of algebraic geometry) applied to coding theory The polynomial degree of the Grassmannian \(G(1,n,q)\) of lines in finite projective space \(PG(n,q)\) | 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 goal of this extend paper is to give a geometric explanation of the problem posed by Ginzburg of finding a geometric interpretation of Broer's covariant theorem in the context of geometric Satake. For a simply-connected simple algebraic group \(G\) over \(\mathbb{C}\), the authors show a subvariety of its affine Grassmannian \(\mathbf{Gr}\) that is closely related to the nilpotent cone \(\mathcal{N}\) of \(G\), generalizing a well-known result about \(\mathrm{GL}_n\). By using this subvariety, they construct a sheaf-theoretic functor that leads, when combined with the geometric Satake equivalence and the Springer correspondence, to a geometric explanation for a number of known facts, previously obtained by Broer and Reeder, about small representations of the dual group. As the main result, they prove that there is an action of \(\mathbb{Z} / 2 \mathbb{Z}\) on \(\mathcal{M}\) commuting with the \(G\)-action, and a finite \(G\)-equivariant map \(\mathcal{M} \mapsto \mathcal{N}\) that induces a bijection between \(\mathcal{M} / (\mathbb{Z} / 2 \mathbb{Z})\) and a certain closed subvariety \(\mathcal{N}_{sm}\) of \(\mathcal{N}\), where \(\mathbf{Gr}_{sm}\) is the closed subvariety of \(\mathbf{Gr}\) corresponding to small representation under geometric Satake and \(\mathcal{M} \in \mathbf{Gr}\) is the intersection of \(\mathbf{Gr}_{sm}\) with the opposite Bruhal cell. affine Grassmannian; nilpotent orbits; Springer correspondence Achar, P., Henderson, A.: Geometric Satake, Springer correspondence, and small representations. Preprint. arXiv:1108.4999 (2012) Coadjoint orbits; nilpotent varieties, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Geometric Satake, Springer correspondence and small representations | 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 Suppose some complex veector bundles are given over a manifold \(M\), together with some (sufficiently generic) vector bundle maps among them. Over a point in \(M\) then one has a quiver: some vector spaces with linear maps among them. Quivers may degenerate -- e.g. dimensions of intersections of images, kernels may drop, and more general degenerations also happen. A quiver degeneracy locus is the set a points in \(M\) over which the quiver degenerates in a particular way.
It is known that the fundamental class of quiver degeneracy loci can be expressed by a universal polynomial (the quiver polynomial) in the charactersitic classes of the bundles involved, both in cohomology and in \(K\)-theory. The paper under review presents such universal polynomials in \(K\)-theory if the quiver is of Dynkin type.
The formula presented is a non-conventional generating function (named Iterated Residue generating function, pioneered by Bérczi-Szenes, Kazarian, Feher-Rimányi). Advantages of the presented formula for quiver loci include that stabilization properties are explicit, and the expansion in terms of Grothendieck polynomials is well motivated.
The paper ends with interesting comments comparing \(K\)-theory and cohomology Iterated Residue formulas, as well as remarks on Buch's positivity conjecture on the coefficients of the Grothendieck polynomial expansion. quiver polynomials; Grothendieck polynomials; iterated residues; equivariant K theory Allman, Justin, Grothendieck classes of quiver cycles as iterated residues, Michigan Math. J., 63, 4, 865-888, (2014) Grassmannians, Schubert varieties, flag manifolds, Equivariant \(K\)-theory, Symmetric functions and generalizations, Representations of quivers and partially ordered sets Grothendieck classes of quiver cycles as 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 We establish effective uniform degree bounds for the generalized Gauss map images of an embedded projective variety \(X\subset\mathbb{P}^N\) in terms of numerical invariants such as \(\dim \)X, \(\deg X\) and \(N\). This can be seen as a generalization of a classical Castelnuovo type bound. Heier, G.; Takayama, S., Effective degree bounds for generalized Gauss map images \textit{Adv. Stud. Pure Math.}, 74, 203-235, (2017) Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, \(n\)-folds (\(n>4\)) Effective degree bounds for generalized Gauss map images | 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 Every morphism from \(\mathbb{P}^n\) to \(\mathbb{G}(k,m)\) is constant if \(m<n\), and nonconstant morphisms from \(\mathbb{P}^n\) to \(\mathbb{G}(k,n)\) rarely appear when \(0<k<n-1\). In this setting, \textit{H. Tango} [J. Math. Kyoto Univ. 16, 201--207 (1976; Zbl 0326.14015); Bull. Kyoto Univ. Educ., Ser. B 64, 1--20 (1984; Zbl 0642.14012)] proved that a morphism from \(\mathbb{P}^n\) to \(\mathbb{G}(1,n)\) is constant if \(n\notin\{3,5\}\). Here we focus on the case \(n=3\) and show that if \(\phi :\mathcal{O}_{\mathbb{P}^3}^{\oplus 4}\rightarrow E\) is the surjection onto a rank \(2\) vector bundle \(E\) inducing a morphism \(\varphi :\mathbb{P}^3\rightarrow\mathbb{G}(1,3)\), then \(h^1(E^*)\le 1\). Furthermore, a complete classification is given if equality holds. Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry On morphisms from \(\mathbb{P}^3\) to \(\mathbb{G}(1,3)\) | 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 article is to give explicit formulae for various generating functions, including the generating function of torus-invariant primitive ideals in the big cell of the quantum minuscule Grassmannian of type \(B_n\).
Let \(\mathfrak g\) be a simple Lie algebra of rank \(n\) over the field of complex numbers, and let \(\pi:=\{\alpha_1,\ldots,\alpha_n\}\) be the set of simple roots associated to a triangular decomposition \(\mathfrak g=\mathfrak n^-\oplus\mathfrak h\oplus\mathfrak n^+\). Let \(W\) be the Weyl group associated to \(\mathfrak g\).
The aim of this article is to study the prime spectrum of so-called quantum Schubert cells from the point of view of algebraic combinatorics. Quantum Schubert cells have been introduced by \textit{C. De Concini, V. G. Kac} and \textit{C. Procesi} [Stud. Math., Tata Inst. Fundam. Res. 13, 41--65 (1995; Zbl 0878.17014)] as quantisations of enveloping algebras of nilpotent Lie algebras \(\mathfrak n_w:=\mathfrak n^+\cap\mathrm{Ad}_w(\mathfrak n^-)\), where \(\mathrm{Ad}\) stands for the adjoint action and \(w\in W\). These noncommutative algebras are defined thanks to the braid group action of \(W\) on the quantised enveloping algebra \(U_q(\mathfrak g)\) induced by Lusztig automorphisms. The resulting (quantum) algebra associated to a chosen \(w\in W\) is denoted by \(U_q[w]\). Here \(q\) denotes a nonzero element of the base field \(\mathbb K\), and we assume that \(q\) is not a root of unity. It was recently shown by \textit{M. Yakimov} [Proc. Am. Math. Soc. 138, No. 4, 1249--1261 (2010; Zbl 1245.16030)] that these algebras can be seen as the Schubert cells of the quantum flag varieties. Our aim is to study combinatorially the prime spectrum of the algebras \(U_q[w]\). quantum algebras; quantized enveloping algebras; primitive ideals; quantum Schubert cells; quantum flag varieties; algebraic combinatorics Ring-theoretic aspects of quantum groups, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Ideals in associative algebras Enumeration of torus-invariant strata with respect to dimension in the big cell of the quantum minuscule Grassmannian of type \(B_n\). | 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 \(G_0\) be a real semisimple Lie group of Hermitian type. Associated to a \(G_0\)-orbit \(\gamma\) in an arbitrary manifold \(Z=G/Q\) there is a natural group theoretically defined cycle space \(C(\gamma)\). The main goal of this survey is to show that a given orbit \(\gamma\) is either of special Hermitian type and \(C(\gamma)\) is just the associated bounded domain, or \(C(\gamma)\) is biholomorphically equivalent to a certain universal domain, which is a remarkable neighborhood \(\mathcal U\) of the Riemannian symmetric space \(G_0/K_0\) and its complexification. The first part of the work presents necessary background that should be useful to motivate complex geometers to work on open questions. Homogeneous complex manifolds; flag manifolds; cycle spaces Homogeneous complex manifolds, Grassmannians, Schubert varieties, flag manifolds, Algebraic cycles Actions on flag manifolds: related cycle spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author looks at algebraic-geometric codes arising from vector bundles over a finite field. In particular, his investigation focuses upon those bundles constructed via finite coverings of the curve over a finite field. The main result provides conditions for the existence of weakly stable bundles of certain rank and degree having a specified dimension and minimum distance, from which the code generated can be used to correct a fixed number of errors. AG codes; stable vector bundles; algebraic curves; finite coverings Nakashima T.: AG codes from vector bundles. Des. Codes Cryptogr. \textbf{57}, 107-115 (2010). Grassmannians, Schubert varieties, flag manifolds, Applications to coding theory and cryptography of arithmetic geometry AG codes from vector bundles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple adjoint group over the field \(\mathbb C\) of complex numbers. Let \(T\) be a maximal torus of \(G\). Let \(P\) be a parabolic subgroup of \(G\). In this article, we give a survey on the Geometric Invariant Theory related problems for the left action of \(T\) on \(G/P\). Schubert varieties; line bundles; semi-stable points S. S. Kannan, ``GIT related problems of the flag variety for the action of a maximal torus'' in Groups of Exceptional Type, Coxeter Groups and Related Geometries , Springer Proc. Math. Stat. 82 , Springer, New Delhi, 2014, 189-203. Grassmannians, Schubert varieties, flag manifolds, Geometric invariant theory GIT related problems of the flag variety for the action of a maximal torus | 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 indicate how Hecke algebras, the Yang-Baxter equation, Hall-Littlewood polynomials, and Macdonald polynomials are related to the parameter \(y\) that Hirzebruch introduced in his study of the Riemann-Roch theorem. \(\chi_y\)-characteristic; flag variety; Riemann-Roch; Hall-Littlewood polynomial; Yang-Baxter equation; Hecke algebra; Macdonald polynomial; characteristic of Hirzebruch; spaces of cohomology; generating function; manifold; chern classes; cohomology ring; Grothendieck ring Combinatorial aspects of representation theory, Riemann-Roch theorems, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups About the ``\(y\)'' in the \(\chi_y\)-characteristic of Hirzebruch | 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 in model theory that it is far from being true that elementary equivalence of two fields implies their isomorphism. But this question becomes interesting when restricting to certain classes of fields. For example, it follows from work by Duret and Pierce that if \(K_1\) and \(K_2\) are function fields of curves over an algebraically closed field, then either both are of genus one (with \(K_1\) and \(K_2\) isomorphic in many but not all cases), or both function fields are of genus \(\neq 1\) and isomorphic. Pop has shown that two elementarily equivalent fields that are finitely generated over an algebraically closed field (resp. their prime field) are of the same transcendence degree, from which one can conclude that elementarily equivalent function fields, one of them being of so-called general type, over an algebraically closed field, a number field or a finite field are always isomorphic (Corollary 4 in the present paper in the case of number fields and finite fields).
Here, the author studies this question for function fields of quadrics and Severi-Brauer varieties. He introduces the notion of two fields \(K_1\) and \(K_2\) being (\(k\)-)isogenous, i.e. there exist field homomorphisms \(K_1\to K_2\) and \(K_2\to K_1\) (over a common base field \(k\)). In Theorem 7, he shows that function fields \(K_1\), \(K_2\) over some base field \(k\) of Severi-Brauer varieties of central simple \(k\)-algebras of the same degree and with cyclic division parts are isomorphic iff they are isogenous iff \(\text{Br}(K_1/k)=\text{Br}(K_2/k)\) holds for the Brauer kernels. For function fields \(K_1\) and \(K_2\) of \(n\)-dimensional quadrics (\(n\leq 2\)) over some field \(k\) of characteristic \(\neq 2\), he obtains the same results except for the condition on the Brauer kernels which has to be replaced by \(\text{Br}(lK_1/k)=\text{Br}(lK_2/k)\) for all quadratic extensions \(l/k\). If \(K_1\) is the function field over \(k\) (\(\text{char}(k)\neq 2\)) of an \(n\)-dimensional Severi-Brauer variety and \(K_2\) that of an \(n\)-dimensional quadric (\(n>1\)), then \(K_1\) and \(K_2\) are isomorphic iff they are isogenous iff they have the same Brauer kernels iff they are both rational function fields. He deduces (Corollary 8) that in all these situations, elementary equivalence implies isomorphism whenever \(k\) is algebraic over its prime field. In Theorem 10, he considers the case \(K=k(C)\) of a genus \(1\) curve \(C\) over a number field \(k\) and shows that under certain assumptions on the Jacobian \(J(C)\), any finitely generated field elementarily equivalent to \(K\) is actually isomorphic to \(K\). elementary equivalence; isomorphism; isogeny; function field; Severi-Brauer variety; quadric; elliptic curve; Jacobian Transcendental field extensions, Quadratic forms over general fields, Elliptic curves over global fields, Arithmetic theory of algebraic function fields, Model theory of fields, Grassmannians, Schubert varieties, flag manifolds, Brauer groups (algebraic aspects) On elementary equivalence, isomorphism and isogeny | 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 structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minor of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studies by \textit{R. P. Stanley} [Higher Comb., Proc. NATO Adv. Study Inst., Berlin (West) 1976, 49 (1977; Zbl 0359.05001)], \textit{B. Sturmfels} [Grőbner bases and convex polytopes. University Lecture Series. 8. Providece, RI: American Mathematical Society (AMS) (1996; Zbl 0856.13020)], \textit{T. Lam} and \textit{A. Postnikov} [Discrete Comput. Geom. 38, No. 3, 453--478 (2007; Zbl 1134.52019)]. Here we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with arrangement of \(t\)th largest minors. Finally, we show that arrangements of largest minors induce a structure of partially ordered set on the entire collection of minors. We use the Lam and Postnikov [loc.cit.] circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset. positive Grassmannian; sorted sets; triangulations; alcoved polytope; positroid stratification Farber, M., Mandelshtam, Y.: Arrangements of equal minors in the positive Grassmannian. In: DMTCS Proceedings of the 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), pp. 499-510 (2015) Combinatorial aspects of simplicial complexes, Grassmannians, Schubert varieties, flag manifolds, Special polytopes (linear programming, centrally symmetric, etc.) Arrangements of minors in the positive Grassmannian and a triangulation of the hypersimplex | 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 \(\mathrm{SU}(p, q)\)-flag domain is an open orbit of the real Lie group \(\mathrm{SU}(p, q)\) acting on the complex flag manifold associated to its complexification \(\mathrm{SL}(p + q, \mathbb{C})\). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and representation theoretical properties of the flag domain. This article is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed \(\mathrm{SU}(p, q)\)-orbit in the ambient manifold. Equivalently, the Borel group contains the \(AN\)-factor of some Iwasawa decomposition. We consider the Schubert varieties of this type which are of complementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements. Much of our work is of an algorithmic nature, but, for example in the case of maximal parabolics, i.e. Grassmannians, formulas are derived. flag domains; cycles; Schubert varieties; intersection pairing Grassmannians, Schubert varieties, flag manifolds On the intersection pairing between cycles in \(\mathrm{SU}(p,q)\)-flag domains and maximally real 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 \(k\) be an algebraically closed field of characteristic zero. The Grothendieck ring \(K_0({\mathcal{V}}/k)\) of algebraic varieties over \(k\) is generated (as an abelian group) by the isomorphism classes of schemes of finite type over \(k\) subject to the relations \([X]=[X\backslash Z] + [Z],\) where \(Z\subset X\) is a closed subscheme with the reduced structure. The product is defined as \([X]\cdot [Y]=[X\times Y].\) The main result of the paper asserts that for a pair of closed subschemes cut out (in certain way depending on a non-zero global section \(s\) of the appropriate homogenous variety) from the pair of Grassmanians of type \(G_2\) one has \(([X]-[Y])\cdot {\mathbb L} =0.\) Moreover, for the general choice of \(s\) one has \([X]\neq [Y] \) and both \(X\) and \(Y\) are smooth Calabi-Yau \(3\)-folds. Grothendieck ring; Grassmanian; Dynkin diagram; global section Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups and \(K_0\), Varieties and morphisms, Grassmannians, Schubert varieties, flag manifolds, Calabi-Yau manifolds (algebro-geometric aspects) The class of the affine line is a zero divisor in the Grothendieck ring: via \(G_2\)-Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We will use the combinatorics of the \(G\)-stable pieces to describe the closure relation of the partition of partial flag varieties in [\textit{G. Lusztig}, Represent. Theory 11, 122--171, electronic only (2007; Zbl 1163.20030), section 4]. He, X.: \(G\)-stable pieces and partial flag varieties. In: Representation Theory, 61-70. Contemp. Math., 478, Am. Math. Soc., Providence, RI (2009) Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over finite fields \(G\)-stable pieces and partial 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 We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac-Moody algebra. This naturally transplants the result of Kumar-Mathieu-Schwede about the Frobenius splitting of thin flag varieties to the thick case. As a consequence, we provide a description of the space of global sections of a line bundle of a thick Schubert variety as conjectured in Kashiwara-Shimozono [13]. This also yields the existence of a compatible basis of thick Demazure modules and the projective normality of the thick Schubert varieties. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Frobenius splitting of thick flag manifolds of Kac-Moody 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 Let \(G\) be a connected reductive group over an algebraically closed field \(k\) of positive characteristic and \(B\) be a Borel subgroup of \(G\). Let \(X\) be an equivariant embedding of the group \(G\). A \(G\)-Schubert variety in \(X\) is a subvariety of the form \(\text{diag}(G)\cdot V\), where \(V\) is a \(B\times B\)-orbit closure in \(X\). If \(X\) is the wonderful compactification of a semisimple group of adjoint type, then \(G\)-Schubert varieties coincide with the closures of \textit{G.~Lusztig}'s \(G\)-stable pieces [Parabolic character sheaves. I. Mosc. Math. J. 4, No. 1, 153--179 (2004; Zbl 1102.20030); II. Mosc. Math. J. 4, No. 4, 869--896 (2004; Zbl 1103.20041)].
The authors prove that \(X\) admits a Frobenius splitting which is compatible with all \(G\)-Schubert varieties. Moreover, when \(X\) is smooth, projective and toroidal, then any \(G\)-Schubert variety in \(X\) admits a stable Frobenius splitting along an ample divisor. On the other hand, an example of a nonnormal \(G\)-Schubert variety in the wonderful compactification of a group of type \(G_2\) is given. In the last section, a generalization of the Frobenius splitting results to the more general class of \(R\)-Schubert varieties is obtained. reductive groups; equivariant embeddings; Frobenius splitting; wonderful compactification P. Achinger, N. Perrin, \textit{Spherical multiple flag varieties}, preprint arXiv:1307.7236. Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds Frobenius splitting and geometry of \(G\)-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 Grassmann manifolds have the well-known Schubert cell decomposition. Here the volumes of some Schubert cells standardly embedded into a Grassmann manifold with a natural Riemannian structure are calculated. Grassmann manifolds; volume of Schubert cells Projective differential geometry, Grassmannians, Schubert varieties, flag manifolds Volumes of Schubert cells of Grassmann 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 finite stratification of the Grassmanninan \(\mathrm{Gr}(k,n)\) to so-called positroid varieties \(\Pi_f\) is a refinement of the usual Schubert stratification. Knutson-Lam-Speyer computed that the cohomology class of such a variety \(\Pi_f\) is represented by the affine Stanley symmetric function \(\tilde{F}_f\). The paper under review extends this result to the image of a positroid variety under the (rational) map \(Z_{\mathrm{Gr}}:\mathrm{Gr}(k,n)\to \mathrm{Gr}(k,k+m)\) induced by a linear map \(Z\) from \(n\) dimensions to \(k+m\) dimensions. The closure of the image of \(\Pi_f\) is denoted by \(Y_f\), and is called an amplituhedron variety (when the dimension does not drop under the map). The result of the present paper is that the cohomology class of \(Y_f\) is a truncated version of \(\tilde{F}_f\). Truncation is defined through the shape of the partitions occuring in a Schur expansion. positroid varieties; Schubert calculus; Stanley symmetric functions; amplituhedron variety Lam, T., Amplituhedron cells and Stanley symmetric functions, Commun. Math. Phys., 343, 1025, (2016) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Amplituhedron cells and Stanley symmetric 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 show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher-Kolmogorov-Petrovskii-Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems. nonlocal Riccati-type nonlinearities; Stiefel manifold; Grassmann manifold; Marchenko equation; Fisher-Kolmogorov-Petrovskii-Piskunov equations Nonlinear higher-order PDEs, Grassmannians, Schubert varieties, flag manifolds, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Grassmannian flows and applications to nonlinear partial differential equations | 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 Wenn Grassmann's fundamentale Arbeiten bis auf die neueste Zeit ziemlich unbeachtet waren und man jetzt erst eigentlich beginnt, auf dieselben aufmerksam zu werden und die mannigfachen tiefen Gedanken zu erfassen, die er in denselben niederlegte, so ist das zum Theil in der ausserordentlich abstracten, allgemeinen und eigenartigen Form begründet, welche Grassmann für seine Darstellung gewählt hat, und die das Verständniss derselben für Jeden, der nicht von Vornherein mit der Absicht des Autors bekannt ist, ausserordentlich erschwert. Um so nützlicher scheint ein Versuch, wie ihn das vorliegende Werk bringt, dem Publikum die Grundvorstellungen Grassmann's leichterer Form zugänglich zu machen. Der Verfasser sucht dies zu erreichen, indem er, den abstracten Standpunkt Grassmann's verlassend, von den einfachsten Fällen ausgeht und geradezu nur Mannigfaltigkeit von einer und von zwei Dimensionen betrachtet, die ihr Gegenbild in der Geometrie der Geraden (resp. des Punktes) und der Ebene finden. Er entwickelt also im Wesentlichen und zwar der Reihe nach für Grössen erster, zweiter, dritter Stufe d. h. Punkt resp. Strecke (die den unendlich fernen Punkt vertritt), Linien- und Ebenen-Theil die Gesetze der algebraischen Verknüpfung im Grassmann'schen Sinne, die Addition, die äussere und innere Multiplication. Die Darstellung lehnt sich im Ganzen an die von Grassmann in der zweiten Auflage seiner Ausdehnungslehre befolgte an; jedoch wird durch die strenge Sonderung der verschiedenen Gebiete die Einführung in die symbolische Bezeichnungsweise und die Verknüpfung der geometrischen Vorstellung mit ihr erleichtert. Andererseits sind dadurch freilich manche Wiederholungen bedingt, indem für jedes Gebiet die den früheren analogen Betrachtungen mit derselben Ausführlichkeit durchgeführt werden. Es hängt das wohl mit der Absicht des Verfassers zusammen, sich mit seinem Buche auch an das pädagogische Publikum wenden zu wollen. Mag man über die Zweckmässigkeit einer Einführung grade der Grassmann'schen Ideen in den Unterricht denken, wie man will, das Buch ist durch den doppelten Zweck weniger übersichtlich geworden, als es sonst hätte der Fall sein können. Der Verfasser unternimmt es nämlich, die meisten Lehrsätze der elementaren Planimetrie, wie sie gewöhnlich vorgetragen werden, mit Hülfe der Grassmann'schen Methoden zu entwickelt. So interessant es ist, diese bekannten Beziehungen aus allgemeineren Gesetzen entstehen zu sehen, wie z. B. den Pythagoreischen Lehrsatz aus Betrachtungen über die Differenz zweier Linientheile, so wird dabei doch häufig ein Formelrechnen nöthig, das dem in der Vorrede bezeichneten Zwecke der Ausdehnungslehre, ein Zusammengehen der räumlichen Anschauung und der Rechnung zu vermitteln, kaum entsprechen dürfte. Es tritt dies besonders ein, wo Winkelgrössen in's Spiel kommen; schon die Art, wie dieselben definirt werden, nämlich als Product einer Geraden in eine Potenz der imaginären Einheit, zeigt, dass diese Entwickelungen das Gebiet der linealen Ausdehnungslehre verlassen, wie sie denn auch in Grassmann's eigenem Werke nicht ausgeführt sind. An diese Einführung der Winkel sind dann die Sätze über die Winkel im Dreiecke, über Centriwinkel und Peripheriewinkel geknüpft. Während hierbei die Drehung der Geraden um einen ihrer Punkte benutzt wird, entstehen die Sätze über Flächeninhalt durch beliebige Bewegung einer auf einer Geraden gelegenen Strecke. Indem der Verfasser sodann von festen in der Ebene gegebenen Strecken ausgeht, also insbesondere von den drei Verbindungsgeraden dreier fester Punkte, kommt er zu den Grassmann'schen Coordinaten, welche den homogenen Dreiecks Coordinaten der neueren Geometrie genau entsprechen. An sie schliesst sich die eingehende Betrachtung der Dreiecke naturgemäss an, wobei denn alle jene bekannten Sätze über Transversalen, harmonische Punkte, Aehnlichkeit u. s. w. mit grosser Ausführlichkeit gegeben werden. Den Schwerpunkt dieser ganzen Darstellung bildet stets die Verknüpfung mit der formalen Algebra; die Erzeugung der algebraischen Curven findet (wie übrigens in Grassmann's Ausdehnungslehre selbst, nicht in dessen sonstigen Publicationen) nur eine untergeordnete Stelle, was wohl durch den elementaren Charakter des ganzen Buches begründet ist. Wirklich aufgestellt wird nur die Gleichung des Kegelschnittes (der auf p. 108 angeführte Satz über den einen Kegelschnitt rechtwinklig schneidenden Kreis enthält wohl einen Irrthum), während die Ueberführung Grassmann'scher planimetrischer Producte in Gleichungen mit gewöhnlichen Coordinaten ausführlich durchgeführt wird. Andererseits wird die Bildung regressiver Producte benutzt, um den Inhalt der sogenanten rechnenden Geometrie und der Trigonometrie abzuleiten. Hier nun, wie in dem schon erwähnten Abschnitte über Winkelrelationen entwickelt der Verfasser einen Formalismus, der wesentlich mit demjenigen stimmt, der aus der Interpretation von \(x+iy\) in der Ebene hervorgeht (Equipollenzen-Calcul). So fruchtbar diese Mathode ist, so wenig scheint sie in diejenigen Gesichtspunkte, welche Grassmann wenigstens in seiner linealen Ausdehnungslehre verfolgt, zu passen. Freilich lässt sich mit ihr die Lehre von den metrischen Eigenschaften auch nicht unmittelbar verbinden. Denn hierzu bedarf sie einer Ergänzung, der Einführung imaginärer Elemente, und diese wird weder von Schlegel noch eigentlich von Grassmann berührt. Ueberhaupt werden die eigentlichen Grundgedanken der projectivischen Anschauung nur wenig entwickelt; erst im Schlusse wird der Lehre von den Doppelverhältnissen etc. und deren Polartheorie gedacht, während z. B. das Prinzip der Dualität nirgends deutlich hervortritt.
Vielleicht wäre es wichtiger gewesen, statt Grassmann's Ideengang eben als solchen, nur in elementarer Form, darzustellen, ihn im Zusammenhange und Vergleiche mit dem, was die Wissenschaft unabhängig von Grassmann in ähnlicher Richtung späterhin geleistet hat, aufzufassen. Soweit sie in dem vorliegenden Buche in Betracht kommen, sind der Hauptverdienste von Grassmann wohl wesentlich drei. Ihm verdankt einmal die formale Algebra eine ungeahnte Vertiefung, indem er das Wesen der Additions- und Multiplications-Operationen in sehr viel allgemeinerer Weise erfassen lehrte, als das vor ihm geschehen war, und in dieser Beziehung stellt sich Grassmann neben die englischen Forscher, wie etwa Hamilton. Bei ihm ist ferner die Lehre von den mehrfach ausgedehnten Mannigfaltigkeiten, als deren specieller Fall die Raumlehre erscheint, insbesondere die Lehre von den linealen (projectivischen) Mannigfaltigkeiten wohl zum ersten Male entwickelt. Er hat endlich durch seine Erzeugungsweise aller algebraischen Gebilde vermöge linealer Mechanismen ein neues weit aussehendes Feld der Untersuchung geöffnet.
Aber für zufällige (obgleich an sich interessante) Form müssen wir es halten, wenn Grassmann die (linealen) Verknüpfungsweisen der Elemente einer Mannigfaltigkeit mit den formal-algebraischen Ueberlegungen in Beziehung setzt. Er behandelt auf diese Weise, allerdings in sehr geschickter Art, Probleme, welche die neuere Algebra ihrerseits durch Determinanten- und Polaren-Bildung beherrscht. Dabei mag gern hervorgehoben werden, dass das sog. gemischte Product Grassmann's, das in seinem Satze von der Erzeugnung algebraischer Gebilde eine fundamentale Rolle spielt, von der neueren Algebra seither noch nicht in allgemeinem Sinne untersucht wurde. Die beiden Darstellungsweisen sind im Grunde kaum verschieden und die Formeln, in denen beide ihren Ausdruck finden, häufig geradezu auch dem Aeusseren nach identisch. Wir möchten ferner Grassmann's Lehre von den mehrfach ausgedehnten Mannigfaltigkeiten auf dem heutigen Standpunkte nur als einen Anfang bezeichnen. Denn Grassmann's Mannigfaltigkeit ist nur die durch Vermehrung der Dimensionen erzielte directe Verallgemeinerung des gewöhnlichen Raumes mit seinen Eigenschaften der Lage und des Maasses, während z. B. Riemann's Untersuchungen eine viel allgemeinere Richtung einschlagen und neuere Arbeiten auch diese wieder nach verschiedenen Seiten hin wesentlich erweitert haben.
Wenn man, wie der Verfasser, Grassmann's Ideen abgelöst von solchen Vergleichungen vorträgt, so wird, nach Meinung des Referenten, der Leser eher abgestossen als angezogen; ihm wird zugemuthet, Grassmann's Methoden als die absolut vortrefflichen zu betrachten, und eine solche Zumuthung fordert immer etwas Unwahrscheinliches. Grassmannian geometry; differential geometry History of mathematics in the 19th century, History of differential geometry, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Grassmannians, Schubert varieties, flag manifolds A system to study space according to Grassmann' geometric 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 \(X= \mathbb{G}(k,n) \subseteq \mathbb{P}^N\), \(N={n\choose k}-1\), be the Grassmannian of \(k\)-subspaces of \(\mathbb{C}^n\), in its Plücker embedding; it is an open problem to determine if \(\sigma_s(X)\), the variety of \((s-1)\)-planes \(s\)-secant to \(X\), has the expected dimension \(\min\{sk(n.k)+s-1,N\}\) (recall that \(\dim X = k(n-k)\)) or not, i.e. if \(\sigma_s(X)\) is \textit{defective} or not.
A conjecture states that when \(k\geq 3\) (the case \(k=2\) is well known and almost always defective), the only defective cases occour for \((k,n,s)=(3,7,3); (4,8,3); (4,8,4)\) and \((3,9,4)\).
Several partial results are known (the conjecture is true for \(n\leq 15\) or \(s\leq 6\)); in this paper it is proved that the conjecture holds true for \(s\leq 12\); more scattered results are given also for smaller values of \(s\).
The main methods involve the use of Terracini Lemma and the explicit computation of the form of tangent spaces to \(X\). secant varieties; Terracini Lemma; Grassmannian Boralevi, A., A note on secants of Grassmannians, Rendiconti dell'Istituto Matematico dell'Università di Trieste, 45, 67-72, (2013) Grassmannians, Schubert varieties, flag manifolds, Multilinear algebra, tensor calculus A note on secants of 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 Horn conjecture is concerned with the possible eigenvalues of triples of Hermitian matrices that add up to the 0-matrix. A natural generalization is considering the triples of adjoint orbits of a compact connected Lie group, such that the sum of the orbits contain 0. The set of these triples parameterize a closed convex polyhedral cone. The challenge is to find a (minimal) set of inequalities defining this cone. This has been achieved by works of Klyachko, Knutson, Tao, Woodward, Belkale, Kumar and others. The first achievement of the paper is a description of a minimal set of inequalities defined inductively and without using cohomology. The second part of the paper deals with generalizations to connected reductive complex groups. For this case earlier works of \textit{P. Belkale}, \textit{Sh. Kumar} and \textit{N. Ressayre}
(see, e.g., [Math. Ann. 354, No. 2, 401--425 (2012; Zbl 1258.14008)]) provide a minimal set of inequalities. In the present paper the key condition of the Belkale-Kumar description is showed to be equivalent with the fact that two ordinary LR coefficients are both 1. The author derives this theorem from his cohomology free description. Horn conjecture; eigencone; cohomology-free description N. Ressayre, \textit{A cohomology-free description of eigencones in types A, B, and C}, Inter. Math. Res. Not. (2012), no. 21, 4966-5005. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Eigenvalues, singular values, and eigenvectors, Hermitian, skew-Hermitian, and related matrices, Lie algebras and Lie superalgebras A cohomology-free description of eigencones in types A, B, and C | 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 formula for degeneracy loci of a map of flagged vector bundles. Let \(h:E\to F\) be a map of vector bundles on a variety \(X\) and consider flags \(E_ 1\subset E_ 2\subset\cdots\subset E_ s=E\) (resp., \(F=F_ t\twoheadrightarrow F_{t- 1}\twoheadrightarrow\cdots\twoheadrightarrow F_ 1)\) of subbundles (resp., quotient bundles) of \(E\) (resp., \(F)\). One can consider the degeneracy loci: \(\Omega_ r(h):=\{x\in X|\text{rk}(E_ p(x)\to F_ q(x))\leq r(p,q)\), for all \(p,q\}\), where \(r\) is a collection of rank numbers satisfying certain conditions, which guarantee that, for generic \(h\), \(\Omega_ r(h)\) is irreducible, reduced, Cohen-Macaulay. The author gives a formula for the class \([\Omega_ r(h)]\) of this locus in the Chow ring of \(X\), as a polynomial in the Chern classes of the vector bundles. When expressed in terms of Chern roots, these polynomials are the ``double Schubert polynomials'' introduced and studied by Lascoux and Schützenberger.
Special cases of this formula recover the Kempf-Laksov determinantal formula, the Giambelli-Thom-Porteous formula, as well as a formula of \textit{P. Pragacz} [Ann. Sci. Éc. Norm Supér. IV. Ser. 21, No. 3, 413- 454 (1988; Zbl 0687.14043)]. When specialized to the flag manifold of flags in an \(n\)-dimensional vector space, the formula implies that of \textit{I. N. Bernstein}, \textit{M. Gel'fand} and \textit{S. T. Gel'fand} [Russ. Math. Surveys, 28, No. 3, 1-26 (1973; Zbl 0289.57024)] and \textit{M. Demazure} [Ann. sci. Ec. Norm. Super., IV. Ser. 7, 53-88 (1974; Zbl 0312.14009)]. Doing the general case makes the proof easier. The simplicity of the proof arises from the realization of the operators considered in the papers cited above as correspondences (a fact noticed by several people, and, as the author asserts, communicated to him by R. MacPherson). double Schubert polynomials; degeneracy loci; map of flagged vector bundles; Chow ring; Chern roots; determinantal formula W. Fulton, ``Flags, Schubert polynomials, degeneracy loci, and determinantal formulas'', Duke Math. J. 65 (1992), 381--420. Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Linkage, complete intersections and determinantal ideals, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Flags, Schubert polynomials, degeneracy loci, and determinantal 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 The author shows that the crystal structures on Lusztig's canonical basis and the Mirković--Vilonen cycles on the affine Grassmannian agree. As an application, he proves for \(\mathfrak{sl}_n\) a conjecture of Anderson--Mirković that describes the Braberman--Finkelberg--Gaitsgory crystal structure on the level of MV polytopes and gives a counterexample to this conjecture for \(\mathfrak{sp}_6\). He also explains how Kashiwara data can be recovered from MV polytopes. canonical basis; affine Grassmannian; crystal structure; polytope Kamnitzer, J., \textit{the crystal structure on the set of mirković-vilonen polytopes}, Adv. Math., 215, 66-93, (2007) Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Quantum groups (quantized enveloping algebras) and related deformations, Representation theory for linear algebraic groups, Classical groups (algebro-geometric aspects), Cohomology theory for linear algebraic groups The crystal structure on the set of Mirković-Vilonen polytopes | 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 \(X(n)=\text{GL}(n)\), \(\text{SO}(2n+1)\), \(\text{SO}(2n)\) or \(\text{Sp}(2n)\). \textit{A. D. Berenshtejn} and \textit{A. V. Zelevinskij} [J. Geom. Phys. 5, No. 3, 453-472 (1988; Zbl 0712.17006)] described the reduction multiplicities for the reduction \(X(n)\downarrow X(n-k)\times\mathbb{C}^{*k}\) in terms of certain Gelfand-Zetlin patterns. In this paper, the author gives a geometric interpretation of these patterns for \(G=\text{Sp}(2n)\). Newton polytopes; symplectic groups; reduction multiplicities; Gelfand-Zetlin patterns Andreĭ Okounkov, Multiplicities and Newton polytopes, Kirillov's seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 231 -- 244. Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds Multiplicities and Newton polytopes | 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 this very interesting paper asserts that the center of the principal block of Rocha-Caridi's parabolic generalization of the Bernstein-Gelfand-Gelfand category \(\mathcal{O}\) associated with the standard triangular decomposition of the simple Lie algebra \(\mathfrak{sl}_n(\mathbb{C})\) is isomorphic to the cohomology ring of the corresponding Springer fiber. This result is independently proved by \textit{J. Brundan} [Duke Math. J. 143, No. 1, 41--79 (2008; Zbl 1145.20003)] using completely different methods.
The main result of the paper has some remarkable consequences. First of all it allows the author to give a purely combinatorial diagrammatic construction of the associative algebra of the principal block of the parabolic category \(\mathcal{O}\) in the case of a maximal parabolic subalgebra. The latter allows one to construct an explicit isomorphism to Braden's algebra describing Schubert-constructable perverse sheaves on Grassmannians [\textit{T. Braden}, Can. J. Math. 54, No. 3, 493--532 (2002; Zbl 1009.32019)]. Restricting to a basic projective-injective object the author recovers the diagrammatic construction of Khovanov's algebra, which was used by Mikhail Khovanov in his categorification of the Jones polynomial in [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. From there one deduces that Khovanov's tangle invariants are obtained from the more general functorial invariants, constructed by the author in [Duke Math. J. 126, No. 3, 547--596 (2005; Zbl 1112.17010)], by restriction. Khovanov homology; Springer fibre; knot invariant; perverse sheaf; category \(\mathcal{O}\); center Stroppel, C., Parabolic category \(\mathcal{O}\), perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compos. Math., 145, 4, 954-992, (2009) Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Invariants of knots and \(3\)-manifolds, Representations of finite symmetric groups, Representation theory for linear algebraic groups, Homogeneous spaces and generalizations Parabolic category \(\mathcal O\), perverse sheaves on Grassmannians, Springer fibres and Khovanov homology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Author's abstract: ``We prove that the projectivized cotangent bundles of smooth quadrics of dimensions
three and four are \((p-1)\)-th Frobenius split when \(p>10\). Besides, we show that the
cotangent bundles of certain ordinary elliptic \(K3\) surfaces are not Frobenius split.''
The latter result provides more negative answers to a question in [\textit{P. Achinger} et al., J. Algebr. Geom. 26, No. 4, 603--654 (2017; Zbl 1373.14045)]. That is,
one gets examples of F-split varieties that do not have an F-split cotangent bundle.
From the introduction: ``Let \(X\) be an algebraic variety defined over an algebraically
closed field of characteristic \(p>0\). Then \(X\) is said to be Frobenius split or
F-split for short if the morphism of \( \mathcal O_X \)-modules
\( \mathcal O_X \to {F_X}_*\mathcal O_X \) splits, where
\( F_X : X \to X \) is the Frobenius morphism sending a local section of \( \mathcal O_X \) to
its \( p \)-th power. When \(X\) is smooth, by the Grothendieck duality, we have an
isomorphism
\( \phi_X : \mathrm{Hom}( {F_X}_*\mathcal O_X , \mathcal O_X ) \cong H^0( X , \omega_X^{1-p} ) \).
Definition. A smooth algebraic variety \(X\) is said to be \( (p-1) \)-th F-split if
there exists a global section \(s\) of \(\omega^{-1}_X\) such that \( \phi_X^{-1}(s^{p-1}) \)
defines a Frobenius splitting of \(X\).'' Frobenius splitting; cotangent bundle; K3 surface Positive characteristic ground fields in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Some examples of \((p-1)\)-th Frobenius split projectivized bundles | 0 |
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