text stringlengths 571 40.6k | label int64 0 1 |
|---|---|
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 subgroup of \(\operatorname{O}(n)\) and let \(H\) be a closed subgroup of \(G\cap \operatorname{O}(n-1)\), where \(\operatorname{O}(n-1)\subset \operatorname{O}(n)\) is the stabilizer of the first basis vector. In the following, \(\operatorname{Area}_{G/H}^G\) denotes the space of smooth flag area measures that are equivariant with respect to \(G\). These objects are translation invariant valuations that take values in the space of signed measures on \(G/H\) and that can be represented by certain smooth differential forms.
In their main result, the authors show that \(\operatorname{Area}_{G/H}^G\) is finite-dimensional if and only if \(G\) acts transitively on the unit sphere and that in this case there exist local additive kinematic formulas of the form \[\int_G \Phi_i(K+g L,\kappa\cap g \lambda) d g = \sum_{k,l=1}^N c_{k,l}^i \Phi_k(K,\kappa) \Phi_l(L,\lambda).\] Here, \(\{\Phi_1,\ldots,\Phi_N\}\) is a basis of \(\operatorname{Area}_{G/H}^G\), \(\kappa,\lambda\) are Borel subsets of \(G/H\) and \(K\) and \(L\) are convex bodies. Note that these formulas can be encoded by a cocommutative, coassociative coproduct.
For \(G\subset \operatorname{SO}(n)\) the elements of \(\operatorname{Area}_{G/\{1\}}^G\) are also called rotation measures and an explicit construction and classification of such measures is given. Furthermore, the authors give an algebraic description of the kinematic formulas for \(\operatorname{Area}_{G/H}^G\) when \(H\subset \operatorname{SO}(n-1)\) is a closed subgroup. In particular, an explicit description of this algebraic structure is given for the important case \(G=\operatorname{SO}(n)\), \(H=S(\operatorname{O}(p)\times \operatorname{O}(q))\), \(p+q=n-1\), in which \(G/H=\operatorname{Flag}_{1,p+1}\) is the incomplete flag manifold consisting of pairs \((v,E)\), where \(E\) is a \((p+1)\)-plane and \(v\in E\).
Note that the results of this paper generalize previous results by [\textit{T. Wannerer}, Adv. Math 263, 1--44 (2014; Zbl 1296.53150)] and [\textit{A. Bernig}, Geom. Dedicata 203, 85--110 (2019; Zbl 1432.53099)]. flag area measures; kinematic formulas; valuations Convex sets in \(n\) dimensions (including convex hypersurfaces), Dissections and valuations (Hilbert's third problem, etc.), Mixed volumes and related topics in convex geometry, Grassmannians, Schubert varieties, flag manifolds Additive kinematic formulas for flag area measures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 computes the equivariant cohomology and \(K\)-theory of the Bott-Samelson varieties and deduces some results about flag varieties of Kac-Moody groups. For definitions and results about Kac-Moody groups and Kac-Moody algebras see \textit{V. G. Kac} [``Infinite dimensional Lie algebras'', Cambridge Universsity Press (1985; Zbl 0574.17010)], \textit{V. G. Kac} and \textit{D. H. Peterson}, [``Regular functions on certain infinite-dimensional groups'', in: Arihtmetic and Geometry, Vol. II, Prog. Math. 36, 141--166 (1983; Zbl 0578.17014)] and \textit{S. Kumar} [Invent. Math. 123, 471--506 (1996; Zbl 0863.14031)]. The Kac-Moody algebra \(\mathfrak{g=g}(A) \) for a generalized \( n\times n\;\)Cartan matrix \(A\) is the complex Lie algebra generated by a certain vector space \(\mathfrak{h}\) over \(\mathbb{C}\) of dimension \(2n-\)rank\( (A) \) and certain symbols satisfying some well defined relations. The algebra \( \mathfrak{b}\) is the Borel subalgebra of \(\mathfrak{g}\) and satisfies \( \mathfrak{h\subset b\subset g}\); we denote by \(H\subset B\subset G\) the corresponding Lie groups. One defines certain subgroups \(G_{k}\subset G\) for \(k=1,\ldots ,n\) and a Bott-Samelson variety \(\Gamma \)\ is defined to be the orbit space of the product \(G_{1}\times \ldots \times G_{n}\) under the right \(B^{n}\)-action given by \((g_{1},\ldots ,g_{n}) (b_{1},\ldots ,b_{n}) =(g_{1}b_{1},b_{1}^{-1}g_{2}b_{2},\ldots ,b_{n-1}^{-1}g_{n}b_{n}) \). Let \({\mathcal E}=\Gamma^H\) and denote by \(S\) the symmetric algebra of \(\mathfrak{h}^{\ast }\)\ identified inside \(H_{T}^{\ast }(\Gamma) \)\ where \(T\) is the maximal torus of the unitary form \(K\) of \(G\), by \(F(\mathcal{E};S) \) the \(S\)-algebra of the functions \(\mathcal{ E}\to S\) with pointwise addition and multiplication. Let \(X[T ] \) be the character group of T, and let us denote \(R[T] = \mathbb{Z}[X[T] ] \). The main result is that the \(T\)-equivariant cohomology of \(\Gamma ^{T}\) is isomorphic to \(F(\mathcal{E };S) \) and that its \(T\)-equivariant \(K\)-theory is isomorphic to \( F(\mathcal{E};R[T]) \). \(K\)-theory; equivariant cohomology Willems, Matthieu: Cohomologie et K-théorie équivariantes des variétés de Bott -- Samelson et des variétés de drapeaux. Bull. soc. Math. France 132, No. 4, 569-589 (2004) Equivariant \(K\)-theory, Equivariant homology and cohomology in algebraic topology, Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups Equivariant cohomology and \(K\)-theory of Bott-Samelson varieties and 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 introduce the quantum multi-Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations, the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nägelsbach-Kostka and Jacobi-Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove also an analog of the Giambelli and the Billey-Jockusch-Stanley formula for quantum Schubert polynomials. Finally we formulate two conjectures about the structure of quantum double and quantum Schubert polynomials for 321-avoiding permutations. Anatol N. Kirillov, Quantum Schubert polynomials and quantum Schur functions, Internat. J. Algebra Comput. 9 (1999), no. 3-4, 385 -- 404. Dedicated to the memory of Marcel-Paul Schützenberger. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Quantum Schubert polynomials and quantum Schur functions. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review is concerned with certain nice properties enjoyed by semisimple simply connected algebraic groups. If \(G\) is any one such, and \(Z\) is its center, the latter can be identified with the fundamental group of \(G/Z\). Others important interpretations of the center of \(G\), for instance in terms of coroots and coweight lattices, are quickly listed in the introduction. The paper however focuses on some properties studied in an important paper by \textit{P.~Seidel} [Geom. Funct. Anal. 7, No. 6, 1046--1095 (1997; Zbl 0928.53042)], who proved that the fundamental group of the group of hamiltonian symplectomorphisms of a symplectic variety \(X\) can be mapped to the group of invertible elements of the quantum cohomology ring of \(X\) localised in the quantum parameters.
The point is that if \(X\) is a rational homogeneous space, there is a symplectic structure on \(X\) induced by its natural projective structure, and one so gets what the authors baptize as the \textsl{Seidel's representation}, i.e. a map from the fundamental group of \(G/Z\) and the subgroup of the invertibles elements of the quantum cohomology ring of \(X\) localized at the quantum parameter. In a previous paper [Int. Math. Res. Not. 2007, No. 22, Article ID rnm107, 29 p. (2007; Zbl 1142.14033)] the authors study this map in the case when \(X\) a \textsl{minuscule} or \textsl{cominuscule} homogeneous space (see the review of that paper, by the same reviewer, for glossary and terminology).
The main achievement of this work consists in extending the description of the Seidel's representation for all homogeneous spaces, proving in particular that it is faithful. The product structure in \(QH^*_T(X)_{loc}\) is described in the first main theorem of the paper, while the second main theorem focuses on the explicit description of the representation map. The results rely on two previous results gotten by Peterson [Quantum Cohomology of \(G/P\), 1997, unpublished] and P.~Magyar (Notes on Schubert classes of a loop group (2007), \url{arXiv:0705.3826}).
The detailed description of the Seidel's representation is performed in Section 4, the last of the paper, while the second section, after the general introduction, is devoted to describe the crucial results by Peterson and Magjar. A key formula by Magyar is extended in section 3 in a more general situation. This paper is very well written, like some others previous papers on similar subjects by the same authors. The interested and motivated reader is advised to read all of them at once, for the best profit. equivariant quantum cohomology; homogeneous spaces; Schubert Calculus; Gromov-Witten invariant DOI: 10.4310/MRL.2009.v16.n1.a2 Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials By using the Milnor ring structure of the cohomology of Grassmannian, we prove a vanishing theorem of Witten genus for generalized string complete intersections in products of Grassmannian. residue; Grassmannian; Witten genus; theta function 32. J. Zhou and X. Zhuang, Witten genus of generalized complete intersections in products of Grassmannians, Int. J. Math.25(10) (2014), Article ID: 1450095, 25pp. [Abstract] genRefLink(128, 'S0129167X16500762BIB032', '000346174500004'); Elliptic genera, Theta functions and abelian varieties, Grassmannians, Schubert varieties, flag manifolds, Complete intersections Witten genus of generalized complete intersections in products 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 introduce new, combinatorially defined subvarieties of isotropic Grassmannians called symplectic restriction varieties. We study their geometric properties and compute their cohomology classes. In particular, we give a positive, combinatorial, geometric branching rule for computing the map in cohomology induced by the inclusion \(i:SG(k,n)\to G(k,n)\). This rule has many applications in algebraic geometry, symplectic geometry, combinatorics, and representation theory. In the final section of the paper, we discuss the rigidity of Schubert classes in the cohomology of \(SG(k,n)\). Symplectic restriction varieties, in certain instances, give explicit deformations of Schubert varieties, thereby showing that the corresponding classes are not rigid. isotropic Grassmannians; symplectic restriction varieties Coskun, I., Symplectic restriction varieties and geometric branching rules, \textit{Clay Math. Proc.}, 18, 205-239, (2013) Grassmannians, Schubert varieties, flag manifolds, Symplectic geometry, contact geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homogeneous complex manifolds Symplectic restriction varieties and geometric branching rules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Under the assumption that the equivariant quantum cohomology of a Kähler manifold is well defined, associative, and a weighted-homogeneous ordinary equivariant \(q\)-deformation of ordinary cohomology, the author computes the equivariant quantum cohomology of partial flag manifolds. The author also derives a general result for some manifolds which present their quantum cohomology as regular functions on complete intersections. equivariant quantum cohomology of partial flag manifolds Kim B., Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings, Internat. Math. Res. Notices, 1995, 1, 1--15 Grassmannians, Schubert varieties, flag manifolds, Equivariant algebraic topology of manifolds, Quantum field theory; related classical field theories, Global differential geometry of Hermitian and Kählerian manifolds Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 work, the authors revisit the determination of the boundary map coefficients for the cellular homology of real flag manifolds, a problem equivalent to finding the incidence coefficients of the differential map for the cohomology. They prove a new formula for these coefficients with respect to the height of certain roots.
A generalized flag manifold \(\mathbb F \) is a homogeneous space \(G/P\), where \(G\) is a real noncompact semisimple Lie group and \(P\) is a parabolic subgroup. It admits a cellular decomposition called Bruhat decomposition, where the cells are the Schubert cells and parametrized by the Weyl group \(W\). There is the Bruhat Chevalley order on elements of the Weyl group. In this case, there is a root \(\beta\) such that \(w = s_{\beta}\cdot w'\). In both [\textit{R. R. Kocherlakota}, Adv. Math. 110, No. 1, 1--46 (1995; Zbl 0832.22020)] and [\textit{L. Rabelo} and \textit{L. A. B. San Martin}, Indag. Math., New Ser. 30, No. 5, 745--772 (2019; Zbl 1426.57052)], the authors summarized how to compute the coefficient \(c(w,w')\).
The papers [\textit{L. Rabelo}, Adv. Geom. 16, No. 3, 361--379 (2016; Zbl 1414.57018)] and [\textit{J. Lambert} and \textit{L. Rabelo}, Australas. J. Comb. 75, Part 1, 73--95 (2019; Zbl 1429.05005)] apply this procedure in the context of the Isotropic Grassmannians and the results obtained (for instance, see [\textit{J. Lambert} and \textit{L. Rabelo}, ``Integral homology of real isotropic and odd orthogonal Grassmannians'', Preprint, \url{arXiv:1604.02177}, to appear in Osaka J. Math.], Theorem 3.12) suggest a formula for the coefficients in terms of the height of some root.
Overall they prove a new formula for the cellular homology coefficients of real flag manifolds in terms of the height of certain roots. For real flag manifolds of type \(A\), they get simple expressions for the coefficients that allow us to compute the first and second integral homology groups exhibiting their generators. real flag manifolds; symmetric group; root systems; Schubert cells; homology; height of roots; boundary coefficients Homology and cohomology of homogeneous spaces of Lie groups, Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds, Root systems A correspondence between boundary coefficients of real flag manifolds and height of roots | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis.
In the favourable case a basis of the module is parametrized by the lattice points of a normal polytope. The filtrations induce at degenerations of the corresponding ag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules. Toric varieties, Newton polyhedra, Okounkov bodies, Fibrations, degenerations in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Group rings of finite groups and their modules (group-theoretic aspects), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Favourable modules: filtrations, polytopes, Newton-Okounkov bodies and flat degenerations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is well known that a Schur function is the `limit' of a sequence of Schur polynomials in an increasing number of variables, and that Schubert polynomials generalize Schur polynomials. We show that the set of Schubert polynomials can be organized into sequences, whose `limits' we call Schubert functions. A graded version of these Schubert functions can be computed effectively by the application of mixed shift/multiplication operators to the sequence of variables \(x=(x_1,x_2,x_3,\dots)\). This generalizes the Baxter operator approach to graded Schur functions of G. P. Thomas, and allows the easy introduction of skew Schubert polynomials and functions. Since the computation of these operator formulas relies basically on the knowledge of the set of reduced words of permutations, it seems natural that in turn the number of reduced words of a permutation can be determined with the help of Schubert functions: we describe new algebraic formulas and a combinatorial procedure, which allow the effective determination of the number of reduced words for an arbitrary permutation in terms of Schubert polynomials. Schur function; Schubert polynomials; Schur polynomials; Schubert functions; Baxter operator; reduced words of permutations Winkel, R.: Schubert functions and the number of reduced words of permutations, Sém. lothar. Combin. 39, 1-28 (1997) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Schubert functions and the number of reduced words of permutations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For part I see ibid. 71/72(1988; Zbl 0654.14024)]
When the defining field has characteristic 2 on \({\mathbb{P}}^ 5\) there exists an indecomposable vector bundle E of rank 2. Studying the jumping lines and splitting type of this bundle we have a five dimensional non- singular rational subvariety of the Grassmann variety Gr(5,1). This subvariety can be defined over every field of arbitrary characteristic. This subvariety is defined by seven polynomials in Gr(5,1). In this paper we study the structure of this subvariety. And when the defining field has characteristic 2, we reconstruct the vector bundle E from this subvariety. 5-dimensional variety; characteristic 2; vector bundle; jumping lines; splitting type; Grassmann variety Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(n\)-folds (\(n>4\)), Rational and unirational varieties, Finite ground fields in algebraic geometry On a 5-dimensional non-singular rational subvariety of Grassmann variety \(G_ r(5,1)\). II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors give a characterization of arithmetically Gorenstein Schubert varieties in a minuscule generalized flag variety \(G/P\) over an algebraic closed field. Recall that \(G/P\) is minuscule if \(P\) is a maximal parabolic subgroup associated to a minuscule (fundamental) weights. If \(G=\mathrm{SL}_n\), then \(G/P\) is just a Grassmannian.
The minuscule weights plus zero are a set of ``minimal'' representatives for the cosets of the root lattice in the weight lattice.
All Schubert varieties (in a flag variety) are Cohen-Macaulay, but not all of them are smooth. The smooth ones are completely classified. The Gorenstein property is a geometric property between Cohen-Macaulayness and the smoothness property. The problem of classifying the Gorenstein Schubert varieties is an open problem. The minuscule varieties have a canonical closed embedding in a projective space (associated to the ample generator of the Picard group). A Schubert variety is arithmetically Gorenstein (with respect to this embedding) if the affine cone over it is Gorenstein; this property implies the Gorenstein property.
In [Adv. Math. 207, No. 1, 205--220 (2006; Zbl 1112.14058)], \textit{A. Woo} and \textit{A. Yong} have given a characterization of Gorenstein Schubert varieties in \(\mathrm{SL}_n/B\). In this case the Weyl group is isomorphic to \(S_n\) and the characterization can be given in terms of the Young diagram associated to the Schubert variety. This implies a combinatorial characterization of the Gorenstein Schubert varieties in the Grassmannian. The authors of this work give stronger results, namely, a characterization of the arithmetic Gorenstein property (for the Plücker embedding).
The first main result of this work is a generalization of this combinatorial characterization in the case of the orthogonal Grassmannian where \(G=\mathrm{SO}_n\) and \(P\) is associated to one of the right end roots. The Schubert varieties can be again be represented by a ``generalized'' Young diagram and the combinatorial condition is the same.
The idea of the proof is the following: the homogeneous coordinate ring of a Schubert variety \(X\) (with respect to the canonical embedding of \(G/P\)) is a Cohen-Macaulay and graded Hodge algebra with a set of generators indexed by the Bruhat poset of Schubert subvarieties of \(X\). This facts together with a result of Stanley give a characterization of the Gorenstein property for this algebra.
When \(P=P_1\) and \(G\) is \(\mathrm{SO}_{2m}\) or \(\mathrm{Sp}_{2m}\), the authors prove that all Schubert varieties are arithmetically Gorenstein. In particular, \(\mathrm{Sp}_{2m}/P_1\) is a projective space and the Schubert subvarieties are linear subspaces, in particular smooth.
The authors give also a list of the arithmetically Gorenstein Schubert varieties in the exceptional cases. Finally, they described the arithmetically Gorenstein Schubert varieties of the generalized flag varieties of \(\mathrm{SL}_n\), even if they are non minuscule. Schubert varieties; minuscule; Gorenstein Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Arithmetically Gorenstein Schubert varieties in a minuscule \(G/P\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This survey article, which may also serve as background material while reading, for instance, [\textit{E. Arbarello}, Contemp. Math. 312, 9--69 (2002; Zbl 1056.14023), Section 4] and parts of [\textit{M. Mulase}, Perspectives in mathematical physics. Proceedings of the conference on interface between mathematics and physics, held in Taiwan in summer 1992 and the special session on topics in geometry and physics, held in Los Angeles, CA, USA in winter of 1992. Boston, MA: International Press. 151--217 (1994; Zbl 0837.35132), \textit{M. Sato}, Random systems and dynamical systems, Proc. Symp., Kyoto 1981, RIMS Kokyuroku 439, 30--46 (1981; Zbl 0507.58029)], has the purpose to advertise the notion of Schubert derivation on an exterior algebra, introduced by the first author [Asian J. Math. 9, No. 3, 315--322 (2005; Zbl 1099.14045)] (see also the authors [Hasse-Schmidt derivations on Grassmann algebras. With applications to vertex operators. Cham: Springer (2016; Zbl 1350.15001)]), by showing how it provides another approach to look at the quadratic equations describing the Plücker embedding of Grasmannians -- a very classical and widely studied subject.
In particular, it allows i) to ``discover'' the vertex operators generating the fermionic vertex superalgebra (in the sense of [\textit{S. Galkin} and \textit{D. Ben-Zvi}, Lond. Math. Soc. Lect. Note Ser. 308, 46--97 (2004; Zbl 1170.17303), Section 5.3]); ii) to compute their bosonic expressions as by \textit{V. G. Kac} et al. [Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. 2nd ed. Hackensack, NJ: World Scientific (2013; Zbl 1294.17021), Lecture 5]; iii) to interpret them in terms of Schubert derivations and iv) to provide an almost effortless deduction of the celebrated Hirota bilinear form of the KP hierarchy (after Kadomtsev and Petviashvilii) [Kac et al. loc. cit.]. Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Derivations and commutative rings On Plucker equations characterizing Grassmann cones | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study \(K\)-orbits in \(G/P\) where \(G\) is a complex connected reductive group, \(P\subseteq G\) is a parabolic subgroup, and \(K \subseteq G\) is the fixed point subgroup of an involutive automorphism \(\theta\). Generalizing work of \textit{T. A. Springer} [in: Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya 1983, Adv. Stud. Pure Math. 6, 525-543 (1985; Zbl 0628.20036)] we parametrize the (finite) orbit set \(K \setminus G/P\) and we determine the isotropy groups. As a consequence, we describe the closed (respectively, affine) orbits in terms of \(\theta\)-stable (respectively, \(\theta\)-split) parabolic subgroups. We also describe the decomposition of any \((K,P)\)-double coset in \(G\) into \((K,B)\)-double cosets, where \(B\subseteq P\) is a Borel subgroup. Finally, for certain \(K\)-orbit closures \(X\subseteq G/B\), and for any homogeneous line bundle \({\mathfrak L}\) on \(G/B\) having nonzero global sections, we show that the restriction map \(\text{res}_X\) \(H^0(G/B, {\mathfrak L})\to H^0 (X,{\mathfrak L})\) is surjective and that \(H^i(X, {\mathfrak L})=0\) for \(i\geq 1\). Moreover, we describe the \(K\)-module \(H^0(X, {\mathfrak L})\). This gives information on the restriction to \(K\) of the simple \(G\)-module \(H^0(G/B, {\mathfrak L})\). Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal \(K\)-types. complex connected reductive group; involutive automorphism; isotropy groups; parabolic subgroups; Borel subgroup; homogeneous line bundle Brion M., Helminck A.G.: On orbit closures of symmetric subgroups in flag varieties. Can. J. Math. 52(2), 265--292 (2000) Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups On orbit closures of symmetric subgroups in 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 Authors' abstract: Given a 3-vector \(\underline{z}\in \bigwedge ^3 \mathbb {R}^6\) the least distance problem from the Grassmann variety \(\mathrm{G}_3(\mathbb {R}^6)\) is considered. The solution of this problem is related to a decomposition of \(\underline{z}\) into a sum of at most five decomposable orthogonal 3-vectors in \(\bigwedge ^3 \mathbb {R}^6\). This decomposition implies a certain canonical structure for the Grassmann matrix which is a special matrix related to the decomposability properties of \(\underline{z}\). This special structure implies the reduction of the problem to a considerably lower dimension tensor space \(\bigotimes ^3 \mathbb {R}^2\) where the reduced least distance problem can be solved efficiently. exterior algebra; decomposability; best decomposable approximation; Grassmann variety Grassmannians, Schubert varieties, flag manifolds, Vector and tensor algebra, theory of invariants Approximate decomposability in \(\bigwedge^3\mathbb R ^6\) and the canonical decomposition of 3-vectors | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the cohomology ring of any two step flag variety of type \(A\) combinatorially. The main result of the paper establishes a conjecture of \textit{A. Knutson} [``A conjectural rule for \(\mathrm{GL}_n\) Schubert calculus'', Preprint] asserting that the structure constants in the Schubert basis for the cup product in any two step flag variety of type \(A\) are equal to the number of certain puzzles with specified border labels built only from a list of eight puzzle pieces. Thus, by a result of the first author et al. [J. Am. Math. Soc. 16, No. 4, 901--915 (2003; Zbl 1063.53090)] identifying the structure constants for the cup product of a two step flag variety of type \(A\) with the structure constants for the product in small quantum cohomology of \(A\)-type Grassmannians, one gets the same combinatorial rule for the later. One of the important ingredient in their proof is to show that the puzzle rule defines an associative ring.
Recall that \textit{I. Coskun} [Invent. Math. 176, No. 2, 325--395 (2009; Zbl 1213.14088)] proved another positive formula for the structure constants for the cup product in any two step flag variety of type \(A\), though his formula did not establish Knutson's conjecture [loc. cit.]. Also, recall that \textit{A. Knutson} and \textit{K. Purbhoo} [Electron. J. Comb. 18, No. 1, Research Paper P76, 20 p. (2011; Zbl 1232.05239)] established a combinatorial formula in terms of puzzles for the product given by \textit{P. Belkale} and \textit{S. Kumar} [Invent. Math. 166, No. 1, 185--228 (2006; Zbl 1106.14037)] in the cohomology of \(\mathrm{SL}(n)/P\) for any parabolic \(P\) (not merely two step flag varieties). two-step flag manifolds; puzzle conjecture; cohomology product structure constants; small quantum cohomology A. Buch, A. Kresch, K. Purbhoo, and H. Tamvakis. ''The puzzle conjecture for the cohomol ogy of two-step flag manifolds''. 2014. arXiv:1401.1725. Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) The puzzle conjecture for the cohomology of two-step 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 Let \(\mathbb{G}\) be a connected split reductive group over a finite extension \(L\) of \(\mathbb{Q}_p\), denote by \(\mathbb{X}\) the flag variety of \(\mathbb{G}\), and let \(G = \mathbb{G}(L)\). In this paper we prove that formal models \(\mathfrak{X}\) of the rigid analytic flag variety \(\mathbb{X}^{\mathrm{rig}}\) are \(\mathscr{D}^{\dagger}_{\mathfrak{X}, k} \)-affine for certain sheaves of arithmetic differential operators \(\mathscr{D}^{\dagger}_{\mathfrak{X}, k}\). Furthermore, we show that the category of admissible locally analytic \(G\)-representations with trivial central character is naturally anti-equivalent to a full subcategory of the category of \(G\)-equivariant families \((\mathscr{M}_{\mathfrak{X}, k})\) of modules \((\mathscr{M}_{\mathfrak{X}, k})\) over \(\mathscr{D}^{\dagger}_{\mathfrak{X}, k}\) on the projective system of all formal models \(\mathfrak{X}\) of \(\mathbb{X}^{\mathrm{rig}} \). Rigid analytic geometry, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups \(\mathscr{D}^{\dagger}\)-affinity of formal models of flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We correct two formulae in Section 5 of [the authors, ibid. 22, No. 3, 631--643 (2017; Zbl 1401.14030)] in which, as published, the wrong lift of a class on a Grassmannian to the space of matrices was chosen. (Equivariant) Chow groups and rings; motives, Combinatorial aspects of matroids and geometric lattices, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) Erratum to: ``Equivariant Chow classes of 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 Let \(A\) be a finite dimensional commutative semisimple algebra over a field \(k\), and let \(V\) be a finitely generated \(A\)-module. We examine the action of the general linear group \(GL_ A(V)\) on the set of flags of \(k\)-subspaces of \(V\). Also, let \((V,B)\) be a finitely generated `symplectic module' over \(A\). We also investigate the action of the symplectic group \(Sp_ A(V,B)\) on the set of flags of \(B'\)-isotropic \(k\)-subspaces of \(V\), where \(B'=\varphi \circ B\) is the \(k\)-symplectic form induced by a nonzero \(k\)-linear map \(\varphi:A\to k\). Studied earlier were the cases of \(GL_ A(V)\) acting on the set of \(k\)-subspaces of \(V\) and of \(Sp_ A(V,B)\) acting on the set of \(B'\)-isotropic \(k\)- subspaces of \(V\). So this paper is a natural extension of the previous works. In both cases, the orbits are completely classified in terms of certain integer invariants provided that \(\dim_ k A=2\), from which one can determine the precise structure of orbits, compute the exact number of orbits and give typical representatives for each orbit, etc. action of the general linear group on flags; action of the symplectic group; orbits Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields Action on flag varieties: 2-dimensional 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 For \(n\)-dimensional vector spaces \(V\) and \(V'\) and non-degenerate symplectic forms \(\Omega\) and \(\Omega'\) on \(V\) and \(V'\), respectively, let \({\mathcal G}_{k}(\Omega)\) and \({\mathcal G}_{k}(\Omega')\) be the sets of \(k\)-dimensional totally isotropic subspaces of the projective spaces \(\Pi\) and \(\Pi'\) associated to \(V\) and \(V'\). For any symplectic base \(B\) of \(\Pi\) the set of all elements of \({\mathcal G}_ {k}(\Omega)\) spanned by elements of \(B\) is defined to be the base subset of \({\mathcal G}_{k}(\Omega)\) associated with \(B\).
The author proves that for \(3k + 3 \leq n\) any map \(f : {\mathcal G}_{k} (\Omega) \to {\mathcal G}_{k}(\Omega')\) which preserves base subsets is induced by a strong embedding of \(\Pi\) into \(\Pi'\). symplectic Grassmann space; null system; base subset Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Grassmannians, Schubert varieties, flag manifolds Base subsets in symplectic Grassmannians of small indices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(k=(k_ 1,...,k_ d)\) be a d-tuple of nonnegative integers satisfying \(k_ 1\leq...\leq k_ d\leq n\) where n is a positive integer. Let \({\mathcal F}\) denote either the field of real numbers (\({\mathbb{R}})\) or the field of complex numbers (\({\mathbb{C}})\). Then Flag\((k,{\mathcal F}^ n)\) denotes the flag manifold consisting of d-tuples \((s_ 1,...,s_ d)\) where \(s_ i\) is a \(k_ i\) dimensional subspace of \({\mathcal F}^ n\) and \(s_ 1\subseteq...\subseteq s_ d\). Let A be an \(n\times n\) matrix with entries in \({\mathcal F}\). Then Flag\(_ A(k,{\mathcal F}^ n)\) denotes the subvariety of Flag\((k,{\mathcal F}^ n)\) consisting of those flags that are invariant under A. Due to spectral decomposition A is assumed to be a unipotent matrix. If \(p^ i\) is a partition of \(k_ i\), \(i=1,...,d\), then
\[
\text{Flag}_ A(k,p,{\mathcal F}^ n) = \{(s_ 1,...,s_ d)\in \text{Flag}_ A(k,{\mathcal F}^ n),
\]
\[
A| s_ i\text{ has Jordan type \(p^ i,\quad i=1,...,d\)}.
\]
In the present work the authors investigate the topological structure of Flag\(_ A(k,p,{\mathcal F}^ n).\) They prove that Flag\(_ A(k,p,{\mathcal F}^ n)\) has a compact nonsingular projective variety as a strong deformation retract, which they refer to as a biflag manifold. The authors compute the Betti numbers of the biflag manifold. A partition of the biflag manifold into affine spaces is also considered. flag manifold; compact nonsingular projective variety; strong deformation retract; biflag manifold; Betti numbers Helmke, U.; Shayman, M.: The biflag manifold and the fixed point set of a nilpotent transformation on the flag manifold. Linear algebra appl. 92, 125-159 (1987) Algebraic topology on manifolds and differential topology, Homology and cohomology of homogeneous spaces of Lie groups, Grassmannians, Schubert varieties, flag manifolds, Basic linear algebra The biflag manifold and the fixed points of a unipotent transformation on the flag manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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{B. Leclerc} and \textit{A. Zelevinsky} [Transl., Ser. 2, Am. Math. Soc. 181(35), 85--108 (1998; Zbl 0894.14021)] introduced the notion of weakly separated collections of subsets of the ordered \(n\)-element set [\(n\)] (using this notion to give a combinatorial characterization for quasi-commuting minors of a quantum matrix). They conjectured the purity of certain natural domains \({\mathcal {D}}\subseteq 2^{[n]}\) (in particular, of the hypercube \(2^{[n]}\) itself, and the hyper-simplex \(\{X\subseteq [n]:|X|=m\}\) for \(m\) fixed), where \({\mathcal {D}}\) is called pure if all maximal weakly separated collections in \({\mathcal {D}}\) have the same cardinality. These conjectures have been answered affirmatively. In this paper, generalizing those earlier results, we reveal wider classes of pure domains in \(2^{[n]}\). This is obtained as a consequence of our study of a novel geometric-combinatorial model for weakly separated set-systems, so-called combined (polygonal) tilings on a zonogon, which yields a new insight in the area. weakly separated sets; strongly separated sets; quasi-commuting quantum minors; rhombus tiling; Grassmann necklace Danilov, VI; Karzanov, AV; Koshevoy, GA, Combined tilings and separated set-systems, Selecta Math. (N.S.), 23, 1175-1203, (2017) Combinatorial aspects of representation theory, Combinatorial aspects of tessellation and tiling problems, Grassmannians, Schubert varieties, flag manifolds, Finite ground fields in algebraic geometry Combined tilings and separated set-systems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using a recently developed off-shell formulation for general 4D \(\mathcal N=2\) supergravity-matter systems, we propose a construction to generate higher derivative couplings. We address here mainly the interactions of tensor and vector multiplets, but the construction is quite general. For a certain subclass of terms, the action is naturally written as an integral over 3/4 of the Grassmann coordinates of superspace. extended supersymmetry; higher derivative actions; superspace; supergravity Supergravity, Relativistic gravitational theories other than Einstein's, including asymmetric field theories, Grassmannians, Schubert varieties, flag manifolds, Unified quantum theories Generating higher-derivative couplings in \(\mathcal N= 2\) supergravity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 0719.00018.]
This paper, as a continuation of [\textit{M. Kashiwara}, The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 407-433 (1990; Zbl 0727.17013)], completes the proof of the Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras. The proof consists of two parts: (1) the algebraic part --- the correspondence between \({\mathcal D}\)-modules on the flag variety and representations of the Kac-Moody Lie algebra, (2) the topological part --- the description of geometry of Schubert varieties in terms of the Kazhdan-Lusztig polynomials. The algebraic part is already established in the paper cited above and the paper under review is devoted to the topological part. There are two points in the proof except which the proof is similar to the finite dimensional case. The first one is the usage of the theory of mixed Hodge modules and the second one is the interpretation of the inverse Kazhdan-Lusztig polynomials as the coefficients of certain elements in the dual of the Hecke-Iwahori algebra.
Let \({\mathfrak h}\) be the Cartan subalgebra of a symmetrizable Kac-Moody Lie algebra and \(W\) the Weyl group. For \(w\in W\) define the action on \({\mathfrak h}^*\) by \(w\cdot\lambda=w(\lambda+\rho)-\rho\). Let \(P_{z,w}(q)\) be the Kazhdan-Lusztig polynomial and \(Q_{z,w}(q)\) the inverse Kazhdan- Lusztig polynomial. They are related by
\[
\sum_ w(-1)^{\ell(w)- \ell(y)}Q_{y,w}(q)P_{w,z}(q)=\delta_{y,z}.
\]
The main result of the paper is the following. For a dominant integral weight \(\lambda\in{\mathfrak h}^*\), one has
\[
ch L(w\cdot\lambda)=\sum_ z(-1)^{\ell(z)- \ell(w)}Q_{w,z}(1)ch M(z\cdot\lambda),
\]
or equivalently \(ch M(w\cdot\lambda)=\sum_ zP_{w,z}(1)ch L(z\cdot\lambda)\). Kazhdan-Lusztig conjecture; symmetrizable Kac-Moody Lie algebras; \({\mathcal D}\)-modules; flag variety; representations; geometry of Schubert varieties; Kazhdan-Lusztig polynomials; mixed Hodge modules O.J. Ganor, \textit{Supersymmetric interactions of a six-dimensional self-dual tensor and fixed-shape second quantized strings}, \textit{Phys. Rev.}\textbf{D 97} (2018) 041901 [arXiv:1710.06880] [INSPIRE]. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebra. II: Intersection cohomologies 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 We study the problem of feedback control for skew-symmetric and skew-Hamiltonian transfer functions using skew-symmetric controllers. This extends the work of Helmke et al., who studied static symmetric feedback control of symmetric and Hamiltonian linear systems. We identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a Grassmannian, give necessary and sufficient conditions for pole placement by static skew-symmetric complex feedback, and use Schubert calculus for the orthogonal Grassmannian to count the number of complex feedback laws when there are finitely many of them. Finally, we also construct a real skew-symmetric linear system with only real feedback for any set of real poles. pole placement; feedback control; orthogonal Grassmannian; geometry of Grassmannian manifolds; Lagrangian Grassmannian; skew-symmetric matrix Pole and zero placement problems, Grassmannians, Schubert varieties, flag manifolds, Geometric methods Complex static skew-symmetric output feedback control | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 [Eur. J. Comb. 25, No. 8, 1327--1344 (2004; Zbl 1062.14065); C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)], \textit{W. Kraśkiewicz} and \textit{P. Pragacz} introduced representations of the upper-triangular Lie algebra \(\mathfrak{b}\) whose characters are Schubert polynomials. In [Eur. J. Comb. 58, 17--33 (2016; Zbl 1343.05168)], the author studied the properties of Kraśkiewicz-Pragacz modules using the theory of highest weight categories. From the results there, in particular we obtain a certain highest weight category whose standard modules are KP modules. In this paper we show that this highest weight category is self Ringel-dual. This leads to an interesting symmetry relation on Ext groups between KP modules. We also show that the tensor product operation on \(\mathfrak{b}\)-modules is compatible with Ringel duality functor. Schubert polynomials; Kraśkiewicz-Pragacz modules; highest weight categories; ringel duality; B-modules Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Classical problems, Schubert calculus Kraśkiewicz-Pragacz modules and ringel 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 establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the classical Pieri formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to flag manifolds). Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, some of which we express in terms of paths in the Bruhat order on the symmetric group, which in turn yields an enumerative result about the Bruhat order. multiplication by the class of a special Schubert variety; integral cohomology ring of the flag manifold; Pieri formual; Bruhat order Frank Sottile, Pieri's formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 89-110 (English, with English and French summaries). Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Pieri's formula for flag manifolds and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a complex semisimple, simply-connected algebraic group and let \(P\) be a parabolic subgroup. Consider the flag variety \(X:= G/P\) and the Schubert subvarieties \(X_w:= \overline{BwP/P}\subset G/P\) for any \(w\in W/W_P\), where \(W\) is the Weyl group of \(G\), \(W_P\) is the Weyl group of \(P\) and \(B\) is a Borel subgroup of \(G\) contained in \(P\). Then, any subvariety \(V\) of \(X\) is rationally equivalent to a linear combination of Schubert cycles \([X_w]\) with uniquely determined nonnegative integral coefficients. Then, Brion calls \(V\) multiplicity free if these coefficients are 0 or 1. Examples of multiplicity free \(V\) include the Schubert varieties \(X_w\) themselves, \(G\)-stable (irreducible) subvarieties of \(X\times X\) (under the diagonal action of \(G\)), irreducible hyperplane sections of \(X\) in its smallest projective embedding and the irreducible hyperplane sections of Schubert varieties in Grassmannians embedded by the Plücker embedding.
The main theorem of the paper under review asserts that any multiplicity-free subvariety \(V\subset X\) is normal and Cohen-Macaulay. Further, \(V\) admits a flag degeneration inside \(X\) to a reduced Cohen-Macaulay union of Schubert varieties. Hence, for any globally generated line bundle \(G\) on \(X\), the restriction map \(H^0(X,{\mathcal L})\to H^0(V,{\mathcal L})\) is surjective and \(H^i(V,{\mathcal L})= 0\) for all \(i\geq 1\). If \(L\) is ample, then \(H^i(V, {\mathcal L}^{-1})= 0\) for any \(i<\dim V\). Thus, \(V\) is arithmetically normal and Cohen-Macaulay in the projective embedding given by any ample \({\mathcal L}\). Schubert varieties; Cohen-Macaulay; arithmetically normal; projective embedding Brion, M.: Multiplicity-free subvarieties of flag varieties. Commutative algebra (Grenoble/Lyon, 2001), 13-23, Contemp. Math., \textbf{331}, Amer. Math. Soc., Providence, RI, 2003 Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Multiplicity-free subvarieties of flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The theory of Newton-Okounkov polytopes is a generalization of that of Newton polytopes for toric varieties, and gives a systematic method of constructing toric degenerations of projective varieties. In the case of Schubert varieties, their Newton-Okounkov polytopes are deeply connected with representation theory. Indeed, Littelmann's string polytopes and Nakashima-Zelevinsky's polyhedral realizations are obtained as Newton-Okounkov polytopes of Schubert varieties. In this paper, we apply the folding procedure to a Newton-Okounkov polytope of a Schubert variety, which relates Newton-Okounkov polytopes of Schubert varieties of different types. As an application, we obtain a new interpretation of Kashiwara's similarity of crystal bases. crystal basis; fixed point Lie subalgebra; Newton-Okounkov body; orbit Lie algebra; Schubert variety Quantum groups (quantized enveloping algebras) and related deformations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Folding procedure for Newton-Okounkov polytopes 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 We use the theory of Mori dream spaces to prove that the global Okounkov body of a Bott-Samelson variety with respect to a natural flag of subvarieties is rational polyhedral. As a corollary, Okounkov bodies of effective line bundles over Schubert varieties are shown to be rational polyhedral. In particular, it follows that the global Okounkov body of a flag variety \(G / B\) is rational polyhedral. As an application we show that the asymptotic behaviour of dimensions of weight spaces in section spaces of line bundles is given by the volume of polytopes. Bott-Samelson variety; Mori dream space; linear series; Okounkov body Schmitz, D.; Seppänen, H., Global Okounkov bodies for Bott-Samelson varieties, J. Algebra, 490, 518-554, (2017) Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Minimal model program (Mori theory, extremal rays) Global Okounkov bodies for Bott-Samelson 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 author shows that the family of tangent flags to smooth quadric hypersurfaces in \(\mathbb{P}^n\) extends to a flat family parametrized by the variety of complete quadrics. This answers a question asked by \textit{S. L. Kleiman} (with \textit{A. Thorup}) [in: Algebraic Geometry, Proc. Summer Res. Inst., Brunswick 1985, Part 2, Proc. Symp. Pure Math. 46, 321-370 (1987; Zbl 0664.14031)]. flag variety; tangent flags; complete quadrics Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry Flatness of families induced by hypersurfaces on flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(K\) be a local field of characteristic \(p>0\) and \((G,\rho,V)\) be a prehomogeneous vector space, where \(G\) is a connected linear algebraic group, \(\rho\) its rational representation on a finite-dimensional vector space \(V\), all defined over an algebraic closure \(\overline K\) of \(K\). Suppose that \((G,\rho,V)\) and its dual are \(K\)-regular. For a local field of characteristic 0, the functional equations of zeta distributions of prehomogeneous vector spaces have been obtained by M. Sato, Shintani, Iguasa, F. Sato and Gyoja. In this paper, the case of local fields of characteristic \(p>0\) has been considered. local field; prehomogeneous vector space; functional equations of zeta distributions Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Grassmannians, Schubert varieties, flag manifolds Fundamental theorem of prehomogeneous vector spaces of characteristic \(p\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Fix a split connected reductive group \(G\) over a field \(k\), and a positive integer \(r\). For any \(r\)-tuple of dominant coweights \(\mu_i\) of \(G\), we consider the restriction \(m_{\mu_\bullet}\) of the \(r\)-fold convolution morphism of Mirkovic-Vilonen to the twisted product of affine Schubert varieties corresponding to \(\mu_\bullet\). We show that if all the coweights \(\mu_i\) are minuscule, then the fibers of \(m_{\mu_\bullet}\) are equidimensional varieties, with dimension the largest allowed by the semi-smallness of \(m_{\mu_\bullet}\). We derive various consequences: the equivalence of the non-vanishing of Hecke and representation ring structure constants, and a saturation property for these structure constants, when the coweights \(\mu_i\) are sums of minuscule coweights. This complements the saturation results of Knutson-Tao and Kapovich-Leeb-Millson. We give a new proof of the P-R-V conjecture in the ``sums of minuscules'' setting. Finally, we generalize and reprove a result of Spaltenstein pertaining to equidimensionality of certain partial Springer resolutions of the nilpotent cone for \(\text{GL}_n\). split connected reductive groups; dominant coweights; affine Schubert varieties; equidimensional varieties; Springer resolutions; affine Grassmannians; Hecke algebras; structure constants; saturation Grothendieck, A., Diéudonné, J.: Éléments de Géométrie Algeébrique, IV: Étude locale ded schémas e des morphismes de schémas, Seconde partie, Inst.~ Hautes Études Sci.~Publ.~Math.~\textbf{24} (1965) Linear algebraic groups over local fields and their integers, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Hecke algebras and their representations, Representation theory for linear algebraic groups Equidimensionality of convolution morphisms and applications to saturation problems. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\underline a=(0<a_1<a_2< \cdots <a_k<n)\) be a sequence of integers and consider the general flag variety \(\mathbb{F}(\underline a, \mathbb{C}^n)\). Its homology is generated by the Schubert varieties. The Pieri formula gives a description of the cup product when one of the factors is a Chern class of a tautological bundle.
The (small) quantum cohomology of \(\mathbb{F}(\underline a, \mathbb{C}^n)\) is studied. The main approach to compute the quantum products is a quantum Pieri formula from \textit{I. Ciocan-Fontanine} [Duke Math. J. 98, No.3, 485--524 (1999; Zbl 0969.14039)]. This formula is reproven here by a geometrical argument which reduces it to the usual Pieri formula. The algorithm for computing a general quantum product is thus to use a quantum Giambelli formula (which expresses a quantum cohomology class as a polynomial in the Chern classes of tautological bundles) and then applying the quantum Pieri formula. quantum cohomology; flag manifold; quantum Pieri formula; quantum Giambelli formula Buch A.S.: Quantum cohomology of partial flag manifolds. Trans. Am. Math. Soc. 357, 443--458 (2005) (electronic) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of partial 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 prove the abelian-nonabelian correspondence for quasimap \(I\)-functions. That is, if \(Z\) is an affine l.c.i. variety with an action by a complex reductive group \(G\), we prove an explicit formula relating the quasimap \(I\)-functions of the geometric invariant theory quotients \(Z\mathord{/\mkern -6mu/}_{\theta}G\) and \(Z\mathord{/\mkern -6mu/}_{\theta}T\) where \(T\) is a maximal torus of \(G\). We apply the formula to compute the \(J\)-functions of some Grassmannian bundles on Grassmannian varieties and Calabi-Yau hypersurfaces in them. Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds, Stacks and moduli problems, Formal methods and deformations in algebraic geometry, Sheaves in algebraic geometry The abelian-nonabelian correspondence for \(I\)-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 The results of this paper are a generalization of those in the authors' paper [Special functions, Proc. Hayashibara Forum, Okayama/Jap. 1990, ICM-90 Satell. Conf. Proc., 149-168 (1991)]. Let \(k_ q[G]\) be the quantum algebra of functions on a semisimple algebraic group \(G\) of rank \(\ell\) (in [loc. cit.], the considered \(G= \text{SL}(N)\)). Let \(B\) be a Borel subgroup of \(G\) and let \(P\supseteq B\) be a maximal parabolic subgroup of \(G\). Let \(k_ q[B]\) be the quantum Hopf algebra of functions on \(B\). Let \(w\) be an element of the Weyl group and let \(X(w) \subset G/B\) be the corresponding Schubert variety. The authors define the quantum algebras \(k_ q[G/P]\), \(k_ q[G/B]\), \(k_ q[X(w)]\); the first two are subcomodules of \(k_ q[G]\), the last is a quotient of \(k_ q[G/B]\). Each of these algebras has, in the classical case, a basis consisting of standard monomials---compatible with canonical \(\mathbb{Z}\) or \(\mathbb{Z}^ \ell\)-gradations. The authors prove the existence of such basis and gradations in the quantum case and give a presentation for \(k_ q[G/B]\). quantum algebra of functions; semisimple algebraic group; Schubert variety; basis; gradations V. Lakshmibai and N. Reshetikhin. ''Quantum flag and Schubert schemes''. Deformation Theory and Quantum Groups with Applications to Mathematical Physics. Contemp. Math., Vol. 134. American Mathematical Society, 1992, pp. 145--181. Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Quantum flag and Schubert schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author proves the irreducibility of the family of all plane curves of given degree with a prescribed number of nodes and a further singular point of a special type (quasi-ordinary singularity). This result generalizes the irreducibility of Severi varieties [\textit{J. Harris}, Invent. Math. 84, 445-461 (1986; Zbl 0596.14017)] and answers in particular a conjecture of Enriques. The proof uses a special type of reducible surfaces (fans) whose components are blowing ups of projective planes. Fans were already used by the author in his own proof of the irreducibility of Severi varieties [Invent. Math. 86, 529-534 (1986; Zbl 0644.14009)]. Most of the paper is devoted to developing a degeneration theory for fans and families of curves on fans: such theory is the main tool for the proof and should have many other applications. irreducibility of the family of all plane curves of given degree; nodes; quasi-ordinary singularity; Severi varieties; fans Ziv Ran, Families of plane curves and their limits: Enriques' conjecture and beyond, Ann. of Math. (2) 130 (1989), 121-157 Zbl0704.14018 MR1005609 Families, moduli of curves (algebraic), Grassmannians, Schubert varieties, flag manifolds Families of plane curves and their limits: Enriques' conjecture and beyond | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Theta-vexillary signed permutations are elements in the hyperoctahedral group that index certain classes of degeneracy loci of type B and C. These permutations are described using triples of \(s\)-tuples of integers subject to specific conditions. The objective of this work is to present different characterizations of theta-vexillary signed permutations, describing them in terms of corners in the Rothe diagram and pattern avoidance. permutations; Schubert varieties Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds Theta-vexillary signed permutations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Q\) be a quiver with no oriented cycles. For dimension vectors \(\alpha\) and \(\beta\), define \(N(\beta,\alpha)\) as the number of \(\beta\)-dimensional subrepresentations of a general \(\alpha\)-dimensional representation of \(Q\). If \(\langle\beta,\alpha-\beta\rangle=0\) (here \(\langle\cdot,\cdot\rangle\) denotes the Ringel bilinear form), then \(N(\beta,\alpha)\) is finite. Denote by \(M(\beta,\alpha)\) the dimension of the space of semi-invariant polynomial functions with weight \(\langle\beta,\cdot\rangle\) on the space of \((\alpha-\beta)\)-dimensional representations of \(Q\) (note that any non-zero semi-invariant on \(\text{Rep}(Q,\alpha-\beta)\) has weight of such special form).
The main result of this paper is that \(M(\beta,\alpha)=N(\beta,\alpha)\) when \(\langle\beta,\alpha-\beta\rangle=0\). The proof is that the number \(M(\beta,\alpha)\) can be expressed via a Littlewood-Richardson calculation, which is then compared by the authors with the expression of \(N(\beta,\alpha)\) given by \textit{W. Crawley-Boevey} [Bull. Lond. Math. Soc. 28, No. 4, 363-366 (1996; Zbl 0863.16008)] using intersection theory.
Applying results of \textit{A. Schofield} [J. Lond. Math. Soc., II. Ser. 43, No. 3, 383-395 (1991; Zbl 0779.16005)], a basis of the corresponding space of semi-invariants is obtained. The result is generalized to covariants as follows: the cohomology class of the variety of \(\beta\)-dimensional subrepresentations of an \(\alpha\)-dimensional representation in general position can be expressed in terms of multiplicities in isotypic components of the coordinate ring of \(\text{Rep}(Q,\beta-\alpha)\). semi-invariants of quivers; Schubert classes; Littlewood-Richardson coefficients; representations of quivers; Ringel bilinear forms Derksen, H., Schofield, A., Weyman, J.: On the number of subrepresentations of a general quiver representation. J. Lond. Math. Soc. (2) 76(1), 135--147 (2007) Representations of quivers and partially ordered sets, Actions of groups on commutative rings; invariant theory, Grassmannians, Schubert varieties, flag manifolds On the number of subrepresentations of a general quiver representation. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If \(X\) is a variety in characteristic \(p\) and \(F\) is its absolute Frobenius morphism, a splitting of \({\mathcal O}_X\to F_*{\mathcal O}_X\) is equivalent to a section \(\sigma\in H^0(X,\omega_X^{1-p})\) with nonzero residue. If \(X\) is the product of a Grassmannian by itself, the authors show that there exists such a \(\sigma\) with large multiplicity along the diagonal. This fact implies that, for Grassmannians in positive characteristic, Wahl's conjecture on the surjectivity of the Gauss map is true. Grassmannian; positive characteristic; Frobenius map; surjectivity of the Gauss map V. B. Mehta, A. J. Parameswaran, On Wahl's conjecture for the Grassmannians in positive characteristic, Internat. J. Math. 8 (1997), no. 4, 49598. Grassmannians, Schubert varieties, flag manifolds, Finite ground fields in algebraic geometry On Wahl's conjecture for the Grassmannians in positive characteristic | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Fomin-Kirillov algebra \(\mathcal E_n\) is a noncommutative quadratic algebra with a generator for every edge of the complete graph on \(n\) vertices. For any graph \(G\) on \(n\) vertices, we define \(\mathcal E_G\) to be the subalgebra of \(\mathcal E_n\) generated by the edges of \(G\). We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, \(\mathcal E_G\) is a free \(\mathcal E_H\)-module for any \(H\subseteq G\), and if \(\mathcal E_G\) is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for \(\mathcal E_G\) when \(G\) is a simply laced finite Dynkin diagram or a cycle, in particular showing that \(\mathcal E_G\) is finite-dimensional in these cases. We also present conjectures for the Hilbert series of \(\mathcal E_{\tilde D_n}\), \(\mathcal E_{\tilde E_6}\), and \(\mathcal E_{\tilde E_7}\), as well as the graphs \(G\) on six vertices for which \(\mathcal E_G\) is finite-dimensional. Fomin-Kirillov algebra; Coxeter group; nil-Coxeter algebra; Nichols algebra Jonah Blasiak, Ricky Ini Liu, Karola Mészáros, Subalgebras of the Fomin-Kirillov algebra. Preprint, 2012. arxiv:1310.4112. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Subalgebras of the Fomin-Kirillov 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 In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs.
We construct a 2-\((6, 3, 78)_5\) design by computer, which corresponds to a halving \(\operatorname{LS}_5 [2](2, 3, 6)\). The application of the new recursion method to this halving and an already known \(\operatorname{LS}_3 [2](2, 3, 6)\) yields two infinite two-parameter series of halvings \(\operatorname{LS}_3 [2](2, k, v)\) and \(\operatorname{LS}_5 [2](2, k, v)\) with integers \(v \geq 6\), \(v \equiv 2\pmod 4\) and \(3 \leq k \leq v - 3\), \(k \equiv 3 \pmod 4\).
Thus in particular, two new infinite series of nontrivial subspace designs with \(t = 2\) are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with \(t = 2\). \(q\)-analog; combinatorial design; subspace design; large set; halving Braun, M; Kiermaier, M; Kohnert, A; Laue, R, Large sets of subspace designs, J. Comb. Theory Ser. A, 147, 155-185, (2014) Other designs, configurations, Combinatorial aspects of block designs, Grassmannians, Schubert varieties, flag manifolds Large sets of subspace designs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors consider ``generic'' arrangements of high-dimensional Schubert cells in the real flag variety \(\mathbb{P} T^*\mathbb{P}^ n\) of all flags in \(\mathbb{P}^ n\) consisting of a hyperplane and a distinguished point in it. It is shown that the sum of the Betti numbers (with coefficients \(\mathbb{Z}/2\mathbb{Z})\) of those arrangements coincides with the sum of the Betti numbers of their complexifications \((M\)-property). In general, the \(M\)-property does not hold: an example is given, namely arrangements in \(G_{2,4}\). Finally, for some class of configuration spaces, the Mayer-Vietoris spectral sequence is shown to degenerate in the \(E_ 1\) term. flag varieties; Schubert cell decomposition; \(M\)-property; configuration spaces; Mayer-Vietoris spectral sequence Shapiro B. Z., Topology Appl. 43 (1) pp 65-- (1992) Real-analytic manifolds, real-analytic spaces, Complex spaces, Grassmannians, Schubert varieties, flag manifolds, Spectral sequences in algebraic topology The M-property of flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main purpose of the paper under review is the extension of the theory of the generic identiafiability symmetric tensors, already completely studied, to arbitrary tensors. This type of results is useful in many applicarions of mathematics and during the last years several authors have proved the close connection between these problems and modern birational projective geometry. In particular, the tangential weak defectiveness of varieties is used as a main tool.
The authors of this paper develop tools to plug in maximal singularities methods for non-tangentially weakly defective varieties and in this way they are able to prove the non-identiafiability of many partially symmetric tensors as a first step for the study of the arbitrary tensors case. secant varieties; secant defectiveness; weak defectiveness; tangential weak defectiveness; identifiability Secant varieties, tensor rank, varieties of sums of powers, Projective techniques in algebraic geometry, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Multilinear algebra, tensor calculus, Exterior algebra, Grassmann algebras Tangential weak defectiveness and generic identifiability | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review, the authors introduce the notion of residual normal crossing at a closed point of a variety \(X\) and observe that the \((p-1)\)-th power of a section \(s\) of \(K_X^{-1}\) (for a smooth variety \(X\) with the canonical bundle \(K_X)\) with a residual normal crossing at a closed point Frobenius splits the variety \(X\). This generalizes an earlier result due to Mehta and Ramanathan asserting that the \((p-1)\)-th power of a section \(s\) of \(K_X^{-1}\) with normal crossing divisor at some closed point Frobenius splits \(X\). In the present paper, it is further shown that for any parabolic subgroup \(Q\) of a simple group \(G\) of classical type or of type \(G_2\), \(K^{-1}_X\) (where \(X=G/Q)\) admits a section with residual normal crossing at the base point. This provides another proof of the Frobenius splitting of such an \(X\).
Further the authors conjecture that for any homogeneous space \(X\) in characteristic \(p>0\), \(E\) is compatibly split in the blow-up \(B(X\times X)\) of \(X\times X\) along the diagonal \(\Delta\) with exceptional divisor \(E\). This is equivalent to showing that there exists a splitting of \(X\times X\) which vanishes to order \((\dim X)(p-1)\) generically along \(\Delta\). It is shown that this conjecture implies the Wahl conjecture on the surjectivity of the Gaussian map in characteristic \(p\). Recall that this was proved by the reviewer in characteristic 0 earlier [\textit{S. Kumar}, Am. J. Math. 114, No. 6, 1201-1220 (1992; Zbl 0790.14015)]. Frobenius splittings; residual normal crossing; Wahl's conjecture; characteristic \(p\); homogeneous space; Wahl conjecture; surjectivity of the Gaussian map V. Lakshmibai, V. B. Mehta, A. J. Parameswaran, Frobenius splittings and blow-ups, J. Algebra 208 (1998), no. 1, 10128. Divisors, linear systems, invertible sheaves, Finite ground fields in algebraic geometry, Formal methods and deformations in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Frobenius splittings and blow-ups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Robinson-Schensted-Knuth correspondence (RSK for short) is a bijection between permutations of \(d\) letters and pairs of standard Young tableaux of the same shape on \(d\) boxes. This correspondence comes up when considering flag varieties. Namely, the Bruhat decomposition yields that the relative position of two complete flags in a \(d\)-dimensional space \(V\) is given by an element of the symmetric group \(S_d\). Also, given a nilpotent \(x\in\text{End}(V)\), the irreducible components of the subvariety of flags that are preserved by \(x\) are parametrized by the standard tableaux of the shape \(\lambda\), which is the Jordan type of \(x\). Then, for two general flags, their relative position is given by the permutation that is obtained via the RSK correspondence to the standard tableaux associated to the irreducible components in which they lie. Theorem~4.1 of the present paper provides a generalization of this result to the case of partial flags.
The second part of the paper is concerned with generalizing the mirabolic RSK correspondence [\textit{R.~Travkin}, Sel. Math., New Ser. 14, No. 3--4, 727--758 (2009; Zbl 1230.20047)] to the case of two partial flags and a line. partial flag variety; RSK correspondence; Young tableau Daniele Rosso, Classic and mirabolic Robinson-Schensted-Knuth correspondence for partial flags, Canad. J. Math. 64 (2012), no. 5, 1090 -- 1121. Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices Classic and mirabolic Robinson-Schensted-Knuth correspondence for partial 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 In this note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a non-degenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in a holomorphic vector bundle. This description recovers characterisations of Baird and Wood for Euclidean 3-space. The universal objects that play a key role are the orthogonal Grassmannians. holomorphic vector bundle; Grassmannian; harmonic maps; foliations Holomorphic bundles and generalizations, Kähler manifolds, Harmonic maps, etc., Foliations (differential geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac-Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type \(\tilde{A}_2\) on a Euclidean space \(V \cong \mathbb{R}^2\) to study the Bruhat order on \(W\). We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of \textit{B. D. Boe} and \textit{W. Graham} [Am. J. Math. 125, No. 2, 317--356 (2003; Zbl 1074.14045)] (which is a conjectural simplification of the Carrell-Peterson criterion (see [\textit{J. B. Carrell}, Proc. Symp. Pure Math. 56, 53--61 (1994; Zbl 0818.14020)]) for rational smoothness) to type \(\tilde{A}_2\). Computational evidence suggests that the only Schubert varieties in type \(\tilde{A}_2\) where the ``nontrivial'' case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety \(X ( w )\) of type \(\tilde{A}_2\). The proof uses descriptions we obtain of the elements \(x \leq w\) and of the rationally smooth locus of \(X ( w )\) in terms of the \(W\)-action on \(V\). As a consequence we describe the maximal nonrationally smooth points of \(X ( w )\). The results of this paper are used in a sequel to describe the smooth locus of \(X ( w )\), which is different from the rationally smooth locus. Schubert variety; rationally smooth; lookup conjecture Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds The Bruhat order, the lookup conjecture and spiral Schubert varieties of type \(\tilde{A}_2\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is a continuation of some previous works on the dualities of orbits on flag manifolds [Transform. Groups 8, 333--376 (2003; Zbl 1043.22007); J. Math. Soc. Japan 57, 157--165 (2005; Zbl 1076.14067)]. In these works it has been defined a \(G_{R}\) -\(K_{C}\) invariant subset \(C(S)\) of a connected complex semisimple Lie group \(G_{C}\) for each \(K_{C}-\)orbit \(S\) on every flag manifold \(G_{C}/P\), where \(P\) is an arbitrary parabolic subgroup of \(G_{C}\) and \(G_{R}\) is a connected real form of \(G_{C}\). It was proven that the connected component \(C(S)_{0}\) of the identity is equal to the Akhiezer-Gindikin domain \(D\) if \(S\) is of nonholomorphic type.
In this paper the author proves the previous result of equality for all the other orbits when \(G_{R}\) is of non-Hermitian type. If \(G_{R}\) is simple and of non-Hermitian type, then \(C(S)_{0}=D\) for all the \(K_{C}\)-orbits \(S\neq X\) on \(X=G_{C}/P\), where \(D\) denotes the Akhiezer-Gindikin domain defined in \(G_{C}/K_{C}\). This result was also proved by author [preprint, \texttt{RT/0410302}] for all non-closed \(K_{C}\)-orbits in Hermitian cases. This means that the conjecture of equivalence of domains arising from duality of orbits on flag manifolds is completely solved affirmatively. complex Lie groups; flag manifolds; symmetric spaces; Stein extensions Toshihiko Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds. II, Proc. Amer. Math. Soc. 134 (2006), no. 12, 3423 -- 3428. Grassmannians, Schubert varieties, flag manifolds, General properties and structure of real Lie groups, Semisimple Lie groups and their representations, Complex Lie groups, group actions on complex spaces Equivalence of domains arising from duality of orbits on flag manifolds. II. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 totally nonnegative Grassmannian is the set of \(k\)-dimensional subspaces \(V\) of \(\mathbb{R}^n\) whose nonzero Plücker coordinates (i.e. \(k\times k\) minors of a \(k\times n\) matrix whose rows span \(V\)) all have the same sign. Total positivity has been much studied in the past two decades from an algebraic, combinatorial, and topological perspective, but first arose in the theory of oscillations in analysis. It was in the latter context that \textit{F. R. Gantmacher} and \textit{M. G. Kreĭn} [Осцилляционые матрицы и ядра и малые колебания механических систем (Russian). Moskau-Leningrad: Staatsverlag für technisch-theoretische Literatur (1950; Zbl 0041.35502)] and \textit{I. J. Schoenberg} and \textit{A. Whitney} [Compos. Math. 9, 141--160 (1951; Zbl 0043.25204)] independently showed that a subspace \(V\) is totally nonnegative iff every vector in \(V\), when viewed as a sequence of \(n\) numbers and ignoring any zeros, changes sign fewer than \(k\) times. We generalize this result, showing that the vectors in \(V\) change sign fewer than \(l\) times iff certain sequences of the Plücker coordinates of some generic perturbation of \(V\) change sign fewer than \(l-k+1\) times. We give an algorithm which constructs such a generic perturbation. Also, we determine the positroid cell of each totally nonnegative \(V\) from sign patterns of vectors in \(V\). These results generalize to oriented matroids. sign variation; totally nonnegative Grassmannian; oriented matroid; positroid; Grassmann necklace S.N. Karp, \textit{Sign variation, the Grassmannian and total positivity}, arXiv:1503.05622 [INSPIRE]. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Combinatorial aspects of matroids and geometric lattices Sign variation, the Grassmannian, and total positivity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We treat the time evolution of unitary elements in the \(N\) level system and consider the reduced dynamics from the unitary group \(U(N)\) to flag manifolds of the second type (in our terminology). Then we derive a set of differential equations of matrix Riccati types interacting with one another and present an important problem on a nonlinear superposition formula that the Riccati equation satisfies.
Our result is a natural generalization of the paper \textit{S. Chaturvedi} et al. [\url{arXiv:0706.0964}]. \(N\) level system; reduced dynamics; flag manifolds; matrix Riccati equations DOI: 10.1142/S0219887809003680 Open systems, reduced dynamics, master equations, decoherence, Nonlinear ordinary differential equations and systems, Grassmannians, Schubert varieties, flag manifolds, Quantum computation Reduced dynamics from the unitary group to some flag manifolds: interacting matrix Riccati 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 These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014. In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a survey of some more recent work. In the second half of the article, we give a definition of a Grassmann polytope motivated by work of physicists on the amplituhedron. We propose to use Schubert calculus and canonical bases to replace linear algebra and convexity in the theory of polytopes. total nonnegativity; Grassmann polytope; positroid variety; dimers; Grassmann matroid; scattering amplitudes Grassmannians, Schubert varieties, flag manifolds, Planar graphs; geometric and topological aspects of graph theory, Special polytopes (linear programming, centrally symmetric, etc.), Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) Totally nonnegative Grassmannian and Grassmann polytopes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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{P. Schapira} [Lect. Notes Comput. Sci. 948, 427--435 (1995; Zbl 0878.44002)] studied Radon transforms of constructible functions and obtained a formula related to an inversion formula. We generalize this formula to more complicated cases including Radon transformations between any Grassmann manifolds. In particular, we give an inversion formula for the Radon transformation and characterize images of Radon transforms of characteristic functions of Schubert cells. Matsui, Y.: Radon transforms of constructible functions on Grassmann manifolds. Publ. res. Inst. math. Sci. 42, No. 2, 551-580 (2006) Grassmannians, Schubert varieties, flag manifolds, Twistor theory, double fibrations (complex-analytic aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Radon transform Radon transforms of constructible functions on 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 We launch the study of the tropicalization of the symplectic Grassmannian, that is, the space of all linear subspaces isotropic with respect to a fixed symplectic form. We formulate tropical analogues of several equivalent characterizations of the symplectic Grassmannian and determine all implications between them. In the process, we show that the Plücker and symplectic relations form a tropical basis if and only if the rank is at most 2. We provide plenty of examples that show that several features of the symplectic Grassmannian do not hold after tropicalizing. We show exactly when do conormal fans of matroids satisfy these characterizations, as well as doing the same for a valuated generalization. Finally, we propose several directions to extend the study of the tropical symplectic Grassmannian. Grassmannians, Schubert varieties, flag manifolds, Geometric aspects of tropical varieties The tropical symplectic Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this work we discuss some appearances of semi-infinite combinatorics in representation theory. We propose a semi-infinite moment graph theory and we motivate it by considering the (not yet rigorously defined) geometric side of the story. We show that it is possible to compute stalks of the local intersection cohomology of the semi-infinite flag variety, and hence of spaces of quasi maps, by performing an algorithm due to \textit{T. Braden} and \textit{R. MacPherson} [Math. Ann. 321, No. 3, 533--551 (2001; Zbl 1077.14522)] . moment graphs; semi-infinite order; character formulae Infinite-dimensional Lie (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations Semi-infinite combinatorics in representation theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The classical theory of special divisors is being re-examined. This monograph is a general introduction assuming a background knowledge of the theory of differentials on a Riemann surface -- as far as Abel's theorem. It defines the variety \(W_d^r\) of linear systems with degree \(d\) and projective dimension \(r\) as a subvariety of the Jacobian (because of duality \(d <g\) is assumed) and states the main problems and their interrelations with other classical geometric questions.
Some of these principal questions have now been answered in the author's joint paper with \textit{J. Harris} in [Duke Math. J. 47, 233--272 (1980; Zbl 0446.14011). special divisors Jacobians, Prym varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Grassmannians, Schubert varieties, flag manifolds An introduction to the theory of special divisors on algebraic curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected reductive algebaic group over \(\mathbb C\) with Weyl group \(W\) and Lie algebra \(\mathfrak g\), and let \(\mathcal N \subset \mathfrak g\) be its nilpotent cone. In the present paper, the authors study two different actions of \(W\) on the Springer sheaf on \(\mathcal N\) with arbitrary coefficients, one defined by Fourier transform and one by restriction, and show that these actions coincide up to a twist by the sign character of \(W\).
When the ring of coefficients is a field of characteristic zero, \textit{T. A. Springer} [Invent. Math. 36, 173--207 (1976; Zbl 0374.20054)] showed that \(W\) acts in a natural way on the \(\ell\)-adic cohomology of the Springer fiber. Later \textit{G. Lusztig} [Adv. Math. 42, 169--178 (1981; Zbl 0473.20029)] gave another interpretation of this action in terms of perverse sheaves, and the two actions were shown to coincide up to a twist by the sign character of \(W\). The study of Springer theory with coefficients in a local ring or a finite field \(k\) was initiated by the third author in his thesis [``Modular Springer correspondence and decomposition matrices'', PhD thesis, Université Paris 7 (2007), \url{arxiv:0901.3671}], where the \(W\)-action was defined via a kind of Fourier transform, as Springer did in the characteristic zero case, and continued by the other three authors [``Geometric Satake, Springer correspondence, and small representations. II'', \url{arxiv:1205.5089}], who defined a \(W\)-action on the Springer sheaf by restriction following Lusztig's approach. The two construction are compared in the present paper, and they are shown to coincide up to a twist by the sign character of \(W\) as in the case of the actions on the \(\ell\)-adic cohomology of the Springer fiber.
As a first application, the equivalence of the two constructions is used to define a Springer correspondence with arbitrary coefficients. Assume now that \(k\) is a field and that \(G\) is simple and simply connected, and let \(\check G\) be the split connected reductive group over \(k\) whose root datum is dual to that of \(G\). Following [loc. cit.], in their second application the authors consider the representation of \(W\) on the zero weight space of a small representation of \(\check G\), and they describe such representations in terms of the Springer correspondence over \(k\). In the last section of the paper, the authors finally discuss an analogue of their main theorem for the étale topology over a finite field. Weyl groups; nilpotent cones; Springer sheaf; Springer correspondence Vogan, D.: The local Langlands conjecture. In: Adams J, Herb R, Kudla S, Li J-S, Lipsman R, Rosenberg J (eds) Representation Theory of Groups and Algebras, Contemporary Mathematics vol. 145, pp. 305-379. American Mathematical Society, Providence (1993) Coadjoint orbits; nilpotent varieties, Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds Weyl group actions on the Springer sheaf | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Belkale-Kumar product on \(H^*(G/P)\) is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case \(G = GL_n\), it was used by \(N\). Ressayre to determine the regular faces of the Littlewood-Richardson cone.
We show that for \(G/P\, a (d - 1)\)-step flag manifold, each Belkale-Kumar structure constant is a product of d Littlewood-Richardson numbers, for which there are many formulae 2 available, e.g. the puzzles of \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)]. This refines previously known factorizations into \(d - 1\) factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants.
These ``BK-puzzles'' are related to extremal honeycombs, as in [\textit{A. Knutson}, \textit{T. Tao} and \textit{C. Woodward}, J. Am. Math. Soc. 17, No. 1, 19--48 (2004; Zbl 1043.05111)]; using this relation we give another proof of Ressayre's result.
Finally, we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always 1; they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension 1 or 2. Belkale-Kumar product; combinatorial formula for Belkale-Kumar structure constants A. Knutson and K. Purbhoo, ''Product and puzzle formulae for \({ GL}_n\) Belkale-Kumar coefficients,'' Electron. J. Combin., vol. 18, iss. 1, p. 76, 2011. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Exact enumeration problems, generating functions, Combinatorial aspects of tessellation and tiling problems, Grassmannians, Schubert varieties, flag manifolds Product and puzzle formulae for \(GL_n\) Belkale-Kumar coefficients | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We formulate \(\sigma\)-models of a flag manifold with a zero-curvature representation in the form of a theory of linear ``matter fields'' interacting with auxiliary gauge fields. \(\sigma\)-model; integrable model; flag space; Kähler quotient Model quantum field theories, Grassmannians, Schubert varieties, flag manifolds, Other elementary particle theory in quantum theory, Yang-Mills and other gauge theories in quantum field theory, Groups and algebras in quantum theory and relations with integrable systems A gauged linear formulation for flag-manifold \(\sigma\)-models | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(G_{\mathbb{R}}\) be a real classical group. Assume that the Cartan subgroups of \(G_{\mathbb{R}}\) are connected. There exists a correspondence between irreducible representations of \(G_{\mathbb{R}}\) with a given infinitesimal character and the orbits of a flag of \(G_{\mathbb{C}}\) -- a complexification of \(G_{\mathbb{R}}\) on which \(K_{\mathbb{C}}\) operates, where \(K_{\mathbb{C}}\) is a complexification of a maximal compact subgroup \(K\) of \(G_{\mathbb{R}}\). One is therefore interested in finding an algorithm which can yield representatives of orbits for a suitable parametrization. The author considers the case \(G=U(p,q)\), \(n=p+q\), \(U(p,q)= \{g\in GL(n,\mathbb{C})\), \(^tgJg=J\}\), \(J= \left(\begin{smallmatrix} I_p &0\\ 0&-I_q \end{smallmatrix}\right)\), \(\theta\) the Cartan involution of \(G_{\mathbb{R}}\); \(\theta(x)= JxJ\).
In Definition 2.1 the notion of clan is introduced as an equivalence class of indications and an earlier result of Matsuki-Oshima (Theorem 2.2) is stated and applied to show that clans parametrize \(K\)-orbits in the flag variety \(X\). In Theorem 3.7 for each clan \(\gamma\), and signed class \(\delta\) of \(\gamma\), \(\delta= (d_1,\dots, d_n)\) the representative matrix \(g(\delta)= (g_1,g_2,\dots, g_n)\) is defined by explicitly exhibiting its column vectors. In Proposition 3.9 the dimension and co-dimension of the orbit \(Q_\gamma= Kg(\delta)B: \dim Q_\gamma= \ell(\gamma)+ \frac{1}{2} p(p-1)+ \frac{1}{2} q(q-1)\), \(\text{co-dim}= pq-\ell(\gamma)\), is given, where \(\ell(\gamma)\) is a certain integer introduced in Definition 3.8.
Let \(\gamma=(c_1,\dots,c_n)\) be a clan, \(\delta\) a signed clan of \(\gamma\), \(g(\delta)\) the matrix associated to \(\delta\) in Theorem 3.7. An algorithm is described to obtain the image \(g(\delta){\mathfrak E}^\perp g(\delta)^{-1}\cap{\mathfrak P}\) of the moment map of a fiber at \(g(\delta)\) of the conormal bundle of \(Q_\gamma= Kg(\delta)B\) (Theorem 4.1). In Section 5 tables of signed Young diagrams are displayed for clans of \(U(2,1)\) and \(U(2,2)\). \textit{D. Garfinkle} [Am. J. Math. 115, 305-369 (1993; Zbl 0786.22023)] has given a different algorithm for the same purpose. real classical group; representations; character; flag variety; moment map; Young diagrams Atsuko Yamamoto, Orbits in the flag variety and images of the moment map for \?(\?,\?), Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 6, 114 -- 117. Semisimple Lie groups and their representations, Grassmannians, Schubert varieties, flag manifolds Orbits in the flag variety and images of the moment map for \(U(p, q)\) | 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 A flag of type \((\underbrace{1,1,\dots,1}_k,n)\) in \(\mathbb{R}^m\) is a collection \((V_1,\dots,V_k,V_{k+1})\) of mutually orthogonal vector subspaces of \(\mathbb{R}^m\) such that \(k+n = m\), \(\dim V_i = 1,\;i=1,\dots,k\) and \(\dim V_{k+1} = n\). The set of all flags of type \((1,1,\dots,1,n)\) is described by the real flag manifold \(F(1,1,\dots,1,n)\). Since a paper by \textit{A. Borel} [Comment Math. Helv. 27, 165--197 (1953; Zbl 0052.40301)], it is known that the mod 2 cohomology algebra of \(F(1,1,\dots,1,n)\) can be described as a polynomial algebra with as variables the Stiefel-Whitney classes of the canonical vector bundles \(\gamma_1,\dots,\gamma_k\) over the flag manifold. Borel described a set of generators of the ideal defining the polynomial algebra, but in general this is not sufficient to have a procedure to decide whether a polynomial is zero in the quotient algebra, i.e. whether a polynomial is contained in the ideal. In order to positively answer to this question, the generators have to be a Gröbner basis. In the paper under review, the authors construct a Gröbner basis of the ideal with respect to the lexicographic order. Gröbner bases; flag manifolds; mod 2 cohomology Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Grassmannians, Schubert varieties, flag manifolds On Gröbner bases for flag manifolds \(F(1, 1,\dots , 1, n)\) | 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 The author extends standard monomial theory to the wonderful compactification \(X\) of a semisimple group \(G\) of adjoint type. Recall that \textit{R. Chirivì} and \textit{A. Maffei} have already constructed standard monomials for the more general situation of a wonderful compactification of a symmetric space [J. Algebra 261, No. 2, 310-326 (2003; Zbl 1055.14052)]. We fix a dominant weight \(\lambda\) and the corresponding line bundle \(\mathcal L_\lambda\) on \(X\). Then Chirivì and Maffei provide a basis of \(H^0(X,\mathcal L_\lambda)\) consisting of `standard monomials' with certain properties.
The author shows that in the present situation one can do more. First of all, the standard monomials are shown to behave well with respect to restriction to \(B\times B\)-orbit closures, not just \(G\times G\)-orbit closures. Recall that there are finitely many \(B\times B\)-orbits and that they have been classified by \textit{T. A. Springer} [J. Algebra 258, No. 1, 71-111 (2002; Zbl 1110.14047)]. There are degrees of freedom in the construction of Chirivì, Maffei and these the author exploits to arrange more properties familiar from the classical standard monomial theory for flag varieties.
The basis of \(H^0(X,\mathcal L_\lambda)\) is indexed by LS-paths again. And if \(Z\) is a \(G\times G\)-orbit closure, or more generally a \(B\times B\)-orbit closure, then the standard monomials that do not vanish on \(Z\) form a basis of \(H^0(Z,\mathcal L_\lambda)\). They can be characterized combinatorially. The many ingredients that are needed in the proof are explained clearly. standard monomials; group compactifications; wonderful compactifications; semisimple groups of adjoint type; orbit closures Appel, K, Standard monomials for wonderful group compactifications, J. Algebra, 310, 70-87, (2007) Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Standard monomials for wonderful group compactifications. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a new proof to V. B. Mehta and A. Ramanathan's theorem that the Schubert subschemes in a flag scheme are all simultaneously compatible split, using the representation theory of infinitesimal algebraic groups. In particular, the present proof dispenses with the Bott-Samelson schemes. Schubert subschemes; flag scheme; representation theory of infinitesimal algebraic groups Kaneda, M., On the Frobenius morphism of flag schemes, Pacific J. Math., 163, 315-336, (1994) Grassmannians, Schubert varieties, flag manifolds, Other algebraic groups (geometric aspects) On the Frobenius morphism of flag schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A connection between the scheme of unitary quantization (uniquantization) and para-Fermi statistics of order 2 is considered. An appropriate generalization of the Green's ansatz is suggested based on incorporation of the additional operator \(\Omega\) which allows one to transform into the identity the bilinear and trilinear commutation relations of unitary quantization for the creation and annihilation operators of two different para-Fermi fields \(\phi_a\) and \(\phi_b\). The way of incorporating para-Grassmann variables \(\xi_k\) into the general scheme of unitary quantization necessary for the definition of coherent states is suggested. For parastatistics of order 2, the new fact of existence of two alternative definitions of the coherent state for the para-Fermi oscillators is established. para-Fermi statistics; unitary quantization; Green's ansatz; para-Grassmann variables; coherent states Coherent states, Particle exchange symmetries in quantum theory (general), Geometry and quantization, symplectic methods, Schrödinger operator, Schrödinger equation, Grassmannians, Schubert varieties, flag manifolds Parastatistics and uniquantization | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 define symmetric Dellac configurations as the Dellac configurations that are symmetrical with respect to their centers. The even-length symmetric Dellac configurations coincide with the Fang-Fourier symplectic Dellac configurations. Symmetric Dellac configurations generate the Poincaré polynomials of (odd or even) symplectic or orthogonal versions of degenerate flag varieties. We give several combinatorial interpretations of the Randrianarivony-Zeng polynomial extension of median Euler numbers in terms of objects that we call extended Dellac configurations. We show that the extended Dellac configurations generate symmetric Dellac configurations. As a consequence, the cardinalities of odd and even symmetric Dellac configurations are respectively given by two sequences \((1, 1, 3, 21, 267,\ldots)\) and \((1, 2, 10, 98, 1594,\ldots)\), defined as specializations of polynomial extensions of median Euler numbers. Dellac configuration; median Euler number; flag variety Exact enumeration problems, generating functions, Grassmannians, Schubert varieties, flag manifolds Symmetric Dellac configurations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 present paper we extend the Riemann-Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and push-forwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann-Roch theorem for moment graphs. As an application, we obtain the Riemann-Roch-type theorem for the equivariant \(K\)-theory of some Kac-Moody flag varieties. equivariant cohomology; Chern character; Riemann-Roch theorem; moment graph; structure algebra Group actions on varieties or schemes (quotients), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds A Riemann-Roch type theorem for twisted fibrations of moment graphs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{P. Brosnan} and \textit{T. Y. Chow} [Adv. Math. 329, 955--1001 (2018; Zbl 1410.05222)], and independently \textit{M. Guay-Paquet} [``A second proof of the Shareshian-Wachs conjecture, by way of a new Hopf algebra'', Preprint, \url{arXiv:1601.05498}], proved the Shareshian-Wachs conjecture [\textit{J. Shareshian} and \textit{M. L. Wachs}, Adv. Math. 295, 497--551 (2016; Zbl 1334.05177)], which links the Stanley-Stembridge conjecture in combinatorics to the geometry of Hessenberg varieties through Tymoczko's permutation group action [\textit{J. S. Tymoczko}, Contemp. Math. 460, 365--384 (2008; Zbl 1147.14024)] on the cohomology ring of regular semisimple Hessenberg varieties. In previous work, the authors exploited this connection to prove a graded version of the Stanley-Stembridge conjecture in a special case. In this manuscript, we derive a new set of linear relations satisfied by the multiplicities of certain permutation representations in Tymoczko's representation. We also show that these relations are upper-triangular in an appropriate sense, and in particular, they uniquely determine the multiplicities. As an application of these results, we prove an inductive formula for the multiplicity coefficients corresponding to partitions with a maximal number of parts. Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Symmetric functions and generalizations Upper triangular linear relations on multiplicities and the Stanley-Stembridge conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The first attempt to understand vector bundles on the flag varieties by the author is this article. He treats the flag variety F parametrizing two linear spaces in inclusion relation. The first one third of the article is devoted to constructing the component \(\hat X\) of the Hilbert scheme of F corresponding to ''straight lines'' in F. For a given vector bundle E on F, by restricting E to any ''straight line'' contained in F and by studying properties of the restricted bundles, we can analyse global properties of E. The construction of \(\hat X\) is indispensable for this purpose. It turns out that \(\hat X\) carries also natural tautological vector bundles. In the middle part the author shows that Chern classes of tautological vector bundles on F and those on \(\hat X\) have very systematic correspondence. This is an interesting fact. In the last section he shows that the set J of jumping lines coincides with the image of the second Chern class of E by the Gysin homomorphism in the Chow group under a certain hypothesis. Here J is the set of points in \(\hat X\) such that the restriction of E to the corresponding ''line'' is different from that of the general ''line''. vector bundles on flag variety; Hilbert scheme; Chern classes of tautological vector bundles; jumping lines Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Characteristic classes and numbers in differential topology Some properties of vector bundles on the flag variety Fl(r,s;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 Based on Pieri's formula on Schubert varieties [see \textit{F. Sottile}, Can. J. Math. 49, 1281-1298 (1997; Zbl 0933.14031)], the Pieri homotopy algorithm was first proposed by \textit{B. Huber, F. Sottile}, and \textit{B. Sturmfels} [J. Symb. Comput. 26, 767-788 (1998; Zbl 1064.14508)]\ for numerical Schubert calculus to enumerate all \(p\)-planes in \({\mathbb C}^{m+p}\) that meet \(n\) given planes in general position. The algorithm has been improved by \textit{B. Huber} and \textit{J. Verschelde} [SIAM J. Control Optim. 38, 1265-1287 (2000; Zbl 0955.14038)]\ to be more intuitive and more suitable for computer implementations.
A different approach of employing the Pieri homotopy algorithm for numerical Schubert calculus is presented in this paper. A major advantage of our method is that the polynomial equations in the process are all square systems admitting the same number of equations and unknowns. Moreover, the degree of each polynomial equation is always 2, which warrants much better numerical stability when the solutions are being solved. Numerical results for a big variety of examples illustrate that a considerable advance in speed as well as much smaller storage requirements have been achieved by the resulting algorithm. enumerative geometry; Schubert variety; Pieri formula; Pieri homotopy algorithm; Pieri poset; algorithm Li D, Qi L, Zhou S (2002) Descent directions of quasi-Newton methods for symmetric nonlinear equations. SIAM J Numer Anal 40(5): 1763--1774 Grassmannians, Schubert varieties, flag manifolds, Computational aspects of higher-dimensional varieties, Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation, Enumerative problems (combinatorial problems) in algebraic geometry Numerical Schubert calculus by the Pieri homotopy algorithm | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Somewhat analogous to the case of the variety of complexes, the variety \(P\) of projectors, i.e., idempotents (of rank \(d\) on an \(n\)-space), is shown to be a principal affine open subset of the product of the Grassmannian \(\text{Gr}(d,n)\) and its dual \(\text{Gr}(n-d,n)\). Also \(P\) is identified with the affine coset space \(\text{GL}(n)/H\) for a closed reductive subgroup \(H\) of the form \(\text{GL}(d)\times\text{GL}(n-d)\); consequently, \(P\) is nonsingular and of dimension \(2d(n-d)\). The coordinate ring \(R\) of \(P\) is described explicitly by generators and relations as the subring of left translation \(H\)-invariants of \(k[\text{GL}(n)]\) as an immediate consequence of the classical Hodge standard monomial basis readily available for \(R\) just as for the homogeneous coordinate ring of \(\text{Gr}(d,n)\) for its Plücker embedding. The \(\text{GL}(n)\)-module structure of \(R\) is shown to be the direct limit of the filtered family of representations of \(\text{GL}(n)\): \(m\omega_ d\otimes m\omega_{n-d}\otimes(-m)\text{det}\)., \(m\in\mathbb{Z}^ +\), where \(\omega_ d\) and \(\omega_{n-d}\) are the fundamental weights of \(\text{GL}(n)\) corresponding to \(\text{Gr}(d,n)\) and \(\text{Gr}(n-d,n)\), respectively, and det. is the determinant character of \(\text{GL}(n)\). Hodge algebra; projectors; subset of the product of the Grassmannian; affine coset space; Hodge standard monomial basis Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Linear algebraic groups over arbitrary fields, Birational geometry A note on the variety of projectors | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 several papers by the author [``On a 5-dimensional nonsingular rational subvariety of the Grassmann variety Gr(5,1). I, II, III,'' Bull. Kyoto Univ. Educ., Ser. B 71/72, 1-15 (1988; Zbl 0654.14024), ibid. 73, 1-16 (1988; Zbl 0686.14048, ibid. 75, 1-21 (1989; Zbl 0716.14026)]a 5- dimensional subvariety \(M\) of the Grassmann variety representating the lines of a 5-dimensional projective space over an algebraically closed field has been studied. The aim of the present paper is to discuss some vector bundles over \(M\) having \(\sigma\)-transition matrices. 5-dimensional subvariety of the Grassmann variety; vector bundles Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On \(\sigma\)-transition matrices over a 5-dimensional non-singular rational subvariety of Grassmann variety \(GR(5,1)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 define the notion of a separable element in a finite Weyl group, generalizing the well-studied class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat order generated by a separable element are rank-symmetric and rank-unimodal, and that the product of their rank generating functions gives that of the whole group, answering an open problem of \textit{F. Wei} [Eur. J. Comb. 33, No. 4, 572--582 (2012; Zbl 1236.05008)]. We also prove that separable elements are characterized by pattern avoidance in the sense of \textit{S. Billey} and \textit{A. Postnikov} [Adv. Appl. Math. 34, No. 3, 447--466 (2005; Zbl 1072.14065)]. generating function; product decompositions; symmetric group; separable permutations; weak Bruhat order Permutations, words, matrices, Exact enumeration problems, generating functions, Combinatorial aspects of groups and algebras, Algebraic aspects of posets, Root systems, Simple, semisimple, reductive (super)algebras, Grassmannians, Schubert varieties, flag manifolds Separable elements in Weyl groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The classical dilogarithm \(L_ 2\) was shown to be an important tool in many problems of arithmetics, \(K\)-theory, group cohomology and so on. It is the second term of a row \(L_ 1\) (=usual logarithm), \(L_ 2,\ldots\) --- The authors point out that they ``propose a new approach to constructing higher logarithms \(L_ p\), \(p=0,1,2,\ldots\)'' which will produce what they believe ``should be the true generalizations of the logarithm and the dilogarithm.'' Their construction of \(L_ p\) ``appears as a component of an interesting cocycle whose class in the Deligne- Beilinson cohomology \(H_ D^{2p}(G^ p_ \bullet,\mathbb{Q}(p))\) of a certain simplicial space \(G^ p_ \bullet\) (of Zariski open subsets of various Grassmannian manifolds) is a kind of universal \(p\)-th Chern class.'' The content of the paper includes the following sections: multivalued differential forms, the Grassmannian complex, higher Albanese manifolds, rational \(K(\pi,1)\)-spaces and the existence of the 3- logarithm, symmetry of the Grassmannian, non-triviality and indecomposability of the 3-logarithm, real Albanese manifolds, generalized Bloch-Wigner functions. The main difficulty lies in an existence theorem for the functions \(L_ p\). existence of higher logarithms; polylogarithm; dilogarithm; \(K\)-theory; Deligne-Beilinson cohomology; multivalued differential forms; Grassmann complex; higher Albanese manifolds; \(K(\pi,1)\)-spaces; 3-logarithm Hain (R.), MacPherson (R.).â Higher Logarithms, Ill. J. of Math,, vol. 34, N2, p. 392-475 (1990). Picard schemes, higher Jacobians, Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry Higher logarithms | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 discusses some examples of ``twisted'' autoequivalences on certain Calabi-Yau spaces constructed from Grassmannians. These twisted equivalences are endofunctors that can be thought of as mirror to symplectic monodromies.
Consider \(G=\mathrm{Gr}(r,V)\) to be the Grassmannian of \(r\)-dimensional subspaces of a vector space \(V\) and let \(S\) be the tautological vector bundle on \(G\). The total space \(X:=\text{Tot}(\text{Hom}(V,S))\) is a Calabi-Yau variety and is stratified by the rank of the tautological map \(f:V\rightarrow p^*S\), where \(p\) is the projection \(p:X\rightarrow G\). The main result of this paper is, for \(r=2\), a description of an autoequivalence of the derived category \(D^b(X)\) as a twist around the big stratum \(B\), where \(B\) is the locus where the rank of \(f\) is not full.
This construction naturally extends the work of Seidel-Thomas which gives a similar description for the case \(r=1\). In this paper, in order to obtain the main result, the author constructs a desingularization of the big stratum \(B\), which unlike the case \(r=1\) is singular, however the desingularization has a nice geometric description. This allows one then to define the twisted functor that defines the autoequivalence of \(D^b(X)\). Grassmannian; twists; spherical functors; autoequivalences W. Donovan, Grassmannian twists on the derived category via spherical functors, preprint (2011), . Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Grassmannians, Schubert varieties, flag manifolds Grassmannian twists on the derived category via spherical functors | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 conjecture of \textit{M. Finkelberg} and \textit{A. Ionov} [Bull. Inst. Math., Acad. Sin. (N.S.) 13, No. 1, 31--42 (2018; Zbl 1397.05203)] is proved on the basis of a generalization of the Springer resolution and the Grauert-Riemenschneider vanishing theorem. As a corollary, it is proved that the coefficients of the multivariable version of Kostka functions introduced by Finkelberg and Ionov are nonnegative. Kostka-Shoji polynomials; cohomology vanishing; quivers; Lusztig convolution diagram Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Symmetric functions and generalizations, Representations of quivers and partially ordered sets, Vanishing theorems in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Higher cohomology vanishing of line bundles on generalized Springer 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 We consider the variety of \(X_d\) of punctual flags of length \(d\) in dimension 2, defined as the closure of the variety of complete curvilinear zero-dimensional subschemes of length \(\leq d\) with support at the fixed point on a non-singular algebraic surface; this closure is taken in the direct product of punctual Hilbert schemes.
It is known that for \(2\leq d\leq4\) the variety \(X_d\) is smooth and coincides with the projectivization of the rank 2 vector bundle over \(X_{d-1}\), described as the corresponding \(\mathcal{E}xt\)-sheaf. A similar bundle \(\mathcal E\) is also defined over \(X_4\). However, its projectivization \(\mathbb{P}(\mathcal E)\) is only birational isomorphic, but not isomorphic to \(X_5\). M. Gulbrandsen showed that \(X_5\) has an entire curve of singularities. In the present article, we give a precise description of a minimal birational transformation of \(X_5\) into \(\mathbb{P}(\mathcal E)\) and interpret this transformation and the singularities of \(X_5\) in terms of \(\mathcal{E}xt\)-sheaves. Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds, (Co)homology theory in algebraic geometry On the variety of complete punctual flags of length 5 in dimensions 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 This paper is concerned with an explicit description of the basic locus in the reduction modulo \(p\) of Shimura varieties. Recent important works [\textit{I. Vollaard} and \textit{T. Wedhorn}, Invent. Math. 184, No. 3, 591--627 (2011; Zbl 1227.14027); \textit{M. Rapoport} et al., Math. Z. 276, No. 3--4, 1165--1188 (2014; Zbl 1312.14072); \textit{B. Howard} and \textit{G. Pappas}, Algebra Number Theory 8, No. 7, 1659--1699 (2014; Zbl 1315.11049)] show that for some unitary Shimura varieties, the basic locus can be described in terms of the Bruhat-Tits building, and there is a stratification where the strata are isomorphic to a union of classical Deligne-Lusztig varieties. Moreover the closure relation can be also described in a simple manner by group theory. This paper under review gives a rather complete list of group-theoretic data for which such an explicit description can be obtained. Thus, it gives a great contribution to explicit descriptions of basic loci of Shimura varieties.
Let \(F\) be a non-Archimedean local field. Let \(L\) be the completion of the maximal unramified extension of \(F\), and let \(\sigma\) denote the Frobenius of the extension \(L/F\). Fix a datum \((G,\mu)\), where \(G\) is a connected quasi-simple semi-simple algebraic group over \(F\) which splits over a finite tamely ramified extension, and \(\mu\) is a minuscule cocharacter. For each standard rational maximal parahoric subgroup \(P\) of \(G(L)\) and \(b\in G(L)\), define the affine Deligne-Lusztig variety (ADLV)
\[
X(\mu,b)_P:=\{g\in G(L)/P; g^{-1} b \sigma(b)\in \bigcup_{w\in \text{Adm}(\mu)} P w P \}.
\]
Let \(B(G,\mu)\subset G(L)\) be the union of \(\sigma\)-conjugacy classes of elements \(b\) for which \(X(\mu,b)_P\) is non-empty. One has a fiber space \(Z_J\) over \(B(G,\mu)\) with fibers \(X(\mu,b)_P\), where \(J\) is the type of \(P\). Inside \(Z_J\) there is the basic locus (the preimage over the basic \(\sigma\)-conjugacy class), and EO strata (subsets of form \(Z_{J,w}\) for \(w\in EO(G,\mu)^J:=\text{Adm}^J (\mu)\cap {}^J \tilde W \subset \tilde W)\). The authors consider the triple \((G,\mu,J)\) for which the basic locus is a union of EO strata and each EO stratum is a union of classical Deligne-Lusztig varieties. Under a mild condition, they give a complete classification of such triples up to isomorphism (Theorem 5.1.2). They also obtain a explicit description for the basic locus as work of Vollaard-Wedhorn. Moreover, non-basic Newton strata are shown to be zero-dimensional. In the function field case, they prove the closure relation among the Bruhat-Tits strata. The paper is well written and the reader can use the authors' previous papers for necessary background. Shimura varieties; basic loci; Deligne-Lusztig varieties Görtz, U; He, X, Basic loci in Shimura varieties of Coxeter type, Camb. J. Math., 3, 323-353, (2015) Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over local fields and their integers Basic loci of Coxeter type in Shimura 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 consider a uniform \(r\)-bundle \(E\) on a complex rational homogeneous space \(X\) and show that if \(E\) is poly-uniform with respect to all the special families of lines and the rank \(r\) is less than or equal to some number that depends only on \(X\), then \(E\) is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by
\textit{R. Muñoz} et al. [Eur. J. Math. 6, No. 2, 430--452 (2020; Zbl 1442.14131)].
In particular, if \(X\) is a generalized Grassmannian \({\mathcal{G}}\) and the rank \(r\) is less than or equal to some number that depends only on \(X\), then \(E\) splits as a direct sum of line bundles. So we improve the main theorem of
\textit{R. Muñoz} et al. [J. Reine Angew. Math. 664, 141--162 (2012; Zbl 1271.14058), Theorem 3.1]
when \(X\) is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert-Mülich-Barth theorem on rational homogeneous spaces. vector bundle; generalized Grassmannian; rational homogeneous space Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Vector bundles on rational homogeneous spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In 1997, Jean-Claude Hausmann and Allen Knutson introduced a natural and beautiful correspondence between planar \(n\)-gons and the Grassmann manifold of 2-planes in real \(n\)-space. This construction leads to a natural probability distribution and a natural metric on polygons which has been used in shape classification and computer vision. In this paper, we provide an accessible introduction to this circle of ideas by explaining the Grassmannian geometry of triangles. We use this to find the probability that a random triangle is obtuse, which was a question raised by Lewis Carroll. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space of unordered quadrilaterals. planar \(n\)-gons; Grassmann manifold of 2-planes; Grassmannian geometry of triangles; random triangle; Grassmannian geometry of planar quadrilaterals; moduli space of unordered quadrilaterals Random convex sets and integral geometry (aspects of convex geometry), Curves in Euclidean and related spaces, Geometric probability and stochastic geometry, Grassmannians, Schubert varieties, flag manifolds Random triangles and polygons in the plane | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by \textit{A. Postnikov} [``Total positivity, Grassmannians, and networks'', Preprint, \url{arXiv:math/0609764}]. In this paper, we study the facets of its matroid polytope and the independent set polytope. This allows one to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace. We also describe a criterion for determining whether a given cyclic interval is a flat or not using the decorated permutation, then show how it applies to checking the concordancy of positroids. positroid; matroid polytope; flacet; non-crossing partition; decorated permutation Combinatorial aspects of matroids and geometric lattices, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Grassmannians, Schubert varieties, flag manifolds The facets of the matroid polytope and the independent set polytope of a positroid | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Gauss map of an \(r\)-dimensional variety \(V\subseteq \mathbb{P}^n\) is the map from smooth points of \(V\) to the Grassmannian of \(r\)-planes in \(\mathbb{P}^n\) which sends a smooth point to the tangent space at the point. The Gauss map (when the ground field is algebraically closed of characteristic 0) has the property that if it is not birational onto its image then it is not even generically finite. Analogues of the Gauss map can be defined using the higher osculating spaces to \(V\), and one can ask if they have the analogous property. Concretely, the question is:
If the generic \(i\)-th osculating space to \(V\) is \(i\)-th osculating at more than one point, is it necessarily \(i\)-th osculating at infinitely many points?
The principal result of this paper is that the answer is negative for \(i \geq 2\). In addition, a complete classification is given of surfaces in \(\mathbb{P}^n, n>5\), whose second Gauss map is not birational. These results generalize and make rigorous results of an earlier paper [\textit{M. Castellani}, Rend. Lincei (5) 31, No. 1, 347-350 (1922; JFM 48.0854.07)]. Gauss map; osculating space; Grassmannian; JFM 48.0854.07 DOI: 10.1081/AGB-100104999 Grassmannians, Schubert varieties, flag manifolds, Rational and birational maps, Special surfaces On multiosculating 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 proves some combinatorial identities on root systems. Apart from being of interest by themselves, these identities lead to the computation of inverse Euclidean transform of the Plancherel measure for symmetric spaces of the non-compact type. combinatorial identities; root systems; inverse Euclidean transform; Plancherel measure; symmetric spaces of the non-compact type Harmonic analysis on homogeneous spaces, Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc., Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds Combinatorial identities on root systems and the Fourier transform of Harish-Chandra's c-function | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the category of perfect DG modules over the extension algebra of the direct sum of the simple perverse equivariant sheaves. This proves a conjecture of Soergel and Lunts in the case of flag varieties. Schnürer, O.: Equivariant sheaves on flag varieties, DG modules and formality. PhD thesis, Fakultät für Mathematik und Physik Universität Freiburg (2007). http://www.freidok.uni-freiburg.de/volltexte/4662/ Research exposition (monographs, survey articles) pertaining to algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Equivariant sheaves on flag varieties, DG modules and formality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we focus on subspace learning problems on the Grassmann manifold. Interesting applications in this setting include low-rank matrix completion and low-dimensional multivariate regression, among others. Motivated by privacy concerns, we aim to solve such problems in a decentralized setting where multiple agents have access to (and solve) only a part of the whole optimization problem. The agents communicate with each other to arrive at a consensus, i.e., agree on a common quantity, via the gossip protocol. We propose a novel cost function for subspace learning on the Grassmann manifold, which is a weighted sum of several sub-problems (each solved by an agent) and the communication cost among the agents. The cost function has a finite-sum structure. In the proposed modeling approach, different agents learn individual local subspaces but they achieve asymptotic consensus on the global learned subspace. The approach is scalable and parallelizable. Numerical experiments show the efficacy of the proposed decentralized algorithms on various matrix completion and multivariate regression benchmarks. non-linear gossip; stochastic gradients; manifold optimization; matrix completion; multivariate regression Learning and adaptive systems in artificial intelligence, Grassmannians, Schubert varieties, flag manifolds, Estimation in multivariate analysis, Statistics on manifolds, Stochastic programming A Riemannian gossip approach to subspace learning on Grassmann manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 torus equivariant \(K\)-theory ring of a Grassmannian the classes of the structure sheaves of Schubert varieties form a natural, geometric basis. Understanding the structure constants with respect to this basis is equivalent to describing the ring structure. In an earlier work the authors presented the structure constants as the cardinalities of certain combinatorially defined tableaux. In the paper under review the authors show a bijection from those tableaux to certain puzzles (fillings of a triangle with certain permitted puzzle pieces). As a result they obtain a (mild modification of the) puzzle rule for the structure constants as the number of puzzles, originally conjectured by Knutson and Vakil.
For part I, see [\textit{O. Pechenik} and \textit{A. Yong}, Forum Math. Pi 5, Article ID e3, 128 p. (2017; Zbl 1369.14060)]. Grassmannians; equivariant \(K\)-theory; puzzle; Schubert calculus; Littlewood-Richardson rule O. Pechenik and A. Yong. ''Equivariant K-theory of Grassmannians II: The Knutson-Vakil conjecture''. 2015. arXiv:1508.00446. Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Equivariant \(K\)-theory of Grassmannians. II: The Knutson-Vakil conjecture. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study certain bijection between plane partitions and \(\mathbb{N}\)-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating functions are similar to classical MacMahon's formulas; one of these statistics is equidistributed with the usual volume. We also show natural connections with the longest increasing subsequences of words. plane partitions; MacMahon's formulas; dual Grothendieck polynomials; volume generating functions Combinatorial aspects of representation theory, Exact enumeration problems, generating functions, Combinatorial aspects of partitions of integers, Partitions of sets, Grassmannians, Schubert varieties, flag manifolds Enumeration of plane partitions by descents | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One of the best known results of extremal combinatorics is Sperner's theorem, which asserts that the maximum size of an antichain of subsets of an \(n\)-element set equals the binomial coefficient \({n\choose {\langle n/2 \rangle}}\); that is, the maximum of the binomial coefficients. In the last twenty years, Sperner's theorem has been generalized to wide classes of partially ordered sets. It is the purpose of the present note to propose yet another generalization, which strikes in a different direction.
We consider the lattice \(\text{Mod} (n)\) of linear subspaces (through the origin) of the vector space \(\mathbb{R}^n\). This lattice is infinite, so that the usual methods of extremal set theory do not apply to it. It turns out, however, that the set of elements of rank \(k\) of the lattice \(\text{Mod} (n)\), that is, the set of all subspaces of dimension \(k\) of \(\mathbb{R}^n\), or Grassmannian, possesses an invariant measure, which is unique up to a multiplicative constant. Can this multiplicative constant be chosen in such a way that an analogue of Sperner's theorem holds for \(\text{Mod} (n)\), with measures on Grassmannians replacing binomial coefficients? We show that there is a way of choosing such constants for each level of the lattice \(\text{Mod} (n)\), which is natural and unique in the sense defined in the paper, and for which an analogue of Sperner's theorem can be proven.
The methods of the present note indicate that other results of extremal set theory may be generalized to the lattice \(\text{Mod} (n)\) by similar reasoning. lattice of linear subspaces; intrinsic volume; extremal combinatorics; Grassmannian; invariant measure; analogue of Sperner's theorem [4] Klain D. A., Rota G.-C., ''A continuous analogue of Sperner's theorem'', Comm. Pure Appl. Math., 50 (1997), 205--223 Extremal set theory, Factorials, binomial coefficients, combinatorial functions, Basic linear algebra, Grassmannians, Schubert varieties, flag manifolds A continuous analogue of Sperner's theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct for a given arbitrary skew diagram \({\mathcal{A}}\) all partitions \(v\) with maximal principal hook lengths among all partitions with \([v]\) appearing in \([{\mathcal{A}}]\). Furthermore, we show that these are also partitions with minimal Durfee size. We use this to give the maximal Durfee size for \([v]\) appearing in \([{\mathcal{A}}]\) for the cases when \({\mathcal{A}}\) decays into two partitions and for some special cases of \({\mathcal{A}}\). We also deduce necessary conditions for two skew diagrams to represent the same skew character. principal hook lengths; Durfee size; skew characters; symmetric group; skew Schur functions; Schubert Calculus Gutschwager, C.: On principal hook length partitions and durfee sizes in skew characters. Annals Comb. (to appear). arXiv:0802.0417v2 Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups On principal hook length partitions and Durfee sizes in skew characters | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The action of a connected reductive algebraic group \(G\) on \(G/P_-\), where \(P_-\) is a parabolic subgroup, differentiates to a representation of the Lie algebra \(\mathfrak g\) of \(G\) by vector fields on \(U_+\), the unipotent radical of a parabolic opposite to \(P_-\). The classical instances of this setting that we study in detail are the actions of \(\text{GL}_n\) on the Grassmannian of \(k\)-planes (\(1\leq k\leq n\)), of \(\text{SO}_n\) on the quadric of isotropic lines, and of \(\text{SO}_{2n}\) or \(\text{SP}_{2n}\) on their respective Grassmannians of maximal isotropic spaces; in each instance, \(U_+\) is one of the usual affine charts.
We show that both the polynomials on \(U_+\) and the polynomial vector fields on \(U_+\) form \(\mathfrak g\)-modules dual to parabolically induced modules, construct an explicit composition chain of the former module in the case where \(G\) is classical simple and \(U_+\) is Abelian -- these are exactly the cases above -- and indicate how this chain can be used to analyse the module of vector fields, as well.
We present two proofs of our main theorems: one uses the results of Enright and Shelton on classical Hermitian pairs, and the other is independent of their work. The latter proof mixes classical (and briefly reviewed) facts of representation theory with combinatorial and computational arguments, and is accessible to readers unfamiliar with the vast modern literature on category \(\mathcal O\). representation theory; Lie algebras; vector fields; category \(\mathcal O\); connected reductive algebraic groups; Grassmannians; parabolically induced modules; Hermitian pairs Draisma, J., Representation theory on the open Bruhat cell, J. Symb. Comput., 39, 279-303, (2005) Representation theory for linear algebraic groups, Lie algebras of linear algebraic groups, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Lie algebras of vector fields and related (super) algebras, Combinatorial aspects of representation theory Representation theory on the open Bruhat cell. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of \([ n ] \times [ n ]\) to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over descent classes of the symmetric groups. permutation; generating function; descent class; diagram; odd length; Schubert variety Permutations, words, matrices, Exact enumeration problems, generating functions, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Symmetric groups Odd length: odd diagrams and descent 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 Let \(G\) be a simple algebraic group over the field of complex numbers. Fix a maximal torus \(T\) and a Borel subgroup \(B\) of \(G\) containing \(T\). Let \(w\) be an element of the Weyl group \(W\) of \(G\), and let \(Z(\widetilde{w})\) be the Bott-Samelson-Demazure-Hansen (BSDH) variety corresponding to a reduced expression \(\widetilde{w}\) of \(w\) with respect to the data \((G, B, T)\).
In this article we give complete characterization of the expressions \(\widetilde{w}\) such that the corresponding BSDH variety \(Z(\widetilde{w})\) is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results. Grassmannians, Schubert varieties, flag manifolds, Vanishing theorems in algebraic geometry, Fano varieties On Fano and weak Fano Bott-Samelson-Demazure-Hansen 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 paper is arranged in the following way. In \S1 we introduce notations for Schubert cells and we will show what cells and their neighbourhoods give rise to a uniquely constructed Schubert stratification.
In \S2 we show how a flattening corresponds to a Schubert cell.
In \S3 we introduce the concept of stable equivalence of flattenings, which allows us to compare cascades that generally consist of a different number of curves lying in spaces of different dimensions.
The relation of equivalence of flattenings is constructed in \S4.
In \S5 we give a classification of the flattenings occurring in generic three-parameter families of cascades; relative to this equivalence, we study the singularities of their bifurcation diagrams, and give results of V. I. Arnol'd and O. P. Shcherbak on the connection of these singularities with the geometry of the swallowtail, tangential singularities, and the singularities of projections.
In \S6 we give lists of the simple flattenings of curves, and also of cascades, corresponding to complete flags. The methods used in the proof of the classification theorems of \S\S5 and 6 are validated in \S\S7-9. In \S7 we prove a generalization of the Frobenius theorem on integrable distributions to the case of distributions with singularities. In \S8 this result is carried over to the case of the infinite-dimensional space of germs of cascades. Using these results, we prove the finite determinacy and versality theorems in \S9.
Section 10 and 11 are devoted to applications of the theory of flattenings to the study of oscillatory properties of linear differential equations and to the decomposition of Weierstrass points of algebraic curves, respectively. Grassmannians; flag manifolds; Schubert cells; Schubert stratification; flattening; equivalence; singularities; bifurcation; Weierstrass points; algebraic curves DOI: 10.1070/RM1991v046n05ABEH002844 Deformations of complex singularities; vanishing cycles, Formal methods and deformations in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Differentiable maps on manifolds, Theory of singularities and catastrophe theory Flattenings of projective curves, singularities of Schubert stratifications of Grassmannians and flag varieties, and bifurcations of Weierstrass points of algebraic curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the case of generalized flag manifolds which are homogeneous spaces of the form \(G/C(T)\), where \(G\) is a compact Lie group and \(T\) a maximal torus in \(G\), using results of Morse theory about height functions on an adjoint orbit, a Lie algebraic expression for the Duistermaat-Heckman formula, suitable for calculations, is given. generalized flag manifold; compact Lie group; maximal torus; Duistermaat-Heckman formula Applications of global differential geometry to the sciences, Grassmannians, Schubert varieties, flag manifolds, Differential geometry of homogeneous manifolds, Critical points and critical submanifolds in differential topology The Duistermaat-Heckman integration formula on generalized flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main result of this paper, theorem 4.1, states that any Fano threefold of the first kind, which has degree 14, can be embedded in the Grassmannian \(G=G(2,6)\subset\mathbb{P}^{14}\) as an intersection of \(G\) with a subspace \(\mathbb{P}^ 9\subset\mathbb{P}^{14}\) of codimension 5. Let \(V=V_{14}\) be a Fano 3-fold of degree 14. In \S2 the author proves the existence of a nonsingular elliptic quintic \(Z\subset V\subset\mathbb{P}^ 9\). Any elliptic quintic \(Z\) on \(V\) gives rise to a regular bundle \(M\) on \(V\), such that \(M\) is spanned on its global sections (see \S4). The bundle \(M\) defines a morphism \(\varphi:V=V_{14}\to G(2,6)\), which morphism has the properties: \(\text{deg} \varphi=1\), \(\dim\text{Span}(\varphi(V))=9\) (see 4.1.3-4.1.7). This implies theorem 4.1. embedding into the Grassmannian; Fano threefold of the 1-st kind ---, Fano \(3\) -folds of genus \(8,\) Algebra i Analiz 4 (1992), 120-134. Fano varieties, \(3\)-folds, Grassmannians, Schubert varieties, flag manifolds On Fano 3-folds of genus 8 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 are interested in torus \(T\) orbit closures in a flag variety \(G/B\) in connection with retractions of a finite Coxeter group \(W\) onto Coxeter matroids, i.e. subgroups of \(W\) satisfying the maximality property. In particular, a very important case is when \(W\) is the Weyl group of \(G\). They investigate three types of retractions of \(W\):
\begin{itemize}
\item the \emph{matroid retraction} \(\mathcal{R}^m_{\mathcal{M}}\) onto a matroid \(\mathcal{M} \subset W\), which is related to the Bruhat order on \(W\),
\item the \emph{geometric retraction} \(\mathcal{R}^g_Y\) onto the matroid \(Y^T\) of the fixed point set of the closure \(Y\) of a \(T\)-orbit, which is defined via limits of one-parameter torus subgroups,
\item the \emph{algebraic retraction} \(\mathcal{R}^a_{\mathcal{M}}\), which is defined if \(W\) can be decomposed into groups of classical Lie types.
\end{itemize}
The main results are two identities: \(\mathcal{R}^g_Y = \mathcal{R}^m_{Y^T}\) for any \(T\)-orbit closure \(Y\) in \(G/B\) and \(\mathcal{R}^a_{\mathcal{M}} = \mathcal{R}^m_{\mathcal{M}}\) for any Coxeter matroid \(\mathcal{M}\) of \(W\). In particular, for \(Y^T\) all three retractions are the same. flag varieties; toric varieties; Coxeter matroids; retractions Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Toric topology, Reflection and Coxeter groups (group-theoretic aspects), Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) Torus orbit closures in flag varieties and retractions on Weyl groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The themes of the workshop are the Weak Lefschetz Property -- WLP -- and the Strong Lefschetz Property -- SLP. The name of these properties, referring to Artinian algebras, is motivated by the Lefschetz theory for projective manifolds, initiated by S. Lefschetz, and well established by the late 1950's. In fact, Lefschetz properties of Artinian algebras are algebraic generalizations of the Hard Lefschetz property of the cohomology ring of a smooth projective complex variety. The investigation of the Lefschetz properties of Artinian algebras was started in the mid 1980's and nowadays is a very active area of research.
Although there were limited developments on this topic in the 20th century, in the last years this topic has attracted increasing attention from mathematicians of different areas, such as commutative algebra, algebraic geometry, combinatorics, algebraic topology and representation theory. One of the main features of the WLP and the SLP is their ubiquity and the quite surprising and still not completely understood relations with other themes, including linear configurations, interpolation problems, vector bundle theory, plane partitions, splines, \(d\)-webs, differential geometry, coding theory, digital image processing, physics and the theory of statistical designs, etc. among others. Collections of abstracts of lectures, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to commutative algebra, Proceedings, conferences, collections, etc. pertaining to combinatorics, Syzygies, resolutions, complexes and commutative rings, Commutative Artinian rings and modules, finite-dimensional algebras, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Combinatorial aspects of commutative algebra, Combinatorial aspects of simplicial complexes, Combinatorics of partially ordered sets, Algebraic aspects of posets, Graded rings, Actions of groups on commutative rings; invariant theory, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Group actions on varieties or schemes (quotients), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Complete intersections, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Lefschetz properties in algebra, geometry and combinatorics. Abstracts from the workshop held September 27 -- October 3, 2020 (hybrid meeting) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For each infinite series of the classical Lie groups of type \(B\), \(C\) or \(D\), we construct a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's \(Q\)- or \(P\)-functions defined earlier by Ivanov. double Schubert polynomials; equivariant cohomology Ikeda, T.; Mihalcea, L.; Naruse, H., \textit{double Schubert polynomials for the classical groups}, Adv. Math., 226, 840-886, (2011) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Double Schubert polynomials 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 address the question: How should \(N\) \(n\)-dimensional subspaces of \(m\)-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of \(N\), \(n\), \(m\) are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe \(n\)-dimensional subspaces of \(m\)-space as points on a sphere in dimension \({1\over 2}(m-1)(m+2)\), which provides a (usually) lower-dimensional representation than the Plücker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multi-dimensional data via Asimov's grand tour method. packings; Grassmannian space J. H. Conway, R. H. Hardin, and N. J. A. Sloane, \textit{Packing lines, planes, etc.: Packings in Grassmannian spaces}, Experiment. Math., 5 (1996), pp. 139--159. Polyhedra and polytopes; regular figures, division of spaces, Grassmannians, Schubert varieties, flag manifolds, Packing and covering in \(n\) dimensions (aspects of discrete geometry) Packing lines, planes, etc.: Packings in Grassmannian 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 paper gives explicit descriptions of two maximal cones in the Gröbner fan of the Plücker ideal. These two cones correspond to the monomial ideals given by semistandard and PBW-semistandard Young tableaux, respectively.
For the first one, as an intermediate result the author obtains a description of a maximal cone in the Gröbner fan of any Hibi ideal.
For the second one, the author generalizes the notion of a Hibi ideal by associating an ideal with every interpolating polytope. A family of polytopes arises, which generalizes the order and chain polytopes of a poset.
The author proceeds with a description of a maximal cone in the Gröbner fan of each of these ideals. Finally, they prove that PBW-semistandardness provides a new Hodge algebra structure on the Plücker algebra. flag varieties; Gröbner fans; distributive lattices; poset polytopes; Young tableaux; Hodge algebras; tropical geometry Grassmannians, Schubert varieties, flag manifolds, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Gröbner fans of Hibi ideals, generalized Hibi ideals and 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 Given an affine surface \(X\) with rational singularities and minimal resolution \(X^{\prime}\), the covering of the Artin component of the deformation space of \(X\) where simultaneous resolutions are achieved is Galois and the Galois group is the Weyl group \(W\) associated with the configuration of \((-2)\)-curves on \(X^{\prime}\). This gives the existence of actions of \(W\) on polynomial rings over \(\mathbb{Z}\) where the ring of invariants is also polynomial. In turn, this leads to a description of the integral cohomology rings of flag varieties of type \(\mathit{ADE}\) that extends the known description of the rational cohomology rings as rings of coinvariants for actions of \(W\). Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory, Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds, Homology and cohomology of homogeneous spaces of Lie groups Weyl group covers for Brieskorn's resolutions in all characteristics and the integral cohomology of \(G/P\) | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.