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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a vector space of dimension \(n\) on an algebraically closed field \(K\). Let \(G = Gr(\ell,V) = Gr(\ell, n)\) be the Grassmann variety of subspaces of \(V\) of dimension \(\ell\). If \(V'\) is a linear subspace of \(V\), we denote, as usual, by \(\sigma (V') = \{\Lambda \in G | \Lambda \cap V' \neq 0\}\) the special Schubert variety associated to \(V'\). We prove the following:
Theorem. The intersection \(\bigcap \sigma (V_ j)\) of the special Schubert varieties associated to linear subspaces \(V_ j\), \(j = 1, \dots, m\), of dimension \(n - \ell - a_ j + 1\) such that \(\ell (n - \ell) - \sum^{j=m}_{j=1} a_ j > 0\), is connected. Moreover, the intersection is irreducible of dimension \(\ell (n - \ell) - \sum a_ j\) for a general choice of the subspaces \(V_ j\).
The second part of the theorem follows from its first part and the local irreducibility. \textit{J. Hansen} [Am. J. Math. 105, 633-639 (1983; Zbl 0544.14034)] proved a result about the connectedness of intersections of subvarieties of Grassmannians and flag varieties, but under the stronger hypothesis that the expected codimension of the intersection is smaller than \(n - 1\). As initial motivation for this work, let us remark that a particular case \((\dim V_ j = 2\) for all \(j)\) of this statement appears, without a proof, in a book by \textit{F. Enriques} and \textit{O. Chisini} [``Lezioni sulla teoria geometrica delle equazioni: e delle funzioni algebriche'' Vol. III (1924; reprint 1985; Zbl 0571.51001), page 524]. We conjecture that the irreducibility holds for intersections of Schubert varieties (not necessarily special), when they are in general position with nonempty intersection. (We have checked the conjecture for the case of two Schubert varieties.)
As an example of application we provide a new proof (over the field of complex numbers) of the following theorem. Recall that a curve of genus \(g\) is said \((d,r)\)-Brill-Noether if the variety of linear series of degree \(d\) and dimension \(r\) on \(C\) has dimension equal to the Brill- Noether number \(((r + 1)d - rg - r(r + 1) + r)\) and is empty if the number is negative. The general curve of a given genus is \((d,r)\)-Brill- Noether for all \((d,r)\).
Theorem. If the Brill-Noether number is strictly positive, then the varieties of special divisors of degree \(d\) and dimension \(r\) on a smooth, connected, \((d,r)\)-Brill-Noether curve \(C\) of genus \(g\) are connected. connectedness of intersections of special Schubert varieties; \((d,r)\)- Brill-Noether curve Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Topological properties in algebraic geometry Connectedness of intersections of special Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Standard monomial theory allows to construct bases of the space of global sections on a flag variety \(G/P\), which have nice geometric properties. This work deals with the generalization of standard monomial theory to wonderful compactifications of symmetric spaces \(G/H\), using the standard monomials on flag varieties. For the special case of the wonderful compactification of a group, nice results are obtained. K. Appel, Standardmonome für wundervolle Kompaktifizierungen von Gruppen, PhD thesis, Wuppertal, 2006 Research exposition (monographs, survey articles) pertaining to algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) Standard monomials for wonderful compactifications of 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 book aims to describe the beautiful connection between Schubert varieties and their Standard Monomial Theory (SMT) on the one hand and Classical Invariant Theory (CIT) on the other. The roots of SMT are to be found in the work of Hodge, who described nice bases for the homogeneous coordinate ring of Schubert varieties of the Grassmannian in the Plücker embedding (over a field of characteristic zero). Grassmannians being precisely the homogeneous spaces that arise as quotients of special linear groups by maximal parabolic subgroups, it is natural to try to generalize Hodge's approach to projective embeddings of other spaces \(G/Q\), where \(G\) is a semisimple algebraic group and \(Q\) a parabolic subgroup. In the early '70s Seshadri initiated this generalization and called it SMT.
CIT concerns diagonal actions of classical groups on direct sums of the tautological representations and their duals. A description of the algebra of invariants for these actions comprises of two theorems. The First Fundamental Theorem specifies a finite set of generators for the algebra of invariants, and the Second Fundamental Theorem provides a finite set of generators of the ideal of relations among the algebra generators. A central role here plays the article [\textit{C.~De Concini} and \textit{C.~Procesi}, ``A characteristic free approach to Invariant Theory'', Adv. Math. 21, 330--354 (1976; Zbl 0347.20025)], where the First Fundamental Theorem for all classical groups and the Second Fundamental Theorem for general, orthogonal and symplectic linear groups were obtained (the case when the characteristic of the ground field is zero goes back to H.~Weyl). The idea to use SMT in the proof of the First and the Second Fundamental Theorems appeared in [\textit{V.~Lakshmibai} and \textit{C. S.~Seshadri}, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1--54 (1978; Zbl 0447.14011)] and turned out to be very fruitful. More precisely, one should realize the subalgebra of the algebra of invariants generated by ``basic'' invariants (which will in fact coincide with the algebra of invariants) as the algebra of regular functions on an affine variety related to a Schubert variety. Then there is a morphism from the spectrum of the algebra of invariants to this affine variety. Using Zariski's Main Theorem, one shows that this is an isomorphism. A difficult part of this program is to prove that our affine variety is normal. Normality follows from normality of Schubert varieties, and that is a consequence of SMT. Nowadays this approach is realized in complete generality, and the book under review provides an excellent account of there results.
The authors tried to make the presentation self-contained keeping in mind the needs of prospective graduate students and young researchers. After a detailed introduction, generalities on algebraic varieties and algebraic groups are given. Next chapters are devoted to classical, symplectic and orthogonal Grassmannians, determinantal varieties, Geometric Invariant Theory (GIT), basic results of SMT and their interrelations with CIT. The proof of the main theorem of SMT is given in an appendix. The authors also included some important applications of SMT: to the determination of singular loci of Schubert varieties, to the study of some affine varieties related to Schubert varieties --- ladder determinantal varieties, quiver varieties, varieties of complexes, etc. --- and to toric degenerations of Schubert varieties.
The book may be recommended as a nice introduction to SMT and related active research areas. It may be used for a year long course on Invariant Theory and Schubert varieties. Classical Invariant Theory; Grassmannians; Schubert varieties; homogeneous coordinates Lakshmibai, V.\!; Raghavan, K.\,N.\!, Standard monomial theory, Encyclopaedia of Mathematical Sciences (Invariant Theory and Alg. Transform. Groups VIII) 137, (2008), Springer-Verlag, Berlin Grassmannians, Schubert varieties, flag manifolds, Actions of groups on commutative rings; invariant theory, Rings with straightening laws, Hodge algebras, Determinantal varieties, Classical groups (algebro-geometric aspects) Standard monomial theory. Invariant theoretic approach | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The subordination relation between hypergeometric functions defined on Grassmannians or their strata is introduced and investigated. In the theory of classical hypergeometric functions, relations of this sort are called reduction formulas. The classification of strata of small dimension in Grassmannians is given and spaces of hypergeometric functions on these strata are investigated. 9. I. M. Gel'fand, M. I. Graev and V. S. Retakh, Reduction formulas for hypergeometric functions associated with the Grassmannian Gk,n and description of these functions on strata of small codimension in Gk,n,Russian J. Math. Phys.1 (1993) 19-56. Other hypergeometric functions and integrals in several variables, Appell, Horn and Lauricella functions, Grassmannians, Schubert varieties, flag manifolds, Other functions coming from differential, difference and integral equations Reduction formulas for hypergeometric functions associated with the Grassmannian \(G_{k,n}\) and description of these functions on strata of small codimension in \(G_{k,n}\) | 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 authors' goal is to compute the Chern-Schwartz-MacPherson and Segre-Schwartz-MacPherson classes of the orbits of a certain representation called the matrix Schubert cells. More precisely, let \(k\le n\) be nonnegative integers. Let us consider the group \(GL_k(\mathbb C)\times B_n^-\) acting on \(\text{Hom}(\mathbb C^k; \mathbb C^n)\) by \((A,B)\cdot M = BMA^{-1}\), where \(B_n^-\) denotes the Borel subgroup of \(n\times n\) lower triangular matrices. The finitely many orbits of this action are parametrized by \(d\)-element subsets \(J = \{j_1 < \dots < j_d\} \subset \{1, \dots, n\}\) where \(d \le k\). The corresponding orbits, denoted by \(\Omega_J\), are called matrix Schubert cells and their closures are usually called matrix Schubert varieties (see [\textit{L. M. Fehér} and \textit{R. Rimányi}, Cent. Eur. J. Math. 1, No. 4, 418--434 (2003; Zbl 1038.57008)]). Let us fix \(I\subset \{1,\dots,k\}\). The authors prove that the equivariant Chern-Schwartz-MacPherson class of \(\Omega_I\) is equal to the value of the weight function \(W_I(\alpha, \beta)\), considered in [\textit{V. Tarasov} and \textit{A. Varchenko}, Invent. Math. 128, No. 3, 501--588 (1997; Zbl 0877.33013)] or [\textit{R. Rimányi} and \textit{A. Varchenko}, Impanga 15. EMS Series of Congress Reports 225--235 (2018; Zbl 1391.14108)], where \(\alpha = (\alpha_1, \dots, \alpha_k)\) and \(\beta = (\beta_1, \dots, \beta_n)\) are suitable partitions. In turn, the weight function \(W_I(\alpha, \beta)\) can be expressed in terms of appropriated symmetric functions, which can be computed as values of the ``iterated residues'' of some generating functions parameterized by partitions, and so on. In a similar same way it is possible to compute the Segre-Schwartz-MacPherson classes. As an application, the authors perform the corresponding calculations and write out the exact formulas for these classes in the case of \(A_2\) quiver representation. characteristic classes; equivariant cohomology; Borel subgroup; symmetric functions; fundamental class; degeneracy loci; weight functions; Schubert cells; Schur expansion; iterated residues Grassmannians, Schubert varieties, flag manifolds, Global theory of complex singularities; cohomological properties, Global theory and resolution of singularities (algebro-geometric aspects), Classical problems, Schubert calculus, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Characteristic classes and numbers in differential topology Chern-Schwartz-MacPherson classes of degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the connection between the affine degenerate Grassmannians in type \(A\), quiver Grassmannians for one vertex loop quivers, and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type \(\mathrm{GL}_{n}\) and identify it with semi-infinite orbit closure of type \(A_{2n-1}\). We show that principal quiver Grassmannians for the one vertex loop quiver provide finite-dimensional approximations of the degenerate affine Grassmannian. Finally, we give an explicit description of the degenerate affine Grassmannian of type \(A_{1}^{(1)}\), propose a conjectural description in the symplectic case, and discuss the generalization to the case of the affine degenerate flag varieties. affine Kac-Moody algebras; quiver Grassmannians; flag varieties Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Degenerate affine Grassmannians and loop quivers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One way to define a collection of Lie algebras \({\mathfrak g}(t)\), parameterized by \(t\) and each equipped with a representation \(V(t)\), as forming a ``series'' is to require (following Deligne) that the tensor powers of \(V(t)\) decompose into irreducible \({\mathfrak g}(t)\)-modules in a manner independent of \(t\), with formulas for the dimensions of the irreducible components of the form \(P(t)/Q(t)\) where \(P\), \(Q\) are polynomials that decompose into products of integral linear factors. We study such decomposition formulas in this paper, which provides a companion to [\textit{J. M. Landsberg} and \textit{L. Manivel}, Adv. Math. 171, No.~1, 59--85 (2002; Zbl 1035.17016)], where we study the corresponding dimension formulas. We connect the formulas to the geometry of the closed orbits \(X(t)\subset\mathbb PV(t)\) and their unirulings by homogeneous subvarieties. We relate the linear unirulings to work of \textit{B. Kostant} [Topology 3, Suppl. 2, dedicated to Arnold Shapiro, 147--159 (1965; Zbl 0134.03504)]. By studying such series, we determine new modules that, appropriately viewed, are exceptional in the sense of \textit{M. Brion} [Ann. Inst. Fourier 33, No.~1, 1--27 (1983; Zbl 0475.14038) (see, e.g., Theorem 6.2).
The starting point of this paper was the work of Deligne et al. containing uniform decomposition and dimension formulas for the tensor powers of the adjoint representations of the exceptional simple Lie algebras up to \({\mathfrak g}^{\otimes 5}\). Deligne's method for the decomposition formulas was based on comparing Casimir eigenvalues, and he offered a conjectural explanation for the formulas via a categorical model based on bordisms between finite sets. \textit{P. Vogel} [The universal Lie algebra, preprint (1999)] obtained similar formulas for all simple Lie superalgebras based on his universal Lie algebra. We show that all primitive factors in the decomposition formulas of Deligne and Vogel can be accounted for using a pictorial procedure with Dynkin diagrams. (The nonprimitive factors are those either inherited from lower degrees or arising from a bilinear form, so knowledge of the primitive factors gives the full decomposition.) We also derive new decomposition formulas for other series of Lie algebras.
In Section 2, we describe a pictorial procedure using Dynkin diagrams for determining the decomposition of \(V^{\otimes k}\) In Sections 3 and 4 we distinguish and interpret the primitive components in the decomposition formulas of Deligne and Vogel.
The exceptional series of Lie algebras occurs as a line in Freudenthal's magic square. The three other lines each come with preferred representations. Dimension formulas for all representations supported on the cone in the weight lattice generated by the weights of the preferred representations, similar to those of the exceptional series, were obtained in the authors' paper (loc. cit.).
Sections 5--7 we obtain the companion decomposition formulas. A nice property shared by many of these preferred representations is that they are exceptional , in the sense of Brion (loc. cit.) that is, their covariant algebras are polynomial algebras. We prove that, in some cases where this is not naively true, it becomes so when we take the symmetry group of the associated marked Dynkin diagram into account.
In the course of revising the exposition of this paper, we ran across the closely related article by \textit{P. Deligne} and \textit{B. H. Gross} [C. R., Math., Acad. Sci. Paris 335, No.~11, 877--881 (2002; Zbl 1017.22008)]. J.M. Landsberg and L. Manivel, \textit{Series of Lie Groups}, \textit{Michigan Math. J.}\textbf{52} (2004) 453 [math/0203241]. Exceptional (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Series of Lie groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Andrei Okounkov received the Fields Medal \textsl{for his contributions to bridging probability, representation theory and algebraic geometry} at the 25th International Congress of Mathematicians in Madrid, Spain, in 2006. The paper under review is the author's laudatio on the Andrei Okounkov's work. In the paper his contribution to the theory of Gromov-Witten invariants, Donaldson-Thomas invariants, random partitions, connection of planar dimer models with real algebraic geometry, random surfaces and Harnack curves is discussed. Gromov-Witten invariants; Hurwitz correspondence; Donaldson-Thomas invariants; Baik-Deift-Johannson conjexture; dimer configuration; random surfaces; Harnack curve Biographies, obituaries, personalia, bibliographies, Development of contemporary mathematics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Relationships between algebraic curves and physics, Algebraic geometry, Algebraic combinatorics, Quantum field theory; related classical field theories The work of Andrei Okounkov | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Two of the few known examples of hyper-Kähler varieties can be described as varieties of zeroes of a general section of a homogeneous vector bundle over a Grassmannian. Precisely, one is the zero locus of a section of the third symmetric power of the dual of the tautological bundle on \(\mathbb G(1,5)\), the Grassmannian of lines in \(\mathbb P^5\) (Beauville-Donagi). The other one is the zero locus of a section of the third anti-symmetric power of the dual of the tautological bundle on \(\mathbb G(5,9)\) (Debarre-Voisin). Both varieties have dimension \(4\).
In the article under review, the author gives the complete classification of the fourfolds with trivial canonical bundle which arise as zero loci of general sections of homogeneous, completely reducible bundles over ordinary, symplectic or orthogonal Grassmannians. He gets that the only hyper-Kähler varieties in these lists are the two examples quoted above. All the other varieties in the lists are Calabi-Yau, with two exceptions: the abelian fourfold parametrizing the \(3\)-spaces in \(\mathbb P^9\) contained in the complete intersection of two quadrics, and a non-connected fourfold \(Y\), with \(\chi(\mathcal O_Y)=4\). Its connected components are two isomorphic Calabi-Yau fourfolds, complete intersections in an orthogonal Grassmannian.
The proof relies on a suitable extension of the method of \textit{O. Küchle} [Math. Z. 218, No. 4, 563--575 (1995; Zbl 0826.14024)].
The article contains also a similar complete classification for varieties of dimension \(2\) and \(3\). Calabi-Yau; hyper-Kähler; homogeneous bundle; Grassmannian; symplectic Grassmannian; orthogonal Grassmannian \(4\)-folds, Grassmannians, Schubert varieties, flag manifolds, Kähler manifolds, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Differential geometry of homogeneous manifolds Manifolds of low dimension with trivial canonical bundle in Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let G be an n-dimensional Grassmann manifold. In this paper we show the following results. A Cousin-II domain D in G is a Stein manifold if there exists a complex Lie group L with \(H^ 1(D,{\mathcal A}_ L)=0\) and \(H^ q(D,{\mathcal O})=0\) for \(2\leq q\leq n-1,\) where \({\mathcal A}_ L\) is the sheaf of germs of holomorphic mappings from D into L and \({\mathcal O}={\mathcal A}_{{\mathbb{C}}}\). The limit of a monotone increasing sequence of Cousin-I and -II domains over G is a Cousin-I and multiform Cousin-II domain. A domain D in G is a Stein manifold if \(H^ 1(D,{\mathcal O})=H^ 2(D,{\mathbb{Z}})=0\) and every topologically trivial holomorphic vector bundle over D is also holomorphically trivial, where \({\mathbb{Z}}\) is the additive group of all integers. envelopes of holomorphy; Cousin domains; Grassmann manifold; Stein manifold Stein spaces, Envelopes of holomorphy, Grassmannians, Schubert varieties, flag manifolds On the domains over a 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 [For the entire collection see Zbl 0665.00004.]
The authors consider a noncommutative version of Schubert polynomials. This is done by simultaneous lifting of the classical Schubert polynomials into two noncommutative algebras related by the ``Dualité de Cauchy''. As a consequence they obtain the functoriality of Schubert polynomials. noncommutative version of Schubert polynomials; functoriality A. Lascoux and M.-P. Schützenberger, Fonctorialité des polynômes de Schubert, Invariant theory (Denton, TX, 1986) Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 585 -- 598 (French, with English summary). Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics, Representation theory for linear algebraic groups Fonctorialité des polynômes de Schubert. (Functoriality of Schubert polynomials) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The invariant subsets \(C(S)\) of a connected complex semisimple Lie group \(G_C\) are studied for an arbitrary non-closed \(K_C\)-orbit \(S\), with \(K_C\) -- the complexification in \(G_C\) of a maximal subgroup. It is proven that the connected component \(C(S)_0\) of the identity is equal to the Akhiezer-Gindikin domain \(D\) for all non-closed \(K_C\)-orbits. The case when \(G_C=\text{Sp}(2,C)\) having the \(\text{Sp}(2,R)\) as the connected real form is considered as an illustrative example. All \(K_C\)-orbits are obtained and the equivalence of domains arising from duality of orbits on flag manifolds is verified. flag manifolds; non-closed orbits; equivalent domains Toshihiko Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds. III, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4773 -- 4786. 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. III. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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. Caldero} and \textit{A. Zelevinsky} [Mosc. Math. J. 6, No. 3, 411--429 (2006; Zbl 1133.16012)] studied the geometry of quiver Grassmannians for the Kronecker quiver and computed their Euler characteristics by examining natural stratification of quiver Grassmannians. We consider generalized Kronecker quivers and compute virtual Poincaré polynomials of certain varieties which are the images under projections from strata of quiver Grassmannians to ordinary Grassmannians. In contrast to the Kronecker quiver case, these polynomials do not necessarily have positive coefficients. The key ingredient is the explicit formula for noncommutative cluster variables given by \textit{R. Schiffler} and the first author [Compos. Math. 148, No. 6, 1821--1832 (2012; Zbl 1266.16027)]. quiver Grassmannians; ordinary Grassmannians; virtual Poincaré polynomials; Kronecker quiver; noncommutative cluster variables DOI: 10.1090/conm/592/11771 Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets On natural maps from strata of quiver Grassmannians to ordinary Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The real Grassmannian \(Gr_{k,n}\) is the space parametrizing all \(k\)-dimensional subspaces of \(\mathbb R^n\). Equivalently, it can be seen as the set of equivalence classes of \(k\times n\) real matrices of maximal rank. The Plücker embedding, \(Gr_{k,n}\hookrightarrow\mathbb R\mathbb P^{{n\choose k}-1}\) realizes \(Gr_{k,n}\) as a projective variety by sending a matrix to its maximal minors. The totally non-negative Grassmannian \((Gr_{k,n})_{\geq 0}\) is the subset of \(Gr_{k,n}\) represented by \(k\times n\) matrices with all maximal minors non-negative. For any set \(\mathcal M\), consisting of \(k\)-element subsets of \(\{1,\dots,n\}\), the positive Grassmann cell \(C_{\mathcal M}\) is the subset of \((Gr_{k,n})_{\geq 0}\) represented by matrices in \(Gr_{k,n}\) with strictly positive minors corresponding to elements of \(\mathcal M\) and all the other minors equal to \(0\).
The main result shows that the above cell decomposition of \((Gr_{k,n})_{\geq 0}\) is a CW complex. This is a special case of the conjecture posed by \textit{G. Lusztig} [in de Gruyter Expo. Math. 26, 133-145 (1998; Zbl 0929.20035); Represent. Theory 2, 70-78 (1998; Zbl 0895.14014); and in Prog. Math. 123, 531-568 (1994; Zbl 0845.20034)], who introduced the non-negative part of a real flag variety \((\mathbb G/P)_{\geq 0}\) with its cell decomposition and conjectured that it is a finite, regular CW complex homeomorphic to a ball. A CW complex is regular if the closure of each cell is homeomorphic to a ball and the boundary of each cell is homeomorphic to a sphere.
The proof uses the combinatorial description of \((Gr_{k,n})_{\geq 0}\) by the first author, which allows to parametrize cells of \((Gr_{k,n})_{\geq 0}\) with plane-bipartite graphs. For each such graph \(G\) one can construct a toric variety \(X_G\) such that the interior of its moment polytope \(P(G)\) is homeomorphic to the interior of some cell. However, the isomorphisms do not necessarily extend to the boundary. Moreover, each cell in the closure of the cell parametrized by the moment polytope \(P(G)\) corresponds to a certain face of \(P(G)\), which is of the form \(P(H)\), where \(H\) is a subgraph of \(G\). total positivity; Grassmanianns; CW complexes; Birkhoff polytopes; matchings; matroid polytopes; cluster algebras; toric varieties A. Postnikov, D. Speyer and L. Williams, \textit{Matching polytopes, toric geometry and the non-negative part of the Grassmannian}, arXiv:0706.2501. Linear algebraic groups over the reals, the complexes, the quaternions, Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Combinatorial aspects of matroids and geometric lattices, Polyhedral manifolds, Real algebraic and real-analytic geometry, Cluster algebras Matching polytopes, toric geometry, and the totally non-negative Grassmannian. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author studies and classifies the spherical homogeneous spaces of minimal rank. Examples of such spaces are flag manifolds, algebraic tori and reductive groups \(G\) (seen as \(G\times G\)-spaces).
Let \(G\) be be a connected reductive group over an algebraic closed field of characteristic zero and let \(G/B\) be the complete flag variety. An algebraic subgroup \(H\) of \(G\) is spherical if it acts on \(G/B\) with finitely many orbits; in this case also \(G/H\) is said spherical. A pair \((G,H)\) is a spherical pair of minimal rank if there is \(x\in G/B\) such that \(Hx\subset G/B\) is open and \(H_x\) contains a maximal torus of \(H\); in particular \(H\) is spherical. One can define the rank \(rk(G/H)\) of \(G/H\) for any spherical subgroup \(H\). Moreover, \(rk(G/H)\geq rk(G)-rk(H)\) and the equality holds if and only if \((G,H)\) is of minimal rank.
Let \(H(G/B)\) be the set of \(H\)-orbits in \(G/B\). If \(H\) is a parabolic subgroup the elements of \(H(G/B)\) are the Schubert varieties and are indexed by \(W_G/W_H\) (\(W_G\) is the Weyl group of \(G\)). Most of the properties of Schubert varieties cannot be generalized if \(H\) is only spherical. However, \(H(G/B)\) has nice properties if \(H\) is minimal. For example, the Schubert varieties are normal. By a result of Brion, the elements of \(H(G/B)\) are still normal if \(G/H\) is of minimal rank, but for generic spherical space the elements of \(H(G/B)\) are not normal [see \textit{M. Brion}, Comment. Math. Helv. 76, No. 2, 263--299 (2001; Zbl 1043.14012) and \textit{S. Pin}, Sur les singularités des orbites d'un sous-groupe de Borel dans les espaces symétriques. Thèse, Université Grenoble I, (2001)].
In \S2 the author proves some characterizations of the minimality of \(H\) in terms of \(H(G/B)\). Knop has defined an action of \(W_G\) on \(H(G/B)\). In particular, the author proves that \(W_G\) acts transitively on \(H(G/B)\) if and only if \(G/H\) has minimal rank; in this case the isotropy groups are isomorphic to \(W_H\). In the proof is used a graph with vertices the elements of \(H(G/B)\). This graph has been introduced by Brion [Zbl 1043.14012] and was also studied by the author [Bull. Soc. Math. Fr. 132, No. 4, 543--567 (2004; Zbl 1076.14073)].
The inclusion defines an order on \(H(G/B)\) which generalizes the Bruhat order. \textit{F. Knop} [Comment. Math. Helv. 70, No.2, 285-309 (1995; Zbl 0828.22016)] defined an action of a monoid \(\widetilde{W}\) (constructed from the generator of \(W\)), which can be used to describe the Bruhat order. In Corollary 2.1 is proved a generalization of this description. The number of Schubert varieties of dimension \(d\) equals the number of Schubert varieties of codimension \(d\). A similar symmetry is showed in Proposition 2.3.
Another important property of these space is the following one. Let \(T\) be a maximal torus of \(G\). A spherical homogeneous space \(G/H\) is of minimal rank if and only if for any complete toroidal embedding \(X\) of \(G/H\) and any \(x\in X^T\), \(Gx\) is complete. This property seems to play a key role in several works about the (wonderful) embedding of \(G\times G/G\). For example, in [\textit{A. Tchoudjem}, Bull. Soc. Math. Fr. 135, No. 2, 171--214 (2007; Zbl 1181.14027)] it is used to give a description of the sheaf cohomology of the wonderful compactification of \(G\times G/G\).
In \S3 the author uses the above-mentioned characterizations to reduce the classification of spherical pairs \((G,H)\) of minimal rank to the special case where \(G\) is semisimple adjoint and \(H\) is simple. Indeed, any \(G/H\) of minimal rank can be obtained from ones of this type and from tori by products, finite covers and parabolic inductions.
Finally, in \S4 the author prove a classification of these special varieties (see Theorem A). In the proof he uses the following property. Doing some appropriate choices, the restriction of characters from a maximal torus of \(G\) to a maximal torus of \(H\) induces a surjection on the respective sets of simple roots. The fibers of this map have at most two elements, so to any special \((G,H)\) can be associated an involution of the vertices of the Dynkin diagram of \(G\). Remark that in general such involution is not an automorphism of the Dynkin diagram. Moreover, to each pair: Dynkin diagram plus involution, is associated at most a special pair \((G,H)\) (see Proposition 4.1). spherical varieties; Schubert varieties and generalizations N. Ressayre, Spherical homogeneous spaces of minimal rank, Adv. Math. 224 (2010), 1784--1800. Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Spherical homogeneous spaces of minimal rank | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathbf P}\) be a projective space of dimension \(n>1\) with at least three points on each line. A flag in \({\mathbf P}\) is an ascending chain \((S_ 0,S_ 1,\dots,S_{n-1})\) of subspaces \(S_ i\) of dimension \(i\). The authors consider the collection \(S({\mathbf P})\) of all flags and they call the \((n+1)\)-tuple \(\Sigma({\mathbf P})=(S({\mathbf P})\), \({\mathcal L}_ 0({\mathbf P})\), \({\mathcal L}_ 1({\mathbf P}),\dots,{\mathcal L}_{n-1}({\mathbf P}))\) the flag space of \({\mathbf P}\), where \({\mathcal L}_ i({\mathbf P})\) denotes the collection of all sets of flags where all subspaces but the subspace of dimension \(i\) are fixed. (That is, \(S({\mathbf P})\) is the chamber system of the building of type \(A_ n\) associated with \({\mathbf P}\) and \({\mathcal L}_ i({\mathbf P})\) is the collection of all stars of simplices of codimension 1 that lack a vertex of type \(i\), i.e. an \(i\)-dimensional subspace of \({\mathbf P}\).)
The incidence structure \((S({\mathbf P}),{\mathcal L}({\mathbf P}))\), where \({\mathcal L}({\mathbf P})={\mathcal L}_ 0({\mathbf P})\cup{\mathcal L}_ 1({\mathbf P})\cup\cdots\cup{\mathcal L}_{n-1}({\mathbf P})\), is a proper irreducible connected partial linear space. The first author and \textit{C. Somma} [Rend. Math. Appl., VII. Ser. 6, No. 1/2, 59-75 (1986; Zbl 0707.14046)] geometrically characterized these flag spaces as Schubert spaces of index \(n\), that is, a proper irreducible connected partial linear space \((S,{\mathcal L})\) where \({\mathcal L}={\mathcal L}_ 0\cup{\mathcal L}_ 1\cup\cdots\cup{\mathcal L}_{n-1}\) is the union of pairwise disjoint partitions \({\mathcal L}_ i\) of \(S\) such that three axioms on paths in \((S,{\mathcal L})\) are satisfied. (A path in \((S({\mathbf P}),{\mathcal L}({\mathbf P}))\) corresponds to a gallery in the associated building.)
The authors introduce the notion of a topological Schubert space: Each of the sets \(S\), \({\mathcal L}_ 0\), \({\mathcal L}_ 1,\dots,{\mathcal L}_{n-1}\) carries topologies and \({\mathcal L}\) is endowed with the sum topology. Furthermore, the continuity or openness of certain geometric operations is postulated. It is then shown that using \textit{J. Misfeld}'s [Abh. Math. Semin. Univ. Hamburg 32, 232-263 (1968; Zbl 0164.208)] definition of a topological projective space and flag space of a topological irreducible projective space becomes a topological Schubert space and vice versa, each topological Schubert space can be represented in this way.
The correspondence between buildings of type \(A_ n\) and flag geometries of a projective space, e.g. \textit{J. Tits} [Buildings of spherical type and finite \(BN\)-pairs (1974; Zbl 0295.20047)], suggests that a topological Schubert space corresponds to a topological building in the sense of \textit{K. Burns} and \textit{R. Spatzier} [Publ. Math., Inst. Hautes Etud. Sci. 65, 5-34 (1987; Zbl 0643.53036)]. topological Schubert space; topological projective space; flag space BICHARA, A., MISFELD, J. and ZANELLA, C: Tie flag spaceof atopological projective space. Riv. Mat. Pura Appl.11 (1992). Topological linear incidence structures, General theory of linear incidence geometry and projective geometries, Incidence structures embeddable into projective geometries, Buildings and the geometry of diagrams, Grassmannians, Schubert varieties, flag manifolds The flag space of a topological projective space | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(L\) be a simple simply laced Lie algebra. The author describes tensor product and multiplicity varieties, associated to the Dynkin graph of \(L\), that are closely related to Nakajima's quiver varieties. In particular, the author shows that the set of irreducible components of a tensor product variety can be equipped with the structure of an \(L\)-crystal isomorphic to the crystal of the canonical basis of the tensor product of several simple finite-dimensional representations of \(L\), and that the number of irreducible components of a multiplicity variety is equal to the multiplicity of a certain representation in the tensor product of several others. The decomposition of a tensor product into a direct sum is also described geometrically. tensor product varieties; crystals; multiplicity varieties Malkin, A.: Tensor product varieties and crystals: the ADE case. Duke Math. J. \textbf{116}(3), 477-524 (2003). arXiv:0103025 Representation theory for linear algebraic groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds Tensor product varieties and crystals: The \(ADE\) 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 The goal of the present paper is to extend the mitosis algorithm, originally developed by \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] for the case of Schubert polynomials, to the case of Grothendieck polynomials. In addition we will also use this algorithm to construct a short combinatorial proof of Fomin-Kirillov's formula for the coefficients of Grothendieck polynomials. Grothendieck polynomials; pipe dreams; mitosis algorithm Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Linkage, complete intersections and determinantal ideals Mitosis algorithm for Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider a complex vector space \(V\) of dimension \(N\) equipped with a nondegenerate symmetric form. Choose an integer \(m < N/2\) and consider the Grassmannian \(\mathrm{OG}=\mathrm{OG}(m; N)\) parametrizing isotropic \(m\)-dimensional subspaces of \(V\). In this paper the authors prove a Giambelli formula expressing the Schubert classes on OG as polynomials in certain special Schubert classes that generate the cohomology ring \(H^* (\mathrm{OGF};\mathbb Z)\) for the even \(N\). Giambelli formula; orthohogonal Grassmannians; Schubert classes; Dynkin diagrams Buch, A.; Kresch, A.; Tamvakis, H., \textit{A Giambelli formula for even orthogonal Grassmannians}, J. Reine Angew. Math., 708, 17-48, (2015) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Equivariant algebraic topology of manifolds A Giambelli formula for even orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We investigate the number of orbits in a variety \(\Lambda_ V\) associated to Dynkin graphs of type \(A_ n\) as defined by \textit{G. Lusztig} [J. Am. Math. Soc. 4, No. 2, 365-421 (1991; Zbl 0738.17011)]. For \(n<4\), we show that there is only a finite number of indecomposable representations in \(\Lambda_ V\) up to isomorphism. This implies that \(\Lambda_ V\) consists of finitely many orbits for any \(V\). For each \(n>4\), we show that there exist \(V\) for which \(\Lambda_ V\) contains infinitely many orbits. variety; Dynkin graphs; indecomposable representations; orbits J. Chislenko and C. K. Fan, On representations of graphs of type \(A\) , Proc. Roy. Soc. London Ser. A 439 (1992), no. 1907, 687-690. JSTOR: Graph theory, Algebraic combinatorics, Vector spaces, linear dependence, rank, lineability, Varieties and morphisms On representations of graphs of type A | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An element \([\Phi] \in Gr_n \left(\mathcal{H}_+, \mathbf{F}\right)\) of the Grassmannian of \(n\)-dimensional subspaces of the Hardy space \(\mathcal{H}_+ = H^2\), extended over the field \(\mathbf{F} = \mathbf{C} (x_{1},\dots, x_{n})\), may be associated to any polynomial basis \(\phi = \{\phi_0, \phi_1,\dots\}\) for \(\mathbf{C}(x)\). The Plücker coordinates \(S_{\lambda, n}^{\phi}(x_1, \ldots, x_n)\; \text{of}\; [\Phi]\), labeled by partitions \(\lambda\), provide an analog of Jacobi's bi-alternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system \(\phi\) to the analog \(\{h_i^{(0)} \}\) of the complete symmetric functions generates a doubly infinite matrix \(h_i^{(j)}\) of symmetric polynomials that determine an element \([H] \in \mathrm{Gr}_n(\mathcal{H}_+, \mathbf{F})\). This is shown to coincide with \([\Phi]\), implying a set of generalized Jacobi identities, extending a result obtained by \textit{A. N. Sergeev} and \textit{A. P. Veselov} [Mosc. Math. J. 14, No. 1, 161--168 (2014; Zbl 1297.05244)] for the case of orthogonal polynomials. The symmetric polynomials \(S_{\lambda, n}^{\phi}(x_1, \ldots, x_n)\) are shown to be KP (Kadomtsev-Petviashvili) \(\tau\)-functions in terms of the power sums \([x]\) of the \(x_{a}\)'s, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums \(\sum_{\lambda} S_{\lambda, n}^{\phi}([x]) S_{\lambda, n}^{\theta}(\mathbf{t})\) associated to any pair of polynomial bases \((\phi, \theta)\), which are shown to be 2D Toda lattice \(\tau\)-functions. A number of applications are given, including classical group character expansions, matrix model partition functions, and generators for random processes.{
\copyright 2018 American Institute of Physics} Harnad, J.; Lee, E., Symmetric polynomials, generalized Jacobi-trudi identities and \textit{ {\(\tau\)}}-functions, J. Math. Phys., 59, 091411, (2018) Symmetric functions and generalizations, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Polynomials and rational functions of one complex variable, Hardy spaces, Grassmannians, Schubert varieties, flag manifolds, Schur and \(q\)-Schur algebras, KdV equations (Korteweg-de Vries equations) Symmetric polynomials, generalized Jacobi-Trudi identities and \(\tau\)-functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Quiver Grassmannians are projective varieties parametrizing subrepresentations of given dimension in a quiver representation. We define a class of quiver Grassmannians generalizing those which realize degenerate flag varieties. We show that each irreducible component of the quiver Grassmannians in question is isomorphic to a Schubert variety. We give an explicit description of the set of irreducible components, identify all the Schubert varieties arising, and compute the Poincaré polynomials of these quiver Grassmannians. Schubert varieties; quiver Grassmannians; Dynkin quivers Cerulli Irelli, G., Feigin, E., Reineke, M.: Schubert quiver Grassmannians. In: Algebras and Representation Theory (2016). arXiv:1508.00264 Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds Schubert quiver Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathfrak{S}_n\) be the symmetric group on \(n\) elements and let \(M_{\alpha}\) be its irreducible representation corresponding to the partition \(\alpha \vdash n\).
One of the fundamental problems in algebraic combinatorics and representation theory is to explicitly describe the decomposition of the tensor product of two such irreducibles: \[ M_{\alpha}\otimes M_{\beta} = \bigoplus_{\gamma\vdash n} M_{\gamma}^{\oplus g_{\alpha, \beta, \gamma}}. \] The multiplicities \(g_{\alpha, \beta, \gamma}\) are non-negative integers called \textit{Kronecker coefficients}, the main focus of this article.
\textit{F. D. Murnaghan} [Am. J. Math. 60, 761--784 (1938; JFM 64.0070.02)] noticed an interesting asymptotic property that Kronecker coefficients possess: let \(\alpha, \beta, \gamma\) be partitions of \(n\); if one repeatedly increases by one the first part of each of these partitions, the sequence of Kronecker coefficients stabilizes.
More recently, \textit{J. R. Stembridge} [``Generalized stability of Kronecker coefficients'', Preprint, \url{http://www.math.lsa.umich.edu/ jrs/papers/kron.pdf}] introduced two useful notions of stability of a triple of partitions (Definition 1.1) and conjectured that they are in fact equivalent. In [J. Algebr. Comb. 43, No. 1, 1--10 (2016; Zbl 1345.05113)], \textit{S. V. Sam} and \textit{A. Snowden} proved this conjecture.
In this paper, the author gives a new proof of this result (Theorem 1.2). The novelty is on the methods: the author's approach relies on geometric invariant theory. Indeed, Kronecker coefficients are related to dimensions of spaces of invariant global sections of some line bundles on a projective variety. With the same techniques, some known bounds on Kronecker coefficients are recovered as well. Moreover, interesting stability results for plethysm coefficients and for multiplicities of tensor products of two irreducible representations of the hyperoctahedral group are proven. Kronecker coefficients; stability properties; geometric invariant theory Representation theory for linear algebraic groups, Geometric invariant theory, Group actions on varieties or schemes (quotients), Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds A geometric approach to the stabilisation of certain sequences of Kronecker 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 In this paper, we study a variation in a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal codimension. On complex manifolds, this ampleness property implies intermediate hyperbolic properties. We give a natural application of our main result in this context. ampleness; cotangent bundle; complete intersection; flag manifolds; hyperbolicity Complete intersections, Hyperbolic and Kobayashi hyperbolic manifolds, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Sheaves in algebraic geometry Ampleness of Schur powers of cotangent bundles and \(k\)-hyperbolicity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(f:=(f_0,\dots ,f_r)\) an \((r+1)\)-tuple of holomorphic functions, its Wronski matrix is the one whose \(j\)th row is the \((r+1)\)-tuple of the \(j\)th derivatives of the functions, \(j=0,\dots ,r\). The determinant \(W(f)\) of the Wronski matrix is the Wronskian of \(f\). Wronskians are applied in analysis, algebraic geometry, number theory, combinatorics, theory of infinite dimensional dynamical systems etc. The paper is a survey of these applications, with more attention paid to linear ordinary differential equations, theory of ramification loci of linear systems (e.g., Weierstrass points on curves) and intersection theory of complex Grassmannian varieties. Wronski matrix; Wronskian; linear ordinary differential equations; ramification loci of linear systems; complex Grassmanian varieties 8. L. Gatto and I. Scherbak, On generalized Wronskians, in Contributions to Algebraic Geometry, P. Pragacz, ed., Impanga Lecture Notes, EMS Congress Series Report (2012), pp. 257-296, doi: 10.4171/114, http://arxiv.org/pdf/1310.4683v1.pdf. Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Linear ordinary differential equations and systems On generalized Wrońskians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(d=(d_1,d_2,\dots,d_n)\) and \(v=(v_1,v_2,\dots,v_n)\) be two vectors of non-negative integers. Associated to the pair \((d,v)\) (or equivalently to a collection of vector spaces having dimensions \(d_i\) and \(v_i\)), \textit{H. Nakajima} [Duke Math. J. 76, No. 2, 365-416 (1994; Zbl 0826.17026)] introduced certain quiver varieties \(M(d,v)\) and \(M_0(d,v)\) with a map \(\pi\colon M(d,v)\to M_0(d,v)\) (that is equivariant under the action of the product of a certain collection of general linear groups). Let \(M_1(d,v)=\pi(M(d,v))\). These varieties were used by Nakajima to study representations of Kac-Moody algebras. Nakajima conjectured (and proved in a special case) that the varieties \(M(d,v)\) and \(M_1(d,v)\) could be identified with certain varieties defined geometrically using nilpotent matrices, Slodowy slices, and flag varieties (in such a way as to preserve the aforementioned group action). Somewhat more precisely, let \(N=\sum_{i=i}^nid_i\) and consider the Lie algebra of matrices \(M_N(\mathbb{C})\). Using \(v\), a partition of \(N\) is defined, and then these geometric varieties are determined by the nilpotent orbit corresponding to that partition and an associated flag variety.
The main result of this paper is a proof of Nakajima's conjecture. The proof is accomplished by explicitly constructing maps between the varieties and then showing that the maps are isomorphisms. The proof in part makes use of a modified path algebra to give a description of the coordinate ring of \(M_0(d,v)\). quiver varieties; nilpotent cones; Slodowy slices; flag varieties; universal enveloping algebras; moduli spaces; symmetric Kac-Moody algebras; path algebras; dimension vectors A. Maffei, \textit{Quiver varieties of type A}, \textit{Comment. Math. Helv.}\textbf{80} (2005) 1 [math.AG/9812142]. Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Universal enveloping (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Quiver varieties of type \(A\). | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0695.00010.]
Let \(H=Hilb^ 4{\mathbb{P}}^ 2\), A the subscheme of H parametrizing the aligned quadruplets of points in \({\mathbb{P}}^ 2\), G the Grassmannian of pencils of conics and Z the subvariety of G consisting of pencils with a fixed component. If Q is the tautological line bundle on the plane and L the tautological line bundle on the dual plane, then let W be the scheme of zeroes of the map \(L\to Q.\)
In their main theorem the authors show that the double blow-up of G, \(\hat G,\) first along Z and then along an image of W is isomorphic to \(\hat H,\) the blow-up of H along A. Then the authors apply their main theorem to calculate the Chow ring of H. Hilbert scheme; Grassmannian of pencils of conics; blow-up; Chow ring Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Hilb\({}^ 4{\mathbb{P}}^ 2\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors go back to the original motivation of Yang for his introduction of the Yang-Baxter equation. He had been led to study certain elements of the group algebra of the symmetric group, and to expand them on the basis of permutations. They obtain some new results in the study of these objects by replacing elementary transpositions with generators of a Hecke algebra. They define a new basis of the Hecke algebra and a bilinear form on it. In some cases they give formulas for the coefficients of this basis when expanded in the usual basis. This involves Schubert and Grothendieck polynomials which were originally defined as canonical bases of the cohomology and Grothendieck rings of flag manifolds. Yang-Baxter equations; Hecke algebra; flag varieties; Schubert polynomials; Grothendieck polynomials Lascoux, A.; Leclerc, B.; Thibon, J. -Y: Flag varieties and the Yang--Baxter equation. Lett. math. Phys. 40, No. 1, 75-90 (1997) Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Flag varieties and the Yang-Baxter equation | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(G=PSp(2n, \mathbb{C})\) (\(n\geq 3\)) and \(B\) be a Borel subgroup of \(G\) containing a maximal torus \(T\) of \(G\). Let \(w\) be an element of the Weyl group \(W\) and \(X(w)\) be the Schubert variety in the flag variety \(G/B\) corresponding to \(w\). Let \(Z(w,\underline i)\) be the Bott-Samelson-Demazure-Hansen variety (the desingularization of \(X(w)\)) corresponding to a reduced expression \(\underline i\) of \(w\).
In this article, we study the cohomology groups of the tangent bundle on \(Z(w_0, \underline i)\), where \(w_0\) is the longest element of the Weyl group \(W\). We describe all the reduced expressions \(\underline i\) of \(w_0\) in terms of a Coxeter element such that all the higher cohomology groups of the tangent bundle on \(Z(w_0, \underline i)\) vanish. Bott-Samelson-Demazure-Hansen variety; Coxeter element; tangent bundle Vanishing theorems in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Rigidity of Bott-Samelson-Demazure-Hansen variety for \(\mathrm{PSp}(2n, \mathbb{C})\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In 1980, \textit{R. P. Stanley} [SIAM J. Algebraic Discrete Methods 1, 168--184 (1980; Zbl 0502.05004)] proved the following theorem: Let \(R = k[x_1, \dots, x_r]\), where \(k\) has characteristic zero. Let \(I = (x_1^{a_1}, \dots, x_r^{a_r})\). Let \(\ell\) be a general linear form. Then, for any positive integers \(d\) and \(i\), the homomorphism induced by multiplication by \(\ell^d\), \(\times \ell^d: [R/I]_i \rightarrow [R/I]_{i+d}\), has maximal rank. This theorem has been reproved by a number of mathematicians using a variety of techniques and has motivated the study of the weak and strong Lefschetz properties. Let \(A = R/I\) be a graded artinian algebra, where \(k\) is an infinite field, and let \(\ell\) be a general linear form. \(A\) is said to have the \textit{weak Lefschetz property} (WLP) if the homomorphism induced by multiplication by \(\ell\), \(\times \ell: A_i \rightarrow A_{i+1}\), has maximal rank for all \(i\) (i.e., is injective or surjective). Further, we say that \(A\) has the \textit{strong Lefschetz property} if \(\times \ell^d: A_i \rightarrow A_{i+d}\), has maximal rank for all \(i\) and \(d\) (i.e., is injective or surjective).
This paper is a survey of the different directions and research that has resulted from studying these properties. The directions highlight the intersection of commutative algebra, algebraic geometry, and combinatorics. After providing some background, the paper is divided into sections of cases for the study: complete intersections and Gorenstein algebras; monomial level algebras, powers of linear forms, connections between Fröberg's conjecture and the WLP, and positive characteristics and enumerations (involving determinants of certain matrices, lozenge tilings of punctured hexagons, and perfect matchings of bipartite graphs). Throughout the paper, the authors include a number of open questions that drive current research programs of many experts. weak Lefschetz property; strong Lefschetz property Migliore, J.; Nagel, U., A tour of the weak and strong Lefschetz properties, J. Commut. Algebra, 5, 3, 329-358, (2013) Research exposition (monographs, survey articles) pertaining to commutative algebra, Cohen-Macaulay modules, Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Graded rings, Actions of groups on commutative rings; invariant theory, Grassmannians, Schubert varieties, flag manifolds A tour of the weak and strong Lefschetz properties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 0574.00001.]
The aim of this talk is to give a brief introduction to certain combinatorial properties of a Coxeter group. [The details of the results described here are given by the author in Invent. Math. 79, 499-511 (1985; Zbl 0563.14023).] We describe a few natural problems in Lie theory which can be answered in terms of these combinatorial properties. Bruhat ordering; Coxeter group; Bruhat cell decomposition; generalized flag manifold; semisimple algebraic group; Kazhdan-Lusztig polynomials; shellability Other algebraic groups (geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups, Linear algebraic groups over arbitrary fields A study of subexpressions and its applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Isomorphy classes of \(k\)-involutions have been studied for their correspondence with symmetric \(k\)-varieties, also called generalized symmetric spaces. A symmetric \(k\)-variety of a \(k\)-group \(G\) is defined as \(G_k/H_k\) where \(\theta\colon G\to G\) is an automorphism of order 2 that is defined over \(k\) and \(G_k\) and \(H_k\) are the \(k\)-rational points of \(G\) and \(H=G^\theta\), the fixed point group of \(\theta\), respectively. This is a continuation of papers written by \textit{A. G. Helminck} and collaborators [Acta Appl. Math. 90, No. 1-2, 91-119 (2006; Zbl 1100.14040); Classification of involutions of \(\mathrm{SO}(n;k;b)\) (to appear); Adv. Math. 153, No. 1, 1-117 (2000; Zbl 0974.20033); Commun. Algebra 30, No. 1, 193-203 (2002; Zbl 1001.20044)] expanding on his combinatorial classification over certain fields. Results have been achieved for groups of type \(A\), \(B\) and \(D\). Here we begin a series of papers doing the same for algebraic groups of exceptional type. algebraic groups; exceptional groups; involutions; generalized symmetric spaces; composition algebras Hutchens, J, Isomorphy classes of \(k\)-involutions of \({G}_2\), J. Algebra Appl., 13, 1-16, (2014) Linear algebraic groups over arbitrary fields, Exceptional groups, Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Composition algebras Isomorphy classes of \(k\)-involutions of \(G_2\). | 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 A real form \(G_0\) of a complex semisimple Lie group \(G\) has only finitely many orbits in any given compact \(G\)-homogeneous projective algebraic manifold \(Z=G/Q\). A maximal compact subgroup \(K_0\) of \(G_0\) has special orbits \(C\) which are complex submanifolds in the open orbits of \(G_0\). These special orbits \(C\) are characterized as the closed orbits in \(Z\) of the complexification \(K\) of \(K_0\). These are referred to as \textit{cycles}. The cycles intersect Schubert varieties \(S\) transversely at finitely many points. Describing these points and their multiplicities was carried out for all real forms of \(\mathrm{SL} (n,\mathbb{C})\) by \textit{A.-M. Brecan} [``Schubert duality for \(\mathrm{SL}(n,R)\)-flag domains'', Preprint, \url{arXiv:1409.5868}; J. Algebra 492, 324--347 (2017; Zbl 1401.14204)] and for the other real forms by the author [Combinatorial geometry of flag domains in G/B. Bochum: Ruhr University Bochum (PhD Thesis) (2017)] and Huckleberry (Abu-Shoga and Huckleberry). In the present paper, we deal with the real form \(\mathrm{SO}(p,q)\) acting on the \(\mathrm{SO}(2n,C)\)-manifold of maximal isotropic full flags. We give a precise description of the relevant Schubert varieties in terms of certain subsets of the Weyl group and compute their total number. Furthermore, we give an explicit description of the points of intersection in terms of flags and their number. The results in the case of \(G/Q\) for all real forms will be given by Abu-Shoga and Huckleberry. semisimple Lie group; Schubert varieties Grassmannians, Schubert varieties, flag manifolds, \(q\)-convexity, \(q\)-concavity, General properties and structure of complex Lie groups, Semisimple Lie groups and their representations Cycle intersection for \(\mathrm{SO}(p,q)\)-flag 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 We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then placing the structure of a connected positroid on each of the blocks of the partition. This structural result yields several combinatorial facts about positroids. We show that the face poset of a positroid polytope embeds in a poset of weighted non-crossing partitions. We enumerate connected positroids, and show how they arise naturally in free probability. Finally, we prove that the probability that a positroid on \([n]\) is connected equals \(1/e^2\) asymptotically. Postnikov's matroid F. Ardila, R. Rincón, and L. K. Williams. ''Positroids and non-crossing partitions''. Trans. Amer. Math. Soc. 368 (2016), pp. 337--363. Exact enumeration problems, generating functions, Combinatorial aspects of partitions of integers, Combinatorial aspects of matroids and geometric lattices, Grassmannians, Schubert varieties, flag manifolds, Semialgebraic sets and related spaces, Noncommutative probability and statistics Positroids and non-crossing partitions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 generic number of points in the intersection of three Schubert varieties in the complex Grassmann manifold can be calculated using the Littlewood-Richardson rule. When this number is equal to one, it was shown in [\textit{H. Bercovici} et al., J. Funct. Anal. 258, No. 5, 1579--1627 (2010; Zbl 1196.46048)] that an appropriate analogue of Schubert varieties in a \(\text{II}_1\) factor must have nonempty intersection. We show that this intersection consists generically of precisely one point. The argument also works in finite dimensions for real as well as complex Grassmann manifolds. finite factor; Littlewood-Richardson rule; Schubert calculus General theory of von Neumann algebras, Grassmannians, Schubert varieties, flag manifolds Uniqueness in the solution of intersection problems in a factor of type \(\text{II}_1\) | 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 In this paper we try to connect the Grassmannian subcomplex defined over the projective differential map \(d'\) and the variant of Cathelineau's complex. To do this we define some morphisms over the configuration space for both weight 2 and 3. we also prove the commutativity of corresponding diagrams. Grassmannian complex; configuration; vector space; infinitesimal; cross ratio Multiple Dirichlet series and zeta functions and multizeta values, Grassmannians, Schubert varieties, flag manifolds, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Projective configurations and the variant of Cathelineaus complex | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 sketches his construction for the Chern classes of a projective subvariety \(V_d\) \((d= \) dimension) of the complex projective space \(\mathbb{P}_n\subset\mathbb{C})\) [Shuxue Jinzhan 8, 395--409 (1965)]. One uses the cohomology ring (Ehresmann symbols) of the flag manifold \{point \(\subset\) linear subspace of dimension \(d\}\) and the variety determined by the pairs \(\{x,T_x\}\), where \(x\) is a generic point of \(V_d\) and \(T_x\) the corresponding tangent space.
[For the entire collection see Zbl 0534.00009.] singular projective variety; Chern classes; Ehresmann symbols; flag manifold; tangent space Singularities in algebraic geometry, Characteristic classes and numbers in differential topology, Classical real and complex (co)homology in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Chern classes on algebraic varieties with arbitrary singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We relate a certain category of sheaves of \(k\)-vector spaces on a complex affine Schubert variety to modules over the \(k\)-Lie algebra (for \(\text{char\,}k>0\)) or to modules over the small quantum group (for \(k=0\)) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig's conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation-theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig's modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity-one case for all relevant primes.
One of the fundamental problems in representation theory is the calculation of the simple characters of a given group. This problem often turns out to be difficult and there is an abundance of situations in which a solution is out of reach. In the case of algebraic groups over fields of positive characteristic we have a partial, but not yet a full answer.
In 1979, George Lusztig conjectured a formula for the simple characters of a reductive algebraic group defined over a field of characteristic greater than the associated Coxeter number; [cf. \textit{G. Lusztig}, Proc. Symp. Pure Math. 37, 313-317 (1980; Zbl 0453.20005)]. Lusztig outlined in 1990 a program that led, in a combined effort of several authors, to a proof of the conjecture for almost all characteristics. This means that for a given root system \(R\) there exists a number \(N=N(R)\) such that the conjecture holds for all algebraic groups associated to the root system \(R\) if the underlying field is of characteristic greater than \(N\). This number, however, is unknown in all but low rank cases.
One of the essential steps in Lusztig's program was the construction of a functor between the category of intersection cohomology sheaves with complex coefficients on an affine flag manifold and the category of representations of a quantum group (this combines results of \textit{M. Kashiwara} and \textit{T. Tanisaki} [Duke Math. J. 77, No. 1, 21-62 (1995; Zbl 0829.17020)], and \textit{D. Kazhdan} and \textit{G. Lusztig} [J. Am. Math. Soc. 6, No. 4, 905-947, 949-1011 (1993; Zbl 0786.17017); ibid. 7, No. 2, 335-381, 383-453 (1994; Zbl 0802.17007, Zbl 0802.17008)]). This led to a proof of the quantum (i.e. characteristic 0) analog of the conjecture. \textit{H. H. Andersen, J. C. Jantzen} and \textit{W. Soergel} then showed that the characteristic zero case implies the characteristic \(p\) case for almost all \(p\) [cf. Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Astérisque 220 (1994; Zbl 0802.17009)].
One of the principal functors utilized in Lusztig's program was the affine version of the Beilinson-Bernstein localization functor. It amounts to realizing an affine Kac-Moody algebra inside the space of global differential operators on an affine flag manifold. A characteristic \(p\) version of this functor is a fundamental ingredient in Bezrukavnikov's program for modular representation theory [cf. \textit{R. Bezrukavnikov, I. Mirković} and \textit{D. Rumynin}, Ann. Math. (2) 167, No. 3, 945-991 (2008; Zbl 1220.17009)], and recently Frenkel and Gaitsgory used the Beilinson-Bernstein localization idea in order to study the critical level representations of an affine Kac-Moody algebra [cf. \textit{P. Fiebig}, Duke Math. J. 153, No. 3, 551-571 (2010; Zbl 1207.20040)].
There is, however, an alternative approach that links the geometry of an algebraic variety to representation theory. It was originally developed in the case of finite-dimensional complex simple Lie algebras by \textit{W. Soergel} [J. Am. Math. Soc. 3, No. 2, 421-445 (1990; Zbl 0747.17008)]. The idea was to give a ``combinatorial description'' of both the topological and the representation-theoretic categories in terms of the underlying root system using Jantzen's translation functors. This approach gives a new proof of the Kazhdan-Lusztig conjecture, but it is also important in its own right: when taken together with the Beilinson-Bernstein localization it establishes the celebrated Koszul duality for simple finite-dimensional complex Lie algebras [cf. \textit{W. Soergel}, loc. cit., and \textit{A. Beilinson, V. Ginzburg, W. Soergel}, J. Am. Math. Soc. 9, No. 2, 473-527 (1996; Zbl 0864.17006)].
In this paper we develop the combinatorial approach for quantum and modular representations. We relate a certain category of sheaves of \(k\)-vector spaces on an affine flag manifold to representations of the \(k\)-Lie algebra or the quantum group associated to Langlands' dual root datum (the occurrence of Langlands' duality is typical for this type of approach). As a corollary we obtain Lusztig's conjecture for quantum groups and for modular representations for large enough characteristics.
The main tool that we use is the theory of sheaves on moment graphs, which originally appeared in the work on the localization theorem for equivariant sheaves on topological spaces by \textit{M. Goresky, R. Kottwitz} and \textit{R. MacPherson} [Invent. Math. 131, No. 1, 25-83 (1998; Zbl 0897.22009)] and \textit{T. Braden} and \textit{R. MacPherson} [Math. Ann. 321, No. 3, 533-551 (2001; Zbl 1077.14522)]. In particular, we state a conjecture in terms of moment graphs that implies Lusztig's quantum and modular conjectures for all relevant characteristics.
Although there is no general proof of this moment graph conjecture yet, some important instances are known: The smooth locus of a moment graph is determined by \textit{P. Fiebig} [loc. cit.], which yields the multiplicity-one case of Lusztig's conjecture in full generality. Moreover, by developing a Lefschetz theory on a moment graph we obtain in [\textit{P. Fiebig}, J. Reine Angew. Math. 673, 1-31 (2012; Zbl 1266.20059)] an upper bound on the exceptional primes, i.e. an upper bound for the number \(N\) referred to above. Although this bound is huge (in particular, much greater than the Coxeter number), it can be calculated by an explicit formula in terms of the underlying root system. Kazhdan-Lusztig polynomials; irreducible characters; highest weight modules; simple Lie algebras; quantized enveloping algebras; reductive algebraic groups; positive characteristic; root systems; intersection cohomology sheaves; Schubert varieties; character formulae; Coxeter numbers; Lusztig conjecture; affine flag manifolds; affine Kac-Moody algebras; moment graphs Fiebig, Peter, Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture, J. Amer. Math. Soc., 0894-0347, 24, 1, 133\textendash 181 pp., (2011) Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Modular representations and characters, Sheaf cohomology in algebraic topology Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This note corrects an error in the statements of some of the main theorems of the author's paper [ibid. 350, 440--485 (2019; Zbl 1426.14014)]. degeneracy locus; \(K\)-theory; Giambelli formula Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties Corrigendum to: ``K-theoretic Chern class formulas for vexillary degeneracy loci'' | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of ``standard monomials'' for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. Lakshmibai, Musili, and Seshadri gave a standard monomial basis for the smaller vector space generated by the projective coordinates restricted to a Schubert variety. Reiner and Shimozono made this theory more explicit by giving a straightening algorithm for the products of the minors in terms of the right key of a Young tableau. Since then, Willis introduced scanning tableaux as a more direct way to obtain right keys. This paper uses scanning tableaux to give more-direct proofs of the spanning and the linear independence of the standard monomials. In the appendix it is noted that this basis is a weight basis for the dual of a Demazure module for a Borel subgroup of \(GL_n\). This paper contains a complete proof that the characters of these modules (the key polynomials) can be expressed as the sums of the weights for the tableaux used to index the standard monomial bases. standard monomial; Schubert variety; Demazure module; key polynomial; scanning tableau Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Symmetric functions and generalizations, Combinatorial aspects of commutative algebra Accessible proof of standard monomial basis for coordinatization of Schubert sets of 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 Motivated by Buchstaber's and Terzić's work on the complex Grassmannians \(G_\mathbb{C}(2,4)\) and \(G_\mathbb{C}(2,5)\) we describe the moment map and the orbit space of the oriented Grassmannians \(G_\mathbb{R}^+(2,n)\) under the action of a maximal compact torus. Our main tool is the realisation of these oriented Grassmannians as smooth complex quadric hypersurfaces and the relatively simple Geometric Invariant Theory of the corresponding algebraic torus action. oriented Grassmannian; complexity-one T-variety; GIT quotients Grassmannians, Schubert varieties, flag manifolds, Geometric invariant theory, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies Orbit spaces of maximal torus actions on oriented Grassmannians of planes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We find generating functions for the Poincaré polynomials of hyperquot schemes for all partial flag varieties. These generating functions give the Betti numbers of hyperquot schemes, and thus give dimension information for the cohomology ring of every hyperquot scheme. This can be viewed as a step towards understanding a presentation and the structure of the cohomology rings.
Let
\({\mathbf F}(n;{\mathbf s})\) denote the partial flag variety corresponding to flags of the form
\[
V_1\subset V_2\subset \dots\subset V_l\subset V=\mathbb{C}^n; \quad \dim V_i=s_i.
\]
It is classically known that its Poincaré polynomial \({\mathcal P}({\mathbf F}(n;{\mathbf s}))=\sum_Mb_{2M}({\mathbf F}(n;{\mathbf s}))z^M\) is equal to the following generating function for the Betti numbers of the partial flag variety:
\[
{\mathcal P}\bigl({\mathbf F}(n:{\mathbf s})\bigr)= \sum_Mb_{2M}({\mathbf F})z^M=\frac {\prod^n_{i=1}(1-z^i)}{\prod^{l+1}_{j=1}\prod^{s_j-s_{j-1}}_{i=1}(1-z^i)}
\]
with \(s_{l+1}:=n\) and \(s_0:=0\). Defining \(f_k^{i,j}:=1-t_i \dots,t_jz^k\), the main result is:
Theorem 1.
\[
\sum_{d_1,\dots,d_l}{\mathcal P}\biggl( {\mathcal H}{\mathcal Q}_{\mathbf d}\bigl({\mathbf F}(n;{\mathbf s})\bigr)\biggr) t_1^{d_1}\dots t_l^{d_l}=\;{\mathcal P} \bigl({\mathbf F}(n;{\mathbf s})\bigr)\cdot\prod_{1\leq i\leq j\leq l}\prod_{s_{i-1}<k\leq s_i} \left( \frac{1}{f^{i,j}_{s_j-k}}\right) \left(\frac{1}{f^{i,j}_{s_{j+1}-k+1}} \right).
\]
Chen, L, Poincaré polynomials of hyperquot schemes, Math Ann, 321, 235-251, (2001) Grassmannians, Schubert varieties, flag manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Parametrization (Chow and Hilbert schemes) Poincaré polynomials of hyperquot 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 Stable Grothendieck polynomials can be viewed as a \(K\)-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi-Trudi-type identities, and associated Fomin-Greene operators. symmetric functions; Grothendieck polynomials; Schur polynomials Yeliussizov, D., Duality and deformations of stable Grothendieck polynomials, J. Algebraic Combin., 45, 295-344, (2017) Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Duality and deformations of stable Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schur polynomials are a special case of Schubert polynomials. In this paper, we give an algorithm to compute the product of a Schubert polynomial with a Schur polynomial on the basis of Schubert polynomials. This is a special case of the general problem of the multiplication of two Schubert polynomials, where the corresponding algorithm is still missing. The main tools for the given algorithm are a factorization property of a special class of Schubert polynomials and the transition formula for Schubert polynomials. Schur polynomials; Schubert polynomials; transition formula Kohnert, A., Multiplication of a Schubert polynomial by a Schur polynomial, Ann. Comb., 1, 367-375, (1997) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Multiplication of a Schubert polynomial by a Schur polynomial | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a complete homogeneous variety (defined over an algebraically closed field \(k\) of arbitrary characteristic). Then \(X\) is in a canonical way a direct product of an abelian variety and a quotient \(S/P\) where \(S\) is a semisimple algebraic group and \(P\) is a (not necessarily reduced) parabolic subgroup. This is the main result. It generalizes the result of \textit{A. Borel} and \textit{R. Remmert} [Math. Ann. 145, No. 1, 429--439 (1961; Zbl 0111.18001)] who proved this for \(k={\mathbb C}\). A key tool in the proof is a rigidity result for (not necessarily reduced) parabolic subgroups in positive characteristic. As a consequence the classification of complete homogeneous varieties is reduced to the classification of abelian varieties and parabolic subgroups of semisimple algebraic groups. For the latter, a complete classification except in characteristic \(2\) and \(3\) is obtained by \textit{C. Wenzel} [Trans. Am. Math. Soc. 357, No. 1, 211--218 (1993; Zbl 0785.20024)]. abelian variety; flag variety; parabolic subgroup; algebraic group of positive characteristic C. S. de Salas, \textit{Complete homogeneous varieties: structure and classification}, Trans. Amer. Math. Soc. \textbf{355} (2003), no. 9, 3651-3667 (electronic). Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Homogeneous complex manifolds Complete homogeneous varieties: Structure and classification | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) and \(W\) be vector spaces over a common field, and let \(\mathcal{S}\) and \(\mathcal{T}\) be sets of subspaces of \(V\) and \(W\), respectively. A linear function \(\varphi : V \rightarrow W\) is called a (Grassmann) homomorphism from \(\mathcal{S}\) to \(\mathcal{T}\) if \(\varphi(S) \in \mathcal{T}\) holds for every \(S \in \mathcal{S}\).
Coloring of graphs and hypergraphs and flows in graphs can be reformulated in terms of these homomorphisms.
The main results are constructive characterizations of the sets \(\mathcal{S}\) of lines for which there is no homomorphism from \(\mathcal{S}\) to a singleton set \(\mathcal{T}\) with a line as its element. Hajós construction; finite Grassmannian; homomorphism; combinatorics; complexity classes Coloring of graphs and hypergraphs, Grassmannians, Schubert varieties, flag manifolds, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) Grassmann homomorphism and Hajós-type theorems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a split reductive group \(G\) over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated \(G\)-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic \(t\)-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of \(\ell\) in the Langlands correspondence for function fields. Motivic cohomology; motivic homotopy theory, Stacks and moduli problems, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups The intersection motive of the moduli stack of shtukas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some given general figures [\textit{W. Fulton}, ``Introduction to intersection theory in algebraic geometry'' (1996; Zbl 0913.14001)]. For the problem of plane conics tangent to five general (real) conics, the surprising answer is that all 3264 may be real [cf. \textit{F. Ronga}, \textit{A. Tognoli} and \textit{T. Vust}, Rev. Mat. Univ. Complutense Madr. 10, No. 2, 391-421 (1997; Zbl 0921.14036)]. Similarly, given any problem of enumerating \(p\)-planes incident on some given general subspaces, there are general real subspaces such that each of the (finitely many) incident \(p\)-planes is real [cf. \textit{F. Sottile}, Electron. Res. Announce Am. Math. Soc. 5, No. 5, 35-39 (1999; Zbl 0921.14037)]. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions real.
This problem of enumerating rational curves in a Grassmannian arose in at least two distinct areas of mathematics. The number of such curves was predicted by the formula of \textit{C. Vafa} [in: Essays on mirror manifolds, 96-119 (1992; Zbl 0827.58073)] and \textit{K. Intriligator} from mathematical physics. It is also the number of complex dynamic compensators which stabilize a given linear system, and the enumeration was solved in this context. The question of real solutions also arose in systems theory. This application will be discussed in section 4. number of real solutions; enumerative geometry ------, Real rational curves in Grassmannians , J. Amer. Math. Soc. 13 (2000), 333--341. JSTOR: Real algebraic and real-analytic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Relationships between surfaces, higher-dimensional varieties, and physics, Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Pole and zero placement problems Real rational curves in Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The present paper surveys the geometric properties of the Grassmann manifold \(Gr({\mathcal H})\) of an infinite dimensional complex Hilbert space \({\mathcal H}\). \(Gr({\mathcal H})\) is viewed as a set of operators, identifying each closed subspace \({\mathcal S}\subset{\mathcal H}\) with the orthogonal projection \(P_{\mathcal S}\) onto \({\mathcal S}\). Most of the results surveyed here were stated by G. Corach, H. Porta and L. Recht: submanifold structure, homogeneous reductive structure, local minimality of geodesics. Some recent results concerning the existence and uniqueness of a geodesic joining two given projections, which were obtained by the present author, are also presented. subspaces of a Hilbert space; projections; geodesics Andruchow, E., The Grassmann manifold of a Hilbert space, (Proceedings of the XIIth Dr. Antonio A.R. Monteiro Congress, (2014), Univ. Nac. Sur Dep. Mat. Inst. Mat. Bahía Blanca), 41-55 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds, Grassmannians, Schubert varieties, flag manifolds The Grassmann manifold of a Hilbert space | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review categorifies the Casimir element, which is the free generator of the centre of \(U_q(\mathfrak{sl}_2)\) as a \(\mathbb{Z}[q,q^{-1}]\)-algebra. This is a first step towards categorification of Lawrence's universal \(\mathfrak{sl}_2\) invariant of knots, within the context of the categorified representation theory of \(U_q(\mathfrak{sl}_2)\).
The main technical tool to handle the increased combinatorial complexity seen in such categorified structures is the diagrammatic calculus of a certain categorified algebra introduced by the third author [Adv. Math. 225, No. 6, 3327--3424 (2010; Zbl 1219.17012)]. categorified quantum group; graphical calculus; Casimir element; universal invariant; Witten-Reshetikhin-Turaev invariants Beliakova, A.; Khovanov, M.; Lauda, A., A categorification of the Casimir of quantum \(s l(2)\), Adv. Math., 230, 1442-1501, (2012) Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Double categories, \(2\)-categories, bicategories, hypercategories, Invariants of knots and \(3\)-manifolds A categorification of the Casimir of quantum \(\mathfrak{sl}(2)\) | 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 A partial flag variety \(\mathcal{P}_K\) of a Kac-Moody group \(G\) has a natural stratification into projected Richardson varieties. When \(G\) is a connected reductive group, a Bruhat atlas for \(\mathcal{P}_K\) was constructed in [\textit{X. He}, \textit{A. Knutson} and \textit{J.-H. Lu}, ``Bruhat atlases'', unpublished notes (2013)]: \( \mathcal{P}_K\) is locally modelled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruhat atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties.
A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac-Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff-Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the \(J\)-Schubert varieties for a Birkhoff-Bruhat atlas. The notion of the \(J\)-Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac-Moody group). The main result of this paper is the construction of a Birkhoff-Bruhat atlas for any partial flag variety \(\mathcal{P}_K\) of a Kac-Moody group. We also construct a combinatorial atlas for the index set \(Q_K\) of the projected Richardson varieties in \(\mathcal{P}_K\). As a consequence, we show that \(Q_K\) has some nice combinatorial properties. This gives a new proof and generalizes the work of \textit{L. K. Williams} [J. Reine Angew. Math. 609, 1--21 (2007; Zbl 1132.14045)] in the case where the group \(G\) is a connected reductive group. partial flag varieties; projected Richardson varieties; Kac-Moody groups; Bruhat atlas Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Linear algebraic groups over arbitrary fields, Kac-Moody groups A Birkhoff-Bruhat atlas for partial flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A normal variety \(X\) is called \(H\)-\textit{spherical} for the action of the complex reductive group \(H\) if it contains a dense orbit of some Borel subgroup of \(H\). We resolve a conjecture of \textit{R. Hodges} and \textit{A. Yong} [J. Lie Theory 32, No. 2, 447--474 (2022; Zbl 1486.14070)] by showing that their \textit{spherical permutations} are characterized by permutation pattern avoidance. Together with results of \textit{Y. Gao} et al. [``Classification of Levi-spherical Schubert varieties'', Preprint, \url{arXiv:2104.10101}] this implies that the sphericality of a Schubert variety \(X_w\) with respect to the largest possible Levi subgroup is characterized by this same pattern avoidance condition. spherical variety; Schubert variety; permutation pattern avoidance Compactifications; symmetric and spherical varieties, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Permutations, words, matrices Spherical Schubert varieties and pattern avoidance | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds. Realized as the OPE ring of A/2-twisted two-dimensional theories with (0,2) supersymmetry, quantum sheaf cohomology generalizes the notion of quantum cohomology. Complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models (GLSMs) and for nonabelian GLSMs describing Grassmannians. In this paper we continue to explore the quantum sheaf cohomology of nonabelian theories. We first propose a method to compute the generating relations for (0,2) GLSMs with (2,2) locus. We apply this method to derive the quantum sheaf cohomology of products of Grassmannians and flag manifolds. The dual deformation associated with the biholomorphic duality gives rise to an explicit IR duality of two A/2-twisted (0,2) gauge theories. Yang-Mills and other gauge theories in quantum field theory, Model quantum field theories, Supersymmetric field theories in quantum mechanics, Correspondence, duality, holography (AdS/CFT, gauge/gravity, etc.), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Formal methods and deformations in algebraic geometry, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) Quantum sheaf cohomology and duality of flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider the ``anti-dominant'' variants \(\Theta^-_\lambda\) of the elements \(\Theta_\lambda\) occurring in the Bernstein presentation of an affine Hecke algebra \(\mathcal H\). We find explicit formulae for \(\Theta^-_\lambda\) in terms of the Iwahori-Matsumoto generators \(T_w\) (\(w\) ranging over the extended affine Weyl group of the root system \(R\)), in the case (i) \(R\) is arbitrary and \(\lambda\) is a `minuscule' coweight, or (ii) \(R\) is attached to \(\text{GL}_n\) and \(\lambda=me_k\), where \(e_k\) is a standard basis vector and \(m\geq 1\). In the above cases, certain `minimal expressions' for \(\Theta^-_\lambda\) play a crucial role. Such minimal expressions exist in fact for any coweight \(\lambda\) for \(\text{GL}_n\). We give a sheaf-theoretic interpretation of the existence of a minimal expression for \(\Theta^-_\lambda\): the corresponding perverse sheaf on the affine Schubert variety \(X(t_\lambda)\) is the push-forward of an explicit perverse sheaf on the Demazure resolution \(m\colon\widetilde X(t_\lambda)\to X(t_\lambda)\). This approach yields, for a minuscule coweight \(\lambda\) of any \(R\), or for an arbitrary coweight \(\lambda\) of \(\text{GL}_n\), a conceptual albeit less explicit expression for the coefficient \(\Theta^-_\lambda(w)\) of the basis element \(T_w\) in terms of the cohomology of a fiber of the Demazure resolution. Bernstein presentations; affine Hecke algebras; Iwahori-Matsumoto generators; extended affine Weyl groups; root systems; minuscule coweights; standard basis vectors; minimal expressions; perverse sheaves; affine Schubert varieties; Demazure resolutions Haines, T., Pettet, A.: Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra. J. Algebra 252(1), 127--149 (2002) Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(H\) be a separable real Hilbert space. Denote by \({\mathcal{G}}_{\infty}(H)\) the Grassmannian consisting of closed subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every order preserving transformation of \({\mathcal{G}}_{\infty}(H)\) can be extended to an automorphism of the lattice of closed subspaces of \(H\). It follows from a result of \textit{G.\,W.\thinspace Mackey} [Ann.\ Math.\ (2) 43, 244--260 (1942; Zbl 0061.24210)] that automorphisms of this lattice are induced by invertible bounded linear operators. Hilbert Grassmannian; invertible bounded operator Pankov, M.: Order preserving transformations of the Hilbert Grassmannian, Arch. math. 89, 81-86 (2007) Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product), Grassmannians, Schubert varieties, flag manifolds Order preserving transformations of the Hilbert Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a split semi-simple linear algebraic group over a field \(k\) of characteristic not 2. Let \(P\) be a parabolic subgroup and let \(\mathcal L\) be a line bundle on the projective homogeneous variety \(G/P\). We give a simple condition on the class of \(\mathcal L\) in Pic\((G/P)/2\) in terms of Dynkin diagrams implying that the Witt groups \(W^{i}(G/P,\mathcal L)\) are zero for all integers \(i\). In particular, if \(B\) is a Borel subgroup, then \(W^{i}(G/B,\mathcal L)\) is zero unless \(\mathcal L\) is trivial in Pic\((G/B)/2\). flag varieties; split semi-simple linear algebraic group; parabolic subgroup; Dynkin diagrams; Witt groups; Borel subgroup DOI: 10.1016/j.jpaa.2011.07.004 \(K\)-theory of forms, Algebraic theory of quadratic forms; Witt groups and rings, Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Witt groups of rings, Cohomology theory for linear algebraic groups, Linear algebraic groups over arbitrary fields Trivial Witt groups 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 aim of this paper is to establish a new connection between the theory of totally positive Grassmannians and the theory of M-curves using the finite-gap theory for solitons of the KP equation. KP equation denotes the Kadomtsev-Petviashvili 2 equation, which is the first flow from the KP hierarchy. The authors assume that all KP times are real. They associate to any point of the real totally positive Grassmannian \(\mathrm{Gr}^{\mathrm{TP}}(N,M)\) a reducible curve which is a rational degeneration of an M-curve of minimal genus \(g=N(M-N)\), and they reconstruct the real algebraic-geometric data à la Krichever for the underlying real bounded multiline KP soliton solutions. The main result of this paper is that, for soliton solutions associated with the principal cell \(\mathrm{Gr}^{\mathrm{TP}}(N,M)\), the authors can always fix algebraic-geometric data on reducible curves, generating these solutions, with additional requirements that these curves are rational degenerations of some family of regular M-curves and that the divisor points satisfy the reality and regularity conditions of [\textit{B. A. Dubrovin} and \textit{S. M. Natanzon}, Math. USSR, Izv. 32, No. 2, 269--288 (1988; Zbl 0672.35072); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 2, 267--286 (1988)]. From this construction, it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth M-curves. In their approach, the authors rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection \(\mathrm{Gr}^{\mathrm{TP}}(r+1,M-N+r+1)\longmapsto \mathrm{Gr}^{\mathrm{TP}}(r,M-N+r)\). This paper is organized as follows: The first section is an introduction to the subject. In the second section the authors recall some known facts about regular finite gap and multi-soliton solutions of the KP equation. In the third section, the authors associate the rational degeneration of a genus \((M-1)\) hyperelliptic M-curve and construct the effective divisor for soliton data in \(\mathrm{Gr}^{\mathrm{TP}}(1, M)\). Then they present the main ideas of the algebraic-geometric construction for soliton data in \(\mathrm{Gr}^{\mathrm{TP}}(N, M)\). The forth section concerns the reducible M-curve and the effective divisor for soliton data in \(\mathrm{Gr}^{\mathrm{TP}}(N, M)\). The proof of some results is carried out in detail in section 5. Analytic properties of the effective divisor are given in section 6. In this section 7 the authors construct the rational curve and the vacuum pole divisor associated to generic soliton data in \(\mathrm{Gr}^{\mathrm{TP}}(2, 4)\). The paper is supported by two appendices. total positivity; Grassmannians; KP finite-gap theory; real solitons; M-curves Abenda S and Grinevich P 2015 Rational degeneration of M-curves, totally positive Grassmannians and KP-solitons (https://arxiv.org/abs/1506.00563) Relationships between algebraic curves and physics, Relationships between algebraic curves and integrable systems, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Plane and space curves, Grassmannians, Schubert varieties, flag manifolds Rational degenerations of \({\mathtt M}\)-curves, totally positive Grassmannians and KP2-solitons | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 history of Schubert calculus is presented in the connection with Hilbert's XV problem. Moreover its relations to contemporary algebraic geometry with suitable references are given. Hilbert's fifteenth problem; Schubert calculus Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, History of mathematics in the 20th century, History of algebraic geometry The status of Hilbert's Fifteenth Problem in 1993 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 many geometric properties of Schubert varieties of type \(A\) (and others) can be interpreted combinatorially. Given two permutations \(w, x \in S_n\) we give a combinatorial consequence of the property that the smooth locus of the Schubert variety \(X_w\) contains the Schubert cell \(Y_x\). This provides a necessary ingredient for the interpretation of recent representation-theoretic results of \textit{E. Lapid} and \textit{A. Mínguez} [Adv. Math. 339, 113--190 (2018; Zbl 1400.20047)] in terms of identities of Kazhdan-Lusztig polynomials. Schubert varieties; permutations Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over local fields and their integers A tightness property of relatively smooth 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 We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It is illustrated with the specializations of the algebras of quantum matrices at roots of unity and more generally all quantum Schubert cell algebras. noncommutative discriminants; algebra traces; Poisson prime elements; symplectic foliations; quantum groups at roots of unity Nguyen, B.; Trampel, K.; Yakimov, M., Noncommutative discriminants via Poisson primes, Adv. Math., 322, 269-307, (2017) Quantum groups (quantized enveloping algebras) and related deformations, Poisson manifolds; Poisson groupoids and algebroids, Poisson algebras, Grassmannians, Schubert varieties, flag manifolds Noncommutative discriminants via Poisson primes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be an algebraic variety with an action of a group \(G\) such that the derived category \(D(X)\) is generated by an exceptional sequence of sheaves invariant under the action of \(G\). In this paper, the author describes how to construct a semiorthogonal decomposition of the derived category \(D_G(X)\) of \(G\)-equivariant coherent sheaves on \(X\). The components of such a semiorthogonal decomposition are equivalent to the derived categories of twisted representations of \(G\). In the special case where \(G\) is finite or reductive over an algebraically closed field of characteristic \(0\), such a semiorthogonal decomposition is indeed given by exceptional objects.
The description and the proof of the result is detailed in several steps. In this way, the author describes accurately the case of finite groups first and of algebraic groups later. He then provides a list of results, starting from the most particular and ending with the most general one. The second part of the paper consists of some applications of the construction to different classes of varieties. In particular, the author treats the cases of projective spaces, quadrics, del Pezzo surfaces of degree \(\leq 5\) and Grassmannians providing explicit semiorthogonal decompositions for the derived categories of equivariant coherent sheaves. semiorthogonal decomposition; exceptional collection; twisted sheaf Elagin, A.: Semi-orthogonal decompositions for derived categories of equivariant coherent sheaves. Izv. ross. Akad. nauk ser. Mat. 73, No. 5, 37-66 (2009) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Derived categories, triangulated categories Semiorthogonal decompositions of derived categories of equivariant coherent sheaves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 reductive algebraic group of simply-laced type which is split over the real number field \(\mathbb{R}\). Fix an \(\mathbb{R}\)-split maximal torus \(T\) in \(G\), let \(B\) and \(B^-\) be two opposed Borel subgroups of \(G\) containing \(T\). Let \(w_0\) be the longest element of the Weyl group \(W=N(T)/T\) of \(G\). In the paper under review, the author considers the variety \({\mathcal B}^*=Bw_0B\cap B^-w_0B^-\). He defines a certain graph with \({\mathcal I}\times\{\pm 1\}^n\) its vertex set, where \({\mathcal I}\) is the set of all the reduced expressions of \(w_0\) and \(n\) is the length of \(w_0\). Then the main result of the paper is to show that when \(G\) is of type \(A\) or \(D\), the number of connected components of \({\mathcal B}^*\) coincides with the number of connected components of the graph defined. Finally, the author conjectures that this result should also hold in the case when \(G\) is of type \(E\). flag varieties; big cells; complex reductive algebraic groups of simply-laced type; maximal torus; Borel subgroups; Weyl groups; connected components Rietsch, K., The intersection of opposed big cells in real flag varieties, Proc. R. soc. lond. ser. A, 453, 1959, 785-791, (1997), 14M15 (20G20) Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions The intersection of opposed big cells in real flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) denote a semisimple algebraic group over a field of characteristic \(p>0\). Then \textit{R. Bezrukavnikov, I. Mirković}, and \textit{D. A. Rumynin} [Ann. Math. (2) (to appear)] have established a derived localization theorem for the sheaf \(\mathcal D\) of algebras of crystalline differential operators on the flag variety \(X=G/B\).
Let now \(\overline{\mathcal D}=\mathcal End_{X^{(1)}}(\mathcal O_X)\) where \(X^{(1)}\) is the twist by the Frobenius morphism \(F\) on \(X\). Then \(\overline{\mathcal D}\) is a central reduction of \(\mathcal D\). The authors prove in this paper a derived localization theorem for \(\overline{\mathcal D}\) in the case where \(G=\text{Sp}_4\). They do so by checking via explicit calculations of the socle series for baby Verma modules that \(F_*\mathcal O_X\) is tilting (this requires \(p\geq 5\)). This generalizes earlier work of the first author with \textit{Y. Hashimoto} and \textit{D. Rumynin} [Contemp. Math. 413, 43-62 (2006; Zbl 1121.14041)] where the same result is proved for \(G=\text{SL}_3\). crystalline differential operators; localization theorems; symplectic groups; flag varieties; Frobenius morphisms; invertible sheaves; simple Lie algebras; derived equivalences; enveloping algebras Kaneda, M.; Ye, J., Equivariant localization of \(\overline{D}\)-modules on the flag variety of the symplectic group of degree 4, J. Algebra, 309, 236-281, (2007) Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Universal enveloping (super)algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Group actions on varieties or schemes (quotients), Sheaves of differential operators and their modules, \(D\)-modules, Representation theory for linear algebraic groups Equivariant localization of \(\overline D\)-modules on the flag variety of the symplectic group of degree 4. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 row convex tableaux, we study the section rings of Bott-Samelson varieties of type A. We obtain flat deformations and standard monomial type bases of the section rings. In a separate section, we investigate a three-dimensional Bott-Samelson variety in detail and compute its Hilbert polynomial and toric degenerations. Bott-Samelson variety; straight tableaux; toric deformations D. Anderson, \textit{Effective divisors on Bott-Samelson varieties}, arXiv:1501.00034 (2014). Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of commutative algebra, Combinatorial aspects of representation theory Row convex tableaux and 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 We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Plücker algebras. We show that each such family of tableaux leads to a toric ideal, that can be realized as initial of the Plücker ideal, hence a toric degeneration for the flag variety. toric degenerations; SAGBI bases; Khovanskii bases; Grassmannians; flag varieties; semi-standard Young tableaux Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Fibrations, degenerations in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Toric degenerations of flag varieties from matching field tableaux | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 following result: Any smooth path on the projective line \(L\) which has endpoints \(p_i\) and \(p_j\) and does not pass through \(p_k\) determines a Lagrangian embedding of \(S^3\) in \(F^3\). pseudotoric structure; Lagrangian embedding; flag variety; toric variety; Morse function; Poisson bracket; del Pezzo surface; projective space; Hamiltonian isotopy Lagrangian submanifolds; Maslov index, Grassmannians, Schubert varieties, flag manifolds Pseudotoric structures and Lagrangian spheres in the flag variety \(F^3\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials 1. Die Vorgänger von Veronese waren C. Burali-Forti, G. Bellavitis und H. Grassmann.
2. Grundlegende Gesichtspunkte von Veronese zur mehrdimensionalen Geometrie. Beispiele: Zylindrische Hyperflächen in vier Dimensionen (Fig. 1, 2, 3).
3. Die Erzeugung von Hyperräumen, ausgehend von Punkt und Strahlenbüschel (Fig. 4a, 4b).
4. Inzidenzeigenschaften in \(S^4\) (Fig. 5, 6).
5. Die Axiomatisierung: Axiom 1: Es gibt einen Punkt, der nicht in \(S^3\) liegt und weitere vier Axiome. Zehn Sätze über \(S^4\) (Fig. 7-10), die Veronese 1891 bewiesen hat. hyperspaces; C. Burali-Forti; G. Bellavitis; H. Grassmann History of mathematics in the 19th century, History of geometry, History of mathematics in the 20th century, History of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) Foundations of four-dimensional geometry studied by Giuseppe Veronese | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 weave together a tale of two rings, SYM and QSYM, following one gold thread spun by R. P. Stanley. The lesson we learn from this tale is that ``combinatorial objects like to be counted by quasisymmetric functions''. enumerative combinatorics S.C. Billey and P.R.W. McNamara, The contributions of Stanley to the fabric of symmetric and quasisymmetric functions, in The Mathematical Legacy of Richard P. Stanley (P. Hersh, T. Lam, P. Pylyavskyy, V. Reiner, eds.), Amer. Math. Society, Providence, R.I., 2016, pp. 83--104. Symmetric functions and generalizations, Permutations, words, matrices, Coloring of graphs and hypergraphs, Combinatorial aspects of representation theory, Group actions on combinatorial structures, Combinatorics of partially ordered sets, Algebraic aspects of posets, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups The contributions of Stanley to the fabric of symmetric and quasisymmetric functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials As a generalization of skew Schur functions, the refined dual stable Grothendieck polynomials \(\tilde{g}_{\lambda/\mu}(x;t)\) can be easily defined using reverse plane partitions. Motivated by the Jacobi-Trudi formula and its dual for Schur functions, Grinberg conjectured a Jacobi-Trudi type formula for \(\tilde{g}_{\lambda/\mu}(x;t)\) for any skew partition \(\lambda/\mu\). The case for \(\mu=\emptyset\) has been confirmed by \textit{D. Yeliussizov} [J. Algebr. Comb. 45, No. 1, 295--344 (2017; Zbl 1355.05263)]. In the paper under review, the author completely proved Grinberg's conjecture. Comparing the elegant proof of the classical Jacobi-Trudi formula, the author's proof of Grinberg's conjecture is highly nontrivial, which relies on two bijections due to \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)] on RSE-tableaux and requires tedious analysis of the properties of these bijections on extended skew RSE-tableaux. As remarked by the author, Grinberg's conjecture was also independently proved by \textit{A. Amanov} and \textit{D. Yeliussizov} [`Determinantal formulas for dual Grothendieck polynomials'', Preprint, \url{arXiv:2003.03907}]. Jacobi-Trudi formula; plane partition; Grothendieck polynomial; Young tableau Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Jacobi-Trudi formula for refined dual stable Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathcal A}\) be a hyperplane arrangement, and \(k\) a field of arbitrary characteristic. We show that the projective degree-one resonance variety \(|<\|<|R|>\|>|^1({\mathcal A},k)\) of \({\mathcal A}\) over \(k\) is ruled by lines, and identify the underlying algebraic line complex \(L({\mathcal A},k)\) in the Grassmannian \(|<\|<|G|>\|>|(2,k^n)\), \(n=|{\mathcal A}|\). \(L({\mathcal A},k)\) is a union of linear line complexes corresponding to the neighborly partitions of subarrangements of \({\mathcal A}\). Each linear line complex is the intersection of a family of special Schubert varieties corresponding to a subspace arrangement determined by the partition.
In case \(k\) has characteristic zero, the resulting ruled varieties are linear and pairwise disjoint, by results of \textit{A. Libgober} and \textit{S. Yuzvinsky} [Compos. Math. 121, No.~3, 337--361 (2000; Zbl 0952.52020)]. We give examples to show that each of these properties fails in positive characteristic. The \((4,3)\)-net structure on the Hessian arrangement gives rise to a nonlinear component in \(|<\|<|R|>\|>|^1({\mathcal A},\overline{\mathbb Z}_3)\), a cubic hypersurface in \(\mathbb P^4\) with interesting line structure. This provides a negative answer to a question of A. Suciu. The deleted \(B_3\) arrangement has linear resonance components over \(\mathbb Z_2\) that intersect nontrivially. Falk, M.: Resonance varieties over fields of positive characteristic, Int. math. Res. notices 3 (2007) Configurations and arrangements of linear subspaces, Combinatorial aspects of matroids and geometric lattices, Rational and ruled surfaces, Grassmannians, Schubert varieties, flag manifolds, Relations with arrangements of hyperplanes, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Resonance varieties over fields of 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 [For the entire collection see Zbl 0639.00032.]
This is a survey article of a very broad scope concerning the study of configurations. A configuration is a finite collection of objects (points, lines,...,hyperplanes) in a space (vector, affine or projective) over a field.
The author considers many different aspects of this study (combinatorial type, stratification, topological classification, lattice representation - for example) and the relation of this study to different branches of mathematics (algebraic geometry, topology, algebra, lattice theory, matrix theory, theory of ordered sets - for example). He concludes with a number of difficult open problems.
The list of contents is as follows: 0. Introduction. 1. Configurations, combinatorial type. 2. Space of the configurations. 3. Grassmanian ideology. 4. Orientation, oriented combinatorial type. 5. Saturation, representation of lattices, implications. 6. Coordinate-form formulation. 7. Classification problems. Universality. Basic theorems. 8. The principle idea of the method: joint mechanisms and solutions of algebraic equations. 9. Open problems. space of configurations; polytope; stratification; topological classification; lattice representation; ordered sets; Configurations; combinatorial type A. M. Vershik, Topology of the convex polytopes' manifold, the manifold of the projective configurations of a given combinatorial type and representations of lattices, inTopology and Geometry--Rohlin Seminar (O. Y. Viro, ed.), Lecture Notes in Mathematics, Vol. 1346, Springer-Verlag, Berlin, 1988. Research exposition (monographs, survey articles) pertaining to convex and discrete geometry, Other designs, configurations, Stratifications in topological manifolds, Polytopes and polyhedra, Representation theory of lattices, Ordered sets, Grassmannians, Schubert varieties, flag manifolds, Configuration theorems in linear incidence geometry Topology of the convex polytopes' manifolds, the manifold of the projective configurations of a given combinatorial type and representations of lattices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book presents the proceedings of the 20th International Workshop on Hermitian Symmetric Spaces and Submanifolds, which was held at the Kyungpook National University from June 21 to 25, 2016. The Workshop was supported by the Research Institute of Real and Complex Manifolds (RIRCM) and the National Research Foundation of Korea (NRF). The Organizing Committee invited 30 active geometers of differential geometry and related fields from all around the globe to discuss new developments for research in the area. These proceedings provide a detailed overview of recent topics in the field of real and complex submanifolds.
The articles of this volume will be reviewed individually. For the preceding workshop see [Zbl 1333.53004]. Proceedings, conferences, collections, etc. pertaining to differential geometry, Global differential geometry of Hermitian and Kählerian manifolds, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Grassmannians, Schubert varieties, flag manifolds, Differential geometry of homogeneous manifolds, Proceedings of conferences of miscellaneous specific interest Hermitian-Grassmannian submanifolds. Daegu, Korea, July 2016. Proceedings of the 20th international workshop on Hermitian symmetric spaces and submanifolds, IWHSSS 2016, Daegu, South Korea, July 26--30, 2016 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper a projective classification of two-dimensional quartics on the Grassmannian G(1,3) is given corresponding to congruences of straight lines K(1,3) and K(2,2) in \(P_ 3\). Grassmannian Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry Classification of two-dimensional quartics on the Grassmannian G(1,3) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There is an obvious similarity between the two formulae for the permanent of a square matrix given by \textit{H. J. Ryser} [Combinatorial mathematics. Washington, D.C.: Mathematical Association of America (1963; Zbl 0112.24806)] and \textit{D. G. Glynn} [Eur. J. Comb. 31, No. 7, 1887--1891 (2010; Zbl 1209.15012)]. In this paper the author clarifies this relationship, making use of classic algebraic concepts (theorem of P. Serret), dependencies on the Veronesean (Veronese variety) and polarization identities for symmetric tensors. Using this, it is possible to obtain infinitely many such formulae. permanent; matrix; Veronesean; polarization identity; symmetric tensor Determinants, permanents, traces, other special matrix functions, Combinatorial aspects of matrices (incidence, Hadamard, etc.), Algebraic combinatorics, Projective techniques in algebraic geometry, Other types of codes Permanent formulae from the Veronesean | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G(k,n)\) denote the Grassmannian of all \((k+1)\)-dimensional linear subspaces of the space of all degree \(n\) homogeneous polynomials in two variables. Fix \(A\in G(k,n)\) and let \((f_0,\dots ,f_n)\) a base of it. A degree \(d\) syzygy of it is a set \((g_0,\dots ,g_n)\) of homogeneous degree \(d\) polynomials such that \(g_0f_0+ \cdots +g_nf_n = 0\). Let \(X_{k,j,d}\) be the set of all \(A\in G(k,n)\) with at least \(j\) linearly independent degree \(d\) syzygies. This set is irreducible and \(X_{k,j+1,d} \subset X_{k,j,d}\). The authors study the blowing-up of \(X_{k,j+1,d}\) in \(X_{k,j,d}\). They show that their objects are related to very classical ones (e.g. Waring's problem), generalizing the classical notion of apolarity. apolarity; linear systems on the projective line; Waring's problem; Grassmanian; blow up Divisors, linear systems, invertible sheaves, Configurations and arrangements of linear subspaces, Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Linear systems on \(\mathbb P^{1}\) with syzygies | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials With \(k\) an infinite field and \(\tau_1, \tau_2\) endomorphisms of \(k^m\), we provide a dimension bound on an open locus of a determinantal scheme, under which, for a general subspace \(V \subseteq k^m\) of dimension \(n \leq m / 2\), for \(v_1, v_2 \in V\) we have \(\tau_1( v_1) = \tau_2( v_2)\) only if \(v_1 = v_2\). Specializing to permutations composed by coordinate projections, we obtain an abstract proof of the theorem of [\textit{J. Unnikrishnan} et al., IEEE Trans. Inf. Theory 64, No. 5, 3237--3253 (2018; Zbl 1395.94168)]. homomorphic sensing; unlabeled sensing; linear regression without correspondences; Jordan form; determinantal varieties; rational normal scrolls Diagonalization, Jordan forms, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Signal theory (characterization, reconstruction, filtering, etc.) Determinantal conditions for homomorphic sensing | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 answers affirmatively the question in the survey article by \textit{P. A. Griffiths} [``An introduction to the theory of special divisors on algebraic curves'', Regional Conf. Ser. Math. 44 (1980; Zbl 0446.14010)]: Does the variety \(W_d^r\) of linear systems on a general curve of genus \(g\) with degree \(d\) and dimension at least \(r\) have the Brill-Noether dimension \(g - (r +1)(g - d +r)\)? Moreover the authors determine the class of this variety in the cohomology of the Jacobian and show that it is without multiple components.
The proof is by a detailed geometrical analysis of a classical degeneration idea of Castelnuovo's. It is formalized as the Castelnuovo-Severi-Kleiman conjecture (CSK): the family of \(P^k\)'s in \(P^d\) meeting the chords of a rational normal curve in \(P^d\) has the same dimension as if the chords were lines in general position, and the family has no multiple components.
The paper has three parts: (I) The reduction to CSK -- except for the absence of multiple components to \(W_d^r\); (II) The proof of CSK; (III) Proofs of the absence of multiple components.
(I) has been previously achieved by \textit{S. Kleiman} [Adv. Math. 22, 1--31 (1976; Zbl 0342.14012)]. The proof in this paper is by a geometrical argument based on duality of special divisors. (II) uses a degeneration of the chords to span an osculating flag. (III) is again by degeneration. The degenerate case is chosen so that number of intersections of two varieties as a set equals the algebraic intersection number. The techniques are those of classical algebraic geometry and Schubert calculus. variety of linear systems on a general curve; cohomology of Jacobian; Castelnuovo-Severi-Kleiman conjecture; algebraic intersection number; Schubert calculus Griffiths, P. \& Harris, J.,On the variety of special linear systems on a general algebraic curves, Duke Math. J.,47(1980), 233--272. Jacobians, Prym varieties, Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus, Enumerative problems (combinatorial problems) in algebraic geometry On the variety of special linear systems on a general algebraic curve | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this note, we announce the existence of degenerations of Schubert varieties in a minuscule \(G/P\), as well as of the class of Kempf varieties in the flag variety \(SL(n)/B\), to toric varieties. As a consequence, we obtain that determinantal varieties degenerate to toric varieties. Gröbner basis; degenerations of Schubert varieties; flag variety; toric varieties; determinantal varieties Lakshmibai, V., Degenerations of flag varieties to toric varieties.C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1229--1234. Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Determinantal varieties, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Degenerations of flag varieties to toric varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A tropical plane is meant to be a two-dimensional tropical linear subspace in the tropical projective space \(TP^{n-1}\). In this paper, the authors present a bijection between tropical planes and arrangements of metric trees. A Dressian \(Dr(d, n)\) is the tropical prevariety defined by all three term Plucker relation. Various combinatorial results about \(Dr(3, n)\) are given. An extension of the notion of Grassmannians and Dressians from the hypersimplex to arbitrary matroid polytope is given. tropical plane; metric trees; matroid S. Herrmann, A. Jensen, M. Joswig, B. Sturmfels, How to draw tropical planes. Electron. J. Comb. 16(2), Special volume in honor of Anders Björner, Research Paper 6, 26 (2009) , Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Grassmannians, Schubert varieties, flag manifolds, Trees How to draw tropical planes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Modern data analysis is flush with large databases and high-dimensional inference problems. Often the size of these problems directly relates to the fact that given data may have variations that can be difficult to incorporate into well-known, classical methods. One of these sources of variation is that of differing data sources, often called domain adaptation. Many domain adaptation techniques use the notion of a shared representation to attempt to remove domain-specific variation in given observations. One way to obtain these representations is through dimension reduction using a linear projection onto a lower-dimensional subspace of the predictors. Estimating linear projections is intrinsically linked with parameters lying on the Grassmannian. We present some historical approaches and their relationship to recent advances in domain adaptation that exploit the Grassmannian through subspace estimation. Image processing (compression, reconstruction, etc.) in information and communication theory, Directional data; spatial statistics, Estimation in multivariate analysis, Grassmannians, Schubert varieties, flag manifolds, Differential geometry of homogeneous manifolds Domain adaptation using the 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 The author gives necessary and sufficient conditions for the Plücker embeddings of real and complex Grassmannians to be taut. A convexity property of such Plücker embeddings is also examined. convexity; Grassmannian; perfect Morse function; Plücker embedding; taut embedding Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces, Grassmannians, Schubert varieties, flag manifolds, Eigenvalues, singular values, and eigenvectors Morse theoretic aspects of Plücker embeddings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 0487.00004.]
A closed subscheme \(Y\subset {\mathbb{P}}_ n\) is said to be of maximal rank if for all \(k\geq 1\) the restriction map \(H^ 0({\mathbb{P}}_ n,{\mathcal O}_{{\mathbb{P}}_ n}(k))\to H^ 0(Y,{\mathcal O}_ Y(k))\) is injective or surjective. In this pioneering paper it is proved that for all \(n\geq 3\) and all \(r>0\) the union of r disjoint general lines in \({\mathbb{P}}_ n\) has maximal rank. The method of this paper (with technical refinements) is used in many other papers to construct curves of maximal rank and vector bundles with good cohomology [see e.g. the authors, ''Cohomology of a general istanton bundle'', Ann. Sci. Ec. Norm. Super., IV. Sér. 15, 365- 390 (1982; Zbl 0509.14015)]. maximal rank conjecture; quadric surface; semicontinuity; degeneration; subscheme of maximal rank; lines in general position in projective n- space R. Hartshorne and A. Hirschowitz : Droites and position général dans l'espace projectif . In Algebraic Geometry, Proceedings La Rabida, 1981. Lecture Notes in Math. (61, Springer-Verlag: 169-189 (1982). Curves in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds Droites en position générale dans l'espace projectif | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) denote the subvariety of the grassmannian of nets of quadrics consisting of nets spanned by the minors of a \(3\times 2\) matrix of linear forms. Let \(I\) be the subvariety of \(X\) of nets with a fixed component. Denote by \(H\) the Hilbert scheme of twisted cubic curves. There is a natural map \(h:H\to X\) defined by assigning to a (possibly degenerate) twisted cubic the quadratic part of its homogeneous ideal. We prove the following:
Theorem. \(h:H\to X\) is the blowup of \(X\) along \(I\). grassmannian; nets of quadrics; Hilbert scheme of twisted cubic curves I. Vainsencher,A note on the Hilbert scheme of twisted cubics, Bol. S.B.M.18, \#1 (1987), 81-89. Parametrization (Chow and Hilbert schemes), Plane and space curves, Grassmannians, Schubert varieties, flag manifolds A note on the Hilbert scheme of twisted cubics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that the standard motives of a semisimple algebraic group \(G\) with coefficients in a field of order \(p\) are determined by the upper motives of the group \(G\). As a consequence of this result, we obtain a partial version of the motivic rigidity conjecture of special linear groups. The result is then used to construct the higher indexes which characterize the motivic equivalence of semisimple algebraic groups. The criteria of motivic equivalence derived from the expressions of these indexes produce a dictionary between motives, algebraic structures and the birational geometry of twisted flag varieties. This correspondence is then described for special linear groups and orthogonal groups (the criteria associated with other groups being obtained in \textit{C. De Clercq} and \textit{S. Garibaldi} [J. Lond. Math. Soc., II. Ser. 95, No. 2, 567--585 (2017; Zbl 1428.20043)]). The proofs rely on the Levi-type motivic decompositions of isotropic twisted flag varieties due to Chernousov, Gille and Merkurjev, and on the notion of pondered field extensions. motives; semisimple groups; flag varieties (Equivariant) Chow groups and rings; motives, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields Motivic equivalence of semisimple algebraic groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the structure and the CR geometry of the orbits \(M\) of a real form \(G_0\) of a complex semisimple Lie group \(G\) in a complex flag manifold \(X = G/Q\). It is shown that any such orbit \(M\) has a tower of fibrations over a canonically associated real flag manifold \(M_e\) with fibers that are products of Euclidean complex spaces and open orbits in complex flag manifolds. This result is used to investigate some topological properties of \(M\). For example, it is proved that the fundamental group \(\pi_1(M)\) depends only on \(M_e\) and on the conjugacy class of the maximally noncompact Cartan subgroups of the isotropy of the action of \(G_0\) on \(M\). In particular, the fundamental group of a closed orbit \(M\) is isomorphic to that of \(M_e\). Many other deep results about properties of the CR structure of the orbits and its invariants and about \(G_0\)-equivarant maps between orbits are obtained. flag manifolds; orbits of a real form; CR manifolds; Mostow fibrations; homogeneous CR geometry Altomani, A.; Medori, C.; Nacinovich, M., \textit{orbits of real forms in complex flag manifolds}, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9, 69-109, (2010) Differential geometry of homogeneous manifolds, Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras, CR structures, CR operators, and generalizations, Finite-type conditions on CR manifolds, Real submanifolds in complex manifolds, Homotopy groups of topological groups and homogeneous spaces Orbits of real forms in complex 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 that the variety of complete flags for any semisimple algebraic group is rigid in any smooth family of Fano manifolds. Fibrations, degenerations in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Automorphisms of surfaces and higher-dimensional varieties, Fano varieties On rigidity 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 show that the product in the quantum \(K\)-ring of a generalized flag manifold \(G/P\) involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum \(K\)-theory. At the core of the proof is a bound on the asymptotic growth of the \(J\)-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators. \(K\)-theory in geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds On the finiteness of quantum \(K\)-theory of a homogeneous space | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0535.00016.]
Here is considered certain theory of stationary points or loci arising from families of holomorphic sections of holomorphic vector bundles. The notion of Morse functions is generalized to families of holomorphic sections, called quasilinear sections or holomorphic sections in quasilinear position. One of the results contained in the paper shows that this kind of sections exist generically in certain cases. There is defined a particular subset of Schubert cycles, called stationary loci, associated to quasilinear holomorphic sections. They are concerned with a relation between these loci and characteristic classes. As Morse functions give us some information of topology of differential manifolds, it is shown that these loci tell us some complex analytic structure of complex manifolds, for example, arithmetic genus. stationary locus; stationary points; families of holomorphic sections of holomorphic vector bundles; Morse functions; holomorphic sections in quasilinear position; Schubert cycles H. Morimoto: Some Morse theoretic aspects of holomorphic vector bundles. Advanced Studies in Pure Math., 3, Geometry of Geodesies and related Topics. North-Holland, pp. 283-389 (1984). Sheaves and cohomology of sections of holomorphic vector bundles, general results, Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Some Morse theoretic aspects of holomorphic vector bundles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The classification of globally generated vector bundles on the projective spaces has been given by \textit{J. C. Sierra} and \textit{L. Ugaglia} in the case \(c_1=2\) [J. Pure Appl. Algebra 213, No. 11, 2141--2146 (2009; Zbl 1166.14011)]. Huh considered the case \(c_1=3\), obtaining the list of the triple embeddings of projective spaces in a suitable Grassmannian of projective lines [\textit{S. Huh}, Math. Nachr. 284, No. 11--12, 1453--1461 (2011; Zbl 1279.14014)]. Here the authors classify the globally generated rank two vector bundles with \(c_1\leq 5\), on the projective spaces of dimension \(n\geq 3\).
The main theorem says that such a bundle \(E\) always decomposes as a direct sum of two line bundles if \(n\geq 4\), whereas if \(n=3\) and \(E\) is indecomposable, then the pair of its Chern classes \((c_1, c_2)\) is one of the following eight pairs: \((2,2), (4,5), (4,6), (4,7), (4,8), (5,8), (5,10), (5,12)\). The authors describe the curves \(C\) obtained from a general section of \(E\) in the Serre correspondence. Moreover they prove that there exists a rank two globally generated vector bundle on \(\mathbb P^3\) with Chern classes in the list above in all cases except \((5,12)\), when the problem of the existence remains open. Vector bundles; rank two; globally generated; projective space Chiodera, Ludovica; Ellia, Philippe, Rank two globally generated vector bundles with \(c_1\leq5\), Rend. Istit. Mat. Univ. Trieste, 44, 413-422, (2012) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Rank two globally generated vector bundles with \(c_1\leq 5\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe a closed immersion from each representation space of a type \(A\) quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This ``bipartite Zelevinsky map'' restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type \(A\) quivers of arbitrary orientation, we give the same result up to some factors of general linear groups.
These identifications allow us to recover results of \textit{G. Bobiński} and \textit{G. Zwara} [Manuscr. Math. 105, No. 1, 103--109 (2001; Zbl 1031.16012); Colloq. Math. 94, No. 2, 285--309 (2002; Zbl 1013.14011)]; namely, we see that orbit closures of type \(A\) quivers are normal, Cohen-Macaulay and have rational singularities. We also see that each representation space of a type \(A\) quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split. Kinser, Ryan; Rajchgot, Jenna, Type \(A\) quiver loci and Schubert varieties, J. Commut. Algebra, 7, 2, 265-301, (2015) Grassmannians, Schubert varieties, flag manifolds, Linkage, complete intersections and determinantal ideals, Representations of quivers and partially ordered sets Type \(A\) quiver loci and 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 Es wird das Theorem bewiesen: Zu jedem Schubert-Raum (\({\mathcal S},{\mathcal J})\) gibt es einen dreidimensionalen irreduziblen projektiven Raum \({\mathbb{P}}\), so daß (\({\mathcal S},{\mathcal J})\) isomorph ist zum Flaggenraum (\({\mathcal S}({\mathbb{P}}),{\mathcal J}({\mathbb{P}})\) von \({\mathbb{P}}\); dieser ist eine Inzidenzstruktur mit den Flaggen von \({\mathbb{P}}\) als ''Punkten'' und mit dreierlei Arten von ''Geraden'', nämlich den Flaggen mit gemeinsamem Paar (Gerade, Ebene) bzw. (Punkt, Ebene) bzw. (Punkt, Gerade). Diese Strukturen werden axiomatisch eingeführt und untersucht. Schubert manifold; flag Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Grassmannians, Schubert varieties, flag manifolds A characterization of Schubert manifold associatd with a three- dimensional projective space | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V (d, m, k)\) be the variety of plane projective irreducible curves of degree \(d\) with \(m\) nodes and \(k\) cusps as their only singularities. We prove that \(V(d, m, k)\) is non-empty, non-singular and irreducible when \(m+ 2k< \alpha d^2\), where \(\alpha\) is some absolute explicit constant. This estimate is optimal with respect to the exponent of \(d\). irreducible curves; degree; nodes; cusps Eugenii Shustin, Smoothness and irreducibility of varieties of plane curves with nodes and cusps, Bull. Soc. Math. France 122 (1994), no. 2, 235 -- 253 (English, with English and French summaries). Singularities of curves, local rings, Grassmannians, Schubert varieties, flag manifolds Smoothness and irreducibility of varieties of plane curves with nodes and cusps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann variety which we call Wronski map. Our main result is computation of degrees of the real Wronski maps. Connections with real algebraic geometry and control theory are described. Wronski determinants; Grassmann variety; Wronski map; control theory Eremenko, A. and Gabrielov, A., Degrees of real {W}ronski maps, Discrete \& Computational Geometry. An International Journal of Mathematics and Computer Science, 28, 3, 331-347, (2002) Grassmannians, Schubert varieties, flag manifolds, Determinants, permanents, traces, other special matrix functions, Topology of real algebraic varieties, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Degrees of real Wronski maps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(w_0\) denote the permutation \([n,n-1,...,2,1]\). We give a new explicit formula for the Kazhdan--Lusztig polynomials \(P_{w_0w,w_0x}\) in \(S_n\) when \(x\) indexes an irreducible component of the singular locus of the Schubert variety \(X_w\). To do this, we utilize a standard identity that relates \(P_{x,w}\) and \(P_{w_0w,w_0x}\). Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds A formula for certain inverse Kazhdan--Lusztig polynomials in \(S_{n}\) | 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 This paper studies an analogue of the geometric Satake correspondence [\textit{I. Mirković} and \textit{K. Vilonen}, Ann. Math. (2) 166, No. 1, 95--143 (2007; Zbl 1138.22013)] for the mirabolic affine Grassmannian introduced in [\textit{M. Finkelberg} et al., Sel. Math., New Ser. 14, No. 3--4, 607--628 (2009; Zbl 1215.20041)], which can be roughly said to be a semi-infinite analogue of the usual affine Grassmannian \(\mathbf{Gr}=\mathbf{G}_{\mathbb{C}((t))}/\mathbf{G}_{\mathbb{C}[[t]]}\) of \(\mathbf{G} = \mathrm{GL}_N\), and defined to be \(\mathbf{Gr} \times \mathbf{V}\) with \(\mathbf{V}=V \otimes \mathbb{C}((t))\), \(V=\mathbb{C}^N\).
According to the three possible choices of pullback, denoted by \(!\), \(*\) and \(!*\), this paper introduces three versions of equivariant constructible sheaves on the mirabolic affine Grassmannian, each of which is equipped with a monoidal structure by convolution operations. The corresponding other-side categories are defined in terms of the three \(\mathbb{C}[[\hbar]]\)-algebras \(\mathfrak{D}^{\bullet}_{2,0}\), \(\mathfrak{D}^{\bullet}_{0,2}\) and \(\mathfrak{D}^{\bullet}_{1,1}\), which are graded versions of the algebra of differential operators on \(\mathfrak{gl}_N\). They are also equipped with monoidal structures. Among these triangulated monoidal categories, three equivalences are constructed in Theorems 3.6.1 and 5.1.1, which are the first main results.
The second main result of the paper is an equivalence that relates representations of a Lie supergroup to the category of \(\mathrm{GL}(N-1, \mathbb{C}[[t]])\)-equivariant perverse sheaves on the affine Grassmannian. These results fit into a more general framework of conjectures due to the series of works by Gaiotto, and to the on-going project of Ben-Zvi, Sakellaridis and Venkatesh, as explained in \S 1.6, \S 1.7 and \S 2. Satake equivalence; mirabolic affine Grassmannian; supergroups Geometric Langlands program (algebro-geometric aspects), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras, Geometric Langlands program: representation-theoretic aspects, Supervarieties Mirabolic Satake equivalence and supergroups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 combinatorial bases of the \(T\)-equivariant cohomology \(H_{T}^{\bullet}(\Sigma,k)\) of the Bott-Samelson variety \(\Sigma\) under some mild restrictions on the field of coefficients \(k\). These bases allow us to prove the surjectivity of the restrictions \(H_{T}^{\bullet}(\Sigma,k)\rightarrow H_{T}^{\bullet}(\pi^{-1}(x),k)\) and \(H_{T}^{\bullet}(\Sigma,k)\rightarrow H_{T}^{\bullet}(\Sigma\setminus \pi^{-1}(x),k)\), where \(\pi:\Sigma\rightarrow G/B\) is the canonical resolution. In fact, we also construct bases of the targets of these restrictions by picking up certain subsets of certain bases of \(H_{T}^{\bullet}(\Sigma,k)\) and restricting them to \(\pi^{-1}(x)\) or \(\Sigma\setminus \pi^{-1}(x)\) respectively. As an application, we calculate the cohomology of the costalk-to-stalk embedding for the direct image \(\pi_{\ast}{\underline{k}}_{\Sigma}\). This algorithm avoids division by 2, which allows us to re-establish 2-torsion for parity sheaves in Braden's example, [\textit{G. Williamson} and \textit{T. Braden}, Math. Z. 272, No. 3--4, 697--727 (2012; Zbl 1284.14031)]. Bott-Samelson variety; equivariant cohomology; parity sheaves Equivariant homology and cohomology in algebraic topology, Grassmannians, Schubert varieties, flag manifolds Bases of \(t\)-equivariant cohomology of 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 Let \(G\) be a connected, simply connected, simple, complex Lie group and \(B\subset G\) a Borel subgroup. The author recovered in [Math. Res. Lett. 11, 35--48 (2004; Zbl 1062.14069)], in a purely combinatorial fashion, \textit{B.~Kim}'s presentation [Ann. Math. (2) 149, 129--148 (1999; Zbl 1054.14533)] of the quantum cohomology ring of the flag variety \(G/B\).
The main goal of this paper is to construct a combinatorial quantum product on \(H^*(G/B)\otimes\mathbb R[\{q_i\}_{i=1,\dots,b_2(G/B)}]\) which satisfies the usual properties of the quantum product (e.g. commutativity, associativity, Frobenius property). Next, Mare applies his result in [loc. cit.] to describe this ring in terms of relations and generators, and finds explicit quantum representatives for the Schubert classes. Finally, he proves that the combinatorial and the usual quantum product on \(H^*(G/B)\otimes\mathbb R[\{q_i\}]\) agree. flag manifolds; quantum Chevalley formula; quantum Giambelli formula Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds The combinatorial quantum cohomology ring of \(G/B\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The notion of differential graded scheme (DG scheme) was introduced by M. Kontsevich about twenty years ago, in a first approach to what is now called derived algebraic geometry. By definition, a DG scheme is a usual scheme \((X,\mathcal{O}_X)\) together with a sheaf \(\mathcal{O}_X^\bullet\) of differential graded \(\mathcal{O}_X\)-algebras such that the natural map \(\mathcal{O}_X\to H^0(X,\mathcal{O}_X^\bullet)\) is surjective. The theory of DG schemes was further developed by M. Kapranov, I. Ciocan-Fontanine, K. Behrend, and others, mainly in order to study derived moduli spaces and derived moduli stacks in algebraic geometry.
On the other hand, there is the notion of indschem due to A. Beilinson and V. Drinfeld, which is defined as follows: An indscheme is a presheaf on the category of affine schemes that is representable as an inductive limit of closed embeddings of schemes. A prominent example of an indscheme is the infinite Grassmannian Gr\(_G\) corresponding to an algebraic group \(G\), which plays a crucial role in the geometric Langlands program.
The main motivation for the paper under review is to develop a suitable framework for the study of the category QCoh(Gr\(_G\)) of quasi-coherent sheaves on Gr\(_G\), and that in the context of \textit{D. Gaitsgory}'s multi-volume series [``Notes on geometric Langlands'', \url{http://www.math.harvard.edu/~gaitsgde/GL/}]. To this end, the authors introduce the conceptual framework of so-called DG indschemes, in which the afore-mentioned theories of DG schemes and indschemes are combined to create a new and powerful abstract machinery in derived algebraic geometry. As the entire approach is heavily based on the language, the methods, and the results of D. Gaitsgory's ``Notes on geometric Langlands'' ([loc. cit.] as well as of \textit{J. Lurie}'s large preprint series [``Derived algebraic geometry'', \url{http://www.math.harvard.edu/~lurie}], we here confine ourselves to briefly indicate the topics covered in the ten sections, instead of undertaking the hopeless attempt to explain any of the countless technical details. After a comprehensive introduction to the motivation, the structure, and the main results of the present paper, Section 1 is devoted to the definition of DG indschemes and their basic properties. Section 2 provides a detailed study of quasi-coherent sheaves and ind-coherent sheaves on DG indschemes, respectively, together with their fundamental functorial properties. Closed embeddings into a DG indscheme and push-outs are described in Section 3, whereas Sections 4 and 5 elaborate a characterization of DG indschemes via deformation theory. Thereafter, formal completions of DG indschemes are analyzed in Section 6, and the study of quasi-coherent and ind-coherent sheaves on formal completions of DG indschemes follows in Section 7. The notion of formal smoothness of DG indschemes is introduced in Section 8, and a characterization of formally smooth DG indschemes via deformation theory is provided there, too. Section 9 is devoted to a comparison between classical and derived formal smoothness of DG indschemes, with an outlook to applications with respect to loop groups and infinite Grassmannians, thereby coming back to the motivation for entire paper as indicated in the introduction above. Finally, Section 10 establishes a functorial equivalence between the categories of quasi-coherent sheaves and ind-coherent sheaves on a DG indscheme, respectively. As the authors point out, this main result of theirs was also proved by J. Lurie using a different method. DG schemes; indschemes; DG indschemes; quasi-coherent sheaves; deformation theory; formal smoothness; prestacks; stacks; infinite Grassmannians Gaitsgory, D.; Rozenblyum, N.: DG indschemes Generalizations (algebraic spaces, stacks), Geometric Langlands program (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Double categories, \(2\)-categories, bicategories, hypercategories, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) DG indschemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 the number of non-isomorphic unit interval orders on \([n]\) equals the \(n\)-th Catalan number. Using work of \textit{M. Skandera} and \textit{B. Reed} [J. Comb. Theory, Ser. A 103, No. 2, 237--256 (2003; Zbl 1029.06002)] and work of \textit{A. Postnikov} [``Total positivity, Grassmannians, and networks'', Preprint, \url{arXiv:math/0609764}], we show that each unit interval order on \([n]\) naturally induces a rank \(n\) positroid on \([2n]\). We call the positroids produced in this fashion unit interval positroids. We characterize the unit interval positroids by describing their associated decorated permutations, showing that each one must be a \(2n\)-cycle encoding a Dyck path of length \(2n\). positroid; Dyck path; unit interval order; semiorder; decorated permutation; positive Grassmannian Combinatorial aspects of matrices (incidence, Hadamard, etc.), Symmetric functions and generalizations, Combinatorics of partially ordered sets, Grassmannians, Schubert varieties, flag manifolds Dyck paths and positroids from unit interval orders | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 semi-simple Lie group and form its maximal flag manifold \(\mathbb{F} = G \slash P = U \slash T\) where \(P\) is a minimal parabolic subgroup, \(U\) a compact real form and \(T = U \cap P\) a maximal torus of \(U\). The aim of this paper is to study invariant generalized complex structures on \(\mathbb{F}\). We describe the invariant generalized almost complex structures on \(\mathbb{F}\) and classify which one is integrable. The problem reduces to the study of invariant 4-dimensional generalized almost complex structures restricted to each root space, and for integrability we analyze the Nijenhuis operator for a triple of roots such that its sum is zero. We also conducted a study about twisted generalized complex structures. We define a new bracket `twisted' by a closed 3-form \(\Omega\) and also define the Nijenhuis operator twisted by \(\Omega\). We classify the \(\Omega\)-integrable generalized complex structure. flag manifolds; homogeneous space; semi-simple Lie groups; generalized complex structures Generalized geometries (à la Hitchin), Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces Invariant generalized complex structures on flag manifolds | 0 |
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