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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 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum 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 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum 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 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum 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 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum 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 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum 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 The paper generalizes the concept of partition varieties introduced in a preceding paper by the same author. For this purpose, \(\gamma\)-compatible partitions \(\lambda\) are introduced, where \(\gamma\) is a composition of some integers. It is shown that a certain quotient space associated to such composition \(\gamma\) and partition \(\lambda\) is a projective variety. Moreover, this generalized partition variety is a CW-complex which can be described combinatorially in terms of \(\gamma\)-compatible rook placements of the Ferrers board of \(\lambda\). Finally, the Poincaré polynomial \(P_{\lambda,\gamma}(q)\) for the cohomology equals \(RL_{\lambda,\gamma}(q^2)\), where \(RL_{\lambda,\gamma}\) is the \(\gamma\)-rook length polynomial. The combinatorial description of generalized partition varieties applies to the homology and cohomology of flag manifolds and Grassmann manifolds. rook length polynomial; projective variety; partition variety; Grassmannian manifold; flag manifold; Schubert cells; Bruhat order Ding, K., Rook placements and generalized partition varieties, Discrete Math., 176, 1-3, 63-95, (1997) Algebraic combinatorics, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of partitions of integers, Permutations, words, matrices Rook placements and generalized partition varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper deals with partition varieties, i.e., projective varieties associated to some partition \(\lambda=(\lambda_1,\dots,\lambda_n)\). The main result is that the partition variety of \(\lambda\) is a CW-complex which has a cellular decomposition that can be described combinatorially in terms of rook placements of the Ferrers board of \(\lambda\), and that the Poincaré polynomial \(P_\lambda(q)\) for the cohomology equals \(R_\lambda(q^2)\), where \(R_\lambda\) is the rook length polynomial of \(\lambda\). The combinatorial description of the cells also characterizes the geometric situtation, i.e., whether some cell is contained in the closure of another cell. The exposition is self-contained. Many examples and four appendices (concerning Grassmannian manifolds, Schubert varieties, flag manifolds and an embedding theorem) make the paper accessible also to the non-expert in algebraic geometry. Rook length polynomial; projective variety; partition variety; Grassmannian manifold; flag manifold; Schubert cells; Bruhat order K. Ding, Rook placements and cellular decomposition of partition varieties, Discrete Math. 170 (1997), 107-151. Algebraic combinatorics, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of partitions of integers, Permutations, words, matrices Rook placements and cellular decomposition of partition 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 (tree) amplituhedron \(\mathcal A_{n,k,m}\) is the image in the Grassmannian Gr\(_{k,k+m}\) of the totally nonnegative part of Gr\(_{k,n}\), under a (map induced by a) linear map which is totally positive. It was introduced by \textit{N. Arkani-Hamed} and \textit{J. Trnka} [``The amplituhedron'', J. High Energy Phys. 2014, No. 10, Paper No. 030, 33 p. (2014, \url{doi:10.1007/JHEP10(2014)030})] in order to give a geometric basis for the computation of scattering amplitudes in \(\mathcal N=4\) supersymmetric Yang-Mills theory. When \(k+m=n\), the amplituhedron is isomorphic to the totally nonnegative Grassmannian, and when \(k=1\), the amplituhedron is a cyclic polytope. While the case \(m=4\) is most relevant to physics, the amplituhedron is an interesting mathematical object for any \(m\). We study it in the case \(m=1\). We start by taking an orthogonal point of view and define a related ``B-amplituhedron'' \(\mathcal B_{n,k,m}\), which we show is isomorphic to \(\mathcal A_{n,k,m}\). We use this reformulation to describe the amplituhedron in terms of sign variation. We then give a cell decomposition of the amplituhedron \(\mathcal A_{n,k,1}\) using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also show that \(A_{n,k,1}\) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement. We deduce that \(A_{n,k,1}\) is homeomorphic to a ball. total positivity; Grassmannian; hyperplane arrangements; amplituhedra; sign variation S.N. Karp and L.K. Williams, \textit{The m=1 amplituhedron and cyclic hyperplane arrangements}, arXiv:1608.08288 [INSPIRE]. Algebraic combinatorics, Grassmannians, Schubert varieties, flag manifolds, Supersymmetric field theories in quantum mechanics, Yang-Mills and other gauge theories in quantum field theory The \(m=1\) amplituhedron and cyclic hyperplane arrangements | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let $G=\mathrm{GL}_n(\mathbb{K})$ be the general linear group over an algebraically closed field $\mathbb{K}$. Let $G/B$ be the full flag variety consisting of complete flags
\[
F_\bullet:\quad \{0\}= F_0\subset F_1\subset \cdots \subset F_n=\mathbb{K}^n
\]
of subspaces of $\mathbb{K}^n$ so that $\dim F_i=i$. Fix a Hessenberg function $h:\{1, \dots, n\} \to
\{1, \dots, n\}$ which is, by definition, a non-decreasing function such that $h(i) \geq i$ for all $i$. Fix a linear operator $X:\mathbb{K}^n \to \mathbb{K}^n$. Recall that the
corresponding Hessenberg variety is defined by:
\[
\mathcal{Y}_{X, h}:= \{F_\bullet: X(F_i)\subset F_{h(i)} \text{ for all } i\}\subset G/B.
\]
In the paper under review, the authors give an explicit expression for the class of $ \mathcal{Y}_{X, h}$ in the $K$-theory $K(G/B)$ as well as in the cohomology $H^*(G/B)$ in terms of certain Grothendieck (resp. Schubert) polynomial if $X$ is a regular operator. In fact, they generalize the result for any $X$ such that $ \mathcal{Y}_{X, h}$ has the expected dimension $\sum_i(h(i)-i)$.
We recall that a different formula for the cohomology class of $ \mathcal{Y}_{X, h}$ was given by \textit{D. Anderson} and the second author [J. Algebra 323, No. 10, 2605--2623 (2010; Zbl 1218.14041)] using degeneracy arguments. Also, a different formula for the $K$-theory class of $ \mathcal{Y}_{X, h}$ was given by \textit{H. Abe}, \textit{N. Fujita} and \textit{H. Zeng} [``Geometry of regular Hessenberg varieties'', Transform. Groups (to appear; \url{doi:10.1007/s00031-020-09554-8})]. But the formulae in this paper differ from the ones given by Abe-Fujita-Zeng and Anderson-the second author viewed as polynomials (but of course not as $K$-theory or cohomology classes). The authors also reprove that $ \mathcal{Y}_{X, h}$ is Cohen-Macaulay (hence equi-dimensional). Hessenberg variety; $K$-theory class; cohomology class; Cohen-Macaulay schemes Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Computational aspects and applications of commutative rings, Algebraic combinatorics A formula for the cohomology and \(K\)-class of a regular Hessenberg variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give combinatorial descriptions of the restrictions to \( T\)-fixed points of the classes of structure sheaves of Schubert varieties in the \( T\)-equivariant \( K\)-theory of Grassmannians and of maximal isotropic Grassmannians of orthogonal and symplectic types. We also give formulas, based on these descriptions, for the Hilbert series and Hilbert polynomials at \( T\)-fixed points of the corresponding Schubert varieties.
These descriptions and formulas are given in terms of two equivalent combinatorial models: excited Young diagrams and set-valued tableaux. The restriction formulas are positive, in that for a Schubert variety of codimension \( d\), the formula equals \( (-1)^d\) times a sum, with nonnegative coefficients, of monomials in the expressions \( (e^{-\alpha } -1)\), as \( \alpha \) runs over the positive roots. In types \( A_n\) and \( C_n\) the restriction formulas had been proved earlier by \textit{V. Kreiman} [``Schubert classes in the equivariant \(K\)-theory and equivariant cohomology of the Grassmannian'', Preprint, \url{arXiv:math/0512204}; ``Schubert classes in the equivariant \(K\)-theory and equivariant cohomology of the Lagrangian Grassmannian'', Preprint, \url{arXiv:math/0602245}] using a different method. In type \( A_n\), the formula for the Hilbert series had been proved earlier by \textit{L. Li} and \textit{A.Yong} [Adv. Math. 229, No. 1, 633--667 (2012; Zbl 1232.14033)].
The method of this paper, which relies on a restriction formula of \textit{W. Graham} [``Equivariant \(K\)-theory and Schubert varieties'', Preprint] and \textit{M. Willems} [Duke Math. J. 132, No. 2, 271--309 (2006; Zbl 1118.19002)], is based on the method used by \textit{T. Ikeda} and \textit{H. Naruse} [Trans. Am. Math. Soc. 361, No. 10, 5193--5221 (2009; Zbl 1229.05287)] to obtain the analogous formulas in equivariant cohomology. The formulas we give differ from the \( K\)-theoretic restriction formulas given by \textit{T. Ikeda} and \textit{H. Naruse} [Adv. Math. 243, 22--66 (2013; Zbl 1278.05240)], which use different versions of excited Young diagrams and set-valued tableaux. We also give Hilbert series and Hilbert polynomial formulas which are valid for Schubert varieties in any cominuscule flag variety, in terms of the 0-Hecke algebra. set-valued tableaux W. Graham and V. Kreiman, \textit{Excited Young diagrams, equivariant K-theory, and Schubert varieties}, Trans. AMS, 367 (2015), pp. 6597--6645. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Algebraic combinatorics Excited Young diagrams, equivariant \(K\)-theory, 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 The set of conjugacy classes appearing in a product of conjugacy classes in a compact, \(1\)-connected Lie group \(K\) can be identified with a convex polytope in the Weyl alcove. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety \(G/P\), where \(G\) is the complexification of \(K\) and \(P\) is a maximal parabolic subgroup. This generalizes the results for \(SU(n)\) of \textit{S. Agnihotri} and \textit{C. Woodward} [Math. Res. Lett. 5, No. 6, 817--836 (1998; Zbl 1004.14013)] and \textit{P. Belkale} [Compos. Math. 129, No. 1, 67--86 (2001; Zbl 1042.14031)] on the eigenvalues of a product of unitary matrices and quantum cohomology of Grassmannians. conjugacy classes; parabolic bundles; quantum cohomology; generalized flag variety; Grassmannians; Schubert calculus C. Teleman and C. Woodward, ''Parabolic bundles, products of conjugacy classes and Gromov-Witten invariants,'' Ann. Inst. Fourier \((\)Grenoble\()\), vol. 53, iss. 3, pp. 713-748, 2003. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics, Group actions on varieties or schemes (quotients) Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 superb paper under review is quite (and probably unavoidably) technical. That is why the reviewer will not put a special effort to simplify the description of its content. Otherwise this short summary may turn useless for both non experts and specialists: the former because in any case would lack background, and the latter because the essential mathematical content of the paper would be lost behind excess of simplification. In any case, to save a minimum of friendly shape, let us begin this short summary as if it were a tale.
Schubert calculus. At the very beginning was the intersection theory of the complex Grassmann manifolds \(G(k,n)\) parametrizing \(k\)-dimensional subspaces of \({\mathbb C}^n\). Grassmannians, however, are a special kind of flag varieties, which are in turn special kind of homogeneous projective varieties, i.e. quotient \(G/B\) of a complex connected semi-simple Lie group modulo the action of a Borel subgroup \(B\). Thus, nowadays, the location Schubert calculus has acquired a much broader meaning. Not only for Grassmannians, but on general flag varieties, and not only classical cohomology, but also quantum, equivariant or quantum-equivariant, up to its \(K\)-theory and its connective \(K\)-theory (a theory interpolating, in a suitable sense, the \(K\)-theory and the intersection theory of a homogeneous space).
The paper under review puts itself in this very general framework using in a creative manner a new algebraic tool, what the authors call \textsl{formal root polynomials}, with the purpose of studying the elliptic cohomology of the homogeneous space \(G/B\): this means a cohomology theory where all the odd parts vanish and there is an invertible element \(h\in H^2\) inducing a complex orientation with the same formal group law as that of an elliptic curve. There is a correspondence between generalized cohomology theories and formal group laws, and in particular the authors investigate the \textsl{hyperbolic group law} introduced in Section 2.2. The corresponding Schubert calculus is so called by the authors \textsl{hyperbolic Schubert calculus.} and enables to study the elliptic cohomology of homogeneous spaces, extending previous work by Billey and Graham-Willems letting it to work uniformly in all Lie type. The definition of formal root polynomial is quite technical and is not worth to be recalled in the present review. However the idea is that of replacing, or rather extend, the notion of root polynomials heavily used by \textit{S. C. Billey} [Duke Math. J. 96, No. 1, 205--224 (1999; Zbl 0980.22018)] and by \textit{M. Willems} [Bull. Soc. Math. Fr. 132, No. 4, 569--589 (2004; Zbl 1087.19004)].
After introducing the formal root polynomial, whose definition depends on a reduced word for a Weyl group element, the main Theorem 3.10 states that indeed it does not depend on such a word provided that the formal group law is the hyperbolic one. Section 4 is devoted to applications: in particular the authors show how their techniques provide an efficient method to compute the transition matrix between two natural bases of the formal \textsl{Demazure algebra}, another gadget introduced and explained in Section 2. Section 5 is concerned with localization formulas in cohomology and \(K\)-theory, while section 6 is not only devoted to show further applications of root polynomials to compute Bott-Samelson classes, but also to propose a couple of conjectures in the hyperbolic Schubert calculus, based mainly on analogies and experimental evidence. The paper ends with a comprehensive reference list: among the key ones, the paper by Goresky, Kottwitz and MacPherson [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)], the important 1974 paper by \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér. (4) 7, 53--88 (1974; Zbl 0312.14009)] on desingularization of generalized Schubert varieties, a couple of papers by Graham and Graham-Kumar on equivariant \(K\)-theory, and the papers by Billey and Willems, that inspired the research developed in this amazing step forward a generalized cohomology Schubert calculus. Schubert calculus; equivariant oriented cohomology; flag variety; root polynomial; hyperbolic formal group law ] C. Lenart and K. Zainoulline, Towards generalized cohomology Schubert calculus via formal root polynomials, arXiv:1408.5952. Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Equivariant \(K\)-theory, Generalized (extraordinary) homology and cohomology theories in algebraic topology, Bordism and cobordism theories and formal group laws in algebraic topology, Algebraic combinatorics Towards generalized cohomology Schubert calculus via formal root 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 goal of these notes is to explain the combinatorics that arises in the study of \(K\)-theoretic intersection theory of some concrete varieties and degeneracy loci. The material is mostly taken from the papers [the author, Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015); Duke Math. J. 115, No. 1, 75--103 (2002; Zbl 1052.14056)], as well as from our joint work with \textit{A. Kresch}, \textit{H. Tamvakis} and \textit{A. Yong} [Am. J. Math. 127, No. 3, 551--567 (2005; Zbl 1084.14048)]. Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes Combinatorial \(K\)-theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all \(k\) and \(n\) on what we call the minimal kinematics. We prove that on the \textit{minimal kinematics}, the scattering equations on the configuration space of \(n\) points on \(\mathbb{CP}^{k-1}\) has a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is an all \(k\) generalization of the one recently introduced by Early for \(k=2\) and uses a choice of cyclic ordering. We conjecture an explicit formula for \(m_n^{(k)}(\mathbb{I},\mathbb{I})\) which we have checked analytically through \(n=10\) for all \(k\). The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph \(\mathrm{C}_n^{(1,2,\ldots, k-2)}\). Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian \(\mathrm{Tr}^+ \mathrm{G}(k,n)\) in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the \(90\,608\) planar arrays for \(\mathrm{Tr}^+ \mathrm{G}(4,8)\) recently computed by Cachazo, Guevara, Umbert and Zhang. We also introduce another class of special kinematics called \textit{planar-basis kinematics} which generalizes the one introduced by Cachazo, He and Yuan for \(k=2\) and uses the planar basis recently introduced by Early for all \(k\). Based on numerical computations through \(n=8\) for all \(k\), we conjecture that on the planar-basis kinematics \(m_n^{(k)}(\mathbb{I},\mathbb{I})\) evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For \(k=2\) these are the standard Catalan numbers. scattering amplitudes; tropical Grassmannian; generalized biadjoint scalar Combinatorial aspects of tropical varieties, Algebraic combinatorics, Grassmannians, Schubert varieties, flag manifolds, Feynman diagrams Minimal kinematics: an all \(k\) and \(n\) peek into \(\mathrm{Trop}^+ \mathrm{G}(k,n)\) | 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 Let \(G=\mathrm{GL}(n)\) be the general linear group of rank \(n\) over the complex numbers. Let \(T\) be a maximal torus of \(G\), and \(B\), \(B^-\) opposite Borel subgroups of \(G\) with respect to \(T\). A basis of the cohomology of the flag variety of \(G\) is given by the classes of the \(B\)-orbit closures, the Schubert varieties.
Let \(L\) be the Levi subgroup \(\mathrm{GL}(p)\times \mathrm{GL}(q)\) of \(G\), with \(p+q=n\). It is a symmetric (hence spherical) subgroup of \(G\).
In the flag variety the author consider the intersection of a \(B\)-orbit closure and a \(B^-\)-orbit closure, a Richardson variety, in the case it is \(L\)-stable, and determines its cohomology class in terms of the basis given by the Schubert varieties. He therefore gives a so-called positive formula for some structure constants of the cohomology ring, i.e.\ a formula expressing the structure constant as sum of positive integers with combinatorial meaning.
Since the subgroup \(L\) is spherical in \(G\), every (irreducible) \(L\)-stable subvariety of the flag variety has a dense \(L\)-orbit. The main result is thus obtained by reinterpreting, in terms of specific permutations, expressions for the cohomology classes of \(L\)-orbit closures in the flag variety due to \textit{M. Brion} [Comment. Math. Helv. 76, No. 2, 263--299 (2001; Zbl 1043.14012)]. Schubert calculus; Richardson variety; Schubert variety; spherical subgroup; symmetric subgroup; flag variety Wyser, B.J.: Symmetric subgroup orbit closures on flag varieties: Their equivariant geometry, combinatorics, and connections with degeneracy loci. Ph.D. thesis, University of Georgia (2012) Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Schubert calculus of richardson varieties stable under spherical Levi subgroups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \((W, S)\) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let \(J\subseteq S\). Let \(W^J\) denote the set of minimal coset representatives modulo the parabolic subgroup \(W_J\). For \(w\in W^J\), let \(f^{w,J}_i\) denote the number of elements of length \(i\) below \(w\) in Bruhat order on \(W^J\) (with notation simplified to \(f^w_i\) in the case when \(W^J= W\)). We show that
\[
0\leq i< j\leq\ell(w)- i\quad\text{implies}\quad f^{w,J}_i\leq f^{w,J}_j.
\]
Also, the case of equalities \(f^w_i= f^w_{\ell(w)-i}\) for \(i= 1,\dots,k\) is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial \(P_{e,w}(q)\).
We show that if \(W\) is finite then the number sequence \(f^w_0, f^w_1,\dots, f^w_{\ell(w)}\) cannot grow too rapidly. Further, in the finite case, for any given \(k\geq 1\) and any \(w\in W\) of sufficiently great length (with respect to \(k\)), we show
\[
f^w_{\ell(w)-k}\geq f^w_{\ell(w)- k+1}\geq\cdots\geq f^w_{\ell(w)}.
\]
The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if \(\overline X_w\) is a Schubert variety of dimension \(d=\ell(w)\), and \(\lambda= c_1({\mathcal L})\in H^2(\overline X_w)\) is the restriction to \(\overline X_w\) of the Chem class of an ample line bundle, then
\[
(\lambda^k)\cdot: H^{d-k}(\overline X_w)\to H^{d+k}(\overline X_w)
\]
is injective for all \(k\geq 0\). crystallographic Coxeter group; Weyl group; Bruhat order; Schubert variety; \(\ell \)-adic cohomology; intersection cohomology; Kazhdan-Lusztig polynomial Björner, A; Ekedahl, T, On the shape of Bruhat intervals, Ann. Math., 170, 799-817, (2009) Algebraic combinatorics, Algebraic aspects of posets, Étale and other Grothendieck topologies and (co)homologies, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Permutations, words, matrices On the shape of Bruhat intervals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials How to multiply two Schubert polynomials is a notorious open problem in Schubert calculus. The author addresses the special case where a Schubert polynomial is multiplied by a Schur polynomial. His result is a (not very efficient but still beautiful) description of the expansion coefficients in this product as the number of pairs of an RC-graph as introduced by \textit{S. Fomin} and \textit{A. N. Kirillov} [Discrete Math. 153, 123-143 (1996; Zbl 0852.05078)] and a Young tableau, which have to be related in a certain way. The proof of this result is entirely based on the insertion algorithm for RC-graphs due to \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, 257-269 (1993; Zbl 0803.05054)]. Schubert polynomials; Schur functions; Littlewood-Richardson rule; Monk's rule; Pieri's rule; RC-graphs Kogan, M.: RC-graphs and a generalized Littlewood--Richardson rule. Int. Math. Res. Not. 2001(15), 765--782 (2001) Algebraic combinatorics, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus RC-graphs and a generalized Littlewood-Richardson rule | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper deals with the anatomy of a certain subset of \textit{amplituhedra} which are special images of Grassmannians. The authors give a new explicit description of those objects (so-called BCFW cells) and develop some tools. In the introduction, a conjecture about triangulation of the amplituhedron is stated which is going to be understood deeper within the article. Section 2 gives the background on the totally nonnegative Grassmannian and combinatorial objects. Furthermore, some important information on plabic graphs and hook diagrams is given. In Section 3 the background on sign variation, where in section 4 the \(m=2\) amplituhedron is highlighted as warmup for the \(m=4\) amplituhedron. There the recursion in the style of BCFW and the hook diagrams are given. Section 5 deals with binary trees and BCFW plabic graphs. In the following sections the prerequisites given before are applied for the BCFW cells for several cases where the triangulation is picked up in section 12. The appendix gives information on the Dyck paths and BCFW domino bases. amplituhedron; scattering amplitude; totally nonnegative Grassmannian; BCFW recursion; Narayana number; plane partition Yang-Mills and other gauge theories in quantum field theory, Supersymmetric field theories in quantum mechanics, \(S\)-matrix theory, etc. in quantum theory, Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics, Combinatorial identities, bijective combinatorics, Positive matrices and their generalizations; cones of matrices, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Decompositions of amplituhedra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 and clarifies the interdependence between cluster algebra primer, Postnikov arrangements, quadrilateral arrangement, cluster algebra of geometric type, toric charts and positivity, Grassmannians of finite type, Laurent positivity and Schur polynomials.
Main result: The only Grassmannians \(G(k,n)\), within the range of indices \(2<k\leq {1\over 2}n\), whose homogeneous coordinate rings are of finite type, are the Grassmannians \(G(3,6)\), \(G(3,7)\) and \(G(3,8)\). As cluster algebras, their coordinate rings correspond, respectively, to the root systems \(D_4\), \(E_6\) and \(E_8\). The pairing between cluster variables and almost positive roots is explicitly worked out in each case and a geometric description of all cluster variables is presented. Grassmannian; cluster algebra; Laurent positivity J.S. Scott, \textit{Grassmannians and cluster algebras}, \textit{Proc. London Math. Soc.}\textbf{92} (2006) 345 [math.CO/0311148]. Semisimple Lie groups and their representations, Algebraic combinatorics, Grassmannians, Schubert varieties, flag manifolds Grassmannians and cluster algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We characterize the cone of \(\mathrm{GL}\)-equivariant Betti tables of Cohen-Macaulay modules of codimension~1, up to rational multiple, over the coordinate ring of square matrices. This result serves as the base case for ``Boij-Söderberg theory for Grassmannians,'' with the goal of characterizing the cones of \(\mathrm{GL}_k\)-equivariant Betti tables of modules over the coordinate ring of \(k\times n\) matrices, and, dually, cohomology tables of vector bundles on the Grassmannian \(\mathrm{Gr}(k,\mathbb C^n)\). The proof uses Hall's theorem on perfect matchings in bipartite graphs to compute the extremal rays of the cone, and constructs the corresponding equivariant free resolutions by applying Weyman's geometric technique to certain graded pure complexes of Eisenbud-Fløystad-Weyman [\textit{D. Eisenbud} et al., Ann. Inst. Fourier 61, No. 3, 905--926 (2011; Zbl 1239.13023)]. Boij-Söderberg theory; Betti table; cohomology table; Schur functors; Grassmannian; free resolutions; equivariant \(K\)-theory Syzygies, resolutions, complexes and commutative rings, Algebraic combinatorics, Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry Towards Boij-Söderberg theory for Grassmannians: the case of square matrices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 essential set of a permutation was defined by Fulton as the set of southeast corners of the diagram of the permutation. In this paper we determine explicit formulas for the average size of the essential set in the two cases of arbitrary permutations in \(S_ n\) and 321-avoiding permutations in \(S_ n\). Vexillary permutations are discussed too. We also prove that the generalized Catalan numbers \((\begin{smallmatrix} r+k-1\\ n\end{smallmatrix})- (\begin{smallmatrix} r+k-1\\ n-2\end{smallmatrix})\) count \(r\times k\)-matrices dotted with \(n\) dots that are extendable to 321- avoiding permutation matrices. Fulton's essential set; essential set of a permutation; Catalan numbers; matrices Erikson, K.; Linusson, S.: The size of fulton's essential set. The electronic J. Combin. 1, 18 (1995) Exact enumeration problems, generating functions, Algebraic combinatorics, Grassmannians, Schubert varieties, flag manifolds The size of Fulton's essential set | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Lie group and let \(B\) be a Borel subgroup of \(G\). A Bott-Samelson variety is the twisted product of a number of copies of \(\mathbb{C} \mathbb{P}^1\)'s equipped with a map into the flag variety \(G/B\). The author studies a fiber of this map in the case where the map is not birational to the image. Using the moment map of a Bott-Samelson variety, she states the problem in terms of Knutson and Miller's ``subword complexes'' and shows that in some cases, the general fiber, called a ``brick manifold'', is a toric variety. A construction of certain brick polytopes of spherical subword complexes is presented in [\textit{V. Pilaud} and \textit{C. Stump}, Adv. Math. 276, 1--61 (2015; Zbl 1405.05196); \textit{V. Pilaud} and \textit{F. Santos}, European J. Combin. 33, No. 4, 632--662 (2012; Zbl 1239.52026)]. This construction is used in the paper under review for a precise description of the toric varieties of c-generalized associahedra in connection with Bott-Samelson varieties. The root independent subword complexes have interesting connections with Bott-Samelson varieties and symplectic geometry, which have been used by the author to describe the toric varieties of the associated brick polytopes. One of the main results of the paper is that the moment polytope of the brick manifold is the brick polytope. Also, the author gives a nice description of the toric variety of the associahedron. associahedra; Bott-Samelson varieties; toric varieties; brick polytopes Escobar, L., Brick manifolds and toric varieties of brick polytopes, Electron. J. Combin., 23, 2, (2016) Toric varieties, Newton polyhedra, Okounkov bodies, Grassmannians, Schubert varieties, flag manifolds, Group actions on combinatorial structures, Algebraic combinatorics Brick manifolds and toric varieties of brick polytopes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a flag variety \(\text{Fl}_n\), the classes of structure sheaves of Schubert varieties form an integral basis in the Grothendieck ring. A major open problem in the modern Schubert calculus is to determine the \(K\)-theory Schubert structure constants, which express the product of two Schubert classes in terms of this basis.
The authors derive explicit Pieri-type formulae in the Grothendieck ring of a flag variety, which generalize both the \(K\)-theory Monk formula [see \textit{C.~Lenart}, J. Pure Appl. Algebra 179, No. 1--2, 137--158 (2003; Zbl 1063.14060)] and the cohomology Pieri formula [see \textit{F.~Sottile}, Ann. Inst. Fourier 46, No. 1, 89--110 (1996; Zbl 0837.14041)]. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special classes are indexed by cycles of the form \((k-p+1, k-p+2,\dots,k+1)\) or \((k+p, k+p-1, \dots ,k)\), and are pulled back from the projection of \(\text{Fl}_n\) to the Grassmannian of \(k\)-planes.
The formula is expressed in terms of certain labelled chains in the \(k\)-Bruhat order of the symmetric group, and the multiplicities in it are certain binomial coefficients. The proof exploits algebraic-combinatorial setting of Grothendieck polynomials and a Monk-like formula for multiplying a Grothendieck polynomial by a variable. Grothendieck polynomial; Schubert variety; Bruhat order; Pieri's formula DOI: 10.1090/S0002-9947-06-04043-8 Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics, Geometric applications of topological \(K\)-theory A Pieri-type formula for the \({K}\)-theory of a flag manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\text{Gr}(l, {\mathbb C}^n)\) denote the Grassmannian of \(l\)-dimensional subspaces in \({\mathbb C}^n\). The cohomology ring \(H^*(\text{Gr}(l, {\mathbb C}^n), \mathbb Z)\) has an additive basis of Schubert classes \(\sigma_\lambda\), indexed by Young diagrams \(\lambda = (\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_l \geq 0)\) contained in the \(l \times k\) rectangle where \(k = n - l\) (which is denoted by \(\lambda \subset l\times k\)). The product of two Schubert classes in \(H^*({\text{Gr}}(l, \mathbb C^n), \mathbb Z)\) is given by
\[
\sigma_\lambda \cdot \sigma_\mu =\sum_{ \nu \subset l\times k} c^\nu_{\lambda,\mu}\sigma_\nu ,
\]
where \(c^\nu_{\lambda,\mu}\) is the classical Littlewood-Richardson coefficient. This expansion is \textit{multiplicity-free} if \(c^\nu_{\lambda,\mu}\in \{0, 1\}\) for all \(\nu\subset l\times k\). In this paper, the authors give a nonrecursive, combinatorial answer to the following question of W. Fulton.
\textbf{Question.} When is \(\sigma_\lambda \cdot \sigma_\mu\) multiplicity-free?
Their answer exploits the following considerations. For partitions \(\lambda, \mu \subset l\times k\), place \(\lambda\) against the upper left corner of the rectangle. Then rotate \(\mu\) 180 degrees and place it in the lower right corner. The resulting subshape of \(l\times k\) is referred to as \(\text{rotate}(\mu)\).
If \(\lambda \cap \text{rotate}(\mu)\neq \emptyset\), then the product \(\sigma_\lambda \cdot \sigma_\mu\) is zero, and the interesting part is to handle the case of empty intersection. A \textit{Richardson quadruple} is the datum \((\lambda,\mu, l\times k)\), where \(\lambda \cap \text{rotate}(\mu)= \emptyset\). If \(\lambda \cup \text{rotate}(\mu)\) does not contain a full \(l\)-column or \(k\)-row, call this Richardson quadruple \textit{basic}. The main result of the article, Theorem~1.2, gives a criterion for multiplicity-freeness in terms of combinatorial conditions imposed on the basic Richardson quadruples. Grassmannian; Richardson variety; Schubert class Thomas H., Yong A.: Multiplicity-free Schubert calculus. Canad. Math. Bull. 53, 171--186 (2010) Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics, Classical problems, Schubert calculus Multiplicity-free Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected complex semisimple Lie group and \(B\) a Borel subgroup, and consider the corresponding generalized flag manifold \(G/B\) (with maximal torus \(T\).) If \(E^*(\cdot)\) is a complex oriented cohomology theory, one can obtain in certain cases topological Schubert classes, which are elements in \(E^*(G/B)\). This is true for singular cohomology and \(K\)-theory, and has also been done for their torus-equivariant versions \(E_T^*(\cdot)\), but not much is known for other theories.
This work is concerned with defining Schubert classes for the (\(T\)-equivariant) algebraic oriented cohomology theory corresponding to the formal group law of a singular cubic curve in Weierstrass form (here, the Levine-Morel context of algebraic oriented cohomology theories is implicit, which expands on the concept of complex oriented cohomology theories.) The authors consider this, which will be denoted by \(\mathcal{SE}^*(\cdot)\) and is called \textit{elliptic cohomology}, the first interesting case, in terms of complexity, after \(K\)-theory, and had already dealt with the corresponding Schubert classes in a previous paper. The new results here have to do with a different definition of these Schubert classes that is independent of certain choices, as seen below.
Let \(W\) be the Weyl group of \(G\) with respect to \(T\). For each \(w \in W\), define the Schubert variety \(X(w)\) by \(\overline{BwB/B}\). If we consider a reduced word \(\Gamma_w\) for \(w\) and a Bott-Samelson resolution for the Schubert variety, namely \(\Gamma_{I_w} \rightarrow X(w) \rightarrow G/B\), we get a so-called Bott-Samelson class \(\zeta_{I_w}\) in \(E^*_T (G/B)\), for any theory as before, via push-forwarding \(1 \in \Gamma_{I_w}\). If the theory is singular cohomology or \(K\)-theory, this class is independent of the chosen reduced word (and all the \(\zeta_{I_w}\), for \(w \in W\), form a basis for \(E^*_T (G/B)\) over the corresponding formal group algebra \(S\).) This is not the case for other cohomology theories, including \(\mathcal{SE}^*(\cdot)\).
To circumvent this difficulty, the authors propose a new way of defining the Schubert classes. To calculate Bott-Samelson classes, one can consider the recursive formula that appears here on page 717, giving \(\zeta_{I_w}\) as \(Y_{I_1} \cdots Y_{I_l} \zeta_{\emptyset}\), the successive action of elements \(Y_{I_j}\) on the identity element of \(E^*_T (G/B)\) (we consider here that \(I_w = (i_1, \cdots , i_l)\) is a reduced word for \(w\).) These elements belong to the formal Demazure algebra for \(W\), and the action is on the Borel model for \(E^*_T (G/B)\) (details in section 2.3.) The approach of the authors is to modify each operator \(Y_{I_w} = Y_{I_1} \cdots Y_{I_l}\) so that it will not depend on the reduced word chosen for \(w\). This is done by working with the Kazhdan-Lusztig basis for the Hecke algebra corresponding to \(W\), and in definition 3.5. elements \(\mathfrak{G}_w \in \mathcal{SE}^*(G/B)\) are proposed and called \textit{Kazhdan-Lusztig Schubert classes}. Varying \(w \in W\), we obtain a basis for \(\mathcal{SE}^*(G/B)\) (Corollary 3.6.) and the previous Bott-Samelson classes \(\zeta_{I_w}\) are seen to be limits of the \(\mathfrak{G}_w\) (Corollary 3.6.)
With this new definition, two conjectures are proposed. First, a positivity property generalizing the one in Graham's formula for \(K\)-theory (conjecture 3.9.); secondly, the agreement of \(\mathfrak{G}_w\) with the topologically defined Schubert classes \([X(w)]\) (from section 3) whenever the Schubert variety \(X(w)\) is smooth (conjecture 3.12.) This last conjecture is proved, in theorem 3.14., for some cases, including some \(W\) with root systems of type \(A_n\) or \(C_n\). Schubert calculus; elliptic cohomology; flag variety; Hecke algebra Lenart, C., Zainoulline, K.: A Schubert basis in equivariant elliptic cohomology. arXiv:1508.03134 Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Generalized (extraordinary) homology and cohomology theories in algebraic topology, Bordism and cobordism theories and formal group laws in algebraic topology, Equivariant \(K\)-theory, Algebraic combinatorics A Schubert basis in equivariant elliptic cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This rather important paper indicates a precise concrete way to perform computations in the quantum equivariant ``deformation'' of the cohomology ring of \(G(k,n)\), the complex Grassmannian variety parametrizing \(k\)-dimensional vector subspaces of \({\mathbb C}^n\). It relies on the results of another important paper, regarding the same subject, by the same author [Adv. Math. 203, 1--33 (2006; Zbl 1100.14045)]. The usual singular cohomology ring of \(G(k,n)\) is a very well known object, studied since Schubert's time, at the end of the XIX Century. First of all, it is a finite free \({\mathbb Z}\)-module generated by the so-called Schubert cycles. Furthermore, the special Schubert cycles, the Chern classes of the universal quotient bundle over \(G(k,n)\), generates it as a \({\mathbb Z}\)-algebra. Multiplying two Schubert cycles then amounts to know how to multiply a special Schubert cycle with a general one (Pieri's formula) and a way to express any Schubert cycle as an explicit polynomial expression in the special Schubert cycles (Giambelli's formula).
The obvious way to deform the cohomology of a Grassmannian is to consider the cohomology of the total space of a Grassmann bundle, parametrizing \(k\)-planes in the fibers of a rank \(n\) vector bundle, which is a deformation of the cohomology of any fiber of it. In the last few decades, however, other ways to deform the cohomology ring of \(G(k,n)\) have been studied. \textit{E. Witten} [in: Geometry, topology and physics for Raoul Bott. Lectures of a conference in honor of Raoul Bott's 70th birthday, Harvard University, Cambridge, MA, USA 1993. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 4, 357--422 (1995; Zbl 0863.53054)], introduced the small quantum deformation of the cohomology ring of the Grassmannian, whose structure constants were first determined by \textit{A. Bertram} [Adv. Math. 128, No. 2, 289--305 (1997; Zbl 0945.14031)]. Finally, \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)], studied the equivariant deformation of the cohomology of the Grassmannians via the combinatorics of puzzles.
In the beautiful paper under review the author recovers the quantum and equivariant Schubert calculus within a unified framework. Basing on the algebraic properties of the Schur factorial functions, the author realizes the equivariant quantum cohomology ring in terms of generators and relations and gives an explicit basis of polynomial representatives for the equivariant quantum Schubert classes. An alternative approach is offered by \textit{D. Laksov} [Adv. Math. 217, 1869--1888 (2008; Zbl 1136.14042)], where the author proves that the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of a more general and elementary framework, as in [\textit{D. Laksov}, Indiana Univ. Math. J., 56, No. 2, 825--845 (2007; Zbl 1136.14042)], by a change of basis.
The paper is organized as follows. Section 1 is the introduction, where the main results are clearly stated and motivated; Section 2 is a useful and very pleasant review of the algebra of factorial Schur functions. The quantum equivariant cohomology of Grassmannians is treated in Section 3, while the proof of the theorem about the presentation of the quantum equivariant cohomology ring is given in Section 4. Section 5 ends the paper with the discussion and the proof of Giambelli's formula in equivariant quantum cohomology. Giambelli's formulas; quantum equivariant Schubert calculus; factorial Schur functions L.C. Mihalcea, \textit{Giambelli formulae for the equivariant quantum cohomology of the Grassmannian}, \textit{Trans. AMS}\textbf{360} (2008) 2285 [math/0506335]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Giambelli formulae for the equivariant quantum cohomology of the Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials To any connected reductive group \(G\) over a non-archimedean local field \(F\) and to any maximal torus \(T\) of \(G\), we attach a family of extended affine Deligne-Lusztig varieties (and families of torsors over them) over the residue field of \(F\). This construction generalizes affine Deligne-Lusztig varieties of Rapoport, which are attached only to unramified tori of \(G\). Via this construction, we can attach to any maximal torus \(T\) of \(G\) and any character of \(T\) a representation of \(G\). This procedure should conjecturally realize the automorphic induction from \(T\) to \(G\). For \(G = \mathrm{GL}_2\) in the equal characteristic case, we prove that our construction indeed realizes the automorphic induction from at most tamely ramified tori. Moreover, if the torus is purely tamely ramified, then the varieties realizing this correspondence turn out to be (quite complicate) combinatorial objects: they are zero-dimensional and reduced, i.e., just disjoint unions of points. affine Deligne-Lusztig variety; automorphic induction; local Langlands correspondence; supercuspidal representations Langlands-Weil conjectures, nonabelian class field theory, Grassmannians, Schubert varieties, flag manifolds, Representation-theoretic methods; automorphic representations over local and global fields Ramified automorphic induction and zero-dimensional affine Deligne-Lusztig 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 \(n\) be a fixed positive integer and \(h: \{1,2,\dots,n\} \rightarrow \{1,2,\dots,n\}\) a Hessenberg function. The main results of this paper are two-fold. First, we give a systematic method, depending in a simple manner on the Hessenberg function \(h\), for producing an explicit presentation by generators and relations of the cohomology ring \(H^\ast(\mathrm{Hess}(\mathsf{N},h))\) with \(\mathbb{Q}\) coefficients of the corresponding regular nilpotent Hessenberg variety \(\mathrm{Hess}(\mathsf{N},h)\). Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by \textit{A. Mbirika} and \textit{J. Tymoczko} [J. Algebr. Comb. 37, No. 1, 167--199 (2013; Zbl 1263.14050)]. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring \(H^*({\mathrm{Hess}}(\mathsf{N},h))\) of the regular nilpotent Hessenberg variety and the \(\mathfrak{S}_n\)-invariant subring \(H^*({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n} \) of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the \( \mathfrak{S}_n\)-action on \(H^*({\mathrm{Hess}}(\mathsf{S},h))\) defined by Tymoczko). Our second main result implies that \(\dim_{\mathbb{Q}} H^k({\mathrm{Hess}}(\mathsf{N},h)) = \dim_{\mathbb{Q}} H^k({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n}\) for all \(k\) and hence partially proves the Shareshian-Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley-Stembridge conjecture. A proof of the full Shareshian-Wachs conjecture was recently given by \textit{P. Brosnan} and \textit{T. Y. Chow} [Adv. Math. 329, 955--1001 (2018; Zbl 1410.05222)], and independently by \textit{M. Guay-Paquet} [``A second proof of the Shareshian-Wachs conjecture, by way of a new Hopf algebra'', Preprint, \url{arXiv:1601.0549822}], but in our special case, our methods yield a stronger result (i.e., an isomorphism of rings) by more elementary considerations. This article provides detailed proofs of results we recorded previously in a research announcement [``The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: research announcement'', Morfismos 18, No. 2, 51--65 (2014)]. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) The cohomology rings of regular nilpotent Hessenberg varieties in Lie 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 A relative cellular space is a smooth projective variety \(X\) that admits a filtration by closed subvarieties such that each of the successive complements is an affine filtration over a certain smooth projective base \(Y_i\). Examples are Grassmann varieties and completely split quadrics, see \textit{N. A. Karpenko}'s paper [St.\ Petersbg.\ Math.\ J.~12, 1--50 (2001; Zbl 1003.14016)]. The authors compute \(A(X)\) in terms of the \(A(Y_i)\), where \(A\) is any oriented cohomology theory in the sense of \textit{I. Panin}'s paper [K-Theory 30, 265--314 (2003; Zbl 1047.19001)], and they derive a decomposition for the \(A\)-motive of \(X\). If \(A\) is (higher) Chow theory, their results specialize to earlier work by Karpenko [loc. cit.] and by the reviewer [Manuscr. Math.~70, 363--372 (1991; Zbl 0735.14001)].
In the general case, the main new tool developed by the authors is push-forward for cohomology with support. More recent examples of oriented cohomology theories are motivic cohomology and algebraic cobordism MGL. Related results can be found in \textit{S. del Bano}'s paper [J.\ Reine Angew.\ Math.~532, 105--132 (2001; Zbl 1044.14005)]. push-forward with support; correspondences; motive A. Nenashev and K. Zainoulline, Oriented cohomology and motivic decompositions of relative cellular spaces, J. Pure Appl. Algebra 205 (2006), no. 2, 323-340. Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Oriented cohomology and motivic decompositions of relative cellular spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The notion of algebra with straightening laws (ASL for short) was introduced in the early eighties by \textit{C. De Concini, D. Eisenbud} and \textit{C. Procesi} [Hodge algebras, Astérisque 91 (1982; Zbl 0509.13026)]. These algebras give an unified treatment of both algebraic and geometric objects that have a combinatorial nature. The coordinate rings of some classical algebraic varieties, such as determinantal rings (in particular the coordinate ring of Grassmannians) and Pfaffian rings, are examples of ASL's.
One interesting question regarding ASLs is whether their Veronese algebras still have a structure of algebra with straightening laws. The only positive answer to this question was given by Conca in the case of the polynomial ring.
In Section 2, the author extends Conca's result, proving that the Veronese modules of the polynomial ring have the structure of a module with straightening laws (MSL for short). As a corollary, they obtain the result of \textit{A. Aramova, Ş. Bărcănescu} and \textit{J. Herzog} [Rev. Roum. Math. Pures Appl. 40, No. 3--4, 243--251 (1995; Zbl 0883.13013)] which states that the Veronese modules have a linear resolution. Using the results of Bruns on MSL's, they also give an upper bound for the rate of a finitely generated MSL. The bound is given in terms of the degrees of its generators and the degrees of the generators of the ASL.
In Section 3, the author studies whether the Veronese algebra of a homogeneous ASL still has a structure of algebra with straightening laws. The complicated structure of its Veronese algebra as an ASL indicates that this question does not have an easy answer.
In the last section of this paper, the author constructs a new poset starting from a poset of rank three. Then they prove that it has the combinatorial properties to support an ASL structure of the Veronese algebra. Veronese rings; posets; modules Grassmannians, Schubert varieties, flag manifolds, Module categories and commutative rings, Projective techniques in algebraic geometry Veronese algebras and modules of rings with straightening laws | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The totally nonnegative part of a partial flag variety \(G/P\) has been shown by the author [Ph.D. thesis, M.I.T. 1998; J. Algebra 213, No. 1, 144--154 (1999; Zbl 0920.20041)] to be a union of algebraic cells. We show that the closure of a cell is a union of cells and give a combinatorial description of the closure relations. The totally nonegative cells are defined by intersecting \((G/P)_{\leq 0}\) with a certain stratification of \(G/P\) defined by \textit{G. Lusztig} [Represent. Theory 2, 70--78 (1998; Zbl 0895.14014)]. We also verify the same closure relations for these strata. Rietsch, K., Closure relations for totally nonnegative cells in \(G/P\), Math. Res. Lett., 13, 5-6, 775-786, (2006) Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields Closure relations for totally nonnegative cells in \(G/P\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This papers deals with cell decompositions and algebraicity of cohomology for quiver Grassmannians; in particular, the authors prove that the cohomology ring of a quiver Grassmanian associated with a rigid quiver representation satisfies that there is no odd cohomology and the cycle map is an isomorphism. Also, they proved that the corresponding Chow ring admits explicit generators defined over any field. Moreover, they establish the polynomial point count property, after that they consider only quiver to finite or affine type and show that a quiver Grassmanian associated with an indecomposable representation admits a cellular decomposition. Finally, as a particular result, the authors establish a cellular decomposition for quiver Grassmannians associated with representations with rigid regular part. quiver Grassmannians; cellular decomposition; property (S); cluster algebras Representations of quivers and partially ordered sets, Cluster algebras, Derived categories, triangulated categories, Grassmannians, Schubert varieties, flag manifolds, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Cell decompositions and algebraicity of cohomology for 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 We define and study cocycles on a Coxeter group in each degree generalizing the sign function. When the Coxeter group is a Weyl group, we explain how the degree three cocycle arises naturally from geometric representation theory. Coxeter groups; cocycles; Hecke category Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Higher signs for Coxeter 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 \({\mathcal G}_{3,q}\) denotes Klein's quadric representing the lines in PG(3,q), the three-dimensional projective Galois space, q odd. In this paper point sets on \({\mathcal G}_{3,q}\) are characterised which represent the set of all lines either belonging to or tangent to a non-singular quadric in PG(3,q). It is a continuation of an earlier paper by the author [Rend. Mat. Appl., VII, Ser. 3, 9-16 (1983; Zbl 0516.51015)]. ruled set; projective Galois space; quadric Venezia, A.: On a characterization of the set of lines which either belong to or are tangent to a non-singular quadric in \(PG(3,q)\), q odd. Rend. semin. Mat. brescia 7, 617-623 (1984) Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Grassmannians, Schubert varieties, flag manifolds On a characterization of the set of lines which either belong to or are tangent to a non-singular quadric in PG(3,q), q odd | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give the construction of weighted Lagrangian Grassmannians \(w\mathrm{LGr}(3,6)\) and weighted partial \(A_3\) flag variety \(w\mathrm{FL}_{1,3}\) coming from the symplectic Lie group \(\mathrm{Sp}(6,\mathbb{C})\) and the general linear group \(\mathrm{GL}(4,\mathbb{C})\) respectively. We give general formulas for their Hilbert series in terms of Lie theoretic data. We use them as key varieties (Format) to construct some families of polarized 3-folds in codimension 7 and 9. Finally, we list all the distinct weighted flag varieties in codimension \(4 \leq c\leq 10\).
For part I, see [\textit{M. I. Qureshi} and \textit{B. Szendrői}, Bull. Lond. Math. Soc. 43, No. 4, 786--798 (2011; Zbl 1253.14046)]. Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/ Grassmannians, Schubert varieties, flag manifolds, Classical groups (algebro-geometric aspects), \(3\)-folds, Calabi-Yau manifolds (algebro-geometric aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Constructing projective varieties in weighted flag varieties. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Morse functions are useful tool in revealing the geometric formation of its
domain manifolds \(M\). They define handle decompositions of \(M\) from which the
additive homologies \(H_*(M)\) may be constructed. In these lectures two further
questions were emphasized.
\begin{itemize}
\item[(1)] How to find a Morse function on a given manifold?
\item[(2)] From Morse functions can one derive the multiplicative cohomology in addi-
tion to the additive homology?
\end{itemize}
It is not our intention here to make detailed studies of these questions. Instead,
we will illustrate by examples solutions to them for some classical manifolds as
homogeneous spaces. Grassmannians, Schubert varieties, flag manifolds, Algebraic topology on manifolds and differential topology, Homology and cohomology of homogeneous spaces of Lie groups Morse functions and cohomology of homogeneous spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The complex simple Lie group \(\mathbb G_2\) is attached with an adjoint variety \(G_2\). Geometrically, this can be interpreted as a subvariety of the Grassmannian \(\mathrm{Gr}(5,V)\) for some 7-dimensional vector space \(V\); here \(G_2\) consists of 5-spaces isotropic with respect to a fixed non-degenerate 4-form on \(V\), and non-degeneracy refers to the complement of a hypersurface of degree 7 comprising 4-forms decomposable into sums of 3 simple forms.
One can construct a flat deformation \(\hat G_2\) of \(G_2\) by considering a generic 4-form in the above setting. This degeneration is shown to be singular along a plane; its Plücker embedding is the image of \(\mathbb P^5\) under the map defined by quadrics containing a fixed twisted cubic. Further degenerations arise by allowing the twisted cubic to become reducible. While all of them continue to be flat deformations of \(G_2\), only one appears to be a linear section of \(\mathrm{Gr}(5,V)\); the others are toric degenerations, leading to Gorenstein toric Fano varieties. Intersecting with a hyperplane and a quadric, one obtains Calabi-Yau threefolds as small resolutions. These can be studied in the context of mirror symmetry, related to Borcea's Calabi-Yau threefold of degree \(36\).
The paper concludes with applications to \(K3\) surfaces of genus 10, making precise a few special cases of the classification in [\textit{T. Johnsen} and \textit{A. L. Knutsen}, \(K3\) projective models in scrolls. Berlin: Springer (2004; Zbl 1060.14056)]. Lie group; adjoint variety; degeneration; Grassmannian; Calabi-Yau threefold; \(K3\) surface of genus 10 Calabi-Yau manifolds (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Some degenerations of \(G_2\) and Calabi-Yau varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the reduction at a prime \(p>2\) of Shimura varieties of PEL type with parahoric level structure at \(p\), where the underlying group \(G/\mathbb Q\) is a unitary similitude group for an imaginary quadratic number field \(K/\mathbb Q\) ramified at \(p\). Let \((r,s)\) be the signature of \(G_{\mathbb R}\), \(n=r+s\). An important goal is to find ``good'' models of the Shimura varieties over the ring of integers of \(E_w\), where \(E\) is the reflex field, and \(w\) is the prime lying over \(p\), and to analyze their étale-local structure.
In their book [Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies. 141. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)], \textit{M. Rapoport} and \textit{T. Zink} defined a candidate for such a model in the general framework of PEL data, and explained how to study its local properties in terms of a scheme defined in terms of Grassmannians. This scheme is nowadays called the ``naive'' local model \(M^{\text{naive}}_I\), because it turned out (see [\textit{G. Pappas}, J. Algebr. Geom. 9, No. 3, 577--605 (2000; Zbl 0978.14023)] for examples) that \(M^{\text{naive}}_I\) is not in general flat over its base, a serious defect for many purposes. Here \(I\) is an index set which corresponds to the choice of a parahoric subgroup, determined by the level structure imposed at \(p\).
Later, several variants of the naive local model have been defined and investigated, see [\textit{G. Pappas} and \textit{M. Rapoport}, J. Algebr. Geom. 12, No. 1, 107--145 (2003; Zbl 1063.14029); Duke Math. J. 127, No. 2, 193--250 (2005; Zbl 1126.14028)]. One approach is to consider the flat closure \(M^{\text{loc}}_I\) of the generic fiber of \(M^{\text{naive}}_I\). This ``local model'' is flat by definition, and the task then is to describe its geometry, and to give a functorial description, if possible.
As in previously studied cases, an important tool consists of embedding the special fiber \(\overline{M}^{\text{naive}}\) of \(M^{\text{naive}}\), and in particular the special fiber \(\overline{M}^{\text{loc}}\) of \(M^{\text{loc}}\), into an affine flag variety over the residue class field. In the situation at hand, the natural setup is given by the twisted loop groups studied in [\textit{G. Pappas} and \textit{M. Rapoport}, Adv. Math. 219, No. 1, 118--198 (2008; Zbl 1159.22010)].
Let us discuss the main results of the paper at hand. Assuming the coherence conjecture (see below), it is shown that \(\bigcup_{w\in\text{Adm}(\mu)} S_w = \overline{M}^{\text{loc}}\) (schematically), where \(\text{Adm}(\mu)\) denotes the so-called admissible set, a finite subset of the extended affine Weyl group attached to the Shimura datum ``at \(p\)'', and for each \(w\), \(S_w\) denotes the corresponding Schubert cell in the affine flag variety.
The above-mentioned coherence conjecture is a conjectural explicit formula for the dimension of the space of global sections on \(\bigcup_{w\in\text{Adm}(\mu)} S_w\) of the natural ample line bundle. See op.~cit.
Furthermore, the following unconditional results are proved: If \(I = \{0\}\) and \(n\) is odd, or if \(I = \{m\}\) and \(n\) is even, then \(\overline{M}^{\text{loc}}\) is irreducible, normal, Frobenius-split, and has only rational singularities. In these two cases, the underlying parahoric subgroup is special in the sense of Bruhat-Tits theory. The same result in the only remaining case of a special parahoric has been proved in the meantime by \textit{K. Arzdorf} [Mich. Math. J. 58, No. 3, 683--710 (2009; Zbl 1186.14026)].
In the case of Picard modular surfaces, i.e., if \(n=3\), the authors have similar, and more detailed results for all parahoric subgroups.
There remains the question of giving a modular description of \(M^{\text{loc}}\). To this end, the authors define a closed subscheme of \(M^{\text{naive}}\) which contains \(M^{\text{loc}}\) and is conjecturally equal to the latter. Some computational evidence supports this conjecture. Shimura variety; unitary group; parahoric subgroup; local model Kudla, S., Rapoport, M.: Special cycles on the \(\Gamma _0(p^n)\)-moduli curve (unpublished) Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Grassmannians, Schubert varieties, flag manifolds Local models in the ramified case. III. Unitary groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct and study a new topological field theory in three dimensions. It is a hybrid between Chern-Simons and Rozansky-Witten theory and can be regarded as a topologically-twisted version of the \(N=4 d=3\) supersymmetric gauge theory recently discovered by Gaiotto and Witten. The model depends on a gauge group \(G\) and a hyper-Kähler manifold \(X\) with a tri-holomorphic action of \(G\). In the case when \(X\) is an affine space, we show that the model is equivalent to Chern-Simons theory whose gauge group is a supergroup. This explains the role of Lie superalgebras in the construction of Gaiotto and Witten. For general \(X\), our model appears to be new. We describe some of its properties, focusing on the case when \(G\) is simple and \(X\) is the cotangent bundle of the flag variety of \(G\). In particular, we show that Wilson loops are labeled by objects of a certain category which is a quantum deformation of the equivariant derived category of coherent sheaves on \(X\). Chern-Simons theory G. Bonelli, K. Maruyoshi and A. Tanzini, \textit{Quantum Hitchin Systems via {\(\beta\)}-deformed Matrix Models}, arXiv:1104.4016 [INSPIRE]. Topological field theories in quantum mechanics, Supersymmetric field theories in quantum mechanics, Yang-Mills and other gauge theories in quantum field theory, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Superalgebras, Grassmannians, Schubert varieties, flag manifolds, Applications of deformations of analytic structures to the sciences Chern-Simons-Rozansky-Witten topological field theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors consider \textit{proper permutations} \(w \in S_n\), i.e., ones which satisfy \(\ell(w) - \binom{d(w) + 1}{2} \leq n\), where \(\ell(w)\) is the number of inversions and \(d(w)\) is the number of left descents of~\(w\). One of the main results of the article is that the probability that a random permutation \(w \in S_n\) is proper goes to zero in the limit when \(n \rightarrow \infty\).
A very important aspect of this result is its relation to geometry: properness of \(w\) is related to the Schubert variety \(X_w\) being spherical. We say that \(X_w\) is spherical if it has a dense orbit of a Borel subgroup of some \(L_I\), a group of invertible block diagonal matrices, where blocks are determined by a set \(I\) of left descents of~\(w\). The authors conclude that the probability that for a random permutation \(w \in S_n\) the Schubert variety \(X_w\) is spherical goes to zero in the limit when \(n \rightarrow \infty\).
Finally, the authors consider the notion of \(w \in S_n\) being \(I\)-spherical, introduced by \textit{R. Hodges} and \textit{A. Yong} in [J. Lie Theory, 32(2), 447--474 (2022; Zbl 1486.14070)], and show that the probability of \(w\) being \(I\)-spherical goes to zero in the limit when \(n \rightarrow \infty\). This result settles a conjecture from the article cited above. Schubert varieties; spherical varieties; proper permutations Grassmannians, Schubert varieties, flag manifolds, Probabilistic methods in group theory Proper permutations, Schubert geometry, and randomness | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 survey on congruences of lines of low order. We
start by recalling the main properties of the focal locus, and use it to reobtain the classification of congruences of order one in \(\mathbb{P}^3\). We then explain the main ideas of a work in progress with S. Verra, outlining how to classify congruences of lines of order two in \(\mathbb{P}^3\). We end by stating the main problems on this topic. Arrondo, E., \textit{line congruences of low order}, Milan J. Math., 70, 223-243, (2002) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Configurations and arrangements of linear subspaces, Grassmannians, Schubert varieties, flag manifolds, Families, moduli of curves (algebraic) Line congruences of low order | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 congruence of lines is a \((n - 1)\)-dimensional family of lines in \(\mathbb{P}^n \) (over \(\mathbb{C})\), i.e. a variety \(Y\) of dimension (and hence of codimension) \(n - 1\) in the Grassmannian Gr\((1, \mathbb{P}^n)\). A fundamental curve for \(Y\) is a curve \(C \subset \mathbb{P}^n\) which meets all the lines of \(Y\). In this paper the authors classify all smooth congruences with fundamental curve \(C\) generalizing a paper by \textit{E. Arrondo} and \textit{M. Gross} [Manuscr. 79, No. 3-4, 283-298 (1993; Zbl 0803.14019)], where the case \(n = 3\) was treated. An explicit construction for all possible congruences that they found is also given. congruence of lines; Grassmannian; fundamental curve Arrondo, E., M. Bertolini and C. Turrini: Classi cation of smooth congruences with a fundamental curve. Projective Geometry with applications. Number 166 in LN. Marcel Dekker, 1994 Families, moduli of curves (algebraic), Grassmannians, Schubert varieties, flag manifolds Classification of smooth congruences with a fundamental 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 Let \(M\) be a compact oriented manifold with a torus \(T\) acting on it. The Atiyah-Bott-Berline-Vergne localization theorem formula gives the integral of an equivariant cohomology class on \(M\) in terms of an integral over the fixed point set \(M^T\). The authors extend this result by replacing \(T\) by \(G\), a compact oriented group. Their fixed point set is \(M^C\), which is the set of points of \(M\) which are fixed by some element in a conjugacy class \(C\) of some element \(g \in G\).
Let \(V\) be a module over \(H\) where \(H\) is the rational cohomology of \(BG\), which is an integral domain and \(Q\) is its field of fractions. Then the localization of \(V\) is defined by \(\widehat V:=V \otimes_H Q\). The usual equivariant cohomology \(H^*_G(M)\) is an \(H\)-module, so \(\widehat H^*_G(M)\) is called the localized equivariant cohomology.
Now the localized equivariant cohomology satisfies an equivariant Mayer-Vietoris theorem. This is the main technique, and it forces the authors to assume that \(M^C\) is a smooth manifold. They do this casually and leave off hypotheses about the smoothness of \(M\) in their theorems, introducing an unnecessary ambiguity to their exposition.
They prove a Borel type Localization formula stating the inclusion \(i:M^C\rightarrow M\) induces an isomorphism in localized cohomology. If \(t\) is contained in the maximal torus \(T\), then the usual Borel localization formula holds.
This provides a systematic method for calculating the Gysin homomorphism in ordinary cohomology of an equivariant map. As an example, the Alkidiz-Carrell formula is obtained for the Gysin map on flag manifolds. Borel localization formula; push-forward; Gysin map; equivariant Pedroza, A.; Tu, LW, On the localization formula in equivariant cohomology, Topology Appl., 154, 1493-1501, (2007) Homology with local coefficients, equivariant cohomology, Compact Lie groups of differentiable transformations, Grassmannians, Schubert varieties, flag manifolds On the localization formula in equivariant cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We obtain an algorithm computing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in a generalized flag manifold \(G/B\). In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure-Lusztig-type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of \(G/B\). By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold \(G/P\). We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjecture to the torus equivariant setting. Chern-Schwartz-MacPherson class; homogeneous space; Schubert variety; Demazure-Lusztig operator 10.1112/S0010437X16007685 Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Classical groups (algebro-geometric aspects) Chern-Schwartz-MacPherson classes for Schubert cells in 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 study the action of a differential operator on Schubert polynomials. Using this action, we first give a short new proof of an identity of \textit{I. Macdonald} [Notes on Schubert polynomials. Montréal: Publications du LACIM, Université du Québec (1991)]. We then prove a determinant conjecture of \textit{R. Stanley} [``Some Schubert shenanigans'', Preprint, \url{arXiv:1704.00851}]. This conjecture implies the (strong) Sperner property for the weak order on the symmetric group, a property recently established by \textit{C. Gaetz} and \textit{Y. Gao} [Proc. Am. Math. Soc. 148, No. 1, 1--7 (2020; Zbl 07144479)]. Sperner property; weak order; Schubert polynomial; Macdonald identity Combinatorial aspects of representation theory, Symmetric functions and generalizations, Group actions on combinatorial structures, Combinatorics of partially ordered sets, Determinants, permanents, traces, other special matrix functions, Grassmannians, Schubert varieties, flag manifolds Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using the Lagrangian-Grassmannian, a smooth algebraic variety of dimension \(n(n + 1)/2\) that parametrizes isotropic subspaces of dimension \(n\) in a symplectic vector space of dimension \(2n\), we construct a new class of linear codes generated by this variety, the Lagrangian-Grassmannian codes. We explicitly compute their word length, give a formula for their dimension and an upper bound for the minimum distance in terms of the dimension of the Lagrangian-Grassmannian variety. algebraic geometry codes; Grassmann codes; Lagrangian-Grassmannian [1] J. Carrillo-Pacheco and F. Zaldivar, On Lagrangian-Grassmannian Codes, Designs, Codes and Cryptography 60 (2011) 291-268. Geometric methods (including applications of algebraic geometry) applied to coding theory, Permutations, words, matrices, Algebraic coding theory; cryptography (number-theoretic aspects), Finite ground fields in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds On Lagrangian-Grassmannian codes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The categorification program was initiated by I. Frenkel with the aim of extending 3-dimensional topological field theories to dimensions 4 and higher [\textit{L. Crane} and \textit{I. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)]. This program was extended by the first author in his work on categorified tangle invariants [\textit{M. Khovanov}, Algebr. Geom. Topol. 2, 665--741 (2002; Zbl 1002.57006)].
The present paper is a part of ongoing research by the authors on categorification of quantum groups and their representations. A categorification of quantum \(sl(2)\) obtained previously by the second author is generalised to \(sl(n)\). More precisely the authors construct a linear 2-category whose Grothendieck category coincides with the idempotent form of quantum \(sl(n)\). Note that the interpretation of elements of Lustig's canonical basis of the idempotent form as classes of indecomposible ob jects established for \(sl(2)\) is still an open problem for \(sl(n)\). This category has potential applications in representation theory. The authors expect this category to manifest itself as a symmetry of various categories of interest in representation theory, ranging from derived categories of coherent sheaves on quiver varieties to categories of modules over cyclotomic and degenerate affine Hecke algebras. categorification; quantum group; quantum \(sl(n)\); iterated flag variety; 2-representation; 2-category Khovanov, M.; Lauda, A., A categorification of quantum \(\mathfrak{sl}_n\), Quantum Topol., 1, 1, 1-92, (2010) Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Grassmannians, Schubert varieties, flag manifolds, Ring-theoretic aspects of quantum groups A categorification of quantum \(\text{sl}(n)\) | 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 Problems about dimension of secant varieties of many kind of algebraic varieties have been quite a subject of study in the last twenty years or so, mainly related to problems of tensor decompositions, by studying varieties which parameterize tensors of given kind (``structured tensors''), main examples are Segre Varieties (parameterizing all tensors of a given format), Veronese Varieties (parameterizing symmetric tensors, i.e. polynomials), Segre-Veronese Varieties (and partially symmetric tensors), Grassmannians (skew symmetric tensors). Moreover the problem of \(h\)-identifiability of a tensor (i.e. unicity of its decomposition as sum of \(h\) decomposable tensors of the given kind) as been quite studied, also for its relation with applications.
In this paper the author study both problems (dimension of \(h\)-Secant varieties and identifiability) for flag varieties \(\mathbb{ F}(k_1,\dots, k_r; n)\) which are set of flags, that is the set of nested subspaces, \( V_{1} \subset\dots\subset V_{r}\) in a vector space \( V\), where \(V \cong \mathbb{C}^{n+1}\), and \(\dim V_i=k_i\), for \(i=1,\dots,r\) and \(k_1 \leq\cdots \leq k_r\). If \(\mathbb{G}(k_i, n)\) is the Grassmannian of \(k_i\)-dimensional linear subspaces of \(\mathbb{P}(V )\), viewed in its Plucker embedding into \(\mathbb{P}^{N_i}\), where \(N_i =\binom{n+1}{k_i+1}-1\), we have an embedding of the product of these Grassmannians: \[\mathbb{G}(k_1, n)\times\cdots\times\mathbb{G}(k_r, n) \subset \mathbb{P}^{N_1} \times\cdots\times{P}^{N_r} \subset \mathbb{P}^N,\] where \(N = \binom{n+1}{k_1+1}\dots\binom{n+1}{k_r+1}- 1\). So the flag variety \(\mathbb{ F}(k_1,\dots, k_r; n)\), via Plucker and Segre embeddings, can be viewed as a subvariety of \(\Pi_{i=1}^r\mathbb{G}(k_i, n) \subset \mathbb{P}^N\).
The \(h\)-secant varieties of \(\mathbb{ F}(k_1, \dots, k_r; n)\), i.e. the Zariski closure of the union of all the linear spaces generated by \(h\) points of \(\mathbb{ F}(k_1,\dots, k_r; n)\), have expected dimension \(\min\{nh + h-1, N\}\), and if their actual dimension is less that that, they are said to be \(h\)-defective. The main result in the paper enstablishes a bound on \(h\)-defectivity; namely it is proved that if \(n \leq 2k_j + 1\) for some index \(j\) and \(\ell\) is the maximum among these \(j\). Then, for \[h \leq\binom{n + 1}{k_l + 1}^{\log 2(\sum^\ell_{j=1} k_j+\ell-1)}\] \(\mathbb{ F}(k_1, \dots , k_r; n)\) is not \((h +1)\)-defective. Moreover, under the same bound, the generic point of \(\mathbb{ F}(k_1,\dots, k_r; n)\) is identifiable.
In the last section of the paper the case of the flag varieties \(\mathbb{ F}(0,k; n)\) is considered and it is proved that their chordal variety (2-secant variety) is non-defective, except for \(k=n-1\).
One main tool of the proofs is the study of obsulating spaces to products of Grassmannians and projections from them. flag varieties; secant varieties; identifiability Secant varieties, tensor rank, varieties of sums of powers, Projective techniques in algebraic geometry, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Rational and birational maps, Multilinear algebra, tensor calculus On secant dimensions and identifiability 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 Let \(G\) denote an adjoint semisimple group over an algebraically closed field and \(T\) a maximal torus of \(G\). Following \textit{C. Contou-Carrére} [Géométrie des groupes semi-simples, résolutions équivariantes et lieu singulier de leurs variétés de Schubert'', Thése d'état, Univ. Montpellier II, Montpellier (1983)], we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the building associated to the group \(G\). We first determine a cellular decomposition of this variety analogous to the Bruhat decomposition of a Schubert variety and then we describe the fibre of this resolution over a point.
Remark: In ibid. 14, No. 1, 31--33 (2003; Zbl 1052.14058) some counter-examples of two propositions (Propositions 4 and 7 of the paper under review) concerning cellular decompositions occuring in a Bott-Samelson variety were presented, however the description of the fibre of a Bott-Samelson resolution over a point of a Schubert variety is still valid in the Kac-Moody setting.
In an erratum in ibid. 16, No. 2, 179--180 (2005) the author points out a false claim in Theorem 2 on p. 466 that the intersection is given by linear equations, so that this theorem and the extension to the Kac-Moody setting are only valid with a certain restriction. Here the author also cites a result of \textit{M. Härterich} [The \(T\)-equivariant cohomology of Bott-Samelson varieties. Preprint math.AG/0412337] concerning the linearity the equations in question. Gaussent, S.: The fibre of the Bott-Samelson resolution. Indag. Math. N. S. \textbf{12}(4), 453-468 (2001) Grassmannians, Schubert varieties, flag manifolds, Groups with a \(BN\)-pair; buildings, Reflection and Coxeter groups (group-theoretic aspects) The fibre of the Bott-Samelson resolution. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0614.00007.]
A geometrical description of the variety B of complete quadrics of an N- dimensional projective space (over an algebraically closed field of characteristic zero) is given. It is proved that the Schubert conditions of tangency to linear varieties give a basis for the cohomology ring of B. Although the results (as the author says) are contained in earlier work by \textit{C. De Concini} and \textit{C. Procesi} [in ''Invariant theory'', Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)] and by \textit{J. Vaisencher} [in ''Enumerative geometry and classical algebraic geometry'', Proc. Math. 24, 199-235 (1982; Zbl 0501.14032)], they are obtained in a quite different way. Schubert calculus; complete quadrics Finat, J. A., A combinatorial presentation of the variety of complete quadrics. Preprint 1985. Enumerative problems (combinatorial problems) in algebraic geometry, Questions of classical algebraic geometry, Grassmannians, Schubert varieties, flag manifolds A combinatorial presentation of the variety of complete quadrics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We present a class of determinantal ideals (of mixed type) whose singular loci fail to be equidimensional (with arbitrarily large dimension-gap). Since these ideals are the defining ideals of a class of Schubert varieties, we get a family of Schubert varieties (as subvarieties of a variety of full flags or of a Grassmannian) possessing inequidimensional singular locus. Schubert varieties; singularities Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Schubert varieties with inequidimensional singular locus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 0716.00007.]
Let \(G\) be a semi-simple algebraic group over \(\mathbb{C}\), and \(B\) a Borel subgroup of \(G\). Then one knows that the cohomology ring \(H^*(G/B;\mathbb{C})\) is the coinvariant algebra \(A(h)/I^ W\) associated to a certain subalgebra \(h\) of the Lie algebra of \(G\) (here, \(A(h)\) is the coordinate ring of \(h\), and \(I^ W\) is the homogeneous ideal generated by the \(W\)-invariant functions \(f\) on \(h\) such that \(f(0)=0)\).
In this paper, the author proves a similar result for \(H^*(X,\mathbb{C})\), \(X\) being a Schubert variety in \(G/P\), for a parabolic subgroup \(P\supseteq B\). This paper makes a good contribution to the cohomology theory of Schubert varieties. flag manifold; cohomology theory of Schubert varieties Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Cohomology theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Classical real and complex (co)homology in algebraic geometry \(SL_ 2\) actions and cohomology of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let $G$ be a connected complex semisimple Lie group with a real form $G_0\subseteq G$ and a projective algebraic homogeneous space $Z=G/Q$ that is called a flag manifold. Any open $G_0$-orbit $D\subseteq Z$ is called a flag domain. If $K_0\subseteq G_0$ is a maximal compact subgroup, then by a result of \textit{J. A. Wolf} [Bull. Am. Math. Soc. 75, 1121--1237 (1969; Zbl 0183.50901)] there exists exactly one $K_0$-orbit $C_0\subseteq D$ which is a complex manifold, and $C_0$ is called the $K_0$-base cycle. In the paper under review, one proves among other things that if $C=g(C_0)$, with $g\in G$, is a cycle satisfying $C\subseteq D$, then the normal bundle of $C$ in $D$ is trivial if and only if the flag domain $D$ is holomorphically convex, and then it is the product of the cycle under consideration and a Hermitian symmetric space. The authors also prove that if $D$ fails to decompose as such a product, then $D$ is pseudoconcave in the sense of Andreotti. This is a well-written paper and a good illustration of the fruitful interaction between Lie theory and complex analysis. flag domains; Levi curvature; normal bundles Complex Lie groups, group actions on complex spaces, Grassmannians, Schubert varieties, flag manifolds, Noncompact Lie groups of transformations Normal bundles of cycles in 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 Let \(\varpi\) be the fundamental weight of \(\mathrm{SL}_2 (\mathbb{C})\). For a nonnegative integer \(m\), let \(V(m \varpi)\) be the irreducible representation of \(\mathrm{SL}_2 (\mathbb{C})\) with highest weight \(m \varpi\). We let \(V_q(m \varpi)\) denote the corresponding module over the quantum group \(U_q (\mathfrak{sl}_2 (\mathbb{C}))\).
For a nonnegative integer \(n\), the authors construct a basis \(\{ y_w \}_{w \in \mathscr{C}_n}\) of the space \(V(\varpi)^{\otimes n}\) indexed by the set of words on the alphabet \(\{+, -\}\), and then show that this basis is the dual canonical basis of \(V_q(\varpi)^{\otimes n}\) specialized at \(q=1\) and it is the Mirković-Vilonen basis of \(V(\varpi)^{\otimes n}\). It is also shown that the Mirković-Vilonen basis of the space \(V(n_1 \varpi) \otimes \cdots \otimes V(n_r \varpi)\) coincides with the dual canonical basis of this space specialized at \(q=1\). Mirković-Vilonen basis; dual canonical basis Semisimple Lie groups and their representations, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Linear algebraic groups over the reals, the complexes, the quaternions, Geometric Langlands program: representation-theoretic aspects Mirković-Vilonen basis in type \(A_1\) | 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 Let \(P\) be a parabolic subgroup in \(G = \mathrm{SL}_n(\mathbf k)\), for \textbf{k} an algebraically closed field. We show that there is a \(G\)-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle \(T^* G/P\). Restricting this identification to the conormal variety \(N^* X(w)\) of a Schubert divisor \(X(w)\) in \(G/P\), we show that there is a compactification of \(N^* X(w)\) as an affine Schubert variety. It follows that \(N^* X(w)\) is normal, Cohen-Macaulay, and Frobenius split. Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Cotangent bundles of partial flag varieties and conormal varieties of their Schubert divisors | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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(d,n) be the Grassmannian of d-planes in \({\mathbb{P}}_ C^ n\) and let \(V\subset G(d,n)\) be a nonsingular subvariety. The author explores the relations between the Chern classes of G(d,n) and V coming from the exact sequence \(0\to T_ V\to T_{G(d,n)| V}\to N\to 0\), when V is either of codimension one or subcanonical of codimension two. The formulas he obtains are then used to study the problem of the existence in G(d,n) of a given V. In particular he considers the following cases: \((a)\quad V\quad is\) a complete intersection, or \((b)\quad all\) the Chern classes of V are zero, or \((c)\quad V\) is a Segre embedding of \({\mathbb{P}}^ r\times {\mathbb{P}}^ s.\)
Later [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 32, 45-54 (1986) and Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 121, fasc. 5/6 (1987)], following the same idea, the author has examined the existence of 3- scrolls and of other Segre products in G(1,4). Grassmannian; Chern classes; codimension one; codimension two Grassmannians, Schubert varieties, flag manifolds, Low codimension problems in algebraic geometry, Characteristic classes and numbers in differential topology Grassmann varieties and their subvarieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 explicit computation of the intersection cohomology IH(X) à la Goresky-MacPherson of a complex space X is usually difficult. Nevertheless, according to Goresky-MacPherson, if a small resolution of the singularities \(\tilde X\to X\) of X exists, then IH(X) is roughly speaking the same as the cohomology \(H(\tilde X)\) of \(\tilde X.\) The author proves by an explicit construction the existence of a small resolution for any Schubert cell and therefore obtains a combinatorial description of the intersection cohomology. intersection cohomology; small resolution for any Schubert cell Zelevinskiĭ, A. V.: Small resolutions of singularities of Schubert varieties. Funct. anal. Appl. 17, No. 2, 142-144 (1983) Global theory and resolution of singularities (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, (Co)homology theory in algebraic geometry Small resolutions of singularities of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Correction of theorem 2 in the cited paper [ibid. 77, 15-18 (1979; Zbl 0424.14017)]. Grassmannian Grassmannians, Schubert varieties, flag manifolds, Surfaces and higher-dimensional varieties Corrigendum to ``Surfaces in the Grassmann variety 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 Translation from Tr. Mosk. Mat. O.-va 39, 181--211 (1979; Zbl 0438.57013). algebraic or analytic subset of the manifold of all lines in complex n-space; line complexes; decomposition of admissible line complexes; critical points; Grassmannians; subsets of Grassmann bundles as differential equations K. Maius, The structure of admissible line complexes in \(\mathbbCP^n\) , Trans. Mosc. Math. Soc. 39 (1981), 195-226. Differential topology, Integral geometry, Semi-analytic sets, subanalytic sets, and generalizations, Line geometries and their generalizations, Partial differential equations on manifolds; differential operators, Vector distributions (subbundles of the tangent bundles), Grassmannians, Schubert varieties, flag manifolds The structure of admissible line complexes in \(\mathbb{CP}^n\) | 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 We describe recent progress on \(QH^*(G/P)\) with special emphasis on its functoriality property, quantum Pieri rule and their applications. quantum cohomology; homogeneous varieties; Gromov-Witten invariable Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds An update of quantum cohomology of homogeneous 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 Grassmannian has a natural action by an algebraic torus. In this survey we describe the diagonal action of this torus on subvarieties of products of Grassmannians. From this action we describe how to construct an associated moment polytope. The main varieties considered are Schubert varieties, Richardson varieties, and their desingularizations. The moment polytopes of these varieties include the permutahedron and associahedron. We also look at the Barbasch-Evens-Magyar varieties, which are desingularizations of symmetric orbit closures in the flag manifold. moment polytope; Grassmannian; flag manifold; Schubert variety; Bott-Samelson variety; Richardson variety; Brick variety; Barbasch-Evens-Magyar variety; permutahedron; associahedron Grassmannians, Schubert varieties, flag manifolds, Research exposition (monographs, survey articles) pertaining to algebraic geometry A brief survey about moment polytopes of subvarieties of products of Grassmanians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 reductive group over a \(p\)-adic field \(F\). The author considers unramified principal series representations of \(G\) and their Whittaker model. The image, in a Whittaker model, of the \(K\)-spherical vector of an unramified principal series representation is described by the Casselman-Shalika formula which links it to the character of a representation of the dual group to \(G\). By the Borel-Weil theorem, this representation is realized in the global sections of a certain line bundle on the flag manifold associated to the dual group. In the same spirit, the author investigates the images of certain Iwahori fixed vectors and relates them to the Lefschetz traces of various cohomology groups of sheaves on the flag variety. As applications of his results he gives necessary and sufficient conditions for injectivity of the Whittaker map, generalizing results of Bernstein-Zelevinsky and extending his own previous results. The other main result is a non-vanishing result on the maximal \(F\)-split torus of \(G\) for the image of Iwahori fixed vectors in the Whittaker model. split reductive groups; unramified principal series representations; Whittaker models; global sections; line bundles; flag manifolds; Iwahori fixed vectors; Lefschetz traces; cohomology groups of sheaves; maximal \(F\)-split torus Mark Reeder, \?-adic Whittaker functions and vector bundles on flag manifolds, Compositio Math. 85 (1993), no. 1, 9 -- 36. Representations of Lie and linear algebraic groups over local fields, Analysis on \(p\)-adic Lie groups, Harmonic analysis on homogeneous spaces, Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds \(p\)-adic Whittaker functions and vector bundles on flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0728.00006.]
This article gives an informative survey of the combinatorial theory of Schubert polynomials developed by A. Lascoux and M.-P. Schützenberger. Its topic may be described by the titles of the sections and subsections: Permutations (Bruhat order, diagrams and codes, vexillary permutations); divided differences; multi-Schur functions (duality); Schubert polynomials; orthogonality; double Schubert polynomials. In many cases proofs have been omitted. Bruhat order; Schur functions; permutations; divided differences; Schubert polynomials Macdonald, I. G., Notes on Schubert polynomials, (1991), Publications du Laboratoire de Combinatoire et D'informatique Mathématique, Dép. de Mathématiques et D'informatique, Universitédu Québec à Montréal, available at Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Orthogonal polynomials [See also 33C45, 33C50, 33D45], Grassmannians, Schubert varieties, flag manifolds 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 paper is devoted to the study of compactifications of subvarieties of algebraic tori. For a connected closed subvariety~\(X\) of an algebraic torus~\(T\) over an algebraically closed field~\(k\), the author introduces the notion of a tropical compactification: it is the closure \(\overline X\) of~\(X\) in a toric variety \({\mathbb P} \supset T\) such that the multiplication map \(T \times {\overline X} \to {\mathbb P}\) is faithfully flat and \(\overline X\) is proper. If \(\overline X\) and \(\mathbb P\) are as above, then the fan of \(\mathbb P\) is supported on the (non-archimedean) amoeba of \(X(\overline K) \subset T(\overline K)\), where \(\overline K\) is the field of transfinite Puiseux series over~\(k\).
It is proved in the paper that any connected closed subvariety \(X \subset T\) admits a tropical compactification \(\overline X \subset {\mathbb P}\) such that \(\mathbb P\) is smooth. In this case, the boundary \({\overline X} \setminus X\) is divisorial and has ``combinatorial normal crossings'': for any collection \(B_1\), \(\ldots\), \(B_r \subset {\overline X} \setminus X\) of irreducible divisors, the intersection \(\bigcap_{i = 1}^r B_i\) has codimension~\(r\). Moreover, if \(r = \dim X\) and \(p \in \bigcap_{i = 1}^r B_i\), then \(\overline X\) is Cohen-Macaulay at~\(p\).
The author calls a subvariety \(X \subset T\) \textit{schön}, if~\(X\) has a tropical compactification with a smooth multiplication map. He proves that if \(X\) is \textit{schön}, then any of its tropical compactifications \({\overline X} \subset {\mathbb P}\) has a smooth multiplication map, is regularly embedded, normal, has toroidal singularities, and its log canonical line bundle is globally generated and is equal to the determinant of the normal bundle.
The author discusses these notions and results in several situations and considers, in particular, log canonical models of complements of hyperplane arrangements and compact quotients of Grassmannians by a maximal torus. Jenia Tevelev, ``Compactifications of subvarieties of tori'', Am. J. Math.129 (2007) no. 4, p. 1087-1104 , Toric varieties, Newton polyhedra, Okounkov bodies, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Grassmannians, Schubert varieties, flag manifolds, Families, moduli, classification: algebraic theory Compactifications of subvarieties of tori | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Q_0\) and \(Q_1\) be, respectively, the set of non negative integers not bigger than \(n\) and the set of the first \(n\) positive integers. Identify the pair \(Q=(Q_0,Q_1)\) with an oriented graph, where \(Q_0\) is the set of vertices and \(Q_1\) is the set of arrows. Let \(t:Q_1\rightarrow Q_0\) and \(h:Q_1\rightarrow Q_0\) be, respectively, the tail and the head functions on the arrows. Assume that, \(\{t(a),h(a)\}=\{a-1,a\}\), so that one may have leftwards arrows or rightward arrows, depending on \(\delta(a)=h(a)-t(a)\) being negative or positive. Any such a graph \(Q\) is said to be a quiver of type \(A\) (a chain of vertices with arrows between them). To each such a quiver one can associate a suitable set of quiver representations: They form an affine space (direct sum of certain spaces of homomorphisms of \(\mathbb{C}\)-vector spaces) which is naturally acted on by a group \(G\) which is a product of linear groups. As the authors remark, the \(G\)-orbits of such representations are classified by the lace diagrams of \textit{S. Abeasis} and \textit{A. Del Fra} [J. Algebra 93, 376--412 (1985; Zbl 0598.16030)].
The main result of this paper is the proof (Section 2) of a very explicit formula for the \(G\)-equivariant cohomology class of the closure of any orbit of such an action. Nicely, the authors also show how their formula can be interpreted in terms of classes of degeneracy loci defined by a quiver of vector bundles. The proof of the main result is completed in Section 3, while section 4 is devoted to conjectural speculations, regarding the \(T\)-equivariant Grothendieck class of orbits closure with respect to the action referred to above. In spite of being addressed to people already knowing a bit about the subject, the paper is, on the other hand, written in an especially friendly way and, therefore, it may also serve as a stimulus to walk some initial steps into the beautiful theory of quivers and their representations. quiver orbits; equivariant cohomology Buch, Anders Skovsted; Rimányi, Richárd, A formula for non-equioriented quiver orbits of type \(A\), J. Algebraic Geom., 16, 3, 531-546, (2007) Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Singularities of differentiable mappings in differential topology, Grassmannians, Schubert varieties, flag manifolds A formula for non-equioriented quiver orbits 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 A dense orbit in the unipotent radical of a parabolic group under the adjoint action is called a Richardson orbit. The authors define a quiver-graded version of Richardson orbits generalizing the classical definition in the case of the general linear group, which is defined as follows. First, restrict to a finite, simply-laced quiver \(Q\) with a set \(Q_0\) of vertices and a set \(Q_1\) of arrows. Choose a diagonal Levi-subgroup \(\text{GL}_d := \prod_{i\in Q_0} \text{GL}_{d_i} \) of \(\text{GL}_n\) (for some large \(n\)) and view \(\text{R}_{d} := \prod_{(i\to j) \in Q_1} \text{Hom}_k (k^{d_i}, k^{d_j})\) as a \(\text{GL}_d\)-subrepresentation of \(\text{Lie} (\text{GL}_n)=\text{M}_n\), by picking the \(d_j\times d_i\)-matrix blocks in the matrix ring corresponding to \((j,i)\in Q_0^2\) whenever there is an arrow \((i \to j) \in Q_1\) in the quiver. View this as the quiver-graded analogue of the adjoint representation. Let \(\mathbb{P}\subseteq \text{GL}_n\) be the stabilizer of the flag \(\{ 0 \} = F^{(0)} \subset \ldots \subset F^{(s-1)} \subset F^{(s)}=k^n\), and let \(\mathfrak{u}_{\mathbb{P}}\) be the set of endomorphisms of the vector space \(k^n\) mapping each subspace \(F^{(t)}\) to \(F^{(t-1)}\).
Now, \(\text{P}_{\mathbf{d}} :=\mathbb{P}\cap \text{GL}_{d}\) is a parabolic subgroup of \(\text{GL}_d\), i.e., it is the stabilizer of a \(Q_0\)-graded flag of subvector spaces \(\{ 0 \} = F^{(0)} \subset \ldots \subset F^{(s-1)} \subset F^{(s)}=\bigoplus_{i\in Q_0} k^{d_i}\), with \(\underline{\dim} \: F^{(t)} =: \mathbf{d}^{(t)}\), \(\mathbf{d}:= (\mathbf{d}^{(1)},\ldots,\mathbf{d}^{(s)}=d)\). A quiver-graded Richardson orbit is then a dense \(\text{P}_{\mathbf{d}}\)-orbit in \(\text{R}_{d}^{\mathbf{d}} :=\text{R}_{d}\cap \mathfrak{u}_{\mathbb{P}}\); it is a closed subvariety of the representation space of a quiver, which is acted upon by a product of parabolic subgroups of general linear groups. Such dense orbits do not exist in general.
Let \(0 < s\in \mathbb{Z}\). The staircase quiver \(Q^{(s)}\) of \(Q\) has vertices \(i_t\) for \(i \in Q_0\) and \(t=1,\ldots, s\). It has two families of arrows: there is an arrow \(b(i_t) : i_t \rightarrow i_{t+1}\) for each \(i \in Q_0\) and \(t=1,\ldots ,s-1\), which are called vertical arrows, and for each arrow \((a : i \to j) \in Q_1\) and \(t=2,\ldots,s\), there is an arrow \((a_t : i_t \to j_{t-1})\), which are called the diagonal arrows. Consider the following path relations in the path algebra \(kQ^{(s)}\): \(a_{t+1} b(i_t) = b(j_{t-1}) a_{t}\) for each arrow \((a : i \to j) \in Q_1\) and \(1 < t < s\), and \(a_2 b(i_1) = 0\) for each \((a : i \to j) \in Q_1\). Denote the equioriented quiver of type \(A_s\) by \(\mathbb{A}_s\), or more specifically, \(\mathbb{A}_s := 1\longrightarrow 2 \longrightarrow \ldots \longrightarrow s\). Let \(\mathfrak{I} \subset kQ^{(s)}\) be the ideal generated by the two relations above. Define the nilpotent quiver algebra as \(\text{N}_s(Q) = k Q^{(s)} / \mathfrak{I}\). It is a quasi-hereditary algebra, whose isomorphism classes of \(\Delta\)-filtered modules correspond to orbits in their generalized setting (Theorem 4.10, page 4326): given the algebra \(\text{N}_s(Q)\) with the quasi-hereditary structure from Section 3.4 (page 4315), there is a rigid \(\Delta\)-filtered \(\text{N}_s(Q)\)-module of dimension vector \(\mathbf{d}\) if and only if there is a Richardson orbit for \((Q,\mathbf{d})\).
The authors also study an idempotent recollement of this algebra whose associated intermediate extension functor can be used to produce Richardson orbits (Section 4.2, page 4324), and they also give an example where no Richardson orbit exists (Section 5.2, page 4330). quiver Grassmannian; representation variety; quasi-hereditary algebra; recollement Group actions on varieties or schemes (quotients), Representations of associative Artinian rings, Grassmannians, Schubert varieties, flag manifolds Quiver-graded Richardson orbits | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a compact connected Lie group and let \(T\) be a maximal torus of \(G\). In this paper the authors calculate the real \(KO\)-theory of a flag manifold \(G/T\) for the exceptional Lie groups \(G=G_2, F_4, E_6\) using the Atiyah-Hirzebruch spectral sequence and Steenrod squaring cohomology operations. Also they point out the connection between Witt groups and the real \(KO\)-theory of homogeneous spaces such as Grassmannians and flag manifolds. All calculations are done explicitly. real \(K\)-theory; exceptional Lie groups; flag manifolds; Atiyah-Hirzebruch spectral sequence; Witt groups; generalized cohomology Daisuke Kishimoto and Akihiro Ohsita, \?\?-theory of exceptional flag manifolds, Kyoto J. Math. 53 (2013), no. 3, 673 -- 692. Topological \(K\)-theory, Generalized cohomology and spectral sequences in algebraic topology, Grassmannians, Schubert varieties, flag manifolds \(KO\)-theory of exceptional 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 This is an expository article discussing various manifestations of the natural partial ordering \(\leq\) on the set of all partitions of a given natural number: for \(i=(i_ 1,i_ 2,...),\quad j=(j_ 1,j_ 2,...)\) \(i\leq j\) if and only if \(i_ 1+...+i_ p\leq j_ 1+...+j_ p\) for all \(p\geq 1\). In fact the authors deal mainly with the reverse order which they call the specialization order. Besides various classical occurrences of these orderings (permutation representations of the symmetric groups, Gerstenhaber-Hesselink Theorem on orbits of nilpotent matrices, Gale-Ryser Theorem etc.) the authors also discuss their role in control theory (completely reachable systems) and show how some of these manifestations of the orderings are intimately related. partial ordering; partitions; specialization order; permutation representations; symmetric groups; orbits of nilpotent matrices; Gale-Ryser theorem M. Hazewinkel and C. Martin, ''Representations of the symmetric group, the specialization order, systems and the Grassmann manifold,''L'Enscign. Math.,29, 53--87 (1983). Representations of finite symmetric groups, Partial orders, general, Combinatorial aspects of partitions of integers, Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) Representations of the symmetric group, the specialization order, systems and Grassmann manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{Tatsuo Kimura}, An introduction to prehomogeneous vector spaces (Japanese) (1--45); \textit{Fumihiro Sato}, An introduction to zeta functions of prehomogeneous vector spaces (Japanese) (46--60); \textit{Fumihiro Sato}, Report on the convergence of zeta functions associated with prehomogeneous vector spaces (61--73); \textit{Tsuneo Arakawa}, Motives of the Ibukiyama-Saito theory (Japanese) (74--87); \textit{Tomoyoshi Ibukiyama}, An explicit formula for zeta functions associated with quadratic forms (88--101); \textit{Hiroshi Saito}, Classification of algebraic fields over prehomogeneous vector spaces (Japanese) (102--115); \textit{Hiroshi Saito}, The calculation of zeta functions of prehomogeneous vector spaces (Japanese) (116--126); \textit{Tomoyoshi Ibukiyama}, On dimensions of automorphic forms and zeta functions of prehomogeneous vector space (127--133); \textit{Yasuo Ohno}, The history and prospects of zeta functions of binary cubic forms (Japanese) (134--152); \textit{Tatsuo Kimura, Makiko Fujinaga} and \textit{Takeyoshi Kogiso}, On functional equations of prehomogeneous zeta distributions over a local field of characteristic \(p\) (153--163); \textit{Akihiko Gyoja}, Prehomogeneous vector spaces over finite fields (164--187); \textit{Akihiko Gyoja}, Generic quotient varieties (188--197); \textit{Akihiko Gyoja}, \(\mathbb Z\)-forms of representations of reductive groups and prehomogeneous vector spaces (198--207); \textit{Akihiko Gyoja}, Split \(\mathbb Z\)-forms of irreducible prehomogeneous vector spaces (208--253); \textit{Yi Zhu}, Réalisation des modules irréductibles ayant un poids dominant dans des espaces des fonctions analytiques (Realization of irreducible modules with a highest weight in some spaces of analytic functions) (French) (254--262); \textit{Fumihiro Sato}, A bibliography on the subject of prehomogeneous vector spaces (Japanese) (263--296).
\{The papers of this volume will not be reviewed individually\}. Symposium; Proceedings; Kyoto (Japan); Prehomogeneous vector spaces Proceedings, conferences, collections, etc. pertaining to number theory, Proceedings of conferences of miscellaneous specific interest, Prehomogeneous vector spaces, Grassmannians, Schubert varieties, flag manifolds, Other Dirichlet series and zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) Theory of prehomogeneous vector spaces. Proceedings of a symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, October 17--21, 1994 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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: Suppose \(X \subseteq \mathbb{P}^ n\) is an integral nondegenerate local complete intersection variety such that \(\deg X > 2\) and \(h^ 0(X, {\mathcal N}_ X (-1)) \leq n + 1\). If \(X\) is a hyperplane section of a local complete intersection variety \(W \subseteq \mathbb{P}^{n + 1}\), then \(W\) is a cone with base \(X\). The author mentions that the assumption on the normal bundle \({\mathcal N}_ X\) of \(X\) seems to be extremely restrictive, and that finding a nontrivial class of examples would be nice. Then he proceeds to give a weak version of his theorem in the case in which the ambient variety is a Grassmannian. hyperplane section of a local complete intersection variety; cone; normal bundle; Grassmannian Projective techniques in algebraic geometry, Complete intersections, Grassmannians, Schubert varieties, flag manifolds, Surfaces and higher-dimensional varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry On varieties as hyperplane sections | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathfrak G}\) be a Kac-Moody algebra defined by a Cartan matrix. The author studies the Kac-Moody group \(G\) associated with \({\mathfrak G}\). The group \(G\) is generated by automorphisms \(\exp (\text{ad} x)\), where \(x \in {\mathfrak G}_ r\) for \(r \in \Delta^{re}\), where \(\Delta^{re}\) denotes the set of real roots of \({\mathfrak G}\). We denote further by \(B\) the normalizer in \(G\) of the subgroup \(\langle \{\exp (\text{ad} x);\;x \in {\mathfrak G}_ r\), \(r \in (\Delta^{re})^ +\} \rangle\). For \(h \in {\mathfrak H}\) (the Cartan subalgebra), we define \(P_ h\) to be the normalizer in \(G\) of the subgroup \(\langle \{\exp (\text{ad} x);\;x \in {\mathfrak G}_ r, \langle h,r \rangle \geq 0\}\rangle\).
Let \(\Pi = \alpha_ 1,\dots,\alpha_ n\) be a system of simple real roots for \({\mathfrak G}\) of finite growth \(h=\sum \alpha_ i\). The first theorem asserts that \(G\) is a \(BN\)-pair with the Borel subgroup \(P=P_ h\) and normal standard parabolic subgroups \(P_{\alpha_ i}\). In his previous papers [see e.g., On embeddings of some geometries and flag systems in Lie algebras and superalgebras, Akad. Nauk Ukr. SSR Inst. Mat., Preprint 1990, No. 8, 3-17] the author showed that the incidence system of flags \(\Gamma_ P(G)\) in \(G\) can be considered also as a subset \(HG^ -_ \Pi ({\mathfrak G})\) in \({\mathfrak G}\). In the third theorem an isomorphism of incidence systems \(HG^ -_ \Pi ({\mathfrak G}) \to \Gamma_ P(G)\) is introduced, and an equivalence relation \(\sim_ c\) is defined such that each \(\sim_ c\)-equivalence class is mapped onto a small Schubert cell. The second and fourth theorems are only versions of the first and third theorems in the situation where a simple symmetry of \(\Pi\) is given. The proofs of the theorems are not included. geometry of flags; small Schubert cell; Kac-Moody algebra; Kac-Moody group Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds Small Schubert cells as subsets in Lie algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We propose a construction of a Lagrangian torus fibration of the full flag variety in \(\mathbb C^3\). In contrast to the classical fibration obtained from the Gelfand-Zeitlin system, the proposed fibration is special Lagrangian. flag variety; Lagrangian torus; pseudotoric structure; special Lagrangian fibration Tyurin, N A, No article title, Theor. Math. Phys., 167, 567-576, (2011) Symplectic manifolds (general theory), Grassmannians, Schubert varieties, flag manifolds, Milnor fibration; relations with knot theory, Toric varieties, Newton polyhedra, Okounkov bodies Special Lagrangian fibrations on 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 The authors prove the following theorem: if Y is a projective variety such that a (general) hyperplane section is isomorphic to a Grassmann variety (embedded by the usual Plücker embedding), then Y is a projective cone over this Grassmann variety. The result extends a similar result of Scorza for the Segre variety, a result which is used, together with an inductive procedure, in the course of the proof. Without the words in the brackets, the result has recently been extended by \textit{L. Badescu} [in Algebraic geometry, Proc. int. Conf., Bucharest/Rom. 1982, Lect. Notes Math. 1056, 1-33 (1984; Zbl 0539.14004)]. Grassmann variety; Plücker embedding Di Fiore, L., Freni, S.: On varieties cut out by hyperplanes into Grassmann varieties of arbitrary indexes. Rend. Ist. Mat. Univ. Trieste 13, 51--57 (1981) Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry On a variety cut by hyperplanes in Grassmann varieties of arbitrary indices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal{T}\) be an algebraic triangulated category. A sequence of objects \(E_1, \hdots, E_n \in \mathcal{T}\) is called an exceptional sequence if \(\mathsf{Ext}^l(E_i,E_i)\cong \delta_{0,l}K\) for all \(i\) and if there are no morphisms from \(E_i[k]\) to \(E_j\) for any \(i > j\). An exceptional collection is of expected length, if \(n\) is equal to the rank of the Grothendieck group \(K_0(\mathcal{T})\).
Following \textit{M. M. Kapranov}'s result [Invent. Math. 92, No. 3, 479--508 (1988; Zbl 0651.18008)] that the bounded derived category of coherent sheaves on Grassmannian admits exceptional collections of expected length, one may wonder whether the same holds true for any derived category of coherent sheaves on \(G/P\), where \(G\) is a semisimple algebraic group and \(P \subseteq G\) is a parabolic subgroup.
In the article under review the authors construct exceptional collections of expected length for the derived category of coherent sheaves on \(G/P\), where \(G\) is simple of type \(B_n\), \(C_n\) or \(D_n\) and \(P \subseteq G\) is a parabolic subgroup. exceptional collection; derived category of sheaves; isotropic Grassmannian A. Kuznetsov and A. Polishchuk, Exceptional collections on isotropic Grassmannians, J. Eur. Math. Soc. 18 (2016), no. 3, 507--574. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Grassmannians, Schubert varieties, flag manifolds Exceptional collections on isotropic Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the isotropy problem for algebras with involution. Let \(K\) be a field of characteristic different from two, \(A\) -- a central simple algebra over \(K\) and \(\tau\) -- an involution of \(A\) (assumed to be an anti-automorphism of \(A\) ). Let \(F=K^{\tau}\) be the subfield of \(K\) of \({\tau}\)-invariant elements of \(K\). The authors prove the isotropy theorem which asserts that if \({\tau}\) becomes isotropic over any field extension of \(F\) that splits \(A\), then \({\tau}\) becomes isotropic over a finite odd-degree extension of \(F.\) As the authors remark in the case of symplectic \({\tau}\) the word ``splitting'' should be replaced by the phrase ``almost splitting''. The proof uses in essential way the symmetric and Steenrod operations in the appropriate Chow groups as well as the study of the quasi-split unitary Grassmannians. The isotropy theorem generalizes known results on isotropy of orthogonal and symplectic involutions. central simple algebra; involution, symmetric algebra; Steenrod algebra; Grassmannian Karpenko, N.; Zhykhovich, M., Isotropy of unitary involutions, Acta math., 211, 2, 227-253, (2013) (Equivariant) Chow groups and rings; motives, Associative algebras and orders, Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry Isotropy of unitary involutions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Y\) be an \((n-1)\)-dimensional smooth subvariety of the Grassmannian variety \(G(1,n)\) of lines in the complex projective \(n\)-space. In: Projective geometry with applications, Lect. Notes Pure Appl. Math. 166, 43-66 (1994; Zbl 0839.14015), \textit{E. Arrondo}, \textit{M. Bertolini} and \textit{C. Turrini} classified all such varieties admitting a scroll structure over a curve. In this paper we study whether \(Y\) admits a structure of a quadric fibration over a smooth curve. The first observation is that the bigger \(n\) is, the fewer \((n-2)\) varieties of small degree (degree one for studying scrolls and two for quadric bundles) there are in \(G(1,n)\). In particular, we get that, in case \(n\geq 7\), a quadric bundle \(Y\) must have a fundamental curve, so we can use our general result in the mentioned paper. It also happens that the case \(n=3\) turns out to be very difficult. This case has been studied by \textit{M. Gross} [Math. Z. 212, No. 1, 73-106 (1993; Zbl 0812.14033); pp. 89-93], who can only give partial results. Hence we will work out only the cases with \(n\geq 4\).
The main tool to limit the number of possible cases, is to use in a suitable way Castelnuovo's bound for the genus of projective curves, as well as a generalization of it obtained by L. Giraldo for curves in an arbitrary Grassmannian variety, which appears in the Appendix to this work. We give the whole list of possible varieties, although the actual existence of some of them remains still open. This classification, after the one of scrolls over a curve, is another step in the classification of those \(Y\) that do not behave well under the adjunction process (i.e. the adjoint bundle of which is not nef and big). Grassmannian variety; quadric fibration over a smooth curve; Castelnuovo's bound for the genus of projective curves Arrondo, E.; Bertolini, M.; Turrini, C.: Quadric bundle congruences in \(G(1,n)\). Forum math. 12, 649-666 (2000) Grassmannians, Schubert varieties, flag manifolds, Low codimension problems in algebraic geometry, Fibrations, degenerations in algebraic geometry Quadric bundle congruences in \(G(1,n)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a natural filtration \(F\) on quantum cohomology \(QH(G/B)\) of \(G/B\), which respects the quantum product structure. Its associated graded algebra is isomorphic to the tensor product of \(QH(G/P)\) and a corresponding graded algebra of \(QH(P/B)\) after localization. When the quantum parameter goes to zero, this specializes to the filtration on the classical cohomology \(H(G/B)\) from the Leray spectral sequence associated to the fibration \(P/B\to G/B\to G/P\). Leung, N. C.; Li, C.: Functorial relationships between QH\$ast(G/B)\( and QH\$ast(G/P)\). J. differential geom. 86, No. 2, 303-354 (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Functorial relationships between \(QH^*(G/B)\) and \(QH^*(G/P)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [``Polynomials for symmetric orbit closures in the flag variety'', Preprint, \url{arXiv:1308.2632}] we introduced polynomial representatives of cohomology classes of orbit closures in the flag variety, for the symmetric pair \((\mathrm{GL}_{p+q}, \mathrm{GL}_{p} \times \mathrm{GL}_{q})\). We present analogous results for the remaining symmetric pairs of the form \((\mathrm{GL}_{n}, K)\), i.e., \((\mathrm{GL}_{n}, O_{n})\) and \((\mathrm{GL}_{2n}, \mathrm{Sp}_{2n})\). We establish ``well-definedness'' of certain representatives from \textit{B. J. Wyser} [Transform. Groups 18, No. 2, 557--594 (2013; Zbl 1284.14068)]. It is also shown that the representatives have the combinatorial properties of nonnegativity and stability. Moreover, we give some extensions to equivariant \(K\)-theory. Wyser, B.J., Yong, A.: Polynomials for symmetric orbit closures in the flag variety. Transform. Gr. (\textbf{to appear}) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Equivariant \(K\)-theory Polynomials for symmetric orbit closures in the flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author uses the theory of canonical bases of Kashiwara and Lusztig to generalize the notion of a totally positive matrix to the case of elements of an algebraic group. He also shows that the Kazhdan-Lusztig polynomials appear as matrix coefficients in the change of base between the canonical bases and other natural bases. canonical bases; totally positive matrix; Kazhdan-Lusztig polynomials P. Littelmann, ''Bases canoniques et applications,''Séminaire Bourbaki Vol. 1997/98. Astérisque No. 252 (1998), Exp. No. 847, 5, 287--306. Universal enveloping (super)algebras, Quantum groups (quantized enveloping algebras) and related deformations, Linear algebraic groups over the reals, the complexes, the quaternions, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Grassmannians, Schubert varieties, flag manifolds Canonical bases and applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce and study a class of determinantal probability measures generalising the class of discrete determinantal point processes. These measures live on the Grassmannian of a real, complex, or quaternionic inner product space that is split into pairwise orthogonal finite-dimensional subspaces. They are determined by a positive self-adjoint contraction of the inner product space, in a way that is equivariant under the action of the group of isometries that preserve the splitting. determinantal measures; geometric probability; integral geometry; random geometry; enumerative geometry; Grassmannians; Plücker coordinates; graded vector spaces; matroid stratification; uniform spanning tree; quantum spanning forest Point processes (e.g., Poisson, Cox, Hawkes processes), Geometric probability and stochastic geometry, Grassmannians, Schubert varieties, flag manifolds, Quantum measurement theory, state operations, state preparations, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics Determinantal probability measures on Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author conjectures that a Fano manifold X is homogeneous if the tangent bundle TX is uniform w.r.t. some dominating unsplit family of rational curves (see the paper for the definition). He tests it successfully on the known cases: Characterization of projective spaces by \textit{S. Mori} [Ann. Math. 110, 593--606 (1979; Zbl 0423.14006)]; characterization of hyperquadrics by Cho-Sato, Y. Ye, Hwang-Mok and Wierzba; Grassmann manifolds and isotropic Grassmann manifolds by Hwang and Mok. The key new ideas are: (1) a characterization of twisted trivial vector bundles by \textit{M. Andreatta} and the author [Invent. Math. 146, 209-217 (2001; Zbl 1081.14060)], and (2) the Atiyah extension of the short exact sequence with the cotangent bundle on the left and the structure sheaf on the right. The new proofs of old theorems look very elegant. homogeneous Fano manifold; hyperquadrics; isotropic Grassmann manifolds Fano varieties, Grassmannians, Schubert varieties, flag manifolds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Uniform vector bundles on Fano manifolds and an algebraic proof of Hwang-Mok characterization of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a reductive algebraic group and \(P_1,\dots,P_k\) be parabolic subgroups containing a fixed Borel subgroup. If \(G\) has finitely many orbits under the diagonal action in
\[
X=G/P_1 \times\cdots \times G/P_k,
\]
then \(X\) is called a multiple flag variety of finite type. In the paper under review, the authors classify multiple flag varieties of finite type for \(G=\text{Sp}_{2n}\) continuing their earlier work for \(G=\text{GL}_n\). They also give a complete enumeration of the orbits and explicit representatives for them. Their main tool (as in their earlier paper) is the algebraic theory of quiver representations. reductive algebraic group; Borel subgroup; multiple flag variety of finite type; quiver representations Magyar, P.; Weyman, J.; Zelevinsky, A., Symplectic multiple flag varieties of finite type, J. Algebra, 230, 1, 245-265, (2000) Group actions on varieties or schemes (quotients), Linear algebraic groups over arbitrary fields, Grassmannians, Schubert varieties, flag manifolds Symplectic multiple flag varieties of finite type | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a semisimple algebraic group over an algebraically closed field \(k\) of characteristic zero. Further, let \(\sigma\) be a simple involution of \(G\) and \(H\) denote the corresponding fixed point subgroup of \(G\). Of interest here is the coordinate ring \(k[G/H]\) of the symmetric variety \(G/H\). The \(G\)-module structure of \(k[G/H]\) is known in terms of spherical \(G\)-modules (relative to \(H\)). The goal of this paper is to give a more precise description by identifying the relations among the generators. More precisely, the authors construct a standard monomial theory for \(k[G/H]\) and identify the defining relations when the associated \textit{restricted root system} is of type \(A\), \(C\), or \(BC\) for \(G\) simply connected or type \(B\) for \(G\) of adjoint type. For the root system associated to \(G\), the positive roots can be chosen so that for a positive root \(\alpha\) either \(\sigma(\alpha) = \alpha\) or \(\sigma(\alpha)\) is a negative root. With such a choice, the restricted root system then consists of all non-zero roots of the form \(\alpha - \sigma(\alpha)\) for a root \(\alpha\).
The construction of the standard monomial theory for \(k[G/H]\) is accomplished by showing that \(k[G/H]\) is isomorphic as a \(G\)-module to the coordinate ring of a Richardson variety inside a Grassmann variety associated to \(G\). The Grassmann variety is constructed by creating an extended Lie algebra by adding a node to the Dynkin diagram for the Lie algebra of \(G\). The standard monomial theory for the Grassmann variety can then be translated to \(k[G/H]\). When the restricted root system is of type \(A\) an even more specific description of \(k[G/H]\) is obtained.
While the work here is done over a field of characteristic zero, the authors discuss the extent to which the results could be extended to arbitrary characteristic. symmetric varieties; spherical modules; Grassmannians; standard monomial theory; wonderful compactification; Schubert variety; Richardson variety Rocco Chirivì, Peter Littelmann, and Andrea Maffei, Equations defining symmetric varieties and affine Grassmannians, Int. Math. Res. Not. IMRN 2 (2009), 291 -- 347. Homogeneous spaces and generalizations, Group varieties, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Equations defining symmetric varieties and affine Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we find some necessary and sufficient conditions on the dimension vector \(\underline {\mathbf d} = (d_1, \ldots, d_k; n)\) so that the diagonal action of \(\mathbb{P}{GL}(n)\) on \(\prod_{i = 1}^k {Gr}(d_i; n)\) has a dense orbit. Consequently, we obtain some algorithms for finding dense and sparse dimension vectors and classify dense dimension vectors with small length or size. We also classify dimension vectors where \(| d_i - d_j | < 3\) for all \(i, j\) generalizing a theorem of \textit{V. L. Popov}, in: Proceedings of the international colloquium on algebraic groups and homogeneous spaces. New Delhi: Narosa Publishing House. 481--523 (2007; Zbl 1135.14038)]. Grassmannians; \(\mathbb{P}GL(n)\) actions; dense orbits Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Classical groups (algebro-geometric aspects) Dense \(\mathbb{P}{GL}\)-orbits in products of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Authors' abstract: Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open \(G_{\mathbb{R}}\)-orbits in flag varieties \(G / P\). We investigate Hodge-theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths-Yukawa coupling to the variety of lines on \(G / P\) (under a minimal homogeneous embedding), construct a large class of polarized \(G_{\mathbb{R}}\)-orbits in \(G / P\), and compute the associated Hodge-theoretic boundary components. An emphasis is placed throughout on adjoint flag varieties and the corresponding families of Hodge structures of levels two and four. Mumford-Tate domain; variation of Hodge structure; boundary component; Schubert variety M. Kerr and C. Robles, Hodge theory and real orbits in flag varieties , preprint, [math.AG]. arXiv:1407.4507v1 Variation of Hodge structures (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations, Grassmannians, Schubert varieties, flag manifolds, Transcendental methods, Hodge theory (algebro-geometric aspects) Variations of Hodge structure and orbits in flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute the parameters of the linear codes that are associated with \textit{all} projective embeddings of Grassmann varieties. Grassmann codes; projective systems; parameters of a code Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Linear codes (general theory), Grassmannians, Schubert varieties, flag manifolds, Rational points Higher Grassmann codes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A classical problem in invariant theory is to find a precise description of the subring of the coordinate ring for the \(n\times r\) matrices \(k[M_{nr}]\), \(r<n\), consisting of all the invariant elements with respect to the natural right action of the special linear group \(\text{SL}_r(k)\), where \(k\) is an algebraically closed field of characteristic \(0\). In this paper the authors present a solution for the analogous problem in the quantum case. Here \(k[M_{nr}]\) is replaced by the quantum matrix bialgebra and \(\text{SL}_r(k)\) by the quantum special linear group. This leads to the formulation of the first and second fundamental theorem for the quantum special linear group. Similarly to what happens for the commutative case, the first theorem of quantum coinvariant theory states that the ring of quantum coinvariants coincides with the ring generated by certain quantum minors in the quantum matrix bialgebra. This is precisely the ring of the so-called quantum Grassmannian. Using the results in the first author's earlier paper they are able to give a presentation of the ring of quantum coinvariants in terms of generators and relations. This is the content of the second fundamental theorem of quantum coinvariant theory. Both the first and the second theorem of quantum coinvariant theory reduce to the corresponding classical results when the indeterminate \(q\) is specialized to \(1\). At the end they use the given presentation of the quantum Grassmannian to define quantum Schubert varieties and to show that they are quantum homogeneous spaces; that is, they admit a coaction by a suitable quantum group. quantum special linear groups; quantum matrix bialgebras; rings of quantum coinvariants; quantum Grassmannians; presentations; quantum Schubert varieties; coactions R. Fioresi and C. Hacon, Quantum coinvariant theory for the quantum special linear group and quantum Schubert varieties. J. Algebra 242 (2001), 433-446. Quantum groups (quantized function algebras) and their representations, Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds, Actions of groups and semigroups; invariant theory (associative rings and algebras), Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Quantum groups (quantized enveloping algebras) and related deformations Quantum coinvariant theory for the quantum special linear group and quantum 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 Noting the discrepency between the number, 7, of Clay Mathematics Institute Millennium Problems and the number, 23, of problems posed by David Hilbert at the 1900 Paris ICM, the editors of the Australian Mathematics Society Gazette asked some leading Australian mathematicians to make up the difference with their own favourite `Millennium problem'. In this, the first installment of that series, Alexander Molev proposed the Littlewood-Richardson problem for Schubert polynomials. That problem, which is dear to this reviewer's heart, is to give a manifestly positive combinatorial formula for the product of two Schubert classes in the cohomology ring of a flag manifold, in terms of Schubert classes. For flag manifolds of type A, these classes are represented by Schubert polynomials, and the problem is to generalize the Littlewood-Richardson rule for Schur functions, which governs the multiplication of Schubert classes in the cohomology ring of the Grassmannian.
Like Hilbert's 3rd problem (partially solved in 1900) and perhaps the 5th Millennium problem (the Poincaré conjecture), this challenge problem may be solved soon after it was posed. In 2004, Izzet Coskun announced a solution generalizing Ravi Vakil's geometric Littlewood-Richardson rule for the Grassmannian. Coskun now (August 2005) has a manuscript with a proof of this rule in the important special case of two-step flag manifolds, which gives a Littlewood-Richardson rule for the quantum cohomology of a Grassmannian.
While this problem is not as deep as the Clay Mathematical Institute's Millennium Challenge Problems, it is one of the most vexing and important open problems in algebraic combinatorics. Having resisted the efforts of many strong mathematicians in the past 15 years, it is a fine beginning to the Australian Mathematical Society Gazette's light-hearted series of 16 challenges. flag manifolds; cohomology ring; Grassmannian; quantum cohomology Molev, A.: Littlewood-Richardson problem for Schubert polynomials. Austral. math. Soc. gaz. 31, No. 5, 295-297 (2004) Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds The 8th problem: Littlewood-Richardson problem for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Author's abstract: ``Let \(V\) be an \(n\)-dimensional vector space over a division ring \((4\leq n<\infty)\) and let \(\mathcal G_k(V)\) be the Grassmannian formed by all \(k\)-dimensional subspaces of \(V\). The corresponding Grassmann graph will be denoted by \(\Gamma_k(V)\).
The author describes all isometric embeddings of Johnson graphs \(J(l,m)\), \(1<m<l-1\), in \(\Gamma_k(V)\), \(1<k<n-1\) (Theorem 4). As a consequence, the author obtains the following: the image of every isometric embedding of \(J(n,k)\) in \(\Gamma_k(V)\) is an apartment of \(\mathcal G_k(V)\) if and only if \(n=2k\).
His second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in \(\Gamma_k(V)\), \(1<k<n-1\).''
The term `rigid' means that every automorphism of the restriction of \(\Gamma_k(V)\) to the image can be extended to an automorphism of \(\Gamma_k(V)\). Johnson graph; Grassmann graph; building; apartment; rigid isometric embeddings Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Grassmannians, Schubert varieties, flag manifolds Isometric embeddings of Johnson graphs in Grassmann graphs | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(T\) be a compact torus and \((M,\omega)\) a Hamiltonian \(T\)-space. We give a new proof of the \(K\)-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry [\textit{M. Harada} and \textit{G. D. Landweber}, Trans. Am. Math. Soc. 359, No. 12, 6001--6025 (2007; Zbl 1128.53057)] by using the equivariant version of the Kirwan map introduced in [\textit{R. F. Goldin}, Geom. Funct. Anal. 12, No. 3, 567--583 (2002; Zbl 1033.53072)]. We compute the kernel of this equivariant Kirwan map, and hence give a computation of the kernel of the Kirwan map. As an application, we find the presentation of the kernel of the Kirwan map for the \(T\)-equivariant \(K\)-theory of flag varieties \(G/T\), where \(G\) is a compact, connected and simply-connected Lie group. In the last section, we find explicit formulae for the \(K\)-theory of weight varieties. Kirwan surjectivity; flag variety; weight variety; equivariant \(K\)-theory; symplectic quotient Leung, H. H., K-theory of weight varieties, New York J. Math., 17, 251-267, (2011) Momentum maps; symplectic reduction, Grassmannians, Schubert varieties, flag manifolds, Equivariant \(K\)-theory \(K\)-theory of weight varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the notion of affine ringed space, see its meaning in topological, differentiable and algebro-geometric contexts and show how to reduce the affineness of a ringed space to that of a ringed finite space. Then, we characterize schematic finite spaces and affine schematic spaces in terms of combinatorial data. Finally, we prove Serre's criterion of affineness for schematic finite spaces. This yields, in particular, Serre's criterion of affineness on schemes. ringed space; affine space; finite space; quasi-coherent module; Serre's criterion of affineness Sancho~de Salas, F.; Sancho~de Salas, P., Affine ringed spaces and serre's criterion, Rocky mountain J. math., (2017), in press Generalizations (algebraic spaces, stacks), Algebraic combinatorics, Algebraic aspects of posets, (Co)homology theory in algebraic geometry Affine ringed spaces and Serre's criterion | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 smooth complex projective variety of dimension \(2r\) and let \(E\) be a \((2r-4)\)-ample vector bundle of rank two on \(X\).
In the paper under review, the authors prove that if a subvariety of \(X\) isomorphic to the Grassmannian of lines \({\mathbb G}(1,r)\), with \(r \geq 4\), is the zero set of a section of \(E\), then \(X\) is isomorphic to the Grassmannian \({\mathbb G}(1,r+1)\) and \(E\) is the universal quotient of this Grassmannian.
As a consequence, they obtain that the Grassmannian of lines in \({\mathbb P}^r\), with \(r\geq 4\), cannot be the zero locus of a section of an ample vector bundle of rank two on a smooth complex projective variety, extending a classical non-extendability theorem of \textit{T. Fujita} on ample divisors in [J. Math. Soc. Japan 33, 405--414 (1981; Zbl 0475.14014)] to codimension two. Grassmannians; vector bundles Grassmannians, Schubert varieties, flag manifolds, Minimal model program (Mori theory, extremal rays), Fano varieties An extension of Fujita's non-extendability theorem for 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 \(k\) be an algebraically closed field of positive characteristic \(p=\text{char}(k)\).
Let \(X\) be a smooth projective \(k\)-variety,
and let \(\mathcal{T}\) denote the pushforward of the structure sheaf of \(X\) by \textit{the} Frobenius morphism \(\text{Fr}:X\to X\), \(f\mapsto f^p\). It is a natural question to ask when \(\mathcal{T}\) satisfies the following two properties:
1) \(\mathcal{T}\) is a generator for the bounded derived category of cohorent modules on \(X\),
2) \(\text{Ext}^i_X (\mathcal{T},\mathcal{T})=0\) for every \(i>0\).
If these two conditions hold, then \(\mathcal{T}\) is said to be a \textit{tilting bundle} on \(X\).
These are rather stringent conditions, so, in general, \(\mathcal{T}\) will not be a tilting bundle
on every \(k\)-variety. Nevertheless, depending on the characteristic of the field,
for several well-known families of varieties \(\mathcal{T}\) gives a tilting bundle.
The examples include all partial flag varieties, quadrics, as well as some Fano toric varieties.
Let \(\mathbb{G}\) denote the Grassmann variety of 2 dimensional subspaces of \(k^n\), where \(n\) is assumed to be \(\geq 4\).
Let \(\mathcal{T}\) denote the bundle \(\text{Fr}_*(\mathcal{O}_{\mathbb{G}})\).
In this article under review, the authors present several important calculations about \(\mathcal{T}\),
and they formulate a series of general conjectures.
As one of the main results of the article,
in Theorem 16.5 and Corollary 16.6, all indecomposable summands of the bundle \(\mathcal{T}\) are described.
We will not reproduce these summands in here since they form a rather lengthy list;
we refer the reader to the article for details.
As a corollary of this main decomposition result, the authors are able to conclude that
\(\mathcal{T}\) is not a tilting bundle on \(X\) unless \(n=4\) and \(p>3\).
This conclusion is recorded as Theorem 1.1 in the introduction and as Corollary 16.7 in the main body of the paper.
The proof of Theorem 16.5 is obtained from a more general decomposition result described by the authors.
The main ingredients of this more general result can be summarized as follows: let \(G\) denote \(\text{SL}(V)\), where \(\dim_k V =2\).
Let \(S\) denote the symmetric algebra \(\text{Sym}(V^{\oplus n})\) (\(n\geq 4\)), and let \(R\) denote the ring of \(G\)-invariants \(S^G\).
One knows that, coincidentally, the homogeneous coordinate ring of \(\mathbb{G}\) is equal to \(R\).
Roughly speaking, the main result of the paper is a theorem about the decomposition of \(R\) into \(R^p\)-modules, where \(R^p\) is the Frobenius image of \(R\).
It turns out that this decomposition is equivalent to the decomposition of \(S^{G_1}\) as a \((G^{(1)}, S^p)\)-module,
where \(S^p\) is the Frobenius image of \(S\), \(G_1\) is the kernel of the Frobenius map on \(G\), and \(G^{(1)} := G/G_1\).
Once again, since the indexing sets that appear in this decomposition are rather lengthy, we refer the reader to the text, specifically, to Theorem 6.2.
Note that Theorem 6.2 leads to the decomposition theorem for \(R\), which is recorded as Theorem 15.1.
In turn, the decomposition theorem for \(R\) leads to the decomposition theorem for \(\mathcal{T}\), which is Theorem 16.5. Now, let \(Y\) be a singular variety, and set \(\mathcal{M}:= \text{Fr}_*(\mathcal{O}_Y)\).
Then \(\mathcal{M}\) is said to provide a \textit{noncommutative resolution} (abbreviated to NCR)
if \(\mathcal{E}nd_Y(\mathcal{M})\) is a sheaf of algebras on \(Y\) which is locally of finite global dimension.
As an example for such a variety,
the authors consider the cone \(Y\) over the Pücker embedding of \(\mathbb{G}\).
The second main result of the present paper states that, for \(p \geq n-2\),
the bundle \(\text{Fr}_*(\mathcal{O}_Y)\) provides an NCR for \(Y\). The proof of this result, which is essentially a corollary of the authors' decomposition theorems is in Section 17 of the article.
Next, we will discuss some general conjectures that are presented in the article.
Let \(R=k+R_1+\cdots \) be an integral finitely generated \(\mathbb{N}\)-graded algebra, where \(k\) is, as before,
an algebraically closed field of positive characteristic \(p=\text{char}(k)\).
A \textit{higher Frobenius summand} is an indecomposable summand of \(R^{1/p^r}\) for some \(r\geq 1\).
\(R\) is said to have \textit{finite \(F\)-presentation type}, abbreviated to FFRT, if the number of isomorphism classes of higher Frobenius summands is finite.
It is known for sometime that the invariant rings for reductive group actions have FFRT, see
[\textit{K. E. Smith} and \textit{M. Van den Bergh}, Proc. Lond. Math. Soc. (3) 75, No. 1, 32--62 (1997; Zbl 0948.16019)].
The authors' definition of \textit{FFRT in characteristic zero} is as follows:
Assume that \(k\) is algebraically closed and of characteristic zero, and
let \(R=k+R_1+\cdots \) be an integral finitely generated \(\mathbb{N}\)-graded algebra.
Then \(R\) is said to satisfy \textit{FFRT} if there exists a finitely generated \(\mathbb{Z}\)-algebra \(A\subset k\), a finitely generated graded flat \(A\)-algebra \(R_A\) such that \(k\otimes_A R_A = R\)
and such that for every geometric point \(x : \text{Spec}\ l \to \text{Spec}\ A\), where \(\text{char}(l) >0\), \(x^*(R_A)\) satisfies FFRT.
The authors conjecture that if \(W\) is a finite dimensional representation of a reductive group \(G\), then \(k[W]^G\) satisfies FFRT. This conjecture is called the \textit{FFRT conjecture in characteristic zero.}
Another interesting conjecture that they propose is called the \textit{\(p\)-uniformity for invariant rings}:
Assume that \(k\) is an algebraically closed field of characteristic zero.
Let \(R=k+R_1+\cdots \) be an integral finitely generated \(\mathbb{N}\)-graded algebra.
Then \(R\) is said to satisfy \textit{\(p\)-uniformity} if there exists a finitely generated \(\mathbb{Z}\)-algebra
\(A\subset k\), a finitely generated graded flat \(A\)-algebra \(R_A\) such that \(k\otimes_A R_A = R\),
a finite set \((M_{A,i})_{i=1}^m\) of finitely generated graded \(R_A\)-modules such that for every geometric
point \(x : \text{Spec}\ l \to \text{Spec}\ A\), where \(\text{char}(l) >0\),
the Frobenius summands of \(x^*(R_A)\) are given by the shifts of \((x^*(M_{A,i})^{\text{Fr}})_{i=1}^m\).
Then the authors' \textit{\(p\)-uniformity conjecture for invariant rings (in characteristic zero)} states that
if \(W\) is a finite dimensional representation of a reductive group \(G\), then \(k[W]^G\) satisfies \(p\)-uniformity.
Finally, authors' \textit{resolution conjecture} goes as follows:
Let \(k\) be an algebraically closed field of characteristic \(p>0\), and let \(G\) be an algebraic \(k\)-group.
Assume that \(G\) acts on a commutative \(k\)-algebra \(S\).
Let \(M\) be a finitely generated \((G,S)\)-module with a good filtration.
Then there exists a tilting module \(T\) for \(G\) and a surjective \((G,S)\)-morphism \(T\otimes_k S \to M\) whose
kernel has a good filtration.
There is also a graded version of this conjecture, where \(T\) is assumed to be a graded \(G\)-module.
In Section 9 of the article, the authors show that the graded resolution conjecture holds for \(\text{SL}_2\). invariant theory; Frobenius summand; FFRT; Grassmannian; tilting bundle; noncommutative resolution Grassmannians, Schubert varieties, flag manifolds, Actions of groups on commutative rings; invariant theory, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Positive characteristic ground fields in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects), Noncommutative algebraic geometry The Frobenius morphism in invariant theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article comprises the notes of three lectures delivered by the author at the Summer School ``Algebraic Groups'', which took place at the University of Göttingen, Germany, in June/July 2005. Apparently geared toward a wider audience, these lectures were to provide a concise introduction to loop groups of compact Lie groups and some aspects of their representation theory. In view of their introductory character, these notes are organized as follows.
Lecture 1: Review of compact Lie groups and their representations, basics of loop groups of compact Lie groups, and illustrating examples.
Lecture 2: Finer properties of loop groups, root systems and Weyl groups of loop groups, and central extensions of loop groups.
Lecture 3: Representations and homogeneous spaces of loop groups, with a view toward infinite Grassmannians.
The mostly explanatory and survey-like notes are largely based on the excellent standard monograph ``Loop Groups'' by \textit{A. Pressley} and \textit{G. Segal} (1986; Zbl 0618.22011), which the author generally refers to. Apart from providing a quick and lucid introduction to the conceptual framework of loop groups of compact Lie groups, these lectures also touch upon the physical background and significance of the subject, mainly by paying special attention to the so-called representations of positive energy for loop groups. However, the relations of loop groups to what is nowadays understood by the notion of string topology [cf. \textit{M. Chas} and \textit{D. Sullivan}, String Topology, math.GT/9911159] are barely mentioned in these notes, in contrast to what their title might suggest. As for the style of exposition, the author naturally focusses on explaining the basic constructions and fundamental theorems within his theme of discussion, with numerous concrete examples, outlines of proofs of major theorems, and a few related exercises included. In this regard, the present notes provide an inspiring first reading for beginners in the rapidly growing field of loop groups and their applications to various theories in contemporary mathematics and theoretical physics. loop groups; representations of loop groups; infinite-dimensional Lie groups; compact Lie groups; homogeneous spaces; Grassmannians Loop groups and related constructions, group-theoretic treatment, Infinite-dimensional Lie groups and their Lie algebras: general properties, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Compact groups, Homogeneous complex manifolds, Grassmannians, Schubert varieties, flag manifolds Loop groups and string topology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 simple (hence adjoint) algebraic group with wonderful compactification \(\overline G\). The main result of this paper is the explicit determination of the closure \(\overline U\) in \(\overline G\) of the unipotent radical \(U\) of a Borel subgroup \(B\). It confirms a conjecture of Lusztig who expects to use it in his theory of ``parabolic character sheaves'' on \(\overline G\). Starting point is the description by Springer of \(B\times B\)-orbits in \(\overline G\). One also needs a study of Coxeter elements in Weyl groups. Some case by case combinatorics gives a lower bound on \(\overline U\). One shows this lower bound equals an upper bound. The unipotent radical is one of the Steinberg fibers. It turns out that the boundary \(\overline F-F\) is the same for every Steinberg fiber \(F\). unipotent varieties; Steinberg fibers; group compactifications; connected simple algebraic groups; wonderful compactifications He, Xuhua: Unipotent variety in the group compactification Linear algebraic groups over arbitrary fields, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Unipotent variety in the group compactification. | 0 |
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