The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope

In their fundamental paper [14] Kazhdan and Lusztig defined, for every Coxeter group W, a family of polynomials, indexed by pairs of elements of W, which have become known as the Kazhdan-Lusztig polynomials of W (see, e.g., [12], Chap. 7). These polynomials are intimately related to the Bruhat order of W and to the algebraic geometry of Schubert varieties, and have proven to be of fundamental importance in representation theory. Our aim in this work is to give a simple, nonrecursive, combinatorial formula for any Kazhdan-Lusztig polynomial of any Coxeter group (Theorem 4.1), and to study some consequences of it. The main idea involved in the proof and statement of this formula is that of extending the concept of the Rpolynomial (see, e.g., [12], w to any (finite) multichain of W (so that, for multichains of length 1, one obtains, apart from sign, the usual R-polynomials). Once this has been done, then the Kazhdan-Lusztig polynomial of a pair u, v turns out to be just the sum, over all multichains from u to v, of the corresponding (generalized) R-polynomials. The R-polynomial of a multichain can be readily defined, and computed from the ordinary R-polynomials (see (9), (10), and (11)). Since several combinatorial formulas and interpretations are known for these polynomials (see, e.g., [5, 10], and, for the case of symmetric groups, [3]) and simple recurrences exist for them, we feel that this formula is a significant step forward in the understanding of the KazhdanLusztig polynomials. Though combinatorial formulas for Kazhdan-Lusztig polynomials have appeared before in the literature (see, e.g., [16, 21, 8, 6]), none of them hold in complete generality.

The connectedness of Hessenberg varieties

The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a non-recursive combinatorial formula for Kazhdan-Lusztig polynomials $P_{x,w}(q)$ of finite Weyl groups. This generalizes results of Billey-Warrington which identified the $321$-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. We also show that the leading coefficient known as $\mu (x,w)$ for these Kazhdan―Lusztig polynomials is always either $0$ or $1$. Finally, we generalize the simple combinatorial formula for the Kazhdan―Lusztig polynomials of the $321$-hexagon-avoiding permutations to the case when $w$ is hexagon avoiding and maximally clustered. Les polynômes de Kazhdan-Lusztig $P_{x,w}(q)$ des groupes de Weyl finis apparaissent en théorie des représentations, ainsi qu’en géométrie des variétés de Schubert. Il a été démontré peu après leur introduction qu’ils avaient des coefficients entiers positifs, mais on ne connaît toujours pas d’interprétation combinatoire simple de cette propriété dans le cas général. Deodhar a proposé un cadre donnant un algorithme, en général récursif, calculant des formules attractives pour les polynômes de Kazhdan-Lusztig. Billey-Warrington ont démontré que cet algorithme est non récursif lorsque$w$ évite les hexagones et les $321$ et qu’il donne des formules combinatoires simples. Nous introduisons une notion d’évitement de schémas dansles groupes de Coxeter quelconques nous permettant de généraliser les résultats de Billey-Warrington à tout groupe de Weyl fini. Nous montrons que le coefficient de tête $\mu (x,w)$ de ces polynômes de Kazhdan-Lusztig est toujours $0$ ou $1$. Cela généralise aussi des résultats de Fan-Greenqui identifient les groupes de Coxeter complètement serrés. Enfin, en type $A$, nous obtenons une classe plus large de permutations évitant la récursion.

The Kazhdan-Lusztig polynomials for nite Weyl groups arise in the geometry of Schubert varieties and representation theory. It was proved very soon after their introduction that they have nonnegative integer coefcients, but no simple all positive interpretation for them is known in general. Deodhar (16) has given a framework for computing the Kazhdan-Lusztig polynomials which generally involves recursion. We dene embedded factor pattern avoidance for general Coxeter groups and use it to characterize when Deodhar's algorithm yields a simple combinatorial formula for the Kazhdan-Lusztig polynomials of nite Weyl groups. Equivalently, if (W; S) is a Coxeter system for a nite Weyl group, we classify the elements w 2 W for which the Kazhdan-Lusztig basis element C 0 can be written as a monomial of C 0 where s 2 S. This work generalizes results of Billey-Warrington (8) that identied the Deodhar elements in type A as 321-hexagon-avoiding permutations, and Fan-Green (18) that identied the fully-tight Coxeter groups.

Given a stratified variety X X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of “parity sheaves,” which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If X X admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semi-small case. Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several well-known complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig’s conjecture.

Operation of class sums on permutation modules

Schubert polynomials and classes of Hessenberg varieties

Let G be a linear, real, reductive group, and let P be a parabolic subgroup. The Bruhat decomposition of G gives a cellular decomposition of the generalized flag manifold X = G/P. Originating in the work of Schubert on Grassmann manifolds, this cellular decomposition was used by Ehresmann [Eh] to give a proof of the basis theorem for the integral cohomology of Grassmannians. Since Ehresmann it has been clear that, for a generalized flag manifold X, a more detailed understanding of this decomposition by generalized Schubert cells would provide a determination of the integral (co-)homology of X. The aim of this article is to give a representation theoretic determination of the differentials in the Schubert cell decomposition of X and thereby obtain a combinatorial description of the integral (co-)homology of X. Our primary tool will be the infinite-dimensional representation theory of G. If G and P are complex groups, elementary considerations show that all Schubert cells define non-zero integral homology classes and that none are torsion. Moreover, the cohomology has the well-known connection to finitedimensional representation theory ([Ct], [Ko1], [Ko2], [Bt]). For real groups G and P, even in low dimensions, torsion can be present in the homology of real flag manifolds. Thus the major new development in this paper is that the infinite-dimensional representation theory of G detects which real Schubert cells define integral, in particular torsion, classes in the (co-)homology. The innovation on the representation-theoretic side that makes possible the consideration of (co-)homology with integer coefficients is the geometric formulation of representation theory introduced by Beilinson-Bernstein. To compute homology we use a topological technique that applies to spaces with filtration. Let X p be the union of Schubert cells whose dimension

In this paper we determine closed-form expressions for the associated Jacobi polynomials, i.e., the polynomials satisfying the recurrence relation for Jacobi polynomials with n replaced by n + c, for arbitrary real c ≧ 0. One expression allows us to give in closed form the [n — 1/n] Padé approximant for what is essentially Gauss' continued fraction, thus completing the theory of explicit representations of main diagonal and off-diagonal Padé approximants to the ratio of two Gaussian hypergeometric functions and their confluent forms, an effort begun in [2] and [19]. (We actually give only the [n — 1/n] Padé element, although other cases are easily constructed, see [19] for details.)We also determine the weight function for the polynomials in certain cases where there are no discrete point masses. Concerning a weight function for these polynomials, so many writers have obtained so many partial results that our formula should be considered an epitome rather than a real discovery, see the discussion in Section 3.

Given a compact Hamiltonian T-manifold M, some component of whose moment map is a Palais-Smale Morse function, we give a formula for the Duistermaat-Heckman measure (the Fourier transform of the equivariant symplectic volume) as a sum of volumes of simplices. Unlike previous formulae for this measure, this sum has only positive terms. In the case of M a flag manifold, our results are an asymptotic version of Littelmann's multiplicity result in representation theory [9].

Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials and may be viewed as K-theoretic analogs of key polynomials. The latter two polynomials have combinatorial formulas involving tableaux: Lascoux and Schützenberger gave a combinatorial formula for key polynomials using right keys; Buch gave a set-valued tableau formula for Grassmannian stable Grothendieck polynomials. We establish a novel combinatorial description for Lascoux polynomials involving right keys and set-valued tableaux. Our description generalizes the tableaux formulas of key polynomials and Grassmannian stable Grothendieck polynomials. To prove our description, we construct a new abstract Kashiwara crystal structure on set-valued tableaux. This construction answers an open problem of Monical, Pechenik and Scrimshaw.Mathematics Subject Classifications: 05E05Keywords: Lascoux polynomials, set-valued tableaux, crystal operators

We obtain several formulas for the Poincaré series defined by B. Kostant for Klein groups (binary polyhedral groups) and some formulas for Coxeter polynomials (characteristic polynomials of monodromy in the case of singularities). Some of these formulas—the generalized Ebeling formula, the Christoffel-Darboux identity, and the combinatorial formula—are corollaries to the well-known statements on the characteristic polynomial of a graph and are analogous to formulas for orthogonal polynomials. The ratios of Poincaré series and Coxeter polynomials are represented in terms of branched continued fractions, which are q-analogs of continued fractions that arise in the theory of resolution of singularities and in the Kirby calculus. Other formulas connect the ratios of some Poincaré series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. The results obtained by S.M. Gusein-Zade, F. Delgado, and A. Campillo allow one to consider these facts as statements on the Poincaré series of the rings of functions on the singularities of curves, which suggests the following conjecture: the ratio of the Poincaré series of the rings of functions for close (in the sense of adjacency or position in a series) singularities of curves is determined by the Burau representation or by the Milnor invariants of a string link, which is an intermediate object in the transformation of the knot of one singularity into the knot of the other.

On commutator fibers over regular semisimple and central elements of SLn

Flag varieties of reductive Lie groups and their subvarieties play a central role in representation theory. In the early 1980s, V. Deodhar introduced a decomposition of the flag variety which was then used to study the Kazdan-Lusztig polynomials. A Deodhar-type decomposition of the product of the flag variety with itself, referred to as the double flag variety, was introduced in 2007 by B. Webster and M. Yakimov, and each piece of the decomposition was shown to be coisotropic with respect to a naturally defined Poisson structure on the double flag variety. The work of Webster and Yakimov was partially motivated by the theory of cluster algebras in which Poisson structures play an important role.
\n
\nThe Deodhar decomposition of the flag variety is better understood in terms of a cell decomposition of Bott-Samelson varieties, which are resolutions of Schubert varieties inside the flag variety. In the thesis, double Bott-Samelson varieties were introduced and cell decompositions of a Bott-Samelson variety were constructed using shuffles. When the sequences of simple reflections defining the double Bott-Samelson variety are reduced, the Deodhar-type decomposition on the double flag variety defined by Webster and Yakimov was recovered. A naturally defined Poisson structure on the double Bott-Samelson variety was also studied in the thesis, and each cell in the cell decomposition was shown to be coisotropic. For the cells that are Poisson, coordinates on the cells were also constructed and were shown to be log-canonical for the Poisson structure.

Preface. Program. List of participants. Part One: Matrix Models and Graph Enumeration. Matrix Quantum Mechanics V. Kazakov. Introduction to matrix models E. Brezin. A Class of the Multi-Interval Eigenvalue Distributions of Matrix Models and Related Structures V. Buslaev, L. Pastur. Combinatorics and Probability of Maps V.A. Malyshev. The Combinatorics of Alternating Tangles: from theory to computerized enumeration J.L. Jacobsen, P. Zinn-Justin. Invariance Principles for Non-uniform Random Mappings and Trees D. Aldous, J. Pitman. Part Two: Integrable Models (of Statistical Physics and Quantum Field Theory). Renormalization group solution of fermionic Dyson model M.D. Missarov. Statistical Mechanics and Number Theory H.E. Boos, V.E. Korepin. Quantization of Thermodynamics and the Bardeen-Cooper-Schriffer-Bogolyubov Equation V.P. Maslov. Approximate Distribution of Hitting Probabilities for a Regular Surface with Compact Support in 2D D.S. Grebenkov. Part Three: Representation Theory. Notes on homogeneous vector bundles over complex flag manifolds S. Igonin. Representation Theory and Doubles of Yangians of Classical Lie Superalgebras V. Stukopin. Idempotent (asymptotic) Mathematics and the Representation theory G.L. Litvinov, et al. A new approach to Berezin kernels and canonical representations G. van Dijk. Theta Hypergeometric Series V.P. Spiridonov.