We make progress on a question of Skandera by showing that a product of Kazhdan-Lusztig basis elements indexed by maximal elements of parabolic subgroups admits a Kazhdan-Lusztig basis element as a quotient arising from operations in the Schur algebroid if and only if the sequence of parabolic subgroups satisfy both a rigidity condition and a combinatorial criterion. For Weyl groups, the rigidity condition specializes to a necessary condition for smallness of Gelfand-MacPherson resolutions. For Schubert varieties indexed by 4231-avoiding permutations, we derive a stronger necessary condition that, up to an appropriate equivalence, is satisfied by at most one Gelfand-MacPherson resolution, and exactly one if and only if 45312 is also avoided. Moreover, we apply the combinatorial criterion to prove the (essentially unique) resolution is small when 34512 and 45123 are likewise avoided.
We develop the combinatorial criterion as part of a generalization of Deodhar's combinatorial setting for questions in Kazhdan-Lusztig theory, which in the case of Weyl groups we show captures the Byałynicki-Birula decompositions of Gelfand-MacPherson resolutions. In particular, we obtain new combinatorial interpretations of Kazhdan-Lusztig polynomials, new algorithms for computing them, and in the case of Weyl groups, an equivalence of the existence of small Gelfand-MacPherson resolutions with that of certain factorizations and generating-function interpretations of the Poincaré polynomial of intersection cohomology of the Schubert variety.
Motivated by the concepts of the inverse Kazhdan-Lusztig polynomial and the equivariant Kazhdan-Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan-Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse Kazhdan-Lusztig polynomial of a matroid is very useful for determining its equivariant Kazhdan-Lusztig polynomials, and we determine the equivariant inverse Kazhdan-Lusztig polynomials for Boolean matroids and uniform matroids. As an application, we give a new proof of Gedeon, Proudfoot and Young's formula for the equivariant Kazhdan-Lusztig poly…
The purpose of this work is to provide a common combinatorial framework for some of the analogues and generalizations of Kazhdan-Lusztig R-polynomials that have appeared since the introduction of these remarkable polynomials (e.g., parabolic Kazhdan-Lusztig R-polynomials, Kazhdan-Lusztig R-polynomials of zircons, and Kazhdan-Lusztig-Vogan polynomials for fixed point free involutions).
We provide a short proof on the change-of-basis coefficients from the Specht basis to the Kazhdan-Lusztig basis, using Kazhdan-Lusztig theory for parabolic Hecke algebra.
We refine an idea of Deodhar, whose goal is a counting formula for Kazhdan-Lusztig polynomials. This is a consequence of a simple observation that one can use the solution of Soergel's conjecture to make ambiguities involved in defining certain morphisms between Soergel bimodules in characteristic zero (double leaves) disappear.
We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan- Lusztig polynomials and we prove it for finite simply-laced Weyl groups.
Kazhdan and Lusztig define, for an arbitrary Coxeter system $(W,S)$, a family of polynomials indexed by pairs of elements of $W$. Despite their relevance and elementary definition, the explicit computation of these polynomials is still one of the hardest open problems in algebraic combinatorics. In this paper we explicitly compute Kazhdan-Lusztig polynomials for a Coxeter system of type $\tilde{B}_2$.
Using a characterization of a generalized $τ$-invariant for intermediate parameter Hecke algebras in type $B_n$, we verify a conjectural description of Kazhdan-Lusztig cells in this setting due to C. Bonnafé, L. Iancu, M. Geck, and T. Lam.
Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan-Lusztig immanants, which are indexed by permutations, involve $q=1$ specializations of Type A Kazhdan-Lusztig polynomials, and were defined in (Rhoades-Skandera, 2006). Using results of (Haiman, 1993) and (Stembridge, 1991), Rhoades and Skandera showed that Kazhdan-Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan-Lusztig immanants are positive on $k$-positive matrices (matrices whose minors of size $k \times k$ and smaller are positive). We show…
We employ general parabolic recursion methods to demonstrate the recently devised hypercube formula for Kazhdan-Lusztig polynomials of $S_n$, and establish its generalization to the full setting of a finite Coxeter system through algebraic proof. We introduce procedures for positive decompositions of $q$-derived Kazhdan-Lusztig polynomials within this setting, that utilize classical Hecke algebra positivity phenomena of Dyer-Lehrer and Grojnowski-Haiman. This leads to a distinct algorithmic approach to the subject, based on induction from a parabolic subgroup. We propose suitable weak variants…
In this note, we show that certain dual canonical basis elements of $\mathbb{C}[SL_m]$ are positive when evaluated on $k$-positive matrices, matrices whose minors of size $k \times k$ and smaller are positive. Skandera showed that all dual canonical basis elements of $\mathbb{C}[SL_m]$ can be written in terms of Kazhdan-Lusztig immanants, which were introduced by Rhoades and Skandera. We focus on the basis elements which are expressed in terms of Kazhdan-Lusztig immanants indexed by 1324- and 2143-avoiding permutations. This extends previous work of the authors on Kazhdan-Lusztig immanants and…
Special matchings and parabolic Kazhdan-Lusztig polynomials
我们证明了一个特殊的匹配的组合概念可以用来计算抛物Kazhdan-Lusztig多项式的双花边Coxeter群和二面角Coxeter群。特别地，对于包含所有外尔群的这类群，我们的结果推广到了抛物情形[Advances in Math. {202}（2006），555-601]中的主要结果。因此，由$u$和$v$索引的抛物Kazhdan-Lusztig多项式仅取决于从单位元到$v$的Bruhat区间的偏序集结构以及该区间的哪些元素是最小陪集代表。
In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig polynomials for matroids. It is known that both the classical Kazhdan-Lusztig polynomials and the matroid Kazhdan-Lusztig polynomials can be considered as special cases of the Kazhdan-Lusztig-Stanley polynomials for locally finite posets. In the framework of Kazhdan-Lusztig-Stanley polynomials, we study the inverse of Kazhdan-Lusztig-Stanley functions and define the inverse Kazhdan-Lusztig polynomials for matroids. We also compute these polynomi…
Combinatorics of Kazhdan-Lusztig cells in affine type $A$ was originally developed by Lusztig, Shi, and Xi. Building on their work, Chmutov, Pylyavskyy, and Yudovina introduced the affine matrix-ball construction (abbreviated AMBC) which gives an analog of Robinson-Schensted correspondence for affine symmetric groups. An alternative approach to Kazhdan-Lusztig theory in affine type $A$ was developed by Blasiak in his work on catabolism. He introduced sign insertion algorithm and conjectured that if one fixes the two-sided cell, the recording tableau of sign insertion process determines uniquel…
The complex irreducible representations of the symmetric group carry an important canonical basis called the Kazhdan-Lusztig basis. Although it is difficult to express how general permutations act on this basis, some distinguished permutations have beautiful descriptions. In 2010 Rhoades showed that the long cycle $(1, 2,..., n)$ acts by the jeu-de-taquin promotion operator in the case when the irreducible representation is indexed by a rectangular partition. We prove a generalisation of this theorem in two directions: on the one hand we lift the restriction on the shape of the partition, and …
Let $(W,S)$ be an arbitrary Coxeter system. We introduce a family of polynomials, $\{ \tilde{\mathcal{R}}_{u,\underline{v}}(t)\}$, indexed by pairs $(u,\underline{v})$ formed by an element $u\in W$ and a (non-necessarily reduced) word $\underline{v}$ in the alphabet $S$. The polynomial $\tilde{\mathcal{R}}_{u,\underline{v}}(t)$ is obtained by considering a certain subset of Libedinsky's light leaves associated to the pair $(u,\underline{v})$. Given a reduced expression $\underline{v}$ of an element $v\in W$; we show that $\tilde{\mathcal{R}}_{u,\underline{v}}(t)$ coincides with the Kazhdan-- L…
We establish a formalism for working with incidence algebras of posets with symmetries, and we develop equivariant Kazhdan-Lusztig-Stanley theory within this formalism. This gives a new way of thinking about the equivariant Kazhdan-Lusztig polynomial and equivariant Z-polynomial of a matroid.
In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan--Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan--Lusztig, the inverse Kazhdan--Lusztig and the $Z$-polynomial of each matroid with those of its relaxations. As an application of our main theorem, we prove that all matroids having a free basis are non-degenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan--Lusztig, inverse Kazhdan--Lusztig and $Z…
The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of symmetric groups. In this paper, we discover a new formula for these polynomials which is related to the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Based on our new formula, we confirm Gedeon's conjecture by the Pieri rule.
