We give a lower bound for the value at q=1 of a Kazhdan-Lustig polynomial in a Weyl group W in terms of "patterns''. This is expressed by a "pattern map" from W to W' for any parabloic subgroup W'. This notion generalizes the concept of patterns and pattern avoidance for permutations to all Weyl groups. The main tool of the proof is a "hyperbolic localization" on intersection cohomology; see the related paper http://front.math.ucdavis.edu/math.AG/0202251
Motivated by studying the Unitary Dual Problem, a variation of Kazhdan-Lusztig polynomials was defined in [Yee08] which encodes signature information at each level of the Jantzen filtration. These so called signed Kazhdan-Lusztig polynomials may be used to compute the signatures of invariant Hermitian forms on irreducible highest weight modules. The key result of this paper is a simple relationship between signed Kazhdan-Lusztig polynomials and classical Kazhdan-Lusztig polynomials: signed Kahzdan-Lusztig polynomials are shown to equal classical Kazhdan-Lusztig polynomials evaluated at $-q$ ra…
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 study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan-Lusztig basis; this was known as the Smoothness Conjecture. For Grassmannians, we p…
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 associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M, in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincare polynomials. We also introduce a q-deformation of the Mobius algebra of M, and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra. We conjecture that t…
Irreducibility results for parabolic induction of representations of the general linear group over a local non-archimedean field can be formulated in terms of Kazhdan--Lusztig polynomials of type $A$. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan--Lusztig polynomials known as parabolic Kazhdan--Lusztig polynomials satisfy properties analogous to those of the ordinary ones.
We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan- Lusztig polynomials and we prove it for finite simply-laced Weyl groups.
We obtain explicit branching rules for graded cell modules and graded simple modules over the endomorphism algebra of a Bott-Samelson bimodule. These rules allow us to categorify a well-known recursive formula for Kazhdan-Lusztig polynomials.
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 $\tilde{B}_2$.
We study the equivariant oriented cohomology ring $h_T(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott-Samelson classes in $h_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results by Brion, Knutson, Peterson, Tymoczko and others. We then focus on the equivariant oriented cohomology theory corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar's construction of the pa…
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.
We explain a strategy for a proof of the positivity of all coefficients of Kazhdan-Lusztig-polynomials for arbitrary Coxeter groups by constructing spaces whose dimensions we conjecture to be these coefficients.
We use Kazhdan-Lusztig bases of representations of the symmetric group to express Pfaffians with entries $(a_i-a_j) h_{i+j}$. In the case where the parameters $a_i$ are specialized to successive powers of $q$, and the $h_i$ are complete functions, we obtain the $q$-discriminant.
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 report on various results, conjectures, and open problems related to Kazhdan-Lusztig polynomials of matroids. We focus on conjectures about the roots of these polynomials, all of which appear here for the first time.
Motivated by a question on the graded rank of the stalks of the canonical sheaf on a Bruhat graph, we lift some equalities concerning (parabolic) Kazhdan-Lusztig polynomials to this moment graph setting. Our proofs hold also in positive characteristic, under some technical assumptions.
The famous Kazhdan-Lusztig Conjecture of the 1970s states that the multiplicity of an irreducible composition factor of a Verma module can be computed by evaluating Kazhdan-Lusztig polynomials at 1. Thus the character of a Verma module is a linear combination of characters of irreducible highest weight modules where the coefficients in the linear combination are Kazhdan-Lusztig polynomials evaluated at 1. Kazhdan-Lusztig showed that inverting and writing the character of an irreducible highest weight module as a linear combination of characters of Verma modules, the coefficients in the linear …
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 prove a tamely ramified version of the Kazhdan-Lusztig equivalence using factorization algebras. More precisely, we establish an equivalence between the DG category of Iwahori-integrable affine Lie algebra representations and the DG category of representations of the "mixed" quantum group. This confirms a conjecture by D. Gaitsgory in arXiv:1810.09054 [math.RT].
Let $G$ be a reductive group over a finite field with a maximal unipotent subgroup $U$, we consider certain sheaves on $G/U$ defined by Kazhdan and Laumon and show that their cohomology produces the cohomology of the Deligne-Lusztig varieties. We then use this comparison to give a new proof of a result of Dudas.
We study equivalence classes relating to the Kazhdan-Lusztig mu(x,w) coefficients in order to help explain the scarcity of distinct values. Each class is conjectured to contain a "crosshatch" pair. We also compute the values attained by mu(x,w) for the permutation groups S_10 and S_11.
We prove that, for any choice of parameters, the Kazhdan-Lusztig cells of a Weyl group of type $B$ are unions of combinatorial cells (defined using the domino insertion algorithm).
The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and derive some consequences for representation theory.
We provide a non-recursive description for the bounded admissible sets of masks used by Deodhar's algorithm to calculate the Kazhdan--Lusztig polynomials $P_{x,w}(q)$ of type $A$, in the case when $w$ is hexagon avoiding and maximally clustered. This yields a combinatorial description of the Kazhdan--Lusztig basis elements of the Hecke algebra associated to such permutations $w$. The maximally-clustered hexagon-avoiding elements are characterized by avoiding the seven classical permutation patterns $\{3421, 4312, 4321, 46718235, 46781235, 56718234, 56781234\}$. We also briefly discuss the appl…
The aim of this work is to prove a conjecture related to the Combinatorial Invariance Conjecture of Kazhdan-Lusztig polynomials, in the parabolic setting, for lower intervals in every arbitrary Coxeter group. This result improves and generalizes, among other results, the main results of [Advances in Math. {202} (2006), 555-601], [Trans. Amer. Math. Soc. {368} (2016), no. 7, 5247--5269].
We deduce the Kazhdan-Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category O from the equivariant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. We make similar speculations for the affine Lie algebras and W-algebras.
A Kazhdan-Lusztig atlas, introduced by He, Knutson and Lu, on a stratified variety (V,Y) is a way of modeling the stratification Y of V locally on the stratification of Kazhdan-Lusztig varieties X^w_o \cap X_v by their intersection with opposite Schubert varieties X_u. We are interested in classifying smooth toric surfaces with Kazhdan-Lusztig atlases. This involves finding a degeneration of V to a union of Richardson varieties in the flag variety H/B_H of some Kac-Moody group H. We determine which toric surfaces have a chance at having a Kazhdan-Lusztig atlas by looking at their moment polyto…
In this paper we show that the leading coefficient $μ(y,w)$ of some Kazhdan-Lusztig polynomials $P_{y,w}$ with $y,w$ in an affine Weyl group of type $\tilde A_n $ is $n+2$. This fact has some consequences on the dimension of first extension groups of finite groups of Lie type with irreducible coefficients.
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…
Based on empirical evidence obtained using the {\sf CHEVIE} computer algebra system, we present a series of conjectures concerning the combinatorial description of the Kazhdan--Lusztig cells for type $B_n$ with unequal parameters. These conjectures form a far-reaching extension of the results of Bonnafé and Iancu obtained earlier in the so-called ``asymptotic case''. We give some partial results in support of our conjectures.
We study the projective objects in an exact category naturally associated to a Coxeter system. We discuss an analog of the Kazhdan-Lusztig conjecture and show how it follows from a "genericity" conjecture and how the latter follows from a "Hard Lefschetz" conjecture.
The Kazhdan-Lusztig polynomials for finite 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 coefficients, but no completely combinatorial interpretation for them is known in general. Deodhar (1990) has given a framework for computing the Kazhdan-Lusztig polynomials, which generally involves recursion. We define 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-Lu…
We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality.
We prove that the combinatorial concept of a special matching can be used to compute the parabolic Kazhdan-Lusztig polynomials of doubly laced Coxeter groups and of dihedral Coxeter groups. In particular, for this class of groups which includes all Weyl groups, our results generalize to the parabolic setting the main results in [Advances in Math. {202} (2006), 555-601]. As a consequence, the parabolic Kazhdan-Lusztig polynomial indexed by $u$ and $v$ depends only on the poset structure of the Bruhat interval from the identity element to $v$ and on which elements of that interval are minimal co…
From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, 'RSK bases' are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra. Applications to invariant theory, over various base rings, of the general linear group and representation theory of the symmetric group are discussed.
We show that the leading coefficient of the Kazhdan--Lusztig polynomial $P_{x,w}(q)$ known as $μ(x,w)$ is always either 0 or 1 when $w$ is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance by Billey--Warrington (2001) and Billey--Jones (2007). In type $A$, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar's (1990) algorithm, we provide some combinatorial criteria to determine when $μ(x,w) = 1$ for such permutations $w$.
We show that the Littlewood-Richardson coefficients are values at 1 of certain parabolic Kazhdan-Lusztig polynomials for affine symmetric groups. These q-analogues of Littlewood-Richardson multiplicities coincide with those previously introduced in terms of ribbon tableaux.
We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL) polynomials for Lie superalgebra gl(m|n) in our parabolic setup can be identified with the usual parabolic Kazhdan-Lusztig polynomials. We establish some special cases of the BKL conjecture on the parabolic category O of gl(m|n)-modules and additional results which support the BKL conjecture and su…
We develop an algorithm for computing affine Kazhdan-Lusztig polynomials, for all Lie types. This generalizes our previously published algorithm for type A, which in turn is a faster version of an algorithm due to Lascouz, Leclerc and Thibon (proposed in the setting of Hecke algebras of type A, at roots of unity.)
The center of an extended affine Hecke algebra is known to be isomorphic to the ring of symmetric functions associated to the underlying finite Weyl group $W\_0$. The set of Weyl characters ${\sf s}\_\la$ forms a basis of the center and Lusztig showed in [Lus15] that these characters act as translations on the Kazhdan-Lusztig basis element $C\_{w\_0}$ where $w\_0$ is the longest element of $W\_0$, that is we have $C\_{w\_0}{\sf s}\_\la =C\_{w\_0t\_\la}$. As a consequence, the coefficients that appear when decomposing~$C\_{w\_0t\_{\la}}{\sf s}\_τ$ in the Kazhdan-Lusztig basis are tensor multipl…
The equivariant Kazhdan-Lusztig polynomial of a braid matroid may be interpreted as the intersection cohomology of a certain partial compactification of the configuration space of n distinct labeled points in the plane, regarded as a graded representation of the symmetric group. We show that, in fixed cohomological degree, this sequence of representations of symmetric groups naturally admits the structure of an FS-module, and that the dual FS^op-module is finitely generated. Using the work of Sam and Snowden, we give an asymptotic formula for the dimensions of these representations and obtain …
In this paper, we prove a decomposition formula for the Kazhdan-Lusztig basis of affine Hecke algebras of rank 2 with positive weight function. Then we discuss some applications of this kind of decomposition to Lusztig's conjectures P1-P15.
We provide combinatorial rules to compute Kazhdan--Lusztig polynomials for the Hermitian symmetric pair $(B_N,A_{N-1})$ when the Hecke algebra has unequal parameters. They are obtained by filling regions delimited by paths with ballot strips. We also extend the binary tree algorithm introduced by Lascoux and Schützenberger to our case.
A positive level Kazhdan-Lusztig functor is defined using Arkhipov-Gaitsgory duality for affine Lie algebras. The functor sends objects in the DG category of G(O)-equivariant positive level affine Lie algebra modules to objects in the DG category of modules over Lusztig's quantum group at a root of unity. We prove that the semi-infinite cohomology functor for positive level modules factors through the Kazhdan-Lusztig functor at positive level and the quantum group cohomology functor with respect to the positive part of Lusztig's quantum group.
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…
