Frederick M. Goodman, Hans Wenzl
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.)
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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.
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…
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…
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.
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.
In this paper we prove the Kazhdan-Lusztig type character formula for irreducible highest weight modules with positive rational highest weights over symmetrizable Kac-Moody Lie algebras.
We shall give a description of the intersection cohomology groups of the Schubert varieties in partial flag manifolds over symmetrizable Kac-Moody Lie algebras in terms of parabolic Kazhdan-Lusztig polynomials introduced by Deodhar.
Monotonicity of inverse Kazhdan-Lusztig polynomials
本研究针对任意Coxeter系统，基于Soergel猜想与混合反常Hecke范畴中Rouquier复形的单射态射，证明逆Kazhdan-Lusztig多项式的单调性，并将其推广到抛物型情形。
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 derive a formula for the entries of the (unitriangular) transition matrices between the standard monomial and dual canonical bases of the irreducible polynomial representations of U_q(gl_n) in terms of Kazhdan-Lusztig polynomials.
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.
Lusztig proved that the Kazhdan-Lusztig basis of a spherical Hecke algebra can be essentially identified with the Weyl characters of the Langlands dual group. We generalize this result to the unequal parameter case. The new proof is pretty simple and quite different from Lusztig's.
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.
Oriented Temperley--Lieb algebras and combinatorial Kazhdan--Lusztig theory
该研究针对经典埃尔米特对称空间，定义了有向Temperley--Lieb代数，为这类空间的Kazhdan--Lusztig多项式存在闭合组合公式提供了解释。
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.
Equivalence between invariance conjectures for parabolic Kazhdan-Lusztig polynomials
本研究证明Mario Marietti提出的抛物型Kazhdan-Lusztig多项式组合不变性猜想等价于其在极大商上的限制，且该等价性与Barkley、Gaetz近期证明的Kazhdan-Lusztig多项式不变性猜想相关等价性处于相反极端。
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…
Relations between generalised Gelfand-Tsetlin and Kazhdan-Lusztig bases of $S_n$
本文证明Specht模的Kazhdan-Lusztig基在任意无重数标准抛物子群塔构造的广义Gelfand-Tsetlin基下呈上三角形式。
According to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant $K$-group of Steinberg's triple variety. The $K$-group is equipped with a filtration indexed by closed $G$-stable subvarieties of the nilpotent variety, where $G$ is the corresponding reductive algebraic group over $\mathbb{C}$. In this paper we will show in the case of type $A$ that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.
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 show a precise formula, in the form of a monomial, for certain families of parabolic Kazhdan-Lusztig polynomials of the symmetric group. The proof stems from results of Lapid-Minguez on irreducibility of products in the Bernstein-Zelevinski ring. By quantizing those results into a statement on quantum groups and their canonical bases, we obtain identities of coefficients of certain transition matrices that relate Kazhdan-Lusztig polynomials to their parabolic analogues. This affirms some basic cases of conjectures raised recently by Lapid.
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).
We study a quantum version of the Kazhdan-Lusztig functor. Namely, we prove that there exists a fully faithfull exact tensor functor from the category of finite dimensional representations of the quantum affine algebra Uq(sl(n)) (with extended scalars) to the category of finite dimensional representations of Felder's elliptic quantum group E_{τ,\gamma2}(gl(n)).
Monodromy in Kazhdan-Lusztig cells in affine type A
本文利用仿射Robinson-Schensted对应描述仿射A型Kazhdan-Lusztig胞腔中双向边结构，等价于全面刻画仿射置换的Knuth等价类。
We reformulate the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of Lie algebras of type D via the theory of canonical bases arising from quantum symmetric pairs initiated by Weiqiang Wang and the author. This is further applied to formulate and establish for the first time the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$.
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].
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…
Isotropic meta Kazhdan--Lusztig combinatorics I: Ext-quiver presentation for the Hecke category
本文基于杯帽元Kazhdan-Lusztig组合学和Temperley-Lieb图示法，为各向同性Grassmannian的反球面Hecke范畴的基本代数提供Ext-有向图和关系呈现。
Covexillary Schubert varieties and Kazhdan-Lusztig Polynomials
本文针对经典型covexillary元素建立Kazhdan-Lusztig多项式的组合与归纳公式，推广了经典型Grassmann流形上的已有结果，并借助交上同调与Kazhdan-Lusztig簇同构完成证明。
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…
In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters of the irreducible and tilting modules in the BGG category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundan's conjecture and its variants associated to all Borel subalgebras in full generality.
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.
Flipclasses and Combinatorial Invariance for Kazhdan--Lusztig polynomials
本文通过引入翻转类概念构造对称群区间的组合不变量，给出计算Kazhdan--Lusztig $\widetilde{R}$-多项式低次项系数的配方，为组合不变性猜想提供新证据。
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 generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based $\textbf{U}^{\imath}$-module and a based $\textbf{U}$-module is a based $\textbf{U}^{\imath}$-module. This is then used to formulate a Kazhdan-Lusztig theory for an arbitrary parabolic BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebras, extending a main result in [BW13] (arXiv:1310.0103).
Combinatorial invariance for Kazhdan-Lusztig $R$-polynomials of elementary intervals
本文通过调整超立方体分解方法，证明了对称群$S_n$初等区间的Kazhdan-Lusztig $R$-多项式组合不变性猜想，显著推广了已知的下区间情形。
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…
On combinatorial invariance of parabolic Kazhdan-Lusztig polynomials
本文证明 Lusztig-Dyer 的 Kazhdan-Lusztig 多项式组合不变性猜想与 Marietti 的抛物型版本及作者新提出的精细抛物型版本等价，并据此在低 Bruhat 区间情形下重新证明 Marietti 猜想，同时证明抛物型猜想的若干新情形。
A note on Combinatorial Invariance of Kazhdan--Lusztig polynomials
本文引入惊人超立方体分解及双捷径概念，提出蕴含对称群Kazhdan--Lusztig多项式组合不变性猜想的组合性猜想，附录还在余初等区间下证明了双超立方体的类似猜想。
We show that Hecke algebra of type B and a coideal subalgebra of the type A quantum group satisfy a double centralizer property, generalizing the Schur-Jimbo duality in type A. The quantum group of type A and its coideal subalgebra form a quantum symmetric pair. A new theory of canonical bases arising from quantum symmetric pairs is initiated. It is then applied to formulate and establish for the first time a Kazhdan-Lusztig theory for the BGG category O of the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2m+1|2n)$. In particular, our approach provides a new formulation of the Kazhdan-Lu…
We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional irreducible modules of the general linear Lie superalgebra are computed by the usual parabolic Kazhdan-Lusztig polynomials of type A. In addition, we establish closed formulas for canonical and dual canonical bases for the tensor product of any two fundamental representations of type A.
Combinatorial invariance for the coefficient of $q$ in Kazhdan-Lusztig polynomials
该研究证明了Kazhdan-Lusztig多项式中q系数在任意Coxeter群中的组合不变性，并验证了相关猜想。
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.
Hybrid Kazhdan-Lusztig basis and parabolic induction
本文通过构造BGG范畴主块上的新分层结构，为Hecke代数的混合Kazhdan-Lusztig基提供了范畴化。
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