Gregory S. Warrington
Let w_0 denote the permutation [n,n-1,...,2,1]. We give two new explicit formulae for the Kazhdan-Lusztig polynomials P_{w_0w,w_0x} in S_n when x is a maximal element in the singular locus of the Schubert variety X_w. To do this, we utilize a standard identity that relates P_{x,w} and P_{w_0w,w_0x}.
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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…
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.
We give closed combinatorial product formulas for Kazhdan-Lusztig poynomials and their parabolic analogue of type q in the case of boolean elements, introduced in [M. Marietti, Boolean elements in Kazhdan-Lusztig theory, J. Algebra 295 (2006)], in Coxeter groups whose Coxeter graph is a tree.
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 $Z…
The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of symmetric groups. In this paper, we discover a new formula for these polynomials which is related to the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Based on our new formula, we confirm Gedeon's conjecture by the Pieri rule.
The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield, whose properties need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was recently conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of squares of paths. We further obtain explicit formulas of the Kazhdan-Lusztig polynomials of wheel matroids and whirl matroids. We prove the real-rootedness of the Kazhdan-Lusztig polynomials of these matroids, which provide…
Kazhdan-Lusztig polynomials of matroids under deletion
本研究提出关联拟阵$M$与其删除元、收缩及局部化后的Kazhdan-Lusztig多项式的公式，并给出其在图拟阵中的应用，包括并联连接图的对应简单公式。
In this paper we compute the leading coefficients $μ(u,w)$ of the Kazhdan-Lusztig polynomials $P_{u,w}$ for an affine Weyl group of type $\widetilde{A_2}$. We give all the values $μ(u,w)$.
We show that for Bruhat intervals starting from the origin in simply-laced Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [e,u] only depend on the isomorphism type of [e,u].
We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank $n+1$ thagomizer matroid by showing that the coefficient of $t^k$ is equal to the number of Dyck paths of semilength $n$ with $k$ long ascents. We also give a conjecture for the $S_n$-equivariant Kazhdan-Lusztig polynomial of a thagomizer matroid.
We study the way in which equivariant Kazhdan-Lusztig polynomials, equivariant inverse Kazhdan-Lusztig polynomials, and equivariant Z-polynomials of matroids change under the operation of relaxation of a collection of stressed hyperplanes. This allows us to compute these polynomials for arbitrary paving matroids, which we do in a number of examples, including various matroids associated with Steiner systems that admit actions of Mathieu groups.
We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple combinatorial formula we discovered for these polynomials using skew young tableaux. This supports the conjecture that Kazhdan-Lusztig polynomials for all matroids have non-negative coeffiecients. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.
Let (W, S) be a Coxeter system. A W-graph is an encoding of a representation of the corresponding Iwahori-Hecke algebra. Especially important examples include the W-graph corresponding to the action of the Iwahori-Hecke algebra on the Kazhdan-Lusztig basis, as well as this graph's strongly connected components (cells). In 2008, Stembridge identified some common features of the Kazhdan-Lusztig graphs and gave a combinatorial characterization of all W-graphs that have these features. He conjectured, and checked up to n=9, that all such A_n-cells are Kazhdan-Lusztig cells. The current paper provi…
We show that the Kazhdan-Lusztig basis elements $C_w$ of the Hecke algebra of the symmetric group, when $w \in S_n$ corresponds to a Schubert subvariety of a Grassmann variety, can be written as a product of factors of the form $T_i+f_j(v)$, where $f_j$ are rational functions.
Log-concavity of inverse Kazhdan-Lusztig polynomials of paving matroids
本文证明了铺砌拟阵的逆Kazhdan-Lusztig多项式与$(1+t)^n$的Hadamard积只有实根，从而通过牛顿不等式证实了Gao-Xie关于这些拟阵的对数凹性猜想。
Real-rootedness of Kazhdan--Lusztig polynomials of sparse paving matroids
本文证明稀疏铺砌拟阵的非常数Kazhdan--Lusztig多项式仅有简单负零点，并给出均匀拟阵多项式与稀疏铺砌拟阵多项式的严格交错关系。
The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot, and Wakefield [{\it Adv. Math. 2016}]. Let $U_{m,d}$ denote the uniform matroid of rank $d$ on a set of $m+d$ elements. Gedeon, Proudfoot, and Young [{\it J. Combin. Theory Ser. A, 2017}] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of $U_{m,d}$ using equivariant Kazhdan-Lusztig polynomials. In this paper we give two alternative explicit formulas, which allow us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of $U_{m,d}$ for $2\leq m\leq 15$ and all $d$'s.…
The combinatorics behind the leading Kazhdan-Lusztig coefficients of braid matroids
本文基于Ferroni和Larson的工作，进一步研究辫子拟阵的首项Kazhdan-Lusztig系数，给出了偶数秩情形、简单串并联拟阵计数及逆多项式的显式公式，并利用Abel多项式二项式恒等式证明。
In this paper we show that the leading coefficient $μ(y,w)$ of certain Kazhdan-Lusztig polynomials $P_{y,w}$ of the permutation group $\mathfrak S_n$ of 1,2,...,n are not greater than 1. More precisely, we show that the leading coefficients $μ(y,w)$ are not greater than 1 whenever $a(y)< a(w)$, where $a: \mathfrak S_n\to\mathbf N$ is the function defined by Lusztig.
From a combinatorial perspective, we establish three inequalities on coefficients of $R$- and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of $(q-1)$-coefficients of $R$-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of $R$-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of $R$-polynomials and edges of Bruhat graphs.
On the homogeneity problem of the Kazhdan-Lusztig ideals
该数学研究针对Kazhdan-Lusztig理想的齐次性问题，得出了其非齐次的若干充分条件，并尝试推导其为“标准齐次”的充要条件。
Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan-Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling …
Inverse Kazhdan-Lusztig polynomials of matroids under deletion
本研究提出拟阵的逆Kazhdan-Lusztig多项式与逆$Z$-多项式的删除公式，推导其在多类拟阵上的闭式与递推关系，并构造秩19的拟阵证否相关实根猜想。
Let $ρ$ be a non-negative integer. A $ρ$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $ρ$ disjoint bases. We present a combinatorial formula for Kazhdan-Lusztig polynomials of $ρ$-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.
Equivariant Kazhdan--Lusztig Polynomials of Thagomizer Matroids with a Hyperoctahedral Group Action
本文研究了具有超菱形群作用的Thagomizer矩阵的等变Kazhdan-Lusztig多项式，利用回文性与对称群理论计算了多项式及其逆多项式，并通过Frobenius特征表达系数。
We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and is simpler and more explicit than any existing one. We then show that, in a certain sense, this formula cannot be simplified.
Equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids
本文研究了与完全三部图 $K_{1,1,n}$ 相关的 thagomizer 拟阵的等变逆 Kazhdan--Lusztig 多项式，推导了其闭式表达式，并证明了其非等变多项式的对数凹性。
Following the work of Gao and Xie in [2], we state some properties of the inverse Kazhdan-Lusztig polynomial of a matroid. We also give partial answers to a conjecture that states that regular connected matroids are non-degenerate. We link the degeneracy of a matroid to the inverse Kazhdan-Lusztig polynomial and we show that the Conjecture holds for modular matroids, by proving that degenerate modular matroids are not regular.
In this paper we compute the leading coefficients $μ(u,w)$ of the Kazhdan--Lusztig polynomials $P_{u,w}$ for an affine Weyl group of type $\tilde{B}_2$. By using the \textbf{a}-function of a Coxeter group defined by Lusztig (see [L1, \S2]), we compute most $μ(u,w)$ explicitly. With part of these values $μ(u,w)$, we show that a conjecture of Lusztig on distinguished involutions is true for an affine Weyl group of type $\tilde{B}_2$. We also show that the conjectural formula in [L3, (12)] needs a modification.
Kazhdan-Lusztig polynomials of braid matroids
本研究给出$\text{A}_{n-1}$型辫子拟阵Kazhdan-Lusztig多项式的组合解释及等变版本，以串并联网络拟阵为核心对象，证明了相关首项系数猜想并给出显式生成函数。
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…
Unimodality of Kazhdan--Lusztig polynomials of sparse paving matroids
本文证明无环稀疏铺砌拟阵的Kazhdan--Lusztig多项式单峰，并刻画了严格单峰性失效的例外情形，同时确定众数并分类退化情形。
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…
Inverse Kazhdan--Lusztig polynomials of fan matroids
本文研究扇形 matroid 的逆 Kazhdan--Lusztig 多项式和逆 Z 多项式，推导了生成函数并得出显式公式，证明了逆 Kazhdan--Lusztig 多项式系数序列的对数凹性。
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 give two contructions of sets of masks on cograssmannian permutations that can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the Iwahori-Hecke algebra. The constructions are respectively based on a formula of Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott-Samelson resolution. These constructio…
We give a combinatorial formula for the Kazhdan-Lusztig polynomials $P_{x,w}$ in the symmetric group when $w$ is a 321-hexagon-avoiding permutation. Our formula, which depends on a combinatorial framework developed by Deodhar, can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for $w$. We also show that $w$ being 321-hexagon-avoiding is equivalent to several other conditions, such as the Bott-Samelson resolution of the Schubert variety $X_w$ being small. We conclude with a simple method for completely determining the singular locus of $X_w$ wh…
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.
When $Sp(2n,\mathbb{C})$ acts on the flag variety of $SL(2n,\mathbb{C})$, the orbits are in bijection with fixed point free involutions in the symmetric group $S_{2n}$. In this case, the associated Kazhdan-Lusztig-Vogan polynomials $P_{v,u}$ can be indexed by pairs of fixed point free involutions $v\geq u$, where $\geq$ denotes the Bruhat order on $S_{2n}$. We prove that these polynomials are combinatorial invariants in the sense that if $f: [u, w_0 ] \rightarrow [u , w_0]$ is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then $P_{v,u} = P_{f(v),u}…
Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl groups, rational polytopes, and realizable matroids, one can count points over finite fields on flag varieties, toric varieties, or reciprocal planes to obtain cohomological interpretations of these polynomials. We survey these results and unite them under a single geometric framework.
We compare the canonical basis for a generalized Temperley-Lieb algebra of type A or B with the Kazhdan-Lusztig basis for the corresponding Hecke algebra.
We introduce $q$-analogues of uniform matroids, which we call $q$-niform matroids. While uniform matroids admit actions of symmetric groups, $q$-niform matroids admit actions of finite general linear groups. We show that the equivariant Kazhdan-Lusztig polynomial of a $q$-niform matroid is the unipotent $q$-analogue of the equivariant Kazhdan-Lusztig polynomial of the corresponding uniform matroid, thus providing evidence for the positivity conjecture for equivariant Kazhdan-Lusztig polynomials.
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.
The BBDVW Conjecture for Kazhdan-Lusztig polynomials of lower intervals
本文证明了BBDVW猜想在下区间中的成立，为Kazhdan-Lusztig多项式组合不变猜想提供了新的证据。
Real-rootedness of Kazhdan--Lusztig and $Z$-polynomials of thagomizer matroids and graphic matroids of $K_{2,n}$
该研究证明了thagomizer拟阵T_n与K_{2,n}图拟阵的Kazhdan--Lusztig多项式及公共Z-多项式均为实根多项式，通过有理变换将两类多项式的实根性分别归约为切比雪夫多项式零点的交替符号求值和自反多项式的单位圆准则。
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 define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.
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 $\tilde{B}_2$.
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.
Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial P_{id,w}(q)=1+q^h for some h.
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 …
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