Lusztig's $a$ function for Coxeter groups of rank 3

Leading Coefficients of Kazhdan--Lusztig Polynomials in Type $D$

Kazhdan--Lusztig polynomials arise in the context of Hecke algebras associated to Coxeter groups. The computation of these polynomials is very difficult for examples of even moderate rank. In type $A$ it is known that the leading coefficient, $μ(x,w)$ of a Kazhdan--Lusztig polynomial $P_{x,w}$ is either 0 or 1 when $x$ is fully commutative and $w$ is arbitrary. In type $D$ Coxeter groups there are certain "bad" elements that make $μ$-value computation difficult. The Robinson--Schensted correspondence between the symmetric group and pairs of standard Young tableaux gives rise to a way to comput…

Leading coefficients of Kazhdan--Lusztig polynomials and fully commutative elements

Let $W$ be a Coxeter group of type $\widetilde{A}_{n-1}$. We show that the leading coefficient, $μ(x, w)$, of the Kazhdan--Lusztig polynomial $P_{x, w}$ is always equal to 0 or 1 if $x$ is fully commutative (and $w$ is arbitrary).

Structural properties of acyclic heaps of pieces with Kazhdan--Lusztig theory

We introduce the notions of boundary vertex, linear equivalence and effective boundary vertex in the context of Viennot's heaps of pieces. We prove that in the heap of a fully commutative element in a star reducible Coxeter group, every boundary vertex is linearly equivalent to an effective boundary vertex. Using this result, we establish Property W (in the sense of math.QA/0509363) for star reducible Coxeter groups; this corrects a mistake in the latter paper.

渐近Hecke范畴的中心与单位特征sheaves

The center of the asymptotic Hecke category and unipotent character sheaves

本文证明了渐近Hecke范畴的中心为模ularity张量范畴，并研究了其在二面体群及H3、H4部分细胞中的应用。

Star reducible Coxeter groups

We define ``star reducible'' Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan--Lusztig bases of these groups have a nice projection property to the Temperley--Lieb type quotient, and furthermore that the images of the basis elements $C'_w$ (for fully commutative $w$) in the quotient have structure constants in ${\Bbb Z}^{\geq 0}[v, v^{-1}]$. We also classify the star reducible Co…

On 321-avoiding permutations in affine Weyl groups

We introduce the notion of 321-avoiding permutations in the affine Weyl group $W$ of type $A_{n-1}$ by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in $W$ coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of $W$ (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations. Using Shi's characterization of the Kazhdan--Lusz…

Affine crystals, one-dimensional sums and parabolic Lusztig q-analogues

This paper is concerned with one-dimensional sums in classical affine types. We prove a conjecture of the third author and Zabrocki by showing they all decompose in terms of one-dimensional sums related to affine type A provided the rank of the root system considered is sufficiently large. As a consequence, any one-dimensional sum associated to a classical affine root system with sufficiently large rank can be regarded as a parabolic Lusztig q-analogue.