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

TL;DR

This paper investigates the leading coefficients of Kazhdan--Lusztig polynomials in type D Coxeter groups, demonstrating that these coefficients are always 0 or 1 when x is fully commutative, using domino tableaux correspondence.

Contribution

It introduces a method using domino tableaux to compute μ-values in type D Coxeter groups, extending known results from type A.

Findings

μ(x,w) is 0 or 1 when x is fully commutative in type D

A domino tableaux correspondence is used for cell computation in type D

The paper addresses computation difficulties caused by 'bad' elements

Abstract

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, $\mu(x,w)$ 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 $\mu$-value computation difficult. The Robinson--Schensted correspondence between the symmetric group and pairs of standard Young tableaux gives rise to a way to compute cells of Coxeter groups of type $A$. A lesser known correspondence exists for signed permutations and pairs of so-called domino tableaux, which allows us to compute cells in Coxeter groups of types $B$ and $D$. I will use this correspondence in type $D$ to compute $\mu$-values involving bad elements. I will conclude by showing that $\mu(x,w)$ is 0 or 1 when $x$ is fully commutative in type $D$.