Leading coefficients of Kazhdan--Lusztig polynomials for Deodhar elements

TL;DR

This paper proves that for Deodhar elements in finite Weyl groups, the leading coefficient of Kazhdan--Lusztig polynomials is always 0 or 1, and provides combinatorial criteria to identify when it equals 1.

Contribution

It establishes the binary nature of the leading coefficient for Deodhar elements and offers combinatorial methods to determine its value.

Findings

Leading coefficient is always 0 or 1 for Deodhar elements.

Characterization of Deodhar elements via pattern avoidance.

Criteria for when the coefficient equals 1 in type A permutations.

Abstract

We show that the leading coefficient of the Kazhdan--Lusztig polynomial $P_{x,w}(q)$ known as $\mu(x,w)$ 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 $\mu(x,w) = 1$ for such permutations $w$.