Leading coefficients of Kazhdan--Lusztig polynomials for Deodhar elements
Open Access Publications from the University of California

## Leading coefficients of Kazhdan--Lusztig polynomials for Deodhar elements

• Author(s): Jones, Brant C.
• et al.

## Published Web Location

https://arxiv.org/pdf/0711.1391.pdf
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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 $w$.

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