Leading coefficients of Kazhdan--Lusztig polynomials for Deodhar elements

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

2007-11-09

arxiv.org/abs/0711.1391v1

Mathematics

Combinatorics

28 pages

Scientific paper

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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