Leading coefficients of Kazhdan--Lusztig polynomials for Deodhar elements

2007/11/09 by Brant Jones, Jones, Brant C.
Mathematics · #20C08 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0711.1391

openalex publication_date 2007/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the leading coefficient of the Kazhdan--Lusztig polynomial P_x,w(q) known as μ(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 μ(x,w) = 1 for such permutations w.

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