The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group $S_n$

Abstract

In this paper we show that the leading coefficient $\mu (y,w)$ of certain Kazhdan-Lusztig polynomials $P_{y,w}$ of the permutation group ${\mathfrak{S}}_n$ of 1,2,...,n are not greater than 1. More precisely, we show that the leading coefficients $\mu (y,w)$ are not greater than 1 whenever $a(y)<a(w)$, where $a:{\mathfrak{S}}_n\to \mathbf{N}$ is the function defined by Lusztig.

7 pages