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

TL;DR

This paper proves that for certain Kazhdan-Lusztig polynomials in the symmetric group, the leading coefficient is at most 1 when a specific Lusztig function value increases.

Contribution

It establishes an upper bound of 1 for the leading coefficients of Kazhdan-Lusztig polynomials under a new condition involving Lusztig's a-function.

Findings

Leading coefficients are not greater than 1 when a(y)<a(w).

Provides bounds for Kazhdan-Lusztig polynomial coefficients in symmetric groups.

Enhances understanding of the structure of Kazhdan-Lusztig polynomials.

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.