Leading coefficients of Kazhdan--Lusztig polynomials and fully commutative elements

TL;DR

This paper proves that for Coxeter groups of type A_{n-1}, the leading coefficient of Kazhdan--Lusztig polynomials is always 0 or 1 when the lower element is fully commutative, simplifying their structure.

Contribution

It establishes a clear binary value for leading coefficients of Kazhdan--Lusztig polynomials in type A_{n-1} Coxeter groups when the lower element is fully commutative.

Findings

Leading coefficient   0 or 1 for fully commutative x

Simplification of Kazhdan--Lusztig polynomial structure in type A_{n-1}

Applicable to arbitrary w in the Coxeter group

Abstract

Let $W$ be a Coxeter group of type $\widetilde{A}_{n-1}$. We show that the leading coefficient, $\mu(x, w)$, of the Kazhdan--Lusztig polynomial $P_{x, w}$ is always equal to 0 or 1 if $x$ is fully commutative (and $w$ is arbitrary).