Combinatorial Proof of the Inversion Formula on the Kazhdan-Lusztig R-Polynomials

TL;DR

This paper provides a combinatorial proof of the inversion formula for Kazhdan-Lusztig R-polynomials in Coxeter groups, introducing a reflection principle on V-paths and exploring its applications to Bruhat order properties.

Contribution

It introduces a new combinatorial proof of the inversion formula using V-paths and reflection principles, with applications to Bruhat order and permutation properties.

Findings

Established a reflection principle on V-paths leading to a combinatorial proof.

Provided a direct combinatorial interpretation of the equi-distribution property in Bruhat intervals.

Refined the inversion formula for symmetric groups by fixing the last element of permutations.

Abstract

Let $W$ be a Coxeter group, and for $u,v\in W$, let $R_{u,v}(q)$ be the Kazhdan-Lusztig $R$-polynomial indexed by $u$ and $v$. In this paper, we present a combinatorial proof of the inversion formula on $R$-polynomials due to Kazhdan and Lusztig. This problem was raised by Brenti. Based on Dyer's combinatorial interpretation of the $R$-polynomials in terms of increasing Bruhat paths, we reformulate the inversion formula in terms of $V$-paths. By a $V$-path from $u$ to $v$ with bottom $w$ we mean a pair $(\Delta_1,\Delta_2)$ of Bruhat paths such that $\Delta_1$ is a decreasing path from $u$ to $w$ and $\Delta_2$ is an increasing path from $w$ to $v$. We find a reflection principle on $V$-paths, which leads to a combinatorial proof of the inversion formula. Moreover, we give two applications of the reflection principle. First, we restrict this involution to $V$-paths from $u$ to $v$ with maximal length. This provides a direct interpretation for the equi-distribution property that any nontrivial interval $[u,v]$ has as many elements of even length as elements of odd length. This property was obtained by Verma in his derivation of the M\"obius function of the Bruhat order. Second, using the reflection principle for the symmetric group, we obtain a refinement of the inversion formula by restricting the summation to permutations ending with a given element.