On factorization of matrix of Kazhdan-Lusztig polynomials

TL;DR

This paper demonstrates that the matrix of Kazhdan-Lusztig polynomials for a Coxeter group can be factorized into a product of matrices with nonnegative polynomial entries, using hybrid bases and geometric methods.

Contribution

It introduces a novel factorization of Kazhdan-Lusztig polynomial matrices into products of matrices with nonnegative coefficients, utilizing hybrid bases and geometric proofs.

Findings

Matrix factorization into |S| matrices with nonnegative polynomial entries

Transition matrices between hybrid bases correspond to intermediate factor matrices

Geometric proof confirms positivity of coefficients in the factorization

Abstract

Let $\mathcal{H} = \mathcal{H}(W,S)$ be the Hecke algebra of the Coxeter system $(W,S)$ over $\mathbb{Z}[q^{\pm1}]$, where $W$ is the Weyl group of a symmetrizable Kac-Moody algebra. In this paper, we show that the matrix of Kazhdan-Lusztig polynomials of $\mathcal{H}$ factorizes into a product of $|S|$ many matrices, each of which has entries as polynomials in $q$ with nonnegative coefficients. To achieve this goal, we use hybrid basis $TC^J$ for $J\subseteq S$ of $\mathcal{H}$, defined by Grojnowski-Haiman. The intermediate matrices in the aforementioned factorization turn out to be the transition matrices from $TC^J$-basis to $TC^I$-basis for $I\subset J$. Equivalently, these coefficients can be computed using a natural restriction map from $\mathcal{H}$ to the parabolic Hecke algebra $\mathcal{H}_J$. Moreover, following the ideas from Grojnowski-Haiman, we also give a geometric proof of the positivity of these coefficients.