Combinatorial formulas for Kazhdan-Lusztig polynomials with respect to W-graph ideals

TL;DR

This paper develops combinatorial formulas for weighted Kazhdan-Lusztig polynomials within W-graph ideals of Coxeter systems, extending previous results and providing new computational tools.

Contribution

It introduces explicit combinatorial formulas for weighted Kazhdan-Lusztig polynomials in W-graph ideals, expanding on prior theoretical frameworks.

Findings

Derived combinatorial formulas for $P_{x,y}$

Extended Brenti's and Deodhar's results

Enhanced understanding of weighted Coxeter systems

Abstract

Let $(W,S,L)$ be a weighted Coxeter system and $J$ a subset of $S$, Yin [12] introduced the weighted $W$-graph ideal $E_J$ and the weighted Kazhdan-Lusztig polynomials $ \left \{ P_{x,y} \mid x,y\in E_J\right \}$. In this paper, we study the combinatorial formulas for $P_{x,y}$, which will extend the results of Brenti [2] and Deodhar [5].