The combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials of lower intervals

TL;DR

This paper proves a conjecture on the combinatorial invariance of parabolic Kazhdan-Lusztig polynomials for lower intervals across all Coxeter groups, extending previous results in the field.

Contribution

It establishes the conjecture in the parabolic setting for all Coxeter groups, generalizing earlier specific cases and advancing understanding of Kazhdan-Lusztig polynomials.

Findings

Proves the combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials

Extends results to all Coxeter groups in the lower interval case

Generalizes previous partial results in the literature

Abstract

The aim of this work is to prove a conjecture related to the Combinatorial Invariance Conjecture of Kazhdan-Lusztig polynomials, in the parabolic setting, for lower intervals in every arbitrary Coxeter group. This result improves and generalizes, among other results, the main results of [Advances in Math. {202} (2006), 555-601], [Trans. Amer. Math. Soc. {368} (2016), no. 7, 5247--5269].