Combinatorial invariance for the coefficient of $q$ in Kazhdan-Lusztig polynomials

TL;DR

This paper proves the combinatorial invariance of the coefficient of q in Kazhdan-Lusztig polynomials for all Coxeter groups, confirming the Conjecture for Bruhat intervals up to length 6 and establishing related invariance results.

Contribution

It establishes the combinatorial invariance of the q-coefficient in Kazhdan-Lusztig polynomials for arbitrary Coxeter groups and verifies the Conjecture for Bruhat intervals of length at most 6.

Findings

Proves combinatorial invariance of the q-coefficient in Kazhdan-Lusztig polynomials.

Confirms the Conjecture for Bruhat intervals of length ≤ 6.

Proves the Gabber-Joseph conjecture for the second-highest Ext group of Verma modules.

Abstract

We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan--Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture, of Lusztig and of Dyer, also for Bruhat intervals of length at most $6$. We also prove the Gabber--Joseph conjecture for the second-highest $Ext$ group of a pair of Verma modules, as well as the combinatorial invariance of the dimension of this group, and of the numbers of frozen and of mutable variables in the cluster structure on Richardson varieties.