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

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.