Flip Combinatorial Invariance and Weyl groups

Abstract

In this work, we investigate the approach via flipclasses to the Combinatorial Invariance Conjecture for Kazhdan--Lusztig polynomials of all Coxeter groups. We prove the combinatorial invariance of Kazhdan--Lusztig $\widetilde{R}$-polynomials of Weyl groups modulo $q^7$ and of Kazhdan--Lusztig $\widetilde{R}$-polynomials of type $A$ Weyl groups modulo $q^8$. As a consequence, we establish the Combinatorial Invariance Conjecture for all intervals up to length 10 in Weyl groups and up to length 12 in type $A$ Weyl groups.