Combinatorial Invariance Conjecture for $\widetilde {A}_2$

Abstract

The combinatorial invariance conjecture (due independently to Lusztig and Dyer) predicts that if $[x,y]$ and $[x^{\prime},y^{\prime}]$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan–Lusztig polynomials are equal, that is, $P_{x,y}(q)=P_{x^{\prime},y^{\prime}}(q)$. We prove this conjecture for the affine Weyl group of type $\widetilde {A}_2$. This is the first infinite group with non-trivial Kazhdan–Lusztig polynomials where the conjecture is proved.

Combinatorial invariance for elementary intervals

“…The most general results to date have established the case of lower intervals, where both u and u ′ are the identity element e [5,6,11,13]. Other known cases include intervals in S n of length at most 8 [18,19] and all intervals in the rank-three affine Weyl group of type A 2 [9]. Patimo [23] has also shown that the coefficient of q in P u,v (q) is combinatorial for an arbitrary interval in S n .…”

“…Our proof of Theorem 1.5 is the first application of hypercube decompositions to prove a new instance of combinatorial invariance. Previous results have used special matchings [5,6] or classifications of small intervals [9,18]. This new method is especially promising in light of Conjecture 1.2 and the fact that all intervals [u, v] ⊂ S n have hypercube decompositions.…”