Combinatorial invariance conjecture for $\widetilde{A}_2$

TL;DR

This paper proves the combinatorial invariance conjecture for the affine Weyl group of type tenilde{A}_2, establishing that isomorphic Bruhat posets imply equal Kazhdan-Lusztig polynomials, marking a significant advance in the field.

Contribution

It provides the first proof of the conjecture for an infinite group with non-trivial Kazhdan-Lusztig polynomials, specifically for tenilde{A}_2.

Findings

Proof of the conjecture for tenilde{A}_2

First infinite group case with non-trivial Kazhdan-Lusztig polynomials

Establishes isomorphism of Bruhat posets implies polynomial equality

Abstract

The combinatorial invariance conjecture (due independently to G. Lusztig and M. Dyer) predicts that if $[x,y]$ and $[x',y']$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan-Lusztig polynomials are equal, that is, $P_{x,y}(q)=P_{x',y'}(q)$. 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.