Combinatorial Invariance Conjecture for A($)over-tilde₂

The combinatorial invariance conjecture (due independently to Lusztig and 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,P-y(q) = P-x',P-y' (q). We prove this conjecture for the affine Weyl group of type A ($) over tilde (2). This is the first infinite group with non-trivial Kazhdan-Lusztig polynomials where the conjecture is proved.