Long Fully Commutative Elements in Affine Coxeter Groups

Abstract

International audienceAn element of a Coxeter group W is fully commutative if any two of its reduceddecompositions are related by a series of transpositions of adjacent commuting generators.Biagioli, Jouhet and Nadeau proved among other things that, for each irreducibleCoxeter group, the sequence counting fully commutative elementswith respect to length is ultimately periodic. In the present work, we study thissequence in its periodic range for each of these groups, and in particular we determinethe minimal period. We also observe that in type A affine we get an instance of thecyclic sieving phenomenon