Long fully commutative elements in affine Coxeter groups

TL;DR

This paper investigates the periodic nature of fully commutative elements in affine Coxeter groups, determines their minimal periods, and explores the cyclic sieving phenomenon in type A affine groups.

Contribution

It analyzes the periodic part of the counting sequence for fully commutative elements and identifies their minimal periods, extending previous work on affine Coxeter groups.

Findings

Determined the minimal period of the counting sequence for each affine Coxeter group.

Identified the cyclic sieving phenomenon in type A affine Coxeter groups.

Extended understanding of the structure of fully commutative elements in affine Coxeter groups.

Abstract

An element of a Coxeter group $W$ is called fully commutative if any two of its reduced decompositions can be related by a series of transpositions of adjacent commuting generators. In the preprint "Fully commutative elements in finite and affine Coxeter groups" (arXiv:1402.2166), R. Biagioli and the authors proved among other things that, for each irreducible affine Coxeter group, the sequence counting fully commutative elements with respect to length is ultimately periodic. In the present work, we study this sequence in its periodic part for each of these groups, and in particular we determine the minimal period. We also observe that in type $A$ affine we get an instance of the cyclic sieving phenomenon.