Fully commutative elements in affine Coxeter groups

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in the finite case. They index naturally a basis of the generalized Temperley–Lieb algebra of W.

In this talk, we give explicit descriptions of fully commutative elements when W is an affine Coxeter group. Using our characterizations we then enumerate these elements according to their Coxeter length, and we show that the corresponding generating function is ultimately periodic in each type.

Based on joint work with F. Jouhet and P. Nadeau.