Canonical reduced expression in affine Coxeter groups of type ${\tilde{A}}_n$, ${\tilde{B}}_n$, ${\tilde{D}}_n$

Abstract

We classify the elements of $W({\tilde{A}}_n)$ by giving a canonical reduced expression for each, using basic tools among which affine length. We give some direct consequences for such a canonical form: a description of left multiplication by a simple reflection, a study of the right descent set, and a proof that the affine length is preserved along the tower of affine Coxeter groups of type $\tilde{A}$, which implies in particular that the corresponding tower of affine Hecke algebras is a faithful tower regardless of the ground ring. We give a similar canonical reduced expression for the elements of $W({\tilde{B}}_n)$ and $W({\tilde{D}}_n)$.