Canonical reduced expression for elements of affine Coxeter groups Part I -- Type ${\tilde{A}}_n$

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.

arXiv commentV3 : mainly a change in the Introduction, with references added. V2 : 23 pages. Proofs improved and errors corrected. The paragraph on right multiplication, now 4.2, has major changes