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

TL;DR

This paper classifies elements of the affine Coxeter group of type A_n by providing canonical reduced expressions, leading to insights on multiplication, descent sets, and the structure of associated Hecke algebras.

Contribution

It introduces a canonical reduced expression for elements of A_n, enabling new understanding of their algebraic properties and the structure of related Hecke algebras.

Findings

Canonical reduced expressions for A_n elements

Description of left multiplication by simple reflections

Proof of affine length preservation in the tower

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.