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

TL;DR

This paper classifies elements of affine Coxeter groups of types _A_n, _B_n, and _D_n by providing canonical reduced expressions, with implications for affine length preservation and Hecke algebra faithfulness.

Contribution

It introduces a canonical reduced expression for elements in affine Coxeter groups of types _A_n, _B_n, and _D_n, advancing understanding of their structure and algebraic properties.

Findings

Canonical reduced expressions for _A_n, _B_n, _D_n

Description of left multiplication by simple reflections

Proof of affine length preservation in the tower of affine Coxeter groups

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 $W(\tilde{D}_n)$.