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

Sadek Al Harbat

TL;DR

This work develops a canonical reduced expression for all elements of the affine Coxeter groups of type $\tilde{A}_n$, $\tilde{B}_n$, and $\tilde{D}_n$, using affine length as the central invariant and affine bricks to parametrize coset representatives of $W(\tilde{A}_n)/W(A_n)$. It proves existence and uniqueness of the canonical form, analyzes left-multiplication and right-descent structure, and shows that affine length is preserved along the natural tower $W(\tilde{A}_{n-1}) \hookrightarrow W(\tilde{A}_n)$, implying faithfulness of the corresponding tower of affine Hecke algebras. The authors extend the canonical framework to types $\tilde{B}$ and $\tilde{D}$, derive explicit left-multiplication rules and descent behavior, and discuss consequences for traces and representations, including potential applications to Markov traces and light-leaves bases. Together, these canonical forms tame the combinatorics of affine Coxeter groups, enable efficient enumeration by affine length, and provide a coherent foundation for studying towers of affine Hecke algebras across multiple types.

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

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

TL;DR

This work develops a canonical reduced expression for all elements of the affine Coxeter groups of type , , and , using affine length as the central invariant and affine bricks to parametrize coset representatives of . It proves existence and uniqueness of the canonical form, analyzes left-multiplication and right-descent structure, and shows that affine length is preserved along the natural tower , implying faithfulness of the corresponding tower of affine Hecke algebras. The authors extend the canonical framework to types and , derive explicit left-multiplication rules and descent behavior, and discuss consequences for traces and representations, including potential applications to Markov traces and light-leaves bases. Together, these canonical forms tame the combinatorics of affine Coxeter groups, enable efficient enumeration by affine length, and provide a coherent foundation for studying towers of affine Hecke algebras across multiple types.

Abstract

We classify the elements of 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 , 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 and .

Paper Structure

This paper contains 33 sections, 40 theorems, 89 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Normal form in Coxeter groups
  3. Parabolic subgroups of Coxeter groups
  4. Fokko du Cloux's normal form
  5. Canonical form in $W(\tilde{A}_{n})$
  6. Canonical form in $W(A_{n})$
  7. Affine length
  8. Affine length reduced expressions
  9. Proof of Theorem \ref{['AA']}
  10. Skeleton of the proof
  11. Rigidity Lemma
  12. A few more lemmas
  13. The expression $w_m a_{n+1}$ is reduced.
  14. The expression $w_m a_{n+1} \sigma_k$ is reduced for $2\le k \le n-1$.
  15. The expression $w_m a_{n+1} \sigma_1$ is reduced.
  16. ...and 18 more sections

Key Result

Theorem 1

$W(A_n)$ is the set of elements of the following canonical reduced form: with $n \ge j_1 > \dots > j_s \ge 1$ and $j_t \ge i_t \ge 1$ for $s \ge t \ge 1$. Identity is to be considered the case where $s=0$.

Figures (7)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (73)

  • Theorem
  • Theorem 1.1
  • Theorem : Theorem \ref{['lefttimes']}
  • Theorem : Theorem \ref{['towerandcanonical']}
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 3.1
  • ...and 63 more