Computing reflection length in an affine Coxeter group

TL;DR

This paper presents a straightforward formula and proof for calculating the reflection length of any element in affine Coxeter groups, simplifying an important aspect of their algebraic structure.

Contribution

It introduces a simple, uniform method to compute reflection length in all affine Coxeter groups, advancing understanding of their conjugacy and generating properties.

Findings

Provides a universal formula for reflection length

Offers a simple proof applicable to all affine Coxeter groups

Simplifies calculations related to conjugates and generators

Abstract

In any Coxeter group, the conjugates of elements in its Coxeter generating set are called reflections and the reflection length of an element is its length with respect to this expanded generating set. In this article we give a simple formula that computes the reflection length of any element in any affine Coxeter group and we provide a simple uniform proof.