Minimal length elements of finite Coxeter groups

TL;DR

This paper provides a geometric proof demonstrating that minimal length elements within a conjugacy class of a finite Coxeter group exhibit notable properties related to conjugation, powers in the braid group, and centralizers.

Contribution

It introduces a geometric proof revealing special properties of minimal length elements in finite Coxeter groups' conjugacy classes, enhancing understanding of their structure.

Findings

Minimal length elements have unique conjugation properties.

They relate to powers in the associated braid group.

They have specific centralizer characteristics.

Abstract

We give a geometric proof that minimal length elements in a (twisted) conjugacy class of a finite Coxeter group $W$ have remarkable properties with respect to conjugation, taking powers in the associated Braid group and taking centralizer in $W$.