Maximal order Abelian subgroups of Coxeter groups

TL;DR

This paper classifies the maximal order Abelian subgroups of finite irreducible Coxeter groups and proves a Weyl group analogue of Cartan's theorem regarding conjugacy of maximal tori.

Contribution

It provides a complete classification of maximal order Abelian subgroups in Coxeter groups and establishes a conjugacy result analogous to Cartan's theorem for Weyl groups.

Findings

Classification of maximal order Abelian subgroups in Coxeter groups

Proof of conjugacy of maximal tori in Weyl groups

Extension of Cartan's theorem to Weyl group context

Abstract

In this note we give a classification of the Maximal order Abelian subgroups of finite irreducible Coxeter groups. We also prove a Weyl group analogue of Cartan's theorem that all maximal tori in a connected compact Lie group are conjugate.