On reflection subgroups of finite Coxeter groups

TL;DR

This paper classifies reflection subgroups of finite Coxeter groups up to conjugacy and characterizes when the map to conjugacy classes of Coxeter elements is injective, providing a detailed structural understanding.

Contribution

It provides a complete classification of reflection subgroups of finite Coxeter groups and establishes criteria for the injectivity of the Coxeter element conjugacy class map.

Findings

Reflection subgroups classified up to conjugacy

Necessary and sufficient conditions for injectivity of the Coxeter element map

Structural insights into Coxeter group substructures

Abstract

Let $W$ be a finite Coxeter group. We classify the reflection subgroups of $W$ up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup $R$ of $W$ the conjugacy class of its Coxeter elements to be injective, up to conjugacy.