Affine reflection subgroups of Coxeter groups

TL;DR

This paper investigates affine reflection subgroups within infinite Coxeter groups, characterizing their roots and limits, and exploring conditions for their existence, thereby advancing understanding of Coxeter group structures.

Contribution

It provides new characterizations of limit roots from affine reflection subgroups and criteria for their presence in Coxeter groups.

Findings

Characterization of limit roots from affine reflection subgroups

Conditions for Coxeter groups to possess affine reflection subgroups

Relationship between isotropic cone, imaginary cone, and limit roots

Abstract

In this paper we study affine reflection subgroups in arbitrary infinite Coxeter groups of finite rank. In particular, we study the distribution of roots of Coxeter groups in the root subsystems associated with affine reflection subgroups. We give a characterization of limit roots arising from affine reflection subgroups. We also give a characterization of when a Coxeter group may possess affine reflection subgroups. We show that the intersection of the normalized isotropic cone (associated with the Tits representation of a Coxeter group) and the imaginary cone consists of limit roots closely related to affine reflection subgroups.