On paired root systems of Coxeter groups

TL;DR

This paper introduces a systematic method to construct paired root systems for Coxeter groups using non-standard geometric representations, enabling new insights into their structure and subgroups.

Contribution

It presents a novel approach to generate paired root systems for Coxeter groups and characterizes these groups within this framework, including reflection subgroups.

Findings

Constructed paired root systems for various Coxeter groups

Characterized Coxeter groups via paired root systems

Analyzed reflection subgroups using the new method

Abstract

This paper examines a systematic method to construct a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of non-standard geometric representations. This method can be employed to construct generalizations of root systems for a large family of groups generated only by involutions. We then give a characterization of Coxeter groups, among these groups, in terms of such paired root systems. Furthermore, we use this method to construct and study the paired root systems for reflection subgroups of Coxeter groups.