On Reflection Representations of Coxeter Groups over Non-Commutative Rings

TL;DR

This paper investigates the algebraic properties of reflection representations of Coxeter groups over non-commutative rings, revealing their near-domain structure and embedding into matrix rings over complex algebraic extensions.

Contribution

It demonstrates that the path algebra R associated with Coxeter groups is nearly a domain and can be embedded into matrix rings over free products of extension fields and Laurent polynomial rings.

Findings

R is 'almost a domain'

R can be embedded into matrix rings over complex algebraic structures

Provides new insights into the multiplicative properties of Coxeter group representations

Abstract

In this paper, we consider representations of Coxeter groups over a path algebra, R, defined by Dyer. We answer a question posed by Dyer about the multiplicative properties of R, showing that it is "almost a domain". We also show that R cam be embedded in a matrix ring over a free product of extension fields of the rational numbers and rings of Laurent polynomials.