On reflection representations of Coxeter groups over non-commutative rings

Abstract

In this paper, we consider representations of Coxeter groups over a path algebra, R, introduced by Dyer in the study of root systems over non-commutative rings. 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.

KEYWORD:

• Coxeter groups
• domain
• path algebras

2020 Mathematics Subject Classification:

• 20F55
• 17B22
• 20C08

Notes

1 A different basis for $\tilde{R}$, with some very favorable properties, is also considered in [13].

2 The finite rank assumption can also be omitted but we leave the additional arguments to show this to the reader.