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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.

On reflection representations of Coxeter groups over non-commutative rings