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.
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Notes
1 A different basis for , with some very favorable properties, is also considered in [Citation13].
2 The finite rank assumption can also be omitted but we leave the additional arguments to show this to the reader.