Relations in hyperreflection groups

Abstract:

Hyperreflection groups ${G_m}$ are generalizations of groups generated by reflections. A hyperreflection group is generated by hyperreflections if $\dim V$ is finite. A hyperreflection is a simple mapping $\sigma$ such that $\det \sigma = \gamma$, where ${\gamma ^m} = 1$. If the field of scalars is commutative, the order of $\sigma$ is $m$. Our main result states that every relation between hyperreflections and their inverses is a consequence of relations of lengths 2, 4, and $m$. The most interesting special case occurs for $m = 2$. Then our result refers to relations between reflections.