Antisymmetric elements in group rings with an orientation morphism

Abstract

Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $ϕ_σ : RG\to RG$ denote the involution defined by $ϕ_σ (\sum r_{g}g) = \sum r_{g} σ(g) g^{-1}$, where $σ:G\to \{\pm 1\}$ is a group homomorphism (called an orientation morphism). An element $x$ in $RG$ is said to be antisymmetric if $ϕ_σ (x) =-x$. We give a full characterization of the groups $G$ and its orientations for which the antisymmetric elements of $RG$ commute.