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