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 φσ : RG → RG denote the involution defined by φσ(Σrgg) = Σrgσ(g)g–1, where σ : G → {±1} is a group homomorphism (called an orientation morphism). An element x in RG is said to be antisymmetric if φσ(x) = –x. We give a full characterization of the groups G and its orientations for which the antisymmetric elements of RG commute.