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 → RG denote the involution defined by φ σ (Σ r g g ) = Σ r g σ( 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 φ σ ( x ) = – x . We give a full characterization of the groups G and its orientations for which the antisymmetric elements of RG commute.