Antisymmetric elements in group rings with an orientation morphism

TL;DR

This paper characterizes when antisymmetric elements in group rings with an orientation morphism commute, linking group properties and orientations to algebraic behavior of these elements.

Contribution

It provides a complete classification of groups and orientations where antisymmetric elements in the group ring commute.

Findings

Identifies conditions on groups for antisymmetric elements to commute.

Characterizes the role of orientation morphisms in the commutativity.

Establishes a full classification linking group structure and antisymmetric element behavior.

Abstract

Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $\phi_{\sigma} : RG\to RG$ denote the involution defined by $\phi_{\sigma} (\sum r_{g}g) = \sum r_{g} \sigma (g) g^{-1}$, 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 $\phi_{\sigma} (x) =-x$. We give a full characterization of the groups $G$ and its orientations for which the antisymmetric elements of $RG$ commute.