Oriented Involutions, Symmetric and Skew-Symmetric Elements in Group Rings

Abstract

Let $G$ be a group with involution * and $\sigma :G\to \{\pm 1\}$ a group homomorphism. The map $♯$ that sends $\alpha =\sum {\alpha }_{g}g$ in a group ring $RG$ to ${\alpha }^{♯}=\sum \sigma (g){\alpha }_g{g}^{\ast }$ is an involution of $RG$ called an \emph{oriented group involution}. An element $\alpha \in RG$ is \emph{symmetric} if ${\alpha }^{♯}=\alpha$ and \emph{skew-symmetric} if ${\alpha }^{♯}=-\alpha$. The sets of symmetric and skew-symmetric elements have received a lot of attention in the special cases that * is the inverse map on $G$ and/or $\sigma$ is identically 1, but not in general. In this paper, we determine the conditions under which the sets of elements that are symmetric and skew-symmetric, respectively, relative to a general oriented involution form subrings of $RG$. The work on symmetric elements is a modification and correction of previous work.