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

Let $G$ be a group with involution * and $σ\colon G\to\{\pm1\}$ a group homomorphism. The map $\sharp$ that sends $α=\sumα_gg$ in a group ring $RG$ to $α^{\sharp}=\sumσ(g)α_gg^*$ is an involution of $RG$ called an \emph{oriented group involution}. An element $α\in RG$ is \emph{symmetric} if $α^{\sharp}=α$ and \emph{skew-symmetric} if $α^{\sharp}=-α$. 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 $σ$ 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.

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