Oriented Group Involutions and Anticommutativity in Group Rings

Abstract

Let G be a group with involution * and σ: G → {±1} a group homomorphism such that σ(g*) = σ(g) for all g ∈ G. The map that sends in a group ring RG to is an involution of RG called an oriented group involution. In this article, noting that the ♯-symmetric elements of RG form a Jordan ring under the product α ○ β = αβ + βα, we ask when this product is trivial; equivalently, when the ♯-symmetric elements anticommute.