ORIENTED INVOLUTIONS AND SKEW-SYMMETRIC ELEMENTS IN GROUP RINGS

Let G be a group with involution * and sigma : G -> {+/- 1} a group homomorphism. The map # that sends alpha = Sigma alpha(g)g in a group ring RG to alpha(#) = Sigma sigma(g)alpha(g)g* is an involution of RG called an oriented group involution. An element alpha epsilon RG is symmetric if alpha(#) = alpha and 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 or sigma is identically 1, while the general case has been almost ignored. In this paper, we determine the conditions under which the set of elements that are skew-symmetric relative to a general oriented involution form a subring of RG. This is the sequel to another paper where the analogous problem for the symmetric elements was studied, with a small oversight that is corrected here.