Mackey’s theory of $${\tau}$$ τ -conjugate representations for finite groups

Abstract

The aim of the present paper is to expose two contributions of Mackey, together with a more recent result of Kawanaka and Matsuyama, generalized by Bump and Ginzburg, on the representation theory of a finite group equipped with an involutory anti-automorphism (e.g. the anti-automorphism \({g \mapsto g^{-1}}\)). Mackey’s first contribution is a detailed version of the so-called Gelfand criterion for weakly symmetric Gelfand pairs. Mackey’s second contribution is a characterization of simply reducible groups (a notion introduced by Wigner). The other result is a twisted version of the Frobenius–Schur theorem, where “twisted” refers to the above-mentioned involutory anti-automorphism.