Mackey's theory of $\tau$-conjugate representations for finite groups. APPENDIX: On Some Gelfand Pairs and Commutative Association Schemes

TL;DR

This paper explores Mackey's contributions to the representation theory of finite groups with involutory anti-automorphisms, including Gelfand pairs, simply reducible groups, and a twisted Frobenius-Schur theorem, with an appendix on related Gelfand pair conditions.

Contribution

It provides detailed expositions of Mackey's results, extends them with recent generalizations, and introduces a new condition related to Gelfand pairs and automorphisms.

Findings

Gelfand criterion for weakly symmetric Gelfand pairs clarified

Characterization of simply reducible groups provided

Examples of groups satisfying or not satisfying Condition ($igstar$)

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. APPENDIX: We consider a special condition related to Gelfand pairs. Namely, we call a finite group $G$ and its automorphism $\sigma$ satisfy Condition ($\bigstar$) if the following condition is satisfied: if for $x,y\in G$, $x\cdot x^{-\sigma}$ and $y\cdot y^{-\sigma}$ are conjugate in $G$, then they are conjugate in $K=C_G(\sigma)$. We study the meanings of this condition, as well as showing many examples of $G$ and $\sigma$ which do (or do not) satisfy Condition ($\bigstar$).