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

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 ($★$) 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 ($★$).

This consists of a 38 pages paper and a 7 pages APPENDIX. The original version of the appendix appeared in the unofficial proceedings, "Combinatorial Number Theory and Algebraic Combinatorics", November 18--21, 2002, Yamagata University, Yamagata, Japan, pp. 1--8

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