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.

中文翻译：

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Mackey的$$ {\ tau} $$τ理论-有限群的共轭表示

本文的目的是揭露Mackey的两个贡献，以及由Bump和Ginzburg推广的Kawanaka和Matsuyama的最新成果，关于装备有非自愿反自同构性的有限群的表示理论（例如反-automorphism \（{g \ mapsto g ^ {-1}} \））。Mackey的第一个贡献是针对弱对称Gelfand对的所谓Gelfand准则的详细版本。Mackey的第二个贡献是对简单可归约基团的表征（Wigner引入的一个概念）。另一个结果是Frobenius-Schur定理的扭曲形式，其中“扭曲”是指上述非自愿抗自同构性。