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Mackey’s theory of \({\tau}\)-conjugate representations for finite groups

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Japanese Journal of Mathematics Aims and scope

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.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Università del Sannio, C.so Garibaldi 107, 82100, Benevento, Italy

    Tullio Ceccherini-Silberstein

  2. Dipartimento SBAI, Sapienza Università di Roma, via A. Scarpa 8, 00161, Roma, Italy

    Fabio Scarabotti

  3. Dipartimento di Matematica e Fisica, Università Roma TRE, L. San Leonardo Murialdo 1, 00146, Roma, Italy

    Filippo Tolli

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  1. Tullio Ceccherini-Silberstein
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Communicated by: Yasuyuki Kawahigashi

Dedicated to our mentors and friends Toni Machì on his 75th birthday and Pierre de la Harpe on his 70th birthday

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Ceccherini-Silberstein, T., Scarabotti, F. & Tolli, F. Mackey’s theory of \({\tau}\)-conjugate representations for finite groups. Jpn. J. Math. 10, 43–96 (2015). https://doi.org/10.1007/s11537-014-1390-8

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