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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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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DOI: https://doi.org/10.1007/s11537-014-1390-8
Keywords and phrases
- representation theory of finite groups
- Gelfand pair
- Kronecker product
- simply reducible group
- Clifford groups
- Frobenius–Schur theorem