Japanese Journal of Mathematics

, Volume 10, Issue 1, pp 43–96 | Cite as

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

  • Tullio Ceccherini-Silberstein
  • Fabio Scarabotti
  • Filippo Tolli
Original Article

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.

Keywords and phrases

representation theory of finite groups Gelfand pair Kronecker product simply reducible group Clifford groups Frobenius–Schur theorem 

Mathematics Subject Classification (2010)

20C15 43A90 20G40 

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Copyright information

© The Mathematical Society of Japan and Springer Japan 2014

Authors and Affiliations

  • Tullio Ceccherini-Silberstein
    • 1
  • Fabio Scarabotti
    • 2
  • Filippo Tolli
    • 3
  1. 1.Dipartimento di IngegneriaUniversità del SannioBeneventoItaly
  2. 2.Dipartimento SBAISapienza Università di RomaRomaItaly
  3. 3.Dipartimento di Matematica e FisicaUniversità Roma TRERomaItaly

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