Mackey’s theory of $${\tau}$$ τ -conjugate representations for finite groups

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 → 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. 42

The Case of a Normal Subgroup

“…Open problem. In [16,Section 4], there is a quite deep analysis of the Mackey-Gelfand criterion for finite Gelfand pairs. It should be interesting to extend that analysis in the present setting.…”

“…Also, the twisted Frobenius-Schur theorem (cf. [16,Section 9]) deserves to be analyzed in the present setting (cf. Section 4.6).…”

“…As mentioned at the end of Section 4.1, the twisted Frobenius-Schur theorem (cf. [16,Section 9]) deserves to be analyzed in the present setting.…”

“…Finite Gelfand pairs play an important role in mathematics and have been studied from several points of view: in algebra (we refer, for instance, to the work of Bump and Ginzburg [7,8] and Saxl [52]; see also [16]), in representation theory (as witnessed by the new approach to the representation theory of the symmetric groups by Okounkov and Vershik [49], see also [14]), in analysis (with relevant contributions to the theory of special functions by Dunkl [29] and Stanton [61]), in number theory (we refer to the book by Terras [63] for a comprehensive introduction; see also [11,18]), in combinatorics (in the language of association schemes as developed by Bannai and Ito [1]), and in probability theory (with the remarkable applications to the study of diffusion processes by Diaconis [23]; see also [10,11]). Indeed, Gelfand pairs arise in the study of algebraic, geometrical, or combinatorial structures with a large group of symmetries such that the corresponding permutation representations decompose without multiplicities: it is then possible to develop a useful theory of spherical functions with an associated spherical Fourier transform.…”

Hecke Algebras

“…Finite Gelfand pairs play an important role in mathematics and have been studied from several points of view: in algebra (we refer, for instance, to the work of Bump and Ginzburg [7,8] and Saxl [52]; see also [16]), in representation theory (as witnessed by the new approach to the representation theory of the symmetric groups by Okounkov and Vershik [49], see also [14]), in analysis (with relevant contributions to the theory of special functions by Dunkl [29] and Stanton [61]), in number theory (we refer to the book by Terras [63] for a comprehensive introduction; see also [11,18]), in combinatorics (in the language of association schemes as developed by Bannai and Ito [1]), and in probability theory (with the remarkable applications to the study of diffusion processes by Diaconis [23]; see also [10,11]). Indeed, Gelfand pairs arise in the study of algebraic, geometrical, or combinatorial structures with a large group of symmetries such that the corresponding permutation representations decompose without multiplicities: it is then possible to develop a useful theory of spherical functions with an associated spherical Fourier transform.…”