Mackey's theory of $\tau$-conjugate representations for finite groups. APPENDIX: On Some Gelfand Pairs and Commutative Association Schemes

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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 m a p s t o 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. APPENDIX: We consider a special condition related to Gelfand pairs. Namely, we call a finite group

G

and its automorphism

s i g m a

satisfy Condition (

) if the following condition is satisfied: if for

x, y i n G

,

x c d o t x^{- s i g m a}

and

y c d o t y^{- s i g m a}

are conjugate in

G

, then they are conjugate in

K = C_{G} (s i g m a)

. We study the meanings of this condition, as well as showing many examples of

G

and

s i g m a

which do (or do not) satisfy Condition (

).

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