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

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. APPENDIX: We consider a special condition related to Gelfand pairs. Namely, we call a finite group $G$ and its automorphism $σ$ satisfy Condition ($\bigstar$) if the following condition is satisfied: if for $x,y\in G$, $x\cdot x^{-σ}$ and $y\cdot y^{-σ}$ are conjugate in $G$, then they are conjugate in $K=C_G(σ)$. We study the meanings of this condition, as well as showing many examples of $G$ and $σ$ which do (or do not) satisfy Condition ($\bigstar$).

Theorems & Definitions (76)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Remark 3.1
• Proposition 3.2
• proof
• Lemma 3.3
• proof