Mackey's theory of $τ$-conjugate representations for finite groups. APPENDIX: On Some Gelfand Pairs and Commutative Association Schemes
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arXiv
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2013
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| author | Ceccherini-Silberstein, Tullio Scarabotti, Fabio Tolli, Filippo Bannai, Eiichi Tanaka, Hajime |
| author_facet | Ceccherini-Silberstein, Tullio Scarabotti, Fabio Tolli, Filippo Bannai, Eiichi Tanaka, Hajime |
| contents | 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 ($\bigstar$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1311_7252 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Mackey's theory of $τ$-conjugate representations for finite groups. APPENDIX: On Some Gelfand Pairs and Commutative Association Schemes Ceccherini-Silberstein, Tullio Scarabotti, Fabio Tolli, Filippo Bannai, Eiichi Tanaka, Hajime Representation Theory Group Theory 20C15, 43A90, 20G40 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 ($\bigstar$). |
| title | Mackey's theory of $τ$-conjugate representations for finite groups. APPENDIX: On Some Gelfand Pairs and Commutative Association Schemes |
| topic | Representation Theory Group Theory 20C15, 43A90, 20G40 |
| url | https://arxiv.org/abs/1311.7252 |