{
  "id": "1311.7252",
  "version": "v2",
  "published": "2013-11-28T09:45:52.000Z",
  "updated": "2014-07-12T08:07:15.000Z",
  "title": "Mackey's theory of $τ$-conjugate representations for finite groups. APPENDIX: On Some Gelfand Pairs and Commutative Association Schemes",
  "authors": [
    "Tullio Ceccherini-Silberstein",
    "Fabio Scarabotti",
    "Filippo Tolli",
    "Eiichi Bannai",
    "Hajime Tanaka"
  ],
  "comment": "This consists of a 38 pages paper and a 7 pages APPENDIX. The original version of the appendix appeared in the unofficial proceedings, \"Combinatorial Number Theory and Algebraic Combinatorics\", November 18--21, 2002, Yamagata University, Yamagata, Japan, pp. 1--8",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "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 $\\sigma$ satisfy Condition ($\\bigstar$) if the following condition is satisfied: if for $x,y\\in G$, $x\\cdot x^{-\\sigma}$ and $y\\cdot y^{-\\sigma}$ are conjugate in $G$, then they are conjugate in $K=C_G(\\sigma)$. We study the meanings of this condition, as well as showing many examples of $G$ and $\\sigma$ which do (or do not) satisfy Condition ($\\bigstar$).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-12T08:07:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C15",
      "43A90",
      "20G40"
    ],
    "keywords": [
      "finite group",
      "gelfand pairs",
      "commutative association schemes",
      "mackeys theory",
      "conjugate representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.7252C"
    }
  }
}