{
  "id": "2212.13241",
  "version": "v1",
  "published": "2022-12-26T18:06:34.000Z",
  "updated": "2022-12-26T18:06:34.000Z",
  "title": "Generalized characters of the generalized symmetric group",
  "authors": [
    "Omar Tout"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1912.05294",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that $(\\mathbb{Z}_k \\wr \\mathcal{S}_n \\times \\mathbb{Z}_k \\wr \\mathcal{S}_{n-1}, \\text{diag} (\\mathbb{Z}_k \\wr \\mathcal{S}_{n-1}) )$ is a symmetric Gelfand pair, where $\\mathbb{Z}_k \\wr \\mathcal{S}_n$ is the wreath product of the cyclic group $\\mathbb{Z}_k$ with the symmetric group $\\mathcal{S}_n.$ The proof is based on the study of the $\\mathbb{Z}_k \\wr \\mathcal{S}_{n-1}$-conjugacy classes of $\\mathbb{Z}_k \\wr \\mathcal{S}_n.$ We define the generalized characters of $\\mathbb{Z}_k \\wr \\mathcal{S}_n$ using the zonal spherical functions of $(\\mathbb{Z}_k \\wr \\mathcal{S}_n \\times \\mathbb{Z}_k \\wr \\mathcal{S}_{n-1}, \\text{diag} (\\mathbb{Z}_k \\wr \\mathcal{S}_{n-1}) ).$ We show that these generalized characters have properties similar to usual characters. A Murnaghan-Nakayama rule for the generalized characters of the hyperoctahedral group is presented. The generalized characters of the symmetric group were first studied by Strahov in [7].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-26T18:06:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "20C30"
    ],
    "keywords": [
      "generalized characters",
      "generalized symmetric group",
      "symmetric gelfand pair",
      "cyclic group",
      "hyperoctahedral group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}