{
  "id": "2304.11690",
  "version": "v1",
  "published": "2023-04-23T15:44:42.000Z",
  "updated": "2023-04-23T15:44:42.000Z",
  "title": "On algebras of double cosets of symmetric groups with respect to Young subgroups",
  "authors": [
    "Yury A. Neretin"
  ],
  "comment": "10p",
  "categories": [
    "math.GR",
    "math.CO",
    "math.RT"
  ],
  "abstract": "We consider the subalgebra $\\Delta$ in the group algebra of the symmetric group $G=S_{n_1+\\dots+n_\\nu}$ consisting of all functions invariant with respect to left and right shifts by elements of the Young subgroup $H:=S_{n_1}\\times \\dots \\times S_{n_\\nu}$. We discuss structure constants of the algebra $\\Delta$ and construct an algebra with continuous parameters $n_1$ extrapolating algebras $\\Delta$, it can be also can be rewritten as an asymptotic algebra as $n_j\\to\\infty$ (for fixed $\\nu$). We show that there is a natural map from the Lie algebra of the group of pure braids to $\\Delta$ (and therefore this Lie algebra acts in spaces of multiplicities of the quasiregular representation of the group $G$ in functions on $G/H$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-23T15:44:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C30",
      "20N20",
      "20C05",
      "33C20",
      "05E16"
    ],
    "keywords": [
      "symmetric group",
      "young subgroup",
      "double cosets",
      "lie algebra acts",
      "group algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}