{
  "id": "1903.10809",
  "version": "v1",
  "published": "2019-03-26T11:39:09.000Z",
  "updated": "2019-03-26T11:39:09.000Z",
  "title": "The Multiset Partition Algebra",
  "authors": [
    "Sridhar Narayanan",
    "Digjoy Paul",
    "Shraddha Srivastava"
  ],
  "comment": "34 pages. Comments welcome",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "For each partition $\\lambda$, we introduce the multiset partition algebra $\\mathcal{MP}_\\lambda(\\xi)$ over $F[\\xi]$, where $F$ is a field of characteristic $0$. Upon specializing $\\xi$ to a positive integer $n$, the resulting algebra $\\mathcal{MP}_\\lambda(n)$ is in Schur-Weyl duality with the action of the symmetric group $S_n$ on $\\text{Sym}^\\lambda(F^n)$. The generating function of the dimensions of irreducible representations of $\\mathcal{MP}_{\\lambda}(n)$ is given in terms of Schur functions. When $\\lambda$ is a partition of $k$, $\\mathcal{MP}_\\lambda(\\xi)$ is isomorphic to a subalgebra of the partition algebra $\\mathcal{P}_k(\\xi)$ of Jones and Martin. When $\\lambda=(1^k)$, $\\mathcal{MP}_\\lambda(\\xi)$ is isomorphic to $\\mathcal{P}_k(\\xi)$. We identify a subalgebra of $\\mathcal{MP}_{\\lambda}(\\xi)$ called the balanced multiset partition algebra whose structure constants do not depend on $\\xi$. This algebra is in Schur-Weyl duality with the group of monomial matrices acting on $\\text{Sym}^\\lambda(F^n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-26T11:39:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "05E15",
      "20C30"
    ],
    "keywords": [
      "schur-weyl duality",
      "balanced multiset partition algebra",
      "symmetric group",
      "monomial matrices",
      "schur functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}