{
  "id": "2109.00107",
  "version": "v1",
  "published": "2021-08-31T23:13:43.000Z",
  "updated": "2021-08-31T23:13:43.000Z",
  "title": "Doubly stochastic matrices and Schur-Weyl duality for partition algebras",
  "authors": [
    "Stephen Doty"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "In 1916 D\\'{e}nes K\\\"{o}nig proved that the permanent of a doubly stochastic matrix is positive; equivalently, the matrix has a positive diagonal. K\\\"{o}nig's result is equivalent to Birkhoff's 1946 result, that the set of $n \\times n$ doubly stochastic matrices is the convex hull of the $n \\times n$ permutation matrices. We conjecture that K\\\"{o}nig's result extends to doubly stochastic matrices in the $\\mathbb{R}$-span of $r$th tensor powers of $n \\times n$ permutation matrices. The conjecture implies a simple new algorithmic proof that Schur--Weyl duality for partition algebras (originally proved over $\\mathbb{C}$ by V.F.R. Jones) holds over any commutative ring.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-31T23:13:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "20C30"
    ],
    "keywords": [
      "doubly stochastic matrix",
      "schur-weyl duality",
      "partition algebras",
      "permutation matrices",
      "th tensor powers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}