{
  "id": "1810.03448",
  "version": "v1",
  "published": "2018-10-08T13:43:23.000Z",
  "updated": "2018-10-08T13:43:23.000Z",
  "title": "Plethysms of symmetric functions and highest weight representations",
  "authors": [
    "Melanie de Boeck",
    "Rowena Paget",
    "Mark Wildon"
  ],
  "comment": "35 pages",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Let $s_\\nu \\circ s_\\mu$ denote the plethystic product of the Schur functions $s_\\nu$ and $s_\\mu$. In this article we define an explicit polynomial representation corresponding to $s_\\nu \\circ s_\\mu$ with basis indexed by certain `plethystic' semistandard tableaux. Using these representations we prove generalizations of four results on plethysms due to Bruns--Conca--Varbaro, Brion, Ikenmeyer and the authors. In particular, we give a sufficient condition for the multiplicity $\\langle s_\\nu \\circ s_\\mu, s_\\lambda\\rangle$ to be stable under insertion of new parts into $\\mu$ and $\\lambda$. We also characterize all maximal and minimal partitions $\\lambda$ in the dominance order such that $s_\\lambda$ appears in $s_\\nu \\circ s_\\mu$ and determine the corresponding multiplicities using plethystic semistandard tableaux.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-08T13:43:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10",
      "17B10",
      "20C30",
      "22E47"
    ],
    "keywords": [
      "highest weight representations",
      "symmetric functions",
      "plethystic semistandard tableaux",
      "schur functions",
      "dominance order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}