{
  "id": "1503.02109",
  "version": "v1",
  "published": "2015-03-06T23:06:46.000Z",
  "updated": "2015-03-06T23:06:46.000Z",
  "title": "A combinatorial approach to the q,t-symmetry relation in Macdonald polynomials",
  "authors": [
    "Maria Monks Gillespie"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\\widetilde{H}_\\mu(\\mathbf{x};q,t) = \\widetilde{H}_{\\mu^\\ast}(\\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $\\mu$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\\mathbf{x}$ for all shapes $\\mu$. We also provide a proof for the full relation in the case when $\\mu$ is a hook shape, and for all shapes at the specialization $t=1$. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-06T23:06:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05A19"
    ],
    "keywords": [
      "t-symmetry relation",
      "combinatorial approach",
      "transformed macdonald polynomials",
      "cocharge statistic",
      "purely combinatorial proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}