{
  "id": "2211.14093",
  "version": "v1",
  "published": "2022-11-25T13:24:29.000Z",
  "updated": "2022-11-25T13:24:29.000Z",
  "title": "Skew hook Schur functions and the cyclic sieving phenomenon",
  "authors": [
    "Nishu Kumari"
  ],
  "comment": "13 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "Fix an integer $t \\geq 2$ and a primitive $t^{\\text{th}}$ root of unity $\\omega$. We consider the specialized skew hook Schur polynomial $\\text{hs}_{\\lambda/\\mu}(X,\\omega X,\\dots,\\omega^{t-1}X/Y,\\omega Y,\\dots,\\omega^{t-1}Y)$, where $\\omega^k X=(\\omega^k x_1, \\dots, \\omega^k x_n)$, $\\omega^k Y=(\\omega^k y_1, \\dots, \\omega^k y_m)$ for $0 \\leq k \\leq t-1$. We characterize the skew shapes $\\lambda/\\mu$ for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials. Then we give a combinatorial interpretation of $\\text{hs}_{\\lambda/\\mu}(1,\\omega^d,\\dots,\\omega^{d(tn-1)}/1,\\omega^d,\\dots,\\omega^{d(tm-1)})$, for all divisors $d$ of $t$, in terms of ribbon supertableaux. Lastly, we use the combinatorial interpretation to prove the cyclic sieving phenomenon on the set of semistandard supertableaux of shape $\\lambda/\\mu$ for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee--Oh (arXiv: 2112.12394, 2021) for the cyclic sieving phenomenon on the set of skew SSYT conjectured by Alexandersson--Pfannerer--Rubey--Uhlin (Forum Math. Sigma, 2021).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-25T13:24:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E18",
      "05E05",
      "05E10"
    ],
    "keywords": [
      "cyclic sieving phenomenon",
      "skew hook schur functions",
      "smaller skew hook schur polynomials",
      "specialized skew hook schur polynomial",
      "combinatorial interpretation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}