{
  "id": "2010.13930",
  "version": "v1",
  "published": "2020-10-26T22:22:04.000Z",
  "updated": "2020-10-26T22:22:04.000Z",
  "title": "Promotion and Cyclic Sieving on Rectangular $δ$-Semistandard Tableaux",
  "authors": [
    "Tair Akhmejanov",
    "Balázs Elek"
  ],
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "Let $\\delta=(\\delta_1,\\ldots,\\delta_n)$ be a string of letters $h$ and $v$. We define a Young tableau to be $\\delta$-semistandard if the entries are weakly increasing along rows and columns, and the entries $i$ form a horizontal strip if $\\delta_i=h$ and a vertical strip if $\\delta_i=v$. We define $\\delta$-promotion on such tableaux via a modified jeu-de-taquin. The first main result is that $\\delta$-promotion has period $n$ on rectangular $\\delta$-semistandard tableaux, generalizing the result of Haiman for rectangular semistandard tableaux. The second main result states that the set of rectangular $\\delta$-semistandard tableaux for fixed $\\delta$ and content $\\gamma$ exhibits a cyclic sieving phenomenon with the generalized Kostka polynomial. To do so we follow Fontaine-Kamnitzer and associate to $(\\delta,\\gamma)$ an invariant space $(V_{\\lambda^1}\\otimes\\cdots\\otimes V_{\\lambda^n})^{SL_m}$ where each $V_{\\lambda^i}$ is an alternating or symmetric representation. We show that the Satake basis of the corresponding invariant space is indexed by the set of tableaux corresponding to $(\\delta,\\gamma)$ and is permuted by rotation of tensor factors. We then diagonalize the rotation action using the fusion product. This cyclic sieving generalizes the result of Rhoades and Fontaine-Kamnitzer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-26T22:22:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10"
    ],
    "keywords": [
      "cyclic sieving",
      "second main result states",
      "rectangular semistandard tableaux",
      "first main result",
      "corresponding invariant space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}