{
  "id": "2211.10584",
  "version": "v1",
  "published": "2022-11-19T04:26:10.000Z",
  "updated": "2022-11-19T04:26:10.000Z",
  "title": "Chess tableaux, powers of two and affine Lie algebras",
  "authors": [
    "Antoine Labelle",
    "Stoyan Dimitrov"
  ],
  "comment": "14 pages, 3 figures",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "Chess tableaux are a special kind of standard Young tableaux where, in the chessboard coloring of the Young diagram, even numbers always appear in white cells and odd numbers in black cells. If, for $\\lambda$ a partition of $n$, $\\text{Chess}(\\lambda)$ denotes the number of chess tableaux of shape $\\lambda$, then Chow, Eriksson and Fan observed that $\\displaystyle\\sum_{\\lambda \\vdash n} \\text{Chess}(\\lambda)^2$ is divisible by unusually large powers of $2$. In this paper, we give an explanation for this phenomenon, proving a lower bound of $n-O(\\sqrt{n})$ for the $2$-adic valuation of this sum and a generalization of it. We do this by exploiting a connection with a certain representation of the affine Lie algebra $\\widehat{\\mathfrak{sl}_2}$ on the vector space with basis indexed by partitions. Our result about chess tableaux then follows from a study of the basic representation of $\\widehat{\\mathfrak{sl}_2}$ with coefficients taken from the ring of rational numbers with odd denominators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-19T04:26:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "affine lie algebra",
      "chess tableaux",
      "standard young tableaux",
      "unusually large powers",
      "basic representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}