{
  "id": "0912.1512",
  "version": "v3",
  "published": "2009-12-08T15:47:44.000Z",
  "updated": "2015-02-26T12:04:30.000Z",
  "title": "Invariant tensors and the cyclic sieving phenomenon",
  "authors": [
    "Bruce W. Westbury"
  ],
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We construct a large class of examples of the cyclic sieving phenomenon by expoiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\\otimes^rB$; the map $c\\colon X\\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degree of the Frobenius character of a representation of $\\mathfrak{S}_r$ related to the natural action of $\\mathfrak{S}_r$ on the subspace of invariant tensors in $\\otimes^rM$. Taking $M$ to be the defining representation of $\\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-07-07T14:28:44.000Z",
      "abstract": "The problem of finding the orbit structure of the promotion map acting on standard tableaux with rectangular shape was solved by Rhoades using Springer's theory of regular elements and the Khazdan-Lusztig basis. This paper extends this result by replacing the vector representation of $\\SL(n)$ by any highest weight representation of a simple Lie algebra. The promotion map is defined using crystal graphs. The orbit structure is determined using Springer's theory of regular elements and Lusztig's theory of based modules.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-02-26T12:04:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "22E45"
    ],
    "keywords": [
      "cyclic sieving phenomenon",
      "invariant tensors",
      "regular elements",
      "springers theory",
      "orbit structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.1512W"
    }
  }
}