{
  "id": "1106.1408",
  "version": "v1",
  "published": "2011-06-07T18:25:24.000Z",
  "updated": "2011-06-07T18:25:24.000Z",
  "title": "On the adjoint representation of $\\mathfrak{sl}_n$ and the Fibonacci numbers",
  "authors": [
    "Pamela E. Harris"
  ],
  "comment": "9 pages",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We decompose the adjoint representation of $\\mathfrak{sl}_{r+1}=\\mathfrak {sl}_{r+1}(\\mathbb C)$ by a purely combinatorial approach based on the introduction of a certain subset of the Weyl group called the \\emph{Weyl alternation set} associated to a pair of dominant integral weights. The cardinality of the Weyl alternation set associated to the highest root and zero weight of $\\mathfrak {sl}_{r+1}$ is given by the $r^{th}$ Fibonacci number. We then obtain the exponents of $\\mathfrak {sl}_{r+1}$ from this point of view.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-06-07T18:25:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10"
    ],
    "keywords": [
      "fibonacci number",
      "adjoint representation",
      "dominant integral weights",
      "zero weight",
      "highest root"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.1408H"
    }
  }
}