{
  "id": "1111.6648",
  "version": "v2",
  "published": "2011-11-28T23:11:49.000Z",
  "updated": "2019-07-30T04:22:28.000Z",
  "title": "Kostant's Weight Multiplicity Formula and the Fibonacci and Lucas Numbers",
  "authors": [
    "Kevin Chang",
    "Pamela Harris",
    "Erik Insko"
  ],
  "comment": "11 pages",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Consider the weight $\\lambda$ which is the sum of all simple roots of a simple Lie algebra. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity of the zero weight in the representation with highest weight $\\lambda$. We prove that in Lie algebras of type $A$ and $B$, the number of contributing terms to the multiplicity of the zero-weight space in the representation with highest weight $\\lambda$ is given by a Fibonacci number, and that in Lie algebras of type $C$ and $D$, the analogous result is given by a multiple of a Lucas number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-28T23:11:49.000Z",
      "title": "Kostant's weight multiplicity formula and the Fibonacci numbers",
      "abstract": "It is well known that the dimension of a weight space for a finite dimensional representation of a simple Lie algebra is given by Kostant's weight multiplicity formula. We address the question of how many terms in the alternation contribute to the multiplicity of the zero weight for certain, very special, highest weights. Specifically, we consider the case where the highest weight is equal to the sum of all simple roots. This weight is dominant only in Lie types $A$ and $B$. We prove that in all such cases the number of contributing terms is a Fibonacci number. Combinatorial consequences of this fact are provided.",
      "comment": "9 pages",
      "journal": null,
      "doi": null,
      "authors": [
        "Pamela Harris"
      ]
    },
    {
      "version": "v2",
      "updated": "2019-07-30T04:22:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10"
    ],
    "keywords": [
      "kostants weight multiplicity formula",
      "fibonacci number",
      "highest weight",
      "simple lie algebra",
      "finite dimensional representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.6648H"
    }
  }
}