{
  "id": "2312.09986",
  "version": "v1",
  "published": "2023-12-15T17:58:26.000Z",
  "updated": "2023-12-15T17:58:26.000Z",
  "title": "Computing the $q$-Multiplicity of the Positive Roots of $\\mathfrak{sl}_{r+1}(\\mathbb{C})$ and Products of Fibonacci Numbers",
  "authors": [
    "Kimberly J. Harry"
  ],
  "comment": "16 pages, 0 figures",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root $\\mu$ in the adjoint representation of $\\mathfrak{sl}_{r+1}(\\mathbb{C})$, which we denote $L(\\tilde{\\alpha})$, where $\\tilde{\\alpha}$ is the highest root of $\\mathfrak{sl}_{r+1}(\\mathbb{C})$. We prove that the number of terms contributing a nonzero value in the multiplicity of the positive root $\\mu=\\alpha_i+\\alpha_{i+1}+\\cdots+\\alpha_j$ with $1\\leq i\\leq j\\leq r$ in $L(\\tilde{\\alpha})$ is given by the product $F_{i}\\cdot F_{r-j+1}$, where $F_n$ is the $n^{\\text{th}}$ Fibonacci number. Using this result, we show that the $q$-multiplicity of the positive root $\\mu=\\alpha_i+\\alpha_{i+1}+\\cdots+\\alpha_j$ with $1\\leq i\\leq j\\leq r$ in the representation $L(\\tilde{\\alpha})$ is precisely $q^{r-h(\\mu)}$, where $h(\\mu)=j-i+1$ is the height of the positive root $\\mu$. Setting $q=1$ recovers the known result that the multiplicity of a positive root in the adjoint representation of $\\mathfrak{sl}_{r+1}(\\mathbb{C})$ is one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-15T17:58:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10"
    ],
    "keywords": [
      "positive root",
      "fibonacci number",
      "nonzero value",
      "kostants weight multiplicity formula",
      "adjoint representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}