{
  "id": "2412.16820",
  "version": "v1",
  "published": "2024-12-22T01:46:30.000Z",
  "updated": "2024-12-22T01:46:30.000Z",
  "title": "The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order",
  "authors": [
    "Portia X. Anderson",
    "Esther Banaian",
    "Melanie J. Ferreri",
    "Owen C. Goff",
    "Kimberly P. Hadaway",
    "Pamela E. Harris",
    "Kimberly J. Harry",
    "Nicholas Mayers",
    "Shiyun Wang",
    "Alexander N. Wilson"
  ],
  "comment": "24 pages, 3 figures, 2 tables",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "For integral weights $\\lambda$ and $\\mu$ of a classical simple Lie algebra $\\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $\\mu$ in the irreducible representation with highest weight $\\lambda$, which we denote by $m(\\lambda,\\mu)$. Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\\mathcal{A}(\\lambda,\\mu)$ is the set of Weyl group elements that contribute nontrivially to the multiplicity $m(\\lambda,\\mu)$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra $\\mathfrak{sl}_{r+1}(\\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\\mathcal{A}(\\tilde{\\alpha},\\mu)$, where $\\tilde{\\alpha}$ is the highest root and $\\mu$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work where we aim to prove Harry's conjecture that the $q$-analog of Kostant's weight multiplicity formula $m_q(\\tilde{\\alpha},\\mu)=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $\\mu=-(\\alpha_i+\\alpha_{i+1}+\\cdots+\\alpha_{j})$ is a negative root of $\\mathfrak{sl}_{r+1}(\\mathbb{C})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-22T01:46:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "17B10",
      "17B22",
      "06A07"
    ],
    "keywords": [
      "kostants weight multiplicity formula",
      "weak bruhat order",
      "order ideal",
      "weyl alternation set",
      "weyl group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}