{
  "id": "2203.15146",
  "version": "v1",
  "published": "2022-03-28T23:42:57.000Z",
  "updated": "2022-03-28T23:42:57.000Z",
  "title": "A proof of the $\\frac{n!}{2}$ conjecture for hook shapes",
  "authors": [
    "Sam Armon"
  ],
  "comment": "13 pages, 0 figures",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "A well-known representation-theoretic model for the transformed Macdonald polynomial $\\widetilde{H}_\\mu(Z;t,q)$, where $\\mu$ is an integer partition, is given by the Garsia-Haiman module $\\mathcal{H}_\\mu$. We study the $\\frac{n!}{k}$ conjecture of Bergeron and Garsia, which concerns the behavior of certain $k$-tuples of Garsia-Haiman modules under intersection. In the special case that $\\mu$ has hook shape, we use a basis for $\\mathcal{H}_\\mu$ due to Adin, Remmel, and Roichman to resolve the $\\frac{n!}{2}$ conjecture by constructing an explicit basis for the intersection of two Garsia-Haiman modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-28T23:42:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10",
      "05A19"
    ],
    "keywords": [
      "hook shape",
      "conjecture",
      "garsia-haiman module",
      "well-known representation-theoretic model",
      "explicit basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}