{
  "id": "2310.14203",
  "version": "v1",
  "published": "2023-10-22T06:43:30.000Z",
  "updated": "2023-10-22T06:43:30.000Z",
  "title": "On stability and nonvanishing of homomorphism spaces between Weyl modules",
  "authors": [
    "Charalambos Evangelou",
    "Mihalis Maliakas",
    "Dimitra-Dionysia Stergiopoulou"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Consider the general linear group $G=GL_{n}(K)$ defined over an infinite field $K$ of positive characteristic $p$. We denote by $\\Delta(\\lambda)$ the Weyl module of $G$ which corresponds to a partition $\\lambda$. Let $\\lambda, \\mu $ be partitions of $r$ and let $\\gamma$ be partition with all parts divisible by $p$. In the first main result of this paper, we find sufficient conditions on $\\lambda, \\mu$ and $\\gamma$ so that $Hom_G(\\Delta(\\lambda),\\Delta(\\mu))$ $ \\simeq$ $ Hom_G(\\Delta(\\lambda +\\gamma),\\Delta(\\mu +\\gamma))$, thus providing an answer to a question of D. Hemmer. As corollaries we obtain stability and periodicity results for homomorphism spaces. In the second main result we find related sufficient conditions on $\\lambda, \\mu$ and $p$ so that $Hom_G(\\Delta(\\lambda),\\Delta(\\mu))$ is nonzero. An explicit map is provided that corresponds to the sum of all semistandard tableaux of shape $\\mu$ and weight $\\lambda$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-22T06:43:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05"
    ],
    "keywords": [
      "homomorphism spaces",
      "weyl module",
      "sufficient conditions",
      "general linear group",
      "first main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}