{
  "id": "2209.15375",
  "version": "v1",
  "published": "2022-09-30T11:09:47.000Z",
  "updated": "2022-09-30T11:09:47.000Z",
  "title": "Nonrealizability of certain representations in fusion systems",
  "authors": [
    "Bob Oliver"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "For a finite abelian $p$-group $A$ and a subgroup $\\Gamma\\le\\text{Aut}(A)$, we say that the pair $(\\Gamma,A)$ is fusion realizable if there is a saturated fusion system $\\mathcal{F}$ over a finite $p$-group $S\\ge A$ such that $C_S(A)=A$, $\\textrm{Aut}_{\\mathcal{F}}(A)=\\Gamma$ as subgroups of $\\text{Aut}(A)$, and $A$ is not normal in $\\mathcal{F}$. In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $\\Gamma$ one of the Mathieu groups, that the only $\\mathbb{F}_p\\Gamma$-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-30T11:09:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D20",
      "20C20",
      "20D05",
      "20E45"
    ],
    "keywords": [
      "representations",
      "nonrealizability",
      "fusion realizable",
      "todd modules",
      "mathieu groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}