{
  "id": "2304.08388",
  "version": "v1",
  "published": "2023-04-17T15:55:03.000Z",
  "updated": "2023-04-17T15:55:03.000Z",
  "title": "Complete reducibility in bad characteristic",
  "authors": [
    "Alastair J. Litterick",
    "Adam R. Thomas"
  ],
  "comment": "41 pages, comments welcome",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p > 0$. This paper continues a long-standing effort to classify the connected reductive subgroups of $G$. Having previously completed the classification when $p$ is sufficiently large, we focus here on the case that $p$ is bad for $G$. We classify the connected reductive subgroups of $G$ which are not $G$-completely reducible, whose simple components have rank at least $3$. For each such subgroup $X$, we determine the action of $X$ on the adjoint module $L(G)$ and on a minimal non-trivial $G$-module, and the connected centraliser of $X$ in $G$. As corollaries we obtain information on: subgroups which are maximal among connected reductive subgroups; products of commuting $G$-completely reducible subgroups; subgroups with trivial connected centraliser; and subgroups which act indecomposably on an adjoint or minimal module for $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-17T15:55:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G07",
      "20G41"
    ],
    "keywords": [
      "bad characteristic",
      "complete reducibility",
      "connected reductive subgroups",
      "simple algebraic group",
      "exceptional type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}