{
  "id": "2406.20025",
  "version": "v1",
  "published": "2024-06-28T16:14:39.000Z",
  "updated": "2024-06-28T16:14:39.000Z",
  "title": "Monogamous subvarieties of the nilpotent cone",
  "authors": [
    "Simon M. Goodwin",
    "Rachel Pengelly",
    "David I. Stewart",
    "Adam R. Thomas"
  ],
  "comment": "15 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of prime characteristic not $2$, whose Lie algebra is denoted $\\mathfrak{g}$. We call a subvariety $\\mathfrak{X}$ of the nilpotent cone $N \\subset \\mathfrak{g}$ monogamous if for every $e\\in \\mathfrak{X}$, the $\\mathfrak{sl}_2$-triples $(e,h,f)$ with $f\\in \\mathfrak{X}$ are conjugate under the centraliser $C_G(e)$. Building on work by the first two authors, we show there is a unique maximal closed $G$-stable monogamous subvariety $V \\subset N$ and that it is an orbit closure, hence irreducible. We show that $V$ can also be characterised in terms of Serre's $G$-complete reducibility.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-28T16:14:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B45",
      "17B50"
    ],
    "keywords": [
      "nilpotent cone",
      "monogamous subvariety",
      "complete reducibility",
      "reductive algebraic group",
      "unique maximal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}