{
  "id": "2410.20392",
  "version": "v1",
  "published": "2024-10-27T09:39:12.000Z",
  "updated": "2024-10-27T09:39:12.000Z",
  "title": "The McKay Conjecture on character degrees",
  "authors": [
    "Marc Cabanes",
    "Britta Späth"
  ],
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "McKay's conjecture (1971) on character degrees was reduced by Isaacs-Malle-Navarro (2007) to a so-called inductive condition on characters of finite quasisimple groups [IMN07], thus opening the way to a proof of McKay's conjecture using the classification of finite simple groups. After [Ma07], [Ma08], [S12], [CS13], [KS16], [MS16], [CS17a], [CS17b], [CS19], [S23a], [S23b], we complete here the last step of a proof with an analysis of the representations of certain normalizers ${\\mathrm N}_G({\\mathbf S})$ in $G={\\mathbf G}^F$ of maximal $d$-tori ${\\mathbf S}$ ($d\\geq 3$) of the ambient simple simply-connected algebraic group ${\\mathbf G}$ of type ${\\mathrm D}_l$ ($l\\geq 4$) for which $F$ is a Frobenius endomorphism. To establish the so-called local conditions A$(d)$ and B$(d)$, we introduce a certain class of $F$-stable reductive subgroups ${\\mathbf M}\\leq {\\mathbf G}$ of maximal rank where ${\\mathbf M}^\\circ$ is of type ${\\mathrm D}_{l_1}\\times {\\mathrm D}_{l-l_1}$ with ${\\mathbf M}/{\\mathbf M}^\\circ$ of order 2. They are an efficient substitute for ${\\mathrm N}_G({\\mathbf S})$ or the local subgroups in non-defining characteristic relevant to McKay's abstract statement. For a general class of those subgroups ${\\mathbf M}^F$ we describe their characters and the action of $\\operatorname {Aut}({\\mathbf G}^F)_{{\\mathbf M}^F}$ on them, showing in particular that $\\mathrm{Irr}({\\mathbf M}^F)$ and $\\mathrm {Irr}({\\mathbf G}^F)$ share some key features in that regard. With this established, McKay's conjecture is now a theorem stating $\\textit{McKay's equality}$: For any prime $\\ell$, any finite group has as many irreducible complex characters of degree prime to $\\ell$ as the normalizers of its Sylow $\\ell$-subgroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-27T09:39:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C15",
      "20C33",
      "20G40"
    ],
    "keywords": [
      "character degrees",
      "mckay conjecture",
      "mckays conjecture",
      "ambient simple simply-connected algebraic group",
      "finite quasisimple groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}