{
  "id": "2102.09077",
  "version": "v1",
  "published": "2021-02-18T00:02:25.000Z",
  "updated": "2021-02-18T00:02:25.000Z",
  "title": "On Héthelyi-Külshammer's conjecture for principal blocks",
  "authors": [
    "Nguyen Ngoc Hung",
    "A. A. Schaeffer Fry"
  ],
  "comment": "27 pages",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "We prove that the number of irreducible ordinary characters in the principal $p$-block of a finite group $G$ of order divisible by $p$ is always at least $2\\sqrt{p-1}$. This confirms a conjecture of H\\'{e}thelyi and K\\\"{u}lshammer for principal blocks and provides an affirmative answer to Brauer's Problem 21 for principal blocks of bounded defect. Our proof relies on recent works of Mar\\'{o}ti and Malle-Mar\\'{o}ti on bounding the conjugacy class number and the number of $p'$-degree irreducible characters of finite groups, earlier works of Brou\\'{e}-Malle-Michel and Cabanes-Enguehard on the distribution of characters into unipotent blocks and $e$-Harish-Chandra series of finite reductive groups, and known cases of the Alperin-McKay conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-18T00:02:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "principal blocks",
      "héthelyi-külshammers conjecture",
      "finite group",
      "conjugacy class number",
      "irreducible ordinary characters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}