{
  "id": "1708.05646",
  "version": "v1",
  "published": "2017-08-18T15:20:26.000Z",
  "updated": "2017-08-18T15:20:26.000Z",
  "title": "On the character degrees of a Sylow $p$-subgroup of a finite Chevalley group $G(p^f)$ over a bad prime",
  "authors": [
    "Tung Le",
    "Kay Magaard",
    "Alessandro Paolini"
  ],
  "comment": "14 pages",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $\\text{Irr}(U(q))$ of irreducible characters of $U(q)$ when $G(q)$ is of type $\\mathrm{G}_2$. This is uniform for primes $p \\ge 5$, while the bad primes $p=2$ and $p=3$ have to be considered separately. We then use this result and the contribution of several authors to show a general result, namely that if $G(q)$ is any finite Chevalley group with $p$ a bad prime, then there exists a character $\\chi \\in \\text{Irr}(U(q))$ such that $\\chi(1)=q^n/p$ for some $n \\in \\mathbb{Z}_{\\ge_0}$. In particular, for each $G(q)$ and every bad prime $p$, we construct a family of characters of such degree as inflation followed by an induction of linear characters of an abelian subquotient $V(q)$ of $U(q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-18T15:20:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C33"
    ],
    "keywords": [
      "finite chevalley group",
      "bad prime",
      "character degrees",
      "general result",
      "linear characters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}