{
  "id": "2007.12503",
  "version": "v1",
  "published": "2020-07-24T12:42:29.000Z",
  "updated": "2020-07-24T12:42:29.000Z",
  "title": "Toward a Classification of the Supercharacter Theories of $C_p\\times C_p$",
  "authors": [
    "Shawn T. Burkett",
    "Mark L. Lewis"
  ],
  "comment": "19 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "In this paper, we study the superscharacter theories of elementary abelian $p$-groups of order $p^2$. We show that the supercharacter theories that arise from the direct product construction and the $\\ast$-product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian $p$-group of order $p^2$ that has a nonidentity superclass of size $1$ or a nonprincipal linear supercharacter must come from either a $\\ast$-product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when $p = 2, 3, 5$, and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-24T12:42:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C15",
      "20D15"
    ],
    "keywords": [
      "supercharacter theory",
      "classification",
      "general prime",
      "elementary abelian",
      "direct product construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}