{
  "id": "0912.1880",
  "version": "v1",
  "published": "2009-12-09T23:10:37.000Z",
  "updated": "2009-12-09T23:10:37.000Z",
  "title": "Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$",
  "authors": [
    "Stephen Lewis",
    "Nathaniel Thiem"
  ],
  "comment": "28 pages",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "The standard supercharacter theory of the finite unipotent upper-triangular matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of $U_m(q)\\subseteq U_n(q)$ for $m\\leq n$ lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of $U_n(q)$ is a nonnegative integer linear combination of supercharacters of $U_m(q)$ (in fact, it is polynomial in $q$). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of $U_n(q)$, this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-09T23:10:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "20C33"
    ],
    "keywords": [
      "tensor product",
      "nonzero coefficients",
      "restriction",
      "finite unipotent upper-triangular matrices",
      "nonnegative integer linear combination"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.1880L"
    }
  }
}