{
  "id": "1511.00711",
  "version": "v1",
  "published": "2015-11-02T21:14:10.000Z",
  "updated": "2015-11-02T21:14:10.000Z",
  "title": "GL_n(F_q)-analogues of factorization problems in the symmetric group",
  "authors": [
    "Joel Brewster Lewis",
    "Alejandro H. Morales"
  ],
  "comment": "22 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider GL_n(F_q)-analogues of certain factorization problems in the symmetric group S_n: rather than counting factorizations of the long cycle (1, 2, ..., n) given the number of cycles of each factor, we count factorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in S_n, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis. Our work generalizes several recent results on factorizations in GL_n(F_q) and also uses a character-based approach. As an application of our results, we compute the asymptotic growth rate of the number of factorizations of fixed genus of a regular elliptic element in GL_n(F_q) into two factors as n goes to infinity. We end with a number of open questions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-02T21:14:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "factorization problems",
      "symmetric group",
      "regular elliptic element",
      "asymptotic growth rate",
      "fixed space dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151100711B"
    }
  }
}