{
  "id": "math/0407148",
  "version": "v1",
  "published": "2004-07-08T20:19:03.000Z",
  "updated": "2004-07-08T20:19:03.000Z",
  "title": "The Affinity of a Permutation of a Finite Vector Space",
  "authors": [
    "W. Edwin Clark",
    "Xiang-dong Hou",
    "Alec Mihailovs"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a permutation f of an n-dimensional vector space V over a finite field of order q we let k-affinity(f) denote the number of k-flats X of V such that f(X) is also a k-flat. By k-spectrum(n,q) we mean the set of integers k-affinity(f) where f runs through all permutations of V. The problem of the complete determination of k-spectrum(n,q) seems very difficult except for small or special values of the parameters. However, we are able to establish that k-spectrum(n,q) contains 0 in the following cases: (i) q>2 and 0<k<n; (ii) q=2, 2<k<n; (iii) q=2, k=2, odd n>2. The maximum of k-affinity(f) is, of course, obtained when f is any semi-affine mapping. We conjecture that the next to largest value of k-affinity(f) is when f is a transposition and we are able to prove this when q=2, k=2, n>2 and when q>2, k=1, n>1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-07-08T20:19:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A20",
      "05D40",
      "05E20",
      "52C45"
    ],
    "keywords": [
      "finite vector space",
      "permutation",
      "n-dimensional vector space",
      "k-spectrum",
      "integers k-affinity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......7148C"
    }
  }
}