{
  "id": "1409.5634",
  "version": "v1",
  "published": "2014-09-19T12:53:43.000Z",
  "updated": "2014-09-19T12:53:43.000Z",
  "title": "A new family of tight sets in $\\mathcal{Q}^{+}(5,q)$",
  "authors": [
    "Jan De Beule",
    "Jeroen Demeyer",
    "Klaus Metsch",
    "Morgan Rodgers"
  ],
  "comment": "27 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we describe a new infinite family of $\\frac{q^{2}-1}{2}$-tight sets in the hyperbolic quadrics $\\mathcal{Q}^{+}(5,q)$, for $q \\equiv 5 \\mbox{ or } 9 \\bmod{12}$. Under the Klein correspondence, these correspond to Cameron--Liebler line classes of ${\\rm PG}(3,q)$ having parameter $\\frac{q^{2}-1}{2}$. This is the second known infinite family of nontrivial Cameron--Liebler line classes, the first family having been described by Bruen and Drudge with parameter $\\frac{q^{2}+1}{2}$ in ${\\rm PG}(3,q)$ for all odd $q$. The study of Cameron--Liebler line classes is closely related to the study of symmetric tactical decompositions of ${\\rm PG}(3,q)$ (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when $q \\equiv 9 \\bmod 12$ (so $q = 3^{2e}$ for some positive integer $e$), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler in 1982; the nature of these decompositions allows us to also prove the existence of a set of type $\\left(\\frac{1}{2}(3^{2e}-3^{e}), \\frac{1}{2}(3^{2e}+3^{e}) \\right)$ in the affine plane ${\\rm AG}(2,3^{2e})$ for all positive integers $e$. This proves a conjecture made by Rodgers in his PhD thesis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-19T12:53:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51A50",
      "51E20"
    ],
    "keywords": [
      "tight sets",
      "infinite family",
      "nontrivial cameron-liebler line classes",
      "positive integer",
      "klein correspondence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.5634D"
    }
  }
}