{
  "id": "2006.01268",
  "version": "v1",
  "published": "2020-06-01T21:08:39.000Z",
  "updated": "2020-06-01T21:08:39.000Z",
  "title": "Cayley algebras give rise to $q$-Fano planes over certain infinite fields and $q$-covering designs over others",
  "authors": [
    "Vincent van der Noort"
  ],
  "comment": "75 pages, 7 tables, 1 figure",
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "abstract": "Let $F$ be a field. A $2$-$(7, 3, 1)_F$-subspace design, or $q$-Fano plane, over $F$, is a $7$-dimensional vector space $V$ over $F$ together with a collection $\\mathfrak{B}$ of three-dimensional subspaces of $V$ such that every two-dimensional subspace of $V$ is contained in exactly one element $B$ of $\\mathfrak{B}$. The question of existence of $q$-Fano planes over any field has been open since the 1970s and has attracted considerable attention in the special case that $F$ is finite. Here we show the existence of $2$-$(7, 3, 1)_F$-subspace designs over certain infinite fields $F$, including (among others) $\\mathbb{Q}, \\mathbb{R}$ and $\\mathbb{F}_q(x, y, z)$ for $q$ odd. The space $V$ is the 7-dimensional space of imaginary elements in a Cayley division algebra $O$ over $F$ and $\\mathfrak{B}$ consists of the intersections with $V$ of all 4-dimensional subalgebras of $O$. We will present the required background on Cayley algebras in a self-contained fashion. Next we study what happens if we apply the same procedure to split (rather than division) Cayley algebras. By identifying all four-dimensional subalgebras of these, we show that in that case our construction still yields an inclusion minimal $(7, 3, 2)$ $q$-covering design. That is: every two-dimensional subspace of $V$ is contained in at least one element of the resulting set $\\mathfrak{B}$ of three-dimensional subspaces of $V$ and no proper subset of $\\mathfrak{B}$ has this property. However none of these $q$-covering designs are $q$-Fano planes. In the case that $F$ is finite we compute the number of elements of $\\mathfrak{B}$. We also give a purely combinatorial construction of our $q$-Fano planes and $q$-covering designs for an abstract 7-dimensional $F$-vector space $V$ by identifying the collection $\\mathfrak{B}$ as a subvariety of the Grassmanian $Gr_3(V)$ defined entirely in terms of the classical Fano plane.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-01T21:08:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E10",
      "51D20",
      "05B25",
      "17D05"
    ],
    "keywords": [
      "fano plane",
      "covering design",
      "cayley algebras",
      "infinite fields",
      "two-dimensional subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 75,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}