{
  "id": "math/0702448",
  "version": "v2",
  "published": "2007-02-15T11:56:11.000Z",
  "updated": "2008-06-05T17:10:03.000Z",
  "title": "Similar sublattices of the root lattice $A_4$",
  "authors": [
    "Michael Baake",
    "Manuela Heuer",
    "Robert V. Moody"
  ],
  "comment": "17 pages, 1 figure; revised version with several additions and improvements",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "abstract": "Similar sublattices of the root lattice $A_4$ are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form $m^2 + mn - n^2$. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra $\\mathbb{H} (\\mathbb{Q} (\\sqrt{5}))$ and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-06-05T17:10:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C07",
      "11R52",
      "05A15"
    ],
    "keywords": [
      "root lattice",
      "similar sublattices",
      "corresponding dirichlet series generating function",
      "quaternion algebra",
      "zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......2448B"
    }
  }
}