{
  "id": "1708.02783",
  "version": "v1",
  "published": "2017-08-09T11:04:07.000Z",
  "updated": "2017-08-09T11:04:07.000Z",
  "title": "Torsion table for the Lie algebra $\\frak{nil}_n$",
  "authors": [
    "Leon Lampret",
    "Aleš Vavpetič"
  ],
  "comment": "10 pages, 1 table",
  "categories": [
    "math.AT"
  ],
  "abstract": "We study the Lie ring $\\mathfrak{nil}_n$ of all strictly upper-triangular $n\\!\\times\\!n$ matrices with entries in $\\mathbb{Z}$. Its complete homology for $n\\!\\leq\\!8$ is computed. We prove that every $p^m$-torsion appears in $H_\\ast(\\mathfrak{nil}_n;\\mathbb{Z})$ for $p^m\\!\\leq\\!n\\!-\\!2$. For $m\\!=\\!1$, Dwyer proved that the bound is sharp, i.e. there is no $p$-torsion in $H_\\ast(\\mathfrak{nil}_n;\\mathbb{Z})$ when prime $p\\!>\\!n\\!-\\!2$. In general, for $m\\!>\\!1$ the bound is not sharp, as we show that there is $8$-torsion in $H_\\ast(\\mathfrak{nil}_8;\\mathbb{Z})$. As a sideproduct, we derive the known result, that the ranks of the free part of $H_\\ast(\\mathfrak{nil}_n;\\mathbb{Z})$ are the Mahonian numbers (=number of permutations of $[n]$ with $k$ inversions), using a different approach than in [KostantLAHGBWT}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-09T11:04:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55U15",
      "55-04",
      "18G35",
      "17B56",
      "13P20",
      "13D02"
    ],
    "keywords": [
      "lie algebra",
      "torsion table",
      "mahonian numbers",
      "free part",
      "torsion appears"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}