{
  "id": "1005.1949",
  "version": "v1",
  "published": "2010-05-11T21:00:46.000Z",
  "updated": "2010-05-11T21:00:46.000Z",
  "title": "Hyperplane Arrangements and Diagonal Harmonics",
  "authors": [
    "Drew Armstrong"
  ],
  "comment": "27 pages, 12 figures",
  "categories": [
    "math.CO",
    "math.QA"
  ],
  "abstract": "In 2003, Haglund's {\\sf bounce} statistic gave the first combinatorial interpretation of the $q,t$-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type $A$. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement - which we call the Ish arrangement. We prove that our statistics are equivalent to the {\\sf area'} and {\\sf bounce} statistics of Haglund and Loehr. In this setting, we observe that {\\sf bounce} is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to \"extended\" Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to the elementary symmetric functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-05-11T21:00:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "52C35"
    ],
    "keywords": [
      "diagonal harmonics",
      "shi hyperplane arrangement",
      "first combinatorial interpretation",
      "elementary symmetric functions",
      "bergeron-garsia nabla operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.1949A"
    }
  }
}