{
  "id": "1307.6523",
  "version": "v1",
  "published": "2013-07-24T18:34:48.000Z",
  "updated": "2013-07-24T18:34:48.000Z",
  "title": "Bijections for the Shi and Ish arrangements",
  "authors": [
    "Emily Leven",
    "Brendon Rhoades",
    "Andrew Timothy Wilson"
  ],
  "comment": "22 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The {\\sf Shi hyperplane arrangement} Shi(n) was introduced by Shi to study the Kazhdan-Lusztig cellular structure of the affine symmetric group. The {\\sf Ish hyperplane arrangement} Ish(n) was introduced by Armstrong in the study of diagonal harmonics. Armstrong and Rhoades discovered a deep combinatorial similarity between the Shi and Ish arrangements. We solve a collection of problems posed by Armstrong and Armstrong-Rhoades by giving bijections between regions of Shi(n) and Ish(n) which preserve certain statistics. Our bijections generalize to the `deleted arrangements' Shi(G) and Ish(G) which depend on a subgraph G of the complete graph K_n on n vertices. The key tools in our bijections are the introduction of an Ish analog of parking functions called {\\sf rook words} and a new instance of the cycle lemma of enumerative combinatorics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-24T18:34:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19"
    ],
    "keywords": [
      "ish arrangements",
      "bijections",
      "shi hyperplane arrangement",
      "affine symmetric group",
      "deep combinatorial similarity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.6523L"
    }
  }
}