{
  "id": "1910.11740",
  "version": "v1",
  "published": "2019-10-25T14:09:20.000Z",
  "updated": "2019-10-25T14:09:20.000Z",
  "title": "The $0$-Rook Monoid and its Representation Theory",
  "authors": [
    "Gay Joël",
    "Hivert Florent"
  ],
  "comment": "77 pages, 41 figures",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "We show that a proper degeneracy at $q=0$ of the $q$-deformed rook monoid of Solomon is the algebra of a monoid $R_n^0$ namely the $0$-rook monoid, in the same vein as Norton's $0$-Hecke algebra being the algebra of a monoid $H_n^0 = H^0(A_{n-1})$ (in Cartan type~$A_{n-1}$). As expected, $R_n^0$ is closely related to the latter: it contains the $H^0(A_{n-1})$ monoid and is a quotient of $H^0(B_{n})$. We give a presentation for this monoid as well as a combinatorial realization as functions acting on the classical rook monoid itself. On the way we get a Matsumoto theorem for the rook monoid a result which was conjectured by Solomon. The $0$-rook monoid shares many combinatorial properties with the Hecke monoid: its Green right preorder is an actual order, and moreover a lattice (analogous to the right weak order) which has some nice combinatorial, and geometrical features. In particular the $0$-rook monoid is J-trivial. Following Denton-Hivert-Schilling-Thi\\'ery, it allows us to describe its representation theory including the description of the simple and projective modules. We further show that $R_n^0$ is projective on $H_n^0$ and make explicit the restriction and induction functors along the inclusion map. We finally give a (partial) associative tower structures on the family of $(R_n^0)$ and we discuss its representation theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-25T14:09:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "05E15",
      "20C08",
      "05B15",
      "05A05"
    ],
    "keywords": [
      "representation theory",
      "rook monoid shares",
      "right weak order",
      "green right preorder",
      "proper degeneracy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 77,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}