{
  "id": "2104.13632",
  "version": "v1",
  "published": "2021-04-28T08:33:36.000Z",
  "updated": "2021-04-28T08:33:36.000Z",
  "title": "Jucys-Murphy elements and Grothendieck groups for generalized rook monoids",
  "authors": [
    "Volodymyr Mazorchuk",
    "Shraddha Srivastava"
  ],
  "comment": "Comments welcome!",
  "categories": [
    "math.RT"
  ],
  "abstract": "We consider a tower of generalized rook monoid algebras over the field $\\mathbb{C}$ of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys-Murphy elements for generalized rook monoid algebras. Over an algebraically closed field $\\Bbbk$ of positive characteristic $p$, utilizing Jucys-Murphy elements of rook monoid algebras, for $0\\leq i\\leq p-1$, we define the corresponding $i$-restriction and $i$-induction functors along with two extra functors. On the direct sum $\\mathcal{G}_{\\mathbb{C}}$ of the Grothendieck groups of module categories over rook monoid algebras over $\\Bbbk$, these functors induce an action of the tensor product of the universal enveloping algebra $U(\\widehat{\\mathfrak{sl}}_p(\\mathbb{C}))$ and the monoid algebra $\\mathbb{C}[\\mathcal{B}]$ of the bicyclic monoid $\\mathcal{B}$. Furthermore, we prove that $\\mathcal{G}_{\\mathbb{C}}$ is isomorphic to the tensor product of the Fock space representation of $U(\\widehat{\\mathfrak{sl}}_{p}(\\mathbb{C}))$ and the unique infinite-dimensional simple module over $\\mathbb{C}[\\mathcal{B}]$, and also exhibit that $\\mathcal{G}_{\\mathbb{C}}$ is a bialgebra. Under some natural restrictions on the characteristic of $\\Bbbk$, we outline the corresponding result for generalized rook monoids.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-28T08:33:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20M30",
      "16G99"
    ],
    "keywords": [
      "jucys-murphy elements",
      "grothendieck groups",
      "generalized rook monoid algebras",
      "unique infinite-dimensional simple module",
      "tensor product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}