{
  "id": "1912.10737",
  "version": "v1",
  "published": "2019-12-23T11:15:56.000Z",
  "updated": "2019-12-23T11:15:56.000Z",
  "title": "Jucys-Murphy elements of partition algebra for the rook monoid",
  "authors": [
    "Ashish Mishra",
    "Shraddha Srivastava"
  ],
  "comment": "38 pages. Comments are welcome!",
  "categories": [
    "math.RT"
  ],
  "abstract": "Kudryavtseva and Mazorchuk exhibited Schur-Weyl duality between the rook monoid algebra $\\mathbb{C}R_n$ and a subalgebra, denoted by $\\mathbb{C}I_k$, of the partition algebra $\\mathbb{C}A_k(n)$ acting on $(\\mathbb{C}^n)^{\\otimes k}$. In this paper, we consider a subalgebra $\\mathbb{C}I_{k+\\frac{1}{2}}$ of $\\mathbb{C}I_{k+1}$ such that there is Schur-Weyl duality between the actions of $\\mathbb{C}R_{n-1}$ and $\\mathbb{C}I_{k+\\frac{1}{2}}$ on $(\\mathbb{C}^n)^{\\otimes k}$. We call $\\mathbb{C}I_k$ and $\\mathbb{C}I_{k+\\frac{1}{2}}$ totally propagating partition algebras. This paper studies the representation theory of $\\mathbb{C}I_k$ and $\\mathbb{C}I_{k+\\frac{1}{2}}$ inductively by considering the tower ($\\mathbb{C}I_1\\subset \\mathbb{C}I_{\\frac{3}{2}}\\subset \\mathbb{C}I_{2}\\subset \\cdots$) whose Bratteli diagram turns out to be a simple graph. Furthermore, this inductive approach is established as a spectral approach by describing the Jucys-Murphy elements and their actions on the canonical Gelfand-Tsetlin bases of irreducible representations of $\\mathbb{C}I_k$ and $\\mathbb{C}I_{k+\\frac{1}{2}}$. Also, we describe the Jucys-Murphy elements of $\\mathbb{C}R_n$ which play a central role in the demonstration of the actions of Jucys-Murphy elements of $\\mathbb{C}I_k$ and $\\mathbb{C}I_{k+\\frac{1}{2}}$. In addition, we compute the Kronecker product of $\\mathbb{C}^n$ and an irreducible representation of $\\mathbb{C}R_n$ which is further used to specify the decomposition of $(\\mathbb{C}^n)^{\\otimes k}$ inductively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-23T11:15:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "20M30",
      "20C15"
    ],
    "keywords": [
      "jucys-murphy elements",
      "schur-weyl duality",
      "bratteli diagram turns",
      "rook monoid algebra",
      "irreducible representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}