{
  "id": "1206.4576",
  "version": "v2",
  "published": "2012-06-20T18:41:18.000Z",
  "updated": "2012-07-25T17:07:51.000Z",
  "title": "Representations of the Rook-Brauer Algebra",
  "authors": [
    "Elise delMas",
    "Tom Halverson"
  ],
  "comment": "18 pages; Updated references; Changed section 2.2 to include the two parameter rook-Brauer algebra RB_k(x,y) defined by Mazorchuk in reference [Mz]; Added comments around equation (4.5); Accepted for publication in Communications in Algebra",
  "categories": [
    "math.RT"
  ],
  "abstract": "We study the representation theory of the rook-Brauer algebra RB_k(x), also called the partial Brauer algebra. This algebra has a basis of \"rook-Brauer\" diagrams, which are Brauer diagrams that allow for the possibility of missing edges. The Brauer, Temperley-Lieb, Motzkin, rook monoid, and symmetric group algebras are all subalgebras of the rook-Brauer algebra. We prove that RB_k(n) is the centralizer algebra of the complex orthogonal group O(n) acting on the k-fold tensor power of the sum of its 1-dimensional trivial module and its n-dimensional defining module, and thus the rook-Brauer algebra and the orthogonal group are in Schur-Weyl duality on this tensor space. In the case where the parameter x is chosen so that RB_k(x) is semisimple, we use its Bratteli diagram to explicitly construct a complete set of irreducible representations for the rook-Brauer algebra as the span of paths in this diagram. These are analogs of Young's seminormal representations of the symmetric group.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-07-25T17:07:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10"
    ],
    "keywords": [
      "rook-brauer algebra",
      "complex orthogonal group",
      "partial brauer algebra",
      "k-fold tensor power",
      "symmetric group algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.4576D"
    }
  }
}