{
  "id": "1007.3950",
  "version": "v4",
  "published": "2010-07-22T17:31:47.000Z",
  "updated": "2011-08-30T17:02:41.000Z",
  "title": "Degenerate two-boundary centralizer algebras",
  "authors": [
    "Zajj Daugherty"
  ],
  "comment": "45 pages, to appear in Pacific Journal of Mathematics",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie algebra action on a tensor space. This work explores centralizers of the action of a complex reductive Lie algebra $\\mathfrak{g}$ on tensor space of the form $M \\otimes N \\otimes V^{\\otimes k}$. We define the degenerate two-boundary braid algebra $\\mathcal{G}_k$ and show that centralizer algebras contain quotients of this algebra in a general setting. As an example, we study in detail the combinatorics of special cases corresponding to Lie algebras $\\mathfrak{gl}_n$ and $\\mathfrak{sl}_n$ and modules $M$ and $N$ indexed by rectangular partitions. For this setting, we define the degenerate extended two-boundary Hecke algebra $\\mathcal{H}_k^{\\mathrm{ext}}$ as a quotient of $\\mathcal{G}_k$, and show that a quotient of $\\mathcal{H}_k^{\\mathrm{ext}}$ is isomorphic to a large subalgebra of the centralizer. We further study the representation theory of $\\mathcal{H}_k^{\\mathrm{ext}}$ to find that the seminormal representations are indexed by a known family of partitions. The bases for the resulting modules are given by paths in a lattice of partitions, and the action of $\\mathcal{H}_k^{\\mathrm{ext}}$ is given by combinatorial formulas.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-08-30T17:02:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "05E10",
      "17B10"
    ],
    "keywords": [
      "degenerate two-boundary centralizer algebras",
      "tensor space",
      "degenerate two-boundary braid algebra",
      "tensor power centralizer algebras",
      "centralizer algebras contain quotients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.3950D"
    }
  }
}