{
  "id": "1507.01329",
  "version": "v1",
  "published": "2015-07-06T05:34:31.000Z",
  "updated": "2015-07-06T05:34:31.000Z",
  "title": "Invariants of the orthosymplectic Lie superalgebra and super Pfaffians",
  "authors": [
    "G. I. Lehrer",
    "R. B. Zhang"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Given a complex orthosymplectic superspace $V$, the orthosymplectic Lie superalgebra $\\mathfrak {osp}(V)$ and general linear algebra ${\\mathfrak {gl}}_N$ both act naturally on the coordinate super-ring $\\mathcal{S}(N)$ of the dual space of $V\\otimes{\\mathbb C}^N$, and their actions commute. Hence the subalgebra $\\mathcal{S}(N)^{\\mathfrak {osp}(V)}$ of $\\mathfrak {osp}(V)$-invariants in $\\mathcal{S}(N)$ has a ${\\mathfrak {gl}}_N$-module structure. We introduce the space of super Pfaffians as a simple ${\\mathfrak {gl}}_N$-submodule of $\\mathcal{S}(N)^{\\mathfrak {osp}(V)}$, give an explicit formula for its highest weight vector, and show that the super Pfaffians and the elementary (or `Brauer') ${\\rm OSp}$-invariants together generate $\\mathcal{S}(N)^{\\mathfrak {osp}(V)}$ as an algebra. The decomposition of $\\mathcal{S}(N)^{\\mathfrak {osp}(V)}$ as a direct sum of simple ${\\mathfrak {gl}}_N$-submodules is obtained and shown to be multiplicity free. Using Howe's $({\\mathfrak {gl}}(V), {\\mathfrak {gl}}_N)$-duality on $\\mathcal{S}(N)$, we deduce from the decomposition that the subspace of $\\mathfrak{osp}(V)$-invariants in any simple ${\\mathfrak {gl}}(V)$-tensor module is either $0$ or $1$-dimensional. These results also enable us to determine the $\\mathfrak {osp}(V)$-invariants in the tensor powers $V^{\\otimes r}$ for all $r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-06T05:34:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16W22",
      "15A72",
      "17B20"
    ],
    "keywords": [
      "orthosymplectic lie superalgebra",
      "super pfaffians",
      "invariants",
      "complex orthosymplectic superspace",
      "general linear algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150701329L"
    }
  }
}