{
  "id": "1805.00384",
  "version": "v1",
  "published": "2018-05-01T15:08:26.000Z",
  "updated": "2018-05-01T15:08:26.000Z",
  "title": "On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n\\vert n)$",
  "authors": [
    "Thorsten Heidersdorf",
    "Rainer Weissauer"
  ],
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the quotient of $\\mathcal{T}_n = Rep(GL(n|n))$ by the tensor ideal of negligible morphisms. If we consider the full subcategory $\\mathcal{T}_n^+$ of $\\mathcal{T}_n$ of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $Rep(H_n)$ where $H_n$ is a pro-reductive algebraic group. We determine the connected derived subgroup $G_n \\subset H_n$ and the groups $G_{\\lambda} = (H_{\\lambda})_{der}^0$ corresponding to the tannakian subcategory in $Rep(H_n)$ generated by an irreducible representation $L(\\lambda)$. This gives structural information about the tensor category $Rep(GL(n|n))$, including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on $2$-torsion in $\\pi_0(H_n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-01T15:08:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B20",
      "17B55",
      "18D10",
      "20G05"
    ],
    "keywords": [
      "irreducible representation",
      "general linear supergroups",
      "classical tensor categories",
      "tensor category",
      "semisimple tannakian category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}