{
  "id": "2403.11393",
  "version": "v1",
  "published": "2024-03-18T01:01:24.000Z",
  "updated": "2024-03-18T01:01:24.000Z",
  "title": "Branching algebras for the general linear Lie superalgebra",
  "authors": [
    "Soo Teck Lee",
    "Ruibin Zhang"
  ],
  "comment": "35 pages",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We develop an algebraic approach to the branching of representations of the general linear Lie superalgebra $\\mathfrak{gl}_{p|q}({\\mathbb C})$, by constructing certain super commutative algebras whose structure encodes the branching rules. Using this approach, we derive the branching rules for restricting any irreducible polynomial representation $V$ of $\\mathfrak{gl}_{p|q}({\\mathbb C})$ to a regular subalgebra isomorphic to $\\mathfrak{gl}_{r|s}({\\mathbb C})\\oplus \\mathfrak{gl}_{r'|s'}({\\mathbb C})$, $\\mathfrak{gl}_{r|s}({\\mathbb C})\\oplus\\mathfrak{gl}_1({\\mathbb C})^{r'+s'}$ or $\\mathfrak{gl}_{r|s}({\\mathbb C})$, with $r+r'=p$ and $s+s'=q$. In the case of $\\mathfrak{gl}_{r|s}({\\mathbb C})\\oplus\\mathfrak{gl}_1({\\mathbb C})^{r'+s'}$ with $s=0$ or $s=1$ but general $r$, we also construct a basis for the space of $\\mathfrak{gl}_{r|s}({\\mathbb C})$ highest weight vectors in $V$; when $r=s=0$, the branching rule leads to explicit expressions for the weight multiplicities of $V$ in terms of Kostka numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-18T01:01:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "15A75",
      "20G05",
      "22E46"
    ],
    "keywords": [
      "general linear lie superalgebra",
      "branching algebras",
      "branching rule",
      "regular subalgebra isomorphic",
      "highest weight vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}