{
  "id": "2401.17496",
  "version": "v2",
  "published": "2024-01-30T23:02:40.000Z",
  "updated": "2025-02-07T16:15:18.000Z",
  "title": "Tensor invariants for classical groups revisited",
  "authors": [
    "William Q. Erickson",
    "Markus Hunziker"
  ],
  "comment": "25 pages + appendix; version 2 modifies the structure and exposition of the paper by presenting all preliminary results in Sections 2 and 3",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "We reconsider an old problem, namely the dimension of the $G$-invariant subspace in $V^{\\otimes p} \\otimes V^{*\\otimes q}$, where $G$ is one of the classical groups ${\\rm GL}(V)$, ${\\rm SL}(V)$, ${\\rm O}(V)$, ${\\rm SO}(V)$, or ${\\rm Sp}(V)$. Spanning sets for the invariant subspace have long been well known, but linear bases are more delicate. The main contribution of this paper is a combinatorial realization of linear bases via standard Young tableaux and arc diagrams, in a uniform manner for all five classical groups. As a secondary contribution, we survey the many equivalent ways -- some old, some new -- to enumerate the elements in these bases.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-02-07T16:15:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "16W22",
      "05A19"
    ],
    "keywords": [
      "classical groups",
      "tensor invariants",
      "invariant subspace",
      "linear bases",
      "standard young tableaux"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}