{
  "id": "math/0604240",
  "version": "v1",
  "published": "2006-04-11T09:40:33.000Z",
  "updated": "2006-04-11T09:40:33.000Z",
  "title": "Tensor subalgebras and First Fundamental Theorems in invariant theory",
  "authors": [
    "Alexander Schrijver"
  ],
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $V=\\oC^n$ and let $T:=T(V)\\otimes T(V^*)$ be the mixed tensor algebra over $V$. We characterize those subsets $A$ of $T$ for which there is a subgroup $G$ of the unitary group $\\UU(n)$ such that $A=T^G$. They are precisely the nondegenerate contraction-closed graded $*$-subalgebras of $T$. While the proof makes use of the First Fundamental Theorem for $\\GL(n,\\oC)$ (in the sense of Weyl), the characterization has as direct consequences First Fundamental Theorems for several subgroups of $\\GL(n,\\oC)$. Moreover, a Galois connection between linear algebraic $*$-subgroups of $\\GL(n,\\oC)$ and nondegenerate contraction-closed $*$-subalgebras of $T$ is derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-04-11T09:40:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A72",
      "20Gxx"
    ],
    "keywords": [
      "tensor subalgebras",
      "invariant theory",
      "direct consequences first fundamental theorems",
      "nondegenerate",
      "galois connection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......4240S"
    }
  }
}