{
  "id": "1502.04867",
  "version": "v1",
  "published": "2015-02-17T11:39:05.000Z",
  "updated": "2015-02-17T11:39:05.000Z",
  "title": "Highest weight vectors and transmutation",
  "authors": [
    "Rudolf Tange"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G={\\rm GL}_n$ be the general linear group over an algebraically closed field $k$, let $\\mathfrak g=\\mathfrak gl_n$ be its Lie algebra and let $U$ be the subgroup of $G$ which consists of the upper uni-triangular matrices. Let $k[\\mathfrak g]$ be the algebra of polynomial functions on $\\mathfrak g$ and let $k[\\mathfrak g]^G$ be the algebra of invariants under the conjugation action of $G$. In characteristic zero, we give for all dominant weights $\\chi\\in\\mathbb Z^n$ finite homogeneous spanning sets for the $k[\\mathfrak g]^G$-modules $k[\\mathfrak g]_\\chi^U$ of highest weight vectors. This result (with some mistakes) was already given without proof by J.~F.~Donin. Then we do the same for tuples of $n\\times n$-matrices under the diagonal conjugation action. Furthermore we extend our earlier results in positive characteristic and give a general result which reduces the problem to giving spanning sets of the highest weight vectors for the action of ${\\rm GL}_r\\times{\\rm GL}_s$ on tuples of $r\\times s$ matrices. This requires the technique called \"transmutation\" by R.~Brylinsky which is based on an instance of Howe duality. In the cases that $\\chi_{{}_n}\\ge -1$ or $\\chi_{{}_1}\\le 1$ this leads to new spanning sets for the modules $k[\\mathfrak g]_\\chi^U$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-17T11:39:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "highest weight vectors",
      "transmutation",
      "spanning sets",
      "general linear group",
      "diagonal conjugation action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}