{
  "id": "2107.10205",
  "version": "v1",
  "published": "2021-07-21T16:54:36.000Z",
  "updated": "2021-07-21T16:54:36.000Z",
  "title": "Quantum immanants, double Young-Capelli bitableaux and Schur shifted symmetric functions",
  "authors": [
    "Andrea Brini",
    "Antonio Teolis"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1608.06780",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "In this paper are introduced two classes of elements in the enveloping algebra $\\mathbf{U}(gl(n))$: the \\emph{double Young-Capelli bitableaux} $[\\ \\fbox{$S \\ | \\ T$}\\ ]$ and the \\emph{central} \\emph{Schur elements} $\\mathbf{S}_{\\lambda}(n)$, that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element $\\mathbf{S}_{\\lambda}(n)$ is the sum of all double Young-Capelli bitableaux $[\\ \\fbox{$S \\ | \\ S$}\\ ]$, $S$ row (strictly) increasing Young tableaux of shape $\\widetilde{\\lambda}$. The Schur elements $\\mathbf{S}_\\lambda(n)$ are proved to be the preimages - with respect to the Harish-Chandra isomorphism - of the \\emph{shifted Schur polynomials} $s_{\\lambda|n}^* \\in \\Lambda^*(n)$. Hence, the Schur elements are the same as the Okounkov \\textit{quantum immanants}, recently described by the present authors as linear combinations of \\emph{Capelli immanants}. This new presentation of Schur elements/quantum immanants doesn't involve the irreducible characters of symmetric groups. The Capelli elements $\\mathbf{H}_k(n)$ are column Schur elements and the Nazarov-Umeda elements $\\mathbf{I}_k(n)$ are row Schur elements. The duality in $\\boldsymbol{\\zeta}(n)$ follows from a combinatorial description of the eigenvalues of the $\\mathbf{H}_k(n)$ on irreducible modules that is {\\it{dual}} (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the $\\mathbf{I}_k(n)$. The passage $n \\rightarrow \\infty$ for the algebras $\\boldsymbol{\\zeta}(n)$ is obtained both as direct and inverse limit in the category of filtered algebras, via the \\emph{Olshanski decomposition/projection}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-21T16:54:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "05E05",
      "05E15",
      "17B35"
    ],
    "keywords": [
      "schur shifted symmetric functions",
      "double young-capelli bitableaux",
      "combinatorial description",
      "schur elements/quantum immanants",
      "highest weight vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}