{
  "id": "0712.2293",
  "version": "v1",
  "published": "2007-12-14T06:48:53.000Z",
  "updated": "2007-12-14T06:48:53.000Z",
  "title": "Representation theory of the $α$-determinant and zonal spherical functions",
  "authors": [
    "Kazufumi Kimoto"
  ],
  "comment": "9 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We prove that the multiplicity of each irreducible component in the $\\mathcal{U}(\\mathfrak{gl}_n)$-cyclic module generated by the $l$-th power $\\det^{(\\alpha)}(X)^l$ of the $\\alpha$-determinant is given by the rank of a matrix whose entries are given by a variation of the spherical Fourier transformation for $(\\mathfrak{S}_{nl},\\mathfrak{S}_l^n)$. Further, we calculate the matrix explicitly when $n=2$. This gives not only another proof of the result by Kimoto-Matsumoto-Wakayama (2007) but also a new aspect of the representation theory of the $\\alpha$-determinants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-12-14T06:48:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E47",
      "43A90"
    ],
    "keywords": [
      "zonal spherical functions",
      "representation theory",
      "determinant",
      "th power",
      "spherical fourier transformation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.2293K"
    }
  }
}