{
  "id": "math/0603699",
  "version": "v2",
  "published": "2006-03-30T07:25:50.000Z",
  "updated": "2007-02-28T08:42:42.000Z",
  "title": "Invariant theory for singular $α$-determinants",
  "authors": [
    "Kazufumi Kimoto",
    "Masato Wakayama"
  ],
  "comment": "26 pages",
  "journal": "J. Combin. Theory Ser. A 115 (2008), no.1, 1--31",
  "doi": "10.1016/j.jcta.2007.03.008",
  "categories": [
    "math.RT"
  ],
  "abstract": "From the irreducible decompositions' point of view, the structure of the cyclic $GL_n$-module generated by the $\\alpha$-determinant degenerates when $\\alpha=\\pm \\frac1k (1\\leq k\\leq n-1)$. In this paper, we show that $-\\frac1k$-determinant shares similar properties which the ordinary determinant possesses. From this fact, one can define a new (relative) invariant called a wreath determinant. Using $(GL_m, GL_n)$-duality in the sense of Howe, we obtain an expression of a wreath determinant by a certain linear combination of the corresponding ordinary minor determinants labeled by suitable rectangular shape tableaux. Also we study a wreath determinant analogue of the Vandermonde determinant, and then, investigate symmetric functions such as Schur functions in the framework of wreath determinants. Moreover, we examine coefficients which we call $(n,k)$-sign appeared at the linear expression of the wreath determinant in relation with a zonal spherical function of a Young subgroup of the symmetric group $S_{nk}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-02-28T08:42:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "15A72"
    ],
    "keywords": [
      "invariant theory",
      "determinant shares similar properties",
      "corresponding ordinary minor determinants",
      "wreath determinant analogue"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3699K"
    }
  }
}