{
  "id": "math/0407367",
  "version": "v3",
  "published": "2004-07-22T02:00:22.000Z",
  "updated": "2005-05-25T14:15:17.000Z",
  "title": "On Bethe vectors in the sl_{N+1} Gaudin model",
  "authors": [
    "S. Chmutov",
    "I. Scherbak"
  ],
  "comment": "the final version, to appear in the IMRN",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The note deals with the Gaudin model associated with the tensor product of n irreducible finite-dimensional sl_{N+1}-modules marked by distinct complex numbers z_1,..., z_n. The Bethe Ansatz is a method to construct common eigenvectors of the Gaudin hamiltonians by means of chosen singular vectors in the factors and z_j's. These vectors are called Bethe vectors. The question if the Bethe vectors are non-zero vectors is open. By the moment, the only way to verify that was based on a relation to critical points of the master function of the Gaudin model, and non-triviality of a Bethe vector was proved only in the case when the corresponding critical point is non-degenerate ([ScV], [MV1]). However degenerate critical points do appear in the Gaudin model (see Section12 of [ReV]). We believe that the Bethe vectors never vanish, and suggest an approach that does not depend on non-degeneracy of the corresponding critical point. The idea is for a Bethe vector to choose a suitable subspace in the weight space and to check that the projection of the Bethe vector to this subspace is non-zero. We apply this approach to verify non-triviality of Bethe vectors in new examples.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-05-25T14:15:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bethe vector",
      "gaudin model",
      "corresponding critical point",
      "distinct complex numbers",
      "construct common eigenvectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......7367C"
    }
  }
}