{
  "id": "math/0211377",
  "version": "v2",
  "published": "2002-11-24T20:03:22.000Z",
  "updated": "2003-08-30T08:35:22.000Z",
  "title": "A theorem of Heine-Stieltjes, the Wronski map, and Bethe vectors in the sl_p Gaudin model",
  "authors": [
    "I. Scherbak"
  ],
  "comment": "Revised version; 24 pages",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "Heine and Stieltjes in their studies of linear second-order differential equations with polynomial coefficients having a polynomial solution of a preassigned degree, discovered that the roots of such a solution are the coordinates of a critical point of a certain remarkable symmetric function, [He], [St]. Their result can be reformulated in terms of the Schubert calculus as follows: the critical points label the elements of the intersection of certain Schubert varieties in the Grassmannian of two-dimensional subspaces of the space of complex polynomials, [S1]. In a hundred years after the works of Heine and Stieltjes, it was established that the same critical points determine the Bethe vectors in the sl_2 Gaudin model, [G]. Recently it was proved that the Bethe vectors of the sl_2 Gaudin model form a basis of the subspace of singular vectors of a given weight in the tensor product of irreducible sl_2-representations, [SV]. In the present work we generalize the result of Heine and Stieltjes to linear differential equations of order p>2. The function, which determines elements in the intersection of corresponding Schubert varieties in the Grassmannian of p-dimensional subspaces, turns out to be the very function which appears in the sl_p Gaudin model. In the case when the space of states of the Gaudin model is the tensor product of symmetric powers of the standard sl_p-representation, we prove that the Bethe vectors form a basis of the subspace of singular vectors of a given weight.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-08-30T08:35:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaudin model",
      "bethe vectors",
      "wronski map",
      "critical point",
      "linear second-order differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....11377S"
    }
  }
}