Gaudin's model and the generating function of the Wronski map

Abstract

We consider the Gaudin model associated to a point z in C^n with pairwise distinct coordinates and to the subspace of singular vectors of a given weight in the tensor product of irreducible finite-dimensional sl_2-representations, [G]. The Bethe equations of this model provide the critical point system of a remarkable rational symmetric function. Any critical orbit determines a common eigenvector of the Gaudin hamiltonians called a Bethe vector. In [ReV], it was shown that for generic z the Bethe vectors span the space of singular vectors, i.e. that the number of critical orbits is bounded from below by the dimension of the space of singular vectors. The upper bound by the same number is one of the main results of [SV]. In the present paper we get this upper bound in another, ``less technical'', way. The crucial observation is that the symmetric function defining the Bethe equations can be interpreted as the generating function of the map sending a pair of complex polynomials into their Wronski determinant: the critical orbits determine the preimage of a given polynomial under this map. Within the framework of the Schubert calculus, the number of critical orbits can be estimated by the intersection number of special Schubert classes. Relations to the sl_2 representation theory [F] imply that this number is the dimension of the space of singular vectors. We prove also that the spectrum of the Gaudin hamiltonians is simple for generic z.