2002/11/24 by I. Scherbak, Inna Scherbak, Scherbak, I.
Mathematics · #14N #16G #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:14N #msc:16G
paper · pdf · doi:10.48550/arxiv.math/0211377
Revised version; 24 pages
openalex publication_date 2002/11/24 · arxiv created 2003/08/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 sl2 Gaudin model, [G]. Recently it was proved that the Bethe vectors of the sl2 Gaudin model form a basis of the subspace of singular vectors of a given weight in the tensor product of irreducible sl2-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 slp Gaudin model. In the case when the space of states of the Gaudin model is the tensor product of symmetric powers of the standard slp-representation, we prove that the Bethe vectors form a basis of the subspace of singular vectors of a given weight.