Intersections of Schubert varieties and highest weight vectors in tensor products of sl_{N+1}-representations

Abstract

There is a correspondence between highest weight vectors in the tensor product of finite-dimensional irreducible sl(N+1)-modules marked by distinct complex numbers, on the one hand, and elements of the intersection of the Schubert varieties taken with respect to the osculating flags of the normal rational curve at the points corresponding to these complex numbers, on the other hand. The highest weight vectors are the Bethe vectors of the sl(N+1) Gaudin model and the elements are the (N+1)-dimensional non-degenerate planes in the vector space of complex polynomials. In the present paper we exploit this correspondence in order to calculate Bethe vectors is the tensor product of two irreducible finite-dimensional sl(N+1)-representations. We find the Bethe vector in the case when one of the two representations is a symmetric power of the standard one. The idea is to look for the intersection of Schubert varieties related to a Bethe vector. We present explicitly a basis of the corresponding (N+1)-dimensional plane in the space of polynomials.