Spaces of quasi-exponentials and representations of gl_N

Abstract

We consider the action of the Bethe algebra B_K on (\otimes_{s=1}^k L_{λ^{(s)}})_λ, the weight subspace of weight $λ$ of the tensor product of k polynomial irreducible gl_N-modules with highest weights λ^{(1)},...,λ^{(k)}, respectively. The Bethe algebra depends on N complex numbers K=(K_1,...,K_N). Under the assumption that K_1,...,K_N are distinct, we prove that the image of B_K in the endomorphisms of (\otimes_{s=1}^k L_{λ^{(s)}})_λis isomorphic to the algebra of functions on the intersection of k suitable Schubert cycles in the Grassmannian of N-dimensional spaces of quasi-exponentials with exponents K. We also prove that the B_K-module (\otimes_{s=1}^k L_{λ^{(s)}})_λis isomorphic to the coregular representation of that algebra of functions. We present a Bethe ansatz construction identifying the eigenvectors of the Bethe algebra with points of that intersection of Schubert cycles.

Latex, 29 pages