Spaces of quasi-exponentials and representations of gl_N

TL;DR

This paper links the action of the Bethe algebra on tensor products of gl_N-modules to the algebra of functions on intersections of Schubert cycles, revealing a deep geometric and algebraic structure.

Contribution

It establishes an isomorphism between the Bethe algebra's image and functions on Schubert cycle intersections, and constructs eigenvectors via Bethe ansatz.

Findings

Bethe algebra's image is isomorphic to algebra of functions on Schubert intersections

Bethe algebra module is isomorphic to the algebra's coregular representation

Eigenvectors correspond to points in Schubert cycle intersections

Abstract

We consider the action of the Bethe algebra B_K on (\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda, the weight subspace of weight $\lambda$ of the tensor product of k polynomial irreducible gl_N-modules with highest weights \lambda^{(1)},...,\lambda^{(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_{\lambda^{(s)}})_\lambda is 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_{\lambda^{(s)}})_\lambda is 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.