Schubert calculus and representations of general linear group

TL;DR

This paper establishes a deep connection between the Bethe algebra in representation theory and Schubert calculus, showing their isomorphism and implications for spectrum simplicity and intersection transversality.

Contribution

It constructs a canonical isomorphism linking Bethe algebra actions on tensor product modules to Schubert variety intersections, revealing new structural insights.

Findings

Bethe algebra spectrum is simple for real evaluation parameters

The intersection of Schubert varieties is transversal

The multiplicity space is isomorphic to the coregular representation

Abstract

We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation gl_N[t]-modules and the scheme-theoretic intersection of suitable Schubert varieties. Moreover, we prove that the multiplicity space as a module over the Bethe algebra is isomorphic to the coregular representation of the scheme-theoretic intersection. In particular, this result implies the simplicity of the spectrum of the Bethe algebra for real values of evaluation parameters and the transversality of the intersection of the corresponding Schubert varieties.