Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus

TL;DR

This paper establishes lower bounds on the number of real self-dual polynomial spaces in Schubert calculus problems, linking algebraic geometry with the spectral properties of Gaudin Hamiltonians.

Contribution

It introduces a method to estimate the minimum number of real solutions in Schubert problems using the signature of a Hermitian form associated with Gaudin models.

Findings

Lower bounds are derived for real self-dual spaces in Schubert intersections.

The bounds are obtained via the signature of a nondegenerate indefinite Hermitian form.

Results connect algebraic geometry, representation theory, and mathematical physics.

Abstract

The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the numbers of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of this form.