Lower bounds for numbers of real solutions in problems of Schubert calculus

TL;DR

This paper establishes lower bounds on the number of real solutions in Schubert calculus problems on Grassmannians, linking these bounds to the signature of a Hermitian form associated with the Gaudin model.

Contribution

It introduces a method to compute lower bounds for real solutions in Schubert calculus using the signature of a Hermitian form related to the Gaudin model.

Findings

Lower bounds are derived from the signature of the Hermitian form.

The bounds relate to solutions in Schubert calculus involving osculating flags.

Connections between real solutions and Bethe vectors are elucidated.

Abstract

We give lower bounds for the numbers of real solutions in problems appearing in Schubert calculus in the Grassmannian Gr(n,d) related to osculating flags. It is known that such solutions are related to Bethe vectors in the Gaudin model associated to gl(n). The Gaudin Hamiltonians are selfadjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of that form.