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

Abstract

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