A congruence modulo four in real Schubert calculus with isotropic flags

TL;DR

This paper extends a known congruence modulo four for real solutions in Schubert calculus from osculating flags to a broader class of isotropic flags, revealing deeper structural properties in real algebraic geometry.

Contribution

It proves a congruence modulo four for the number of real solutions in Schubert problems with isotropic flags, broadening the class of problems where this congruence holds.

Findings

Congruence modulo four established for a large class of isotropic flag Schubert problems.

Extension of previous results from osculating to isotropic flags.

Deepens understanding of real solutions in algebraic geometry.

Abstract

We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here, we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.