Some lower bounds in the B. and M. Shapiro conjecture for flag varieties

TL;DR

This paper investigates lower bounds on the number of real solutions to certain Schubert calculus problems on flag varieties, especially when the known monotonicity conditions are not met, extending understanding beyond previous conjectures.

Contribution

It provides new lower bounds for real solutions in specific cases of the B. and M. Shapiro conjecture for flag varieties without monotonicity conditions.

Findings

Computed explicit lower bounds for real solutions in special cases

Identified scenarios where solutions are fewer than expected

Extended the understanding of real solutions beyond monotonicity constraints

Abstract

The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov and Varchenko. For flag varieties, Sottile found a counterexample and suggested that all solutions should be real under certain monotonicity conditions. In this paper, we compute lower bounds on the number of real solutions for some special cases of the B. and M. Shapiro conjecture for flag varieties, when Sottile's monotonicity conditions are not satisfied.