Vanishing of Schubert Coefficients

Abstract: Schubert coefficients are nonnegative integers $c^w_{u,v}$ that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation, i.e, whether $c^w_{u,v} \in #{\sf P}$. We study the closely related vanishing problem of Schubert coefficients: ${c^w_{u,v}=^? 0}$. Until this work it was open whether this problem is in the polynomial hierarchy ${\sf PH}$. We prove that ${c^w_{u,v}=^? 0}$ in ${\sf coAM}$ assuming the GRH. In particular, the vanishing problem is in ${\Sigma_2^{{\text{p}}}}$. Our approach is based on constructions lifted formulations, which give polynomial systems of equations for the problem. The result follows from a reduction to Parametric Hilbert's Nullstellensatz, recently studied in arXiv:2408.13027. We extend our results to all classical types. Type $D$ is resolved in the appendix (joint with David Speyer).