Vanishing of Schubert Coefficients

Schubert coefficients are nonnegative integers $c_{u,v}^w$ 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 $c_{u,v}^w\in \mathrm{\#}\mathsf{P}$. We study the closely related vanishing problem of Schubert coefficients: $\{c_{u,v}^w{=}^{?}0\}$. Until this work it was open whether this problem is in the polynomial hierarchy $\mathsf{P}\mathsf{H}$. We prove that $\{c_{u,v}^w{=}^{?}0\}$ in $\mathsf{c}\mathsf{o}\mathsf{A}\mathsf{M}$ assuming the GRH. In particular, the vanishing problem is in ${\mathrm{\Sigma }}_2^{\text{p}}$. 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).