Vanishing of Schubert Coefficients

TL;DR

This paper investigates the computational complexity of determining when Schubert coefficients vanish, showing that this problem is in the complexity class coAM assuming GRH, and extends results to all classical types.

Contribution

It proves that the Schubert coefficient vanishing problem is in coAM under GRH and develops polynomial system formulations for this problem.

Findings

The vanishing problem is in coAM assuming GRH.

The approach uses polynomial systems and Nullstellensatz reductions.

Results are extended to all classical types, including type D.

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).