Vanishing of Schubert Coefficients

2024-12-03

arXiv.org

AI Summary

This paper addresses the vanishing problem for Schubert coefficients $c^w_{u,v}$: given Weyl group elements $u,v,w$, decide whether $c^w_{u,v} = 0$. Long unresolved within the polynomial hierarchy (PH), this problem is shown here—unconditionally—to lie in $mathsf{coAM}$, and hence in $Sigma_2^p$, assuming the Generalized Riemann Hypothesis. The result is unified across all classical types (A, B, C, D), with the type-D case developed jointly with David Speyer. Technically, we lift configuration-based constructions to parametrized polynomial systems and apply the recently established parametrized Hilbert’s Nullstellensatz (arXiv:2408.13027) to achieve complexity-theoretic reduction. This yields the first nontrivial PH upper bound for the Schubert coefficient vanishing problem, bridging algebraic combinatorics, representation theory, and computational complexity theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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 ${Sigma_2^{{ ext{p}}}}$. 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).

Problem

• Determining if Schubert coefficients vanish
• Classifying the problem within the polynomial hierarchy
• Extending results to all classical types

Innovation

• Polynomial systems equations
• Parametric Hilbert's Nullstellensatz
• Classical types extension