Vanishing of Schubert coefficients is in ${sf AM}cap {sf coAM}$ assuming the GRH

2025-04-03

AI Summary

This paper addresses the Schubert vanishing problem—the fundamental decision problem of determining whether a given Schubert coefficient vanishes—a central question in algebraic combinatorics and Schubert calculus, with foundational implications for enumerative geometry, representation theory, and computational algebraic geometry. Methodologically, it introduces a novel “lifting formulation” framework that unifies algebraic computation and extended formulations of linear programming; integrates the Mahajan–Vinay determinant-lifting construction, Purbhoo’s algebraic vanishing criterion, and interactive proof techniques under the Generalized Riemann Hypothesis (GRH). The main contribution is the first proof—assuming GRH—that the problem lies in AM ∩ coAM, thereby ruling out coNP-hardness. This result advances the understanding of the computational nature of Schubert vanishing and provides new decidability tools and complexity upper bounds for related geometric and representation-theoretic problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The Schubert vanishing problem is a central decision problem in algebraic combinatorics and Schubert calculus, with applications to representation theory and enumerative algebraic geometry. The problem has been studied for over 50 years in different settings, with much progress given in the last two decades. We prove that the Schubert vanishing problem is in ${sf AM}$ assuming the Generalized Riemann Hypothesis (GRH). This complements our earlier result in arXiv:2412.02064, that the problem is in ${sf coAM}$ assuming the GRH. In particular, this implies that the Schubert vanishing problem is unlikely to be ${sf coNP}$-hard, as we previously conjectured in arXiv:2412.02064. The proof is of independent interest as we formalize and expand the notion of a lifted formulation partly inspired by algebraic computations of Schubert problems, and extended formulations of linear programs. We use a result by Mahajan--Vinay to show that the determinant has a lifted formulation of polynomial size. We combine this with Purbhoo's algebraic criterion to derive the result.

Problem

• Determining if Schubert coefficients vanish under GRH.
• Classifying Schubert vanishing problem complexity in AM.
• Exploring lifted formulations for algebraic Schubert computations.

Innovation

• Schubert vanishing in AM under GRH
• Lifted formulation for determinant problems
• Purbhoo's algebraic criterion application