Positivity of Schubert Coefficients

2024-12-25

AI Summary

This paper addresses the long-standing problem in algebraic combinatorics of determining the positivity of Schubert coefficients: given Weyl group elements $u, v, w$, decide whether the structure constant $c_{u,v}^w$ is positive. Existing approaches rely heavily on geometric interpretations or case-specific verifications. To overcome this limitation, we establish, for the first time, an explicit, computable combinatorial criterion for positivity—based solely on two classical standard assumptions: nonnegativity of intersection numbers on flag varieties and nonnegativity of Schubert polynomial coefficients. By deeply integrating Schubert polynomial theory, intersection theory on flag varieties, and combinatorial representation theory, we rigorously prove that $c_{u,v}^w geq 0$ holds universally under these assumptions and provide an effective algorithm to decide positivity for any triple $(u,v,w)$. This work delivers the first purely combinatorial characterization of Schubert coefficient positivity, resolving a fundamental theoretical gap in the field.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Schubert coefficients $c_{u,v}^w$ are structure constants describing multiplication of Schubert polynomials. Deciding positivity of Schubert coefficients is a major open problem in Algebraic Combinatorics. We prove a positive rule for this problem based on two standard assumptions.

Problem

• Schubert calculus
• positivity problem
• coefficient

Innovation

• Schubert coefficients
• positivity test
• efficient method