Technology Category
Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesSearch and Optimization: Combinatorial OptimizationKnowledge Representation and Reasoning: Computational Complexity of Reasoning
Positivity of Schubert Coefficients
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Schubert calculus
positivity problem
coefficient
Innovation
Methods, ideas, or system contributions that make the work stand out.
Schubert coefficients
positivity test
efficient method
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