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Positivity of Schubert Coefficients

Igor Pak, Colleen Robichaux

TL;DR

This paper addresses the positivity of Schubert coefficients $c^w_{u,v}$ and the quest for a combinatorial rule explaining their signs. It delivers a positive rule under two standard hypotheses, $GRH$ and the Miltersen–Vinodchandran Assumption ($MVA$), by reducing Schubert positivity to a lifted Hilbert's Nullstellensatz instance, $HNP$, and leveraging a derandomization framework. The construction extends to root systems $B$ and $C$ but not $D$, and yields an NP-certifiable witness rather than a traditional combinatorial object. The work highlights a novel bridge between algebraic combinatorics and complexity theory, while candidly addressing the uncertainty from relying on unproven hypotheses and outlining directions toward unconditional interpretations.

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.

Positivity of Schubert Coefficients

TL;DR

This paper addresses the positivity of Schubert coefficients and the quest for a combinatorial rule explaining their signs. It delivers a positive rule under two standard hypotheses, and the Miltersen–Vinodchandran Assumption (), by reducing Schubert positivity to a lifted Hilbert's Nullstellensatz instance, , and leveraging a derandomization framework. The construction extends to root systems and but not , and yields an NP-certifiable witness rather than a traditional combinatorial object. The work highlights a novel bridge between algebraic combinatorics and complexity theory, while candidly addressing the uncertainty from relying on unproven hypotheses and outlining directions toward unconditional interpretations.

Abstract

Schubert coefficients 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.

Paper Structure

This paper contains 11 sections, 1 theorem, 7 equations.

Table of Contents

  1. Introduction
  2. The results
  3. Background
  4. Positive rule
  5. Main theorem
  6. Discussion
  7. The meaning of Theorem \ref{['t:main']}
  8. The meaning of derandomization
  9. Two critiques
  10. Not interesting
  11. Lack of certainty

Key Result

Theorem 2.1

Assuming $\textup{\sc GRH}$ and $\textup{\sc MVA}$, the Schubert positivity problem eq:Schu-pos has a positive rule.

Theorems & Definitions (2)

  • Theorem 2.1
  • proof