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

Igor Pak, Colleen Robichaux

TL;DR

This paper proves that the Schubert vanishing decision problem lies in $AM\cap coAM$ under the Generalized Riemann Hypothesis, improving earlier results that placed it in $coAM$ under the same assumption. The authors develop and leverage lifted (compact) formulations, anchored by a polynomial-size determinant lemma, to reduce Schubert vanishing to a parametric Hilbert Nullstellensatz instance via Purbhoo’s criterion. The approach hinges on a parametric Nullstellensatz framework and a detailed analysis of generic conjugations in the unipotent subgroup, yielding a polynomial-size certificate system and closing gaps for all classical types $A,B,C,D$. The results suggest Schubert vanishing is unlikely to be $coNP$-hard under standard complexity assumptions and highlight new connections between algebraic geometry, complexity theory, and lifted formulations in algebraic combinatorics.

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.

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

TL;DR

This paper proves that the Schubert vanishing decision problem lies in under the Generalized Riemann Hypothesis, improving earlier results that placed it in under the same assumption. The authors develop and leverage lifted (compact) formulations, anchored by a polynomial-size determinant lemma, to reduce Schubert vanishing to a parametric Hilbert Nullstellensatz instance via Purbhoo’s criterion. The approach hinges on a parametric Nullstellensatz framework and a detailed analysis of generic conjugations in the unipotent subgroup, yielding a polynomial-size certificate system and closing gaps for all classical types . The results suggest Schubert vanishing is unlikely to be -hard under standard complexity assumptions and highlight new connections between algebraic geometry, complexity theory, and lifted formulations in algebraic combinatorics.

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 assuming the Generalized Riemann Hypothesis (GRH). This complements our earlier result in arXiv:2412.02064, that the problem is in assuming the GRH. In particular, this implies that the Schubert vanishing problem is unlikely to be -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.

Paper Structure

This paper contains 30 sections, 8 theorems, 35 equations, 1 figure.

Table of Contents

  1. Schubert vanishing problem
  2. Main result
  3. Schubert vanishing
  4. Prior work: general results
  5. Prior work: special cases
  6. Implications
  7. New type of problem
  8. Positive rule via derandomization
  9. Computational hardness of Schubert vanishing
  10. Notation
  11. Lifted and compact formulations
  12. Hilbert's Nullstellensatz
  13. Lifted formulations
  14. Prior work: extended formulations
  15. Prior work: complexity of the determinant
  16. ...and 15 more sections

Key Result

Theorem 1.1

$\textup{\sc SchubertVanishing} \in \textup{AM} \cap \textup{coAM}$ assuming the $\textup{\sc GRH}$.

Figures (1)

  • Figure A.1: Graphs in ${\text{\rm RC} } (1432)$ and the corresponding Schubert polynomial $\mathfrak{S}_{1432} = x_1x_2x_3+x_1^2x_3+ x_1x_2^2+x_2^2x_3+x_1^2x_2$ with monomials in this order.

Theorems & Definitions (11)

  • Theorem 1.1: Main theorem
  • Conjecture 1.3: Pak and Robichaux PR24o
  • Corollary 1.4
  • Theorem 2.1: Koiran96
  • Lemma 2.2: Determinant lemma
  • Remark 2.3
  • Theorem 3.1: A+24
  • Lemma 3.2: Main lemma
  • Lemma 3.3: Purbhoo's criterion Purbhoo06
  • Theorem 3.4: Schubert vanishing for all types
  • ...and 1 more