Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Vanishing of Schubert Coefficients

Igor Pak, Colleen Robichaux

Abstract

Schubert coefficients are nonnegative integers $c^w_{u,v}$ that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation, i.e, whether $c^w_{u,v} \in \#{\sf P}$. We study the closely related vanishing problem of Schubert coefficients: $\{c^w_{u,v}=^? 0\}$. Until this work it was open whether this problem is in the polynomial hierarchy ${\sf PH}$. We prove that $\{c^w_{u,v}=^? 0\}$ in ${\sf coAM}$ assuming the GRH. In particular, the vanishing problem is in ${Σ_2^{\text{p}}}$. Our approach is based on constructions lifted formulations, which give polynomial systems of equations for the problem. The result follows from a reduction to Parametric Hilbert's Nullstellensatz, recently studied in arXiv:2408.13027. We extend our results to all classical types. Type $D$ is resolved in the appendix (joint with David Speyer).

Vanishing of Schubert Coefficients

Abstract

Schubert coefficients are nonnegative integers that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation, i.e, whether . We study the closely related vanishing problem of Schubert coefficients: . Until this work it was open whether this problem is in the polynomial hierarchy . We prove that in assuming the GRH. In particular, the vanishing problem is in . Our approach is based on constructions lifted formulations, which give polynomial systems of equations for the problem. The result follows from a reduction to Parametric Hilbert's Nullstellensatz, recently studied in arXiv:2408.13027. We extend our results to all classical types. Type is resolved in the appendix (joint with David Speyer).

Paper Structure

This paper contains 48 sections, 17 theorems, 81 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Foreword
  3. Littlewood--Richardson coefficients
  4. Schubert polynomials
  5. Schubert coefficients
  6. Main results
  7. Prior work on the Schubert vanishing problem
  8. Poly-time conditions
  9. $\textup{NP}$ conditions
  10. Algebraic conditions
  11. Hilbert's Nullstellensatz
  12. Schubert vanishing via Hilbert's Nullstellensatz
  13. Proof outline
  14. Discussion and implications
  15. Comparison to other problems
  16. ...and 33 more sections

Key Result

Theorem 1.4

$\textup{\sc SchubertVanishing}$ is in $\textup{coAM}$ assuming $\textup{\sc GRH}$. The result holds for the vanishing of Schubert coefficients in types $A$, $B$, $C$ and $D$.

Figures (2)

  • Figure 1.1: Semistandard Young tableau $A\in \operatorname{SSYT}(4,3,2)$ and the corresponding monomial ${\textbf{x}}^A = x_1^2 x_2^3 x_3 x_4^3$ .
  • Figure 1.2: 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 (31)

  • Conjecture 1.1: Pan23
  • Conjecture 1.2: Pak-OPAC
  • Conjecture 1.3: Pak-OPAC
  • Theorem 1.4: Main theorem
  • Conjecture 1.5
  • Conjecture 1.6: cf. ARY19
  • Theorem 1.7: Koiran96
  • Theorem 1.8: A+24
  • Lemma 1.9: Main lemma
  • Proposition 5.1: HS17
  • ...and 21 more