Vanishing of Schubert Coefficients
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912305816535040 |
|---|---|
| author | Pak, Igor Robichaux, Colleen |
| author_facet | Pak, Igor Robichaux, Colleen |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02064 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vanishing of Schubert Coefficients Pak, Igor Robichaux, Colleen Combinatorics Computational Complexity Discrete Mathematics Algebraic Geometry Primary: 05E14, Secondary: 05E05, 14M15, 14N15, 68Q05, 68Q15, 68Q25 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). |
| title | Vanishing of Schubert Coefficients |
| topic | Combinatorics Computational Complexity Discrete Mathematics Algebraic Geometry Primary: 05E14, Secondary: 05E05, 14M15, 14N15, 68Q05, 68Q15, 68Q25 |
| url | https://arxiv.org/abs/2412.02064 |