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June 2, 2026

Igor Pak, UCLA

Vanishing of Schubert coefficients

9th WACT

University of Copenhagen

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Positivity of Schubert coefficients can be decided

in probabilistic polynomial time

Main Theorem (P.-Robichaux, 2025)

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Positivity of Schubert coefficients can be decided

in probabilistic polynomial time

Main Theorem (P.-Robichaux, 2025)

What?

Why?

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Goals of the talk

1) What is this about?

2) Why should one care?

3) How to prove this?

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Structure constants

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Structure constants

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Littlewood–Richardson coefficients

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Littlewood–Richardson coefficients

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Kronecker coefficients

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Kronecker coefficients

Introduced by Murnaghan (1938)

Generalize LR coefficients

Key role in GCT

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Kronecker coefficients

Introduced by Murnaghan (1938)

Generalize LR coefficients

Key role in GCT

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Schubert coefficients

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Schubert coefficients

Introduced by Schubert (1880s)

Reintroduced by Lascoux and Schützenberger (1982)

Represent cohomology classes in complete flag varieties

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Schubert coefficients

Introduced by Schubert (1880s)

Reintroduced by Lascoux and Schützenberger (1982)

Represent cohomology classes in complete flag varieties

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Schubert coefficients

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Schubert coefficients

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Schubert coefficients

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What happened? A personal story

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What happened? A personal story

2012-20

Combinatorial Interpretation ← # integer points in polytopes

Dozens of papers on Kronecker coefficients & integer points

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What happened? A personal story

Conclusion: This is a wrong definition!

2012-20

Combinatorial Interpretation ← # integer points in polytopes

Dozens of papers on Kronecker coefficients & integer points

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What happened? A personal story

2020-24

Combinatorial Interpretation ← #P

First “not in #P” tools with C. Ikenmeyer, S.H. Chan

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What happened? A personal story

Conclusion: Vanishing is the key to the problem!

2020-24

Combinatorial Interpretation ← #P

First “not in #P” tools with C. Ikenmeyer, S.H. Chan

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Main problems, updated

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Kronecker positivity

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Kronecker positivity

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Kronecker positivity

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Schubert positivity

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Schubert positivity

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Schubert positivity

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Schubert positivity

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Motivation

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Motivation

“For applications (including real-world engineering applications) it is more important to know that [Schubert] structure constant is positive, than it is to know its actual value.”

[Knutson, ICM paper, 2022]

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Prior work: LR coefficients

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Prior work: LR coefficients

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Prior work: LR coefficients

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Prior work: LR coefficients

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No saturation for Schubert coefficients

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Prior work

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Schubert coefficients

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Schubert coefficients

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Schubert coefficients

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Structure constants for Schur P-functions

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Structure constants for Schur P-functions

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Structure constants for Schur P-functions

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Evolution of our work

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Evolution of our work

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Evolution of our work

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Evolution of our work

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Evolution of our work

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First approach

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First approach

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First approach

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First approach

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First approach

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First approach

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First approach

(2017)

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Power of this approach

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Power of this approach

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Power of this approach

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Second approach

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Second approach

Rank

condition

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Polynomial identity testing

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Second approach

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Second approach

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Thank you!

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