MathDB

Reiner–Shimozono conjecture and Polo’s Schubert-filtration conjecture

For every n≥1n\ge 1 and every pair of weak compositions α,β∈Z≥0n\alpha,\beta\in\mathbb{Z}_{\ge 0}^n, the product of key polynomials admits a nonnegative expansion in the basis of Demazure atoms: κακβ=∑γ∈Z≥0ncα,βγ Aγ\kappa_\alpha\kappa_\beta=\sum_{\gamma\in\mathbb{Z}_{\ge 0}^n}c_{\alpha,\beta}^{\gamma}\,\mathcal{A}_\gamma, where cα,βγ∈Z≥0c_{\alpha,\beta}^{\gamma}\in\mathbb{Z}_{\ge 0} and all but finitely many coefficients vanish.

References

Primary source

arXiv

Additional references

algebraic-combinatoricsalgebraic-geometryrepresentation-theory
Posted · edited

Progress summary

Refreshed
Claimed solved

A new preprint claims to disprove both conjectures while proving a more limited positivity result, but the work has not been independently verified.

The problem concerns two linked expectations that certain algebraic positivity phenomena always hold. A new preprint claims that both fail through explicit infinite families of counterexamples, while preserving a separate positivity theory.

September 2026 counterexamples

Reuven Hodges’s new preprint constructs explicit counterexamples, including examples with 28 variables, and gives a polynomial-time sufficient criterion for positivity. If correct, this refutes both conjectures rather than merely advancing them; the preprint is not peer reviewed.

Current status (as of September 2026): Both conjectures are claimed to be false by explicit counterexamples, while the counterexamples and the surviving positivity criterion remain unverified.

Sources

Solutions 0

No solutions have been posted yet.