Counterexamples to the Reiner-Shimozono conjecture and the failure of Schubert filtrations
Abstract
We disprove the Reiner--Shimozono conjecture that products of key polynomials have nonnegative expansions in Demazure atoms. By relaxing the defining relations of Demazure modules, we obtain an exact coefficient-extraction formula that yields an explicit infinite family of counterexamples, including negative coefficients in twenty-eight variables. These examples answer van der Kallen's long-standing open question on relative Schubert filtrations negatively, already for tensor products of two dual Joseph modules. For individual atom coefficients, we give a sufficient criterion for positivity that is checkable in polynomial time in the binary input length, uniformly in rank. For each fixed rank n≥4, this criterion recognizes a nonvanishing proportion of the positive atom coefficients as the bound on the composition entries tends to infinity.
Related
- Multivariate V-filtrations and the Strong Monodromy Conjecture for hyperplane arrangements
- Counterexamples to the Monotone Completeness Conjecture
- Counterexamples to the ordinary Fried conjecture near hyperbolic metrics
- Counterexamples to the inhomogeneous Duffin-Schaeffer Conjecture
- A Note on the OpenAI Construction of Counterexamples to the Kaplansky Direct-Finiteness Conjecture
BibTeX
@misc{hodges2026counterexamples,
title = {{Counterexamples to the Reiner-Shimozono conjecture and the failure of Schubert filtrations}},
author = {Reuven Hodges},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28169},
eprint = {2609.28169},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28169},
}
RIS
TY - UNPB
TI - Counterexamples to the Reiner-Shimozono conjecture and the failure of Schubert filtrations
AU - Reuven Hodges
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.28169
UR - https://arxiv.org/abs/2609.28169
AB - We disprove the Reiner--Shimozono conjecture that products of key polynomials have nonnegative expansions in Demazure atoms. By relaxing the defining relations of Demazure modules, we obtain an exact coefficient-extraction formula that yields an explicit infinite family of counterexamples, including negative coefficients in twenty-eight variables. These examples answer van der Kallen's long-standing open question on relative Schubert filtrations negatively, already for tensor products of two dual Joseph modules. For individual atom coefficients, we give a sufficient criterion for positivity that is checkable in polynomial time in the binary input length, uniformly in rank. For each fixed rank n≥4, this criterion recognizes a nonvanishing proportion of the positive atom coefficients as the bound on the composition entries tends to infinity.
ER -