Year: 2026
Type: preprint
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 also disprove Polo's conjecture on Schubert filtrations of tensor products of section modules on Schubert varieties, even when successive quotients given by spaces of sections over unions of Schubert varieties are permitted. 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 $n\ge4$, this criterion recognizes a nonvanishing proportion of the positive atom coefficients as the bound on the composition entries tends to infinity.
Source: arXiv (Cornell University)
Author: Hodges, Reuven
Institution: University of Kansas
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214165205