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 $n\ge 4$, this criterion recognizes a nonvanishing proportion of the positive atom coefficients as the bound on the composition entries tends to infinity.

arXiv comment23 pages; corrected and expanded the historical discussion of Polo's Schubert-filtration conjecture, van der Kallen's question, and the Reiner--Shimozono conjecture. Clarified the exposition in several places