Counterexamples to the Reiner-Shimozono conjecture and the failure of Schubert filtrations

Source 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.