Low-degree filtrations in ordinary and Lagrangian Grassmannian cohomology

Abstract

We prove the Hilbert-series formulas conjectured by Reiner--Tudose and by the Polymath Jr. Grassmannian group for the subalgebras of ordinary and Lagrangian Grassmannian cohomology generated in bounded degree. We determine the finite coefficient images underlying the proofs, together with the kernels of the coefficient and cancellation maps. We also construct bases adapted to the auxiliary ideal filtrations and identify the cancellation layers, as modules, with shifts of finite complete-intersection algebras. First-order deformations give the derivative inclusions needed to recover the filtration increments. We disprove the two ordinary basis conjectures of the Polymath group. For every $r\ge 2$, the span of each prescribed family has codimension one in weight $3r+2$ of $H^{2*}(\operatorname{Gr}(2,3r+1);\mathbb Q)$; replacing one index gives a basis in each of these components. More generally, each prescribed two-row family has codimension at most one in every weight. Weights are half the cohomological degrees, and all coefficients are rational.