Classification of Ding's Schubert Varieties: Finer Rook Equivalence

Abstract K. Ding studied a class of Schubert varieties ${{X}_{\lambda }}$ in type A partial flag manifolds, indexed by integer partitions $\text{ }\!\!\lambda\!\!\text{ }$ and in bijection with dominant permutations. He observed that the Schubert cell structure of ${{X}_{\lambda }}$ is indexed by maximal rook placements on the Ferrers board ${{B}_{\lambda \text{ }}}$ , and that the integral cohomology groups ${{H}^{*}}\left( {{X}_{\lambda }};\,\mathbb{Z} \right),\,{{H}^{*}}\left( {{X}_{\mu }};\,\mathbb{Z} \right)$ are additively isomorphic exactly when the Ferrers boards ${{B}_{\lambda \text{ }}}$ , ${{B}_{\mu }}$ satisfy the combinatorial condition of rook-equivalence. We classify the varieties ${{X}_{\lambda }}$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.