Classification of Ding’s Schubert Varieties: Finer Rook Equivalence

K. Ding studied a class of Schubert varieties Xλ in type A partial flag manifolds, indexed by integer partitions λ and in bijection with dominant permutations. He observed that the Schubert cell structure of Xλ is indexed by maximal rook placements on the Ferrers board Bλ, and that the integral cohomology groups H∗(Xλ; Z), H ∗(Xμ; Z) are additively isomorphic exactly when the Ferrers boards Bλ, Bμ satisfy the combinatorial condition of rook-equivalence. We classify the varieties Xλ 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.