Classification of Ding's Schubert varieties: finer rook equivalence

Abstract

K. Ding studied a class of Schubert varieties X_λin type A partial flag manifolds, corresponding to integer partitions λand in bijection with dominant permutations. He observed that the Schubert cell structure of X_λis indexed by maximal rook placements on the Ferrers board B_λ, and that the integral cohomology groups H^*(X_λ; Zz), H^*(X_μ; Zz) are additively isomorphic exactly when the Ferrers boards B_λ, B_μsatisfy the combinatorial condition of rook-equivalence. We classify the varieties X_λup 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.