Classification of Ding's Schubert Varieties: Finer Rook Equivalence

Abstract

K.~Ding studied a class of Schubert varieties in type A partial flag manifolds, indexed by integer partitions \lambda 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, and that the integral cohomology groups H^*(X_\lambda;\:\Zz), H^*(X_\mu;\:\Zz) are additively isomorphic exactly when the Ferrers boards B_\lambda, B_\mu satisfy the combinatorial condition of \emph{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.