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.

19 pages. Final version, with some minor mistakes corrected. To appear in Canadian Journal of Mathematics