Classification of ding's schubert varieties: Finer rook equivalence

Mike Develin, Jeremy L. Martin, Victor Reiner

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

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

Original languageEnglish (US)
Pages (from-to)36-62
Number of pages27
JournalCanadian Journal of Mathematics
Volume59
Issue number1
DOIs
StatePublished - Feb 2007

Keywords

  • Cohomology ring
  • Ferrers board
  • Flag manifold
  • Nilpotence
  • Rook placement
  • Schubert variety

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