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# Classification of Ding's Schubert Varieties: Finer Rook Equivalence

Published:2007-02-01
Printed: Feb 2007
• Mike Develin
• Jeremy L. Martin
• Victor Reiner
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## Abstract

K.~Ding studied a class of Schubert varieties $X_\lambda$ 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.
 Keywords: Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotence
 MSC Classifications: 14M15 - Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 05E05 - Symmetric functions and generalizations

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