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Classification of Ding's Schubert Varieties: Finer Rook Equivalence

Published online by Cambridge University Press:  20 November 2018

Mike Develin
American Institute of Mathematics 360 Portage Ave. Palo Alto, CA 94306-2244 U.S.A. e-mail:
Jeremy L. Martin
Department of Mathematics University of Kansas Lawrence, KS 66045 U.S.A. e-mail:
Victor Reiner
School of Mathematics University of Minnesota Minneapolis, MN 55455 U.S.A. e-mail:
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K. Ding studied a class of Schubert varieties ${{X}_{\lambda }}$ in type A partial flag manifolds, indexed by integer partitions $\text{ }\!\!\lambda\!\!\text{ }$ 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 ${{B}_{\lambda \text{ }}}$, and that the integral cohomology groups ${{H}^{*}}\left( {{X}_{\lambda }};\,\mathbb{Z} \right),\,{{H}^{*}}\left( {{X}_{\mu }};\,\mathbb{Z} \right)$ are additively isomorphic exactly when the Ferrers boards ${{B}_{\lambda \text{ }}}$, ${{B}_{\mu }}$ satisfy the combinatorial condition of 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.

Research Article
Copyright © Canadian Mathematical Society 2007


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