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Classification of Ding's Schubert Varieties: Finer Rook Equivalence

Published online by Cambridge University Press:  20 November 2018

Mike Develin
Affiliation:
American Institute of Mathematics 360 Portage Ave. Palo Alto, CA 94306-2244 U.S.A. e-mail: develin@post.harvard.edu
Jeremy L. Martin
Affiliation:
Department of Mathematics University of Kansas Lawrence, KS 66045 U.S.A. e-mail: jmartin@math.ku.edu
Victor Reiner
Affiliation:
School of Mathematics University of Minnesota Minneapolis, MN 55455 U.S.A. e-mail: reiner@math.umn.edu
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2007

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