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Classification of Ding's Schubert Varieties: Finer Rook Equivalence

Published online by Cambridge University Press:  20 November 2018

Mike Develin ,
Jeremy L. Martin  and
Victor Reiner
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.

Keywords

14M1505E05Schubert varietyrook placementFerrers boardflag manifoldcohomology ringnilpotence
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 1 , 01 February 2007 , pp. 36 - 62
Copyright
Copyright © Canadian Mathematical Society 2007

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