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Journal of the London Mathematical Society 2002 66(3):550-562; doi:10.1112/S0024610702003605
© 2002 by London Mathematical Society
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© The London Mathematical Society

Cohomology of Smooth Schubert Varieties in Partial Flag Manifolds

V. Gasharov and V. Reiner

Department of Mathematics, Cornell University Ithaca, NY 14853, USA, vesko{at}math.cornell.edu
School of Mathematics, University of Minnesota Minneapolis, MN 55455, USA, reiner{at}math.umn.edu

Received 18 June 2001. Revision received 6 February 2002.

The fact that smooth Schubert varieties in partial flag manifolds are iterated fiber bundles over Grassmannians is used to give a simple presentation for their integral cohomology ring, generalizing Borel's presentation for the cohomology of the partial flag manifold itself. More generally, such a presentation is shown to hold for a larger class of subvarieties of the partial flag manifolds (which are called subvarieties defined by inclusions). The Schubert varieties which lie within this larger class are characterized combinatorially by a pattern avoidance condition.


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