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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Received 18 June 2001; revised 6 February 2002. 2000 Mathematics Subject Classi cation 05E, 14M15. The second author was partially supported by NSF grant DMS-9877047.
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