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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Chern class formulas for $ G_2$ Schubert loci


Author: Dave Anderson
Journal: Trans. Amer. Math. Soc. 363 (2011), 6615-6646
MSC (2010): Primary 14N15; Secondary 14M15, 20G41, 05E05
Published electronically: July 19, 2011
MathSciNet review: 2833570
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Abstract: We define degeneracy loci for vector bundles with structure group $ G_2$ and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type $ G_2$. We include explicit descriptions of the $ G_2$ flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham.

In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, correcting an error in a paper by Edidin and Graham.


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Additional Information

Dave Anderson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: dandersn@umich.edu, dandersn@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05317-1
PII: S 0002-9947(2011)05317-1
Keywords: Degeneracy locus, equivariant cohomology, flag variety, Schubert variety, Schubert polynomial, exceptional Lie group, octonions
Received by editor(s): August 31, 2009
Received by editor(s) in revised form: February 2, 2010
Published electronically: July 19, 2011
Additional Notes: This work was partially supported by NSF Grants DMS-0502170 and DMS-0902967.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.