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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Skew Schubert polynomials


Authors: Cristian Lenart and Frank Sottile
Journal: Proc. Amer. Math. Soc. 131 (2003), 3319-3328
MSC (2000): Primary 05E05, 14M15, 06A07
Published electronically: February 20, 2003
MathSciNet review: 1990619
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Abstract: We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs.


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Additional Information

Cristian Lenart
Affiliation: Department of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222
Email: lenart@csc.albany.edu

Frank Sottile
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
Email: sottile@math.umass.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06919-3
PII: S 0002-9939(03)06919-3
Keywords: Schubert polynomial, Bruhat order, Littlewood-Richardson coefficient
Received by editor(s): February 13, 2002
Received by editor(s) in revised form: May 28, 2002
Published electronically: February 20, 2003
Additional Notes: Most of this work was done while the first author was supported by the Max-Planck-Institut für Mathematik. The second author was supported in part by NSF grants DMS-9701755 and DMS-0070494.
Communicated by: John R. Stembridge
Article copyright: © Copyright 2003 American Mathematical Society