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

 

A Pieri-type formula for isotropic flag manifolds


Authors: Nantel Bergeron and Frank Sottile
Journal: Trans. Amer. Math. Soc. 354 (2002), 2659-2705
MSC (2000): Primary 14M15, 05E15, 05E05, 06A07, 14N10
Published electronically: February 20, 2002
MathSciNet review: 1895198
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Abstract: We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from the Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the infinite Coxeter group ${\mathcal B}_\infty$, identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show that most of these identities follow from the Pieri-type formula, and our analysis leads to a new partial order on the Coxeter group ${\mathcal B}_\infty$ and formulas for many of these structure constants.


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Additional Information

Nantel Bergeron
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada
Email: bergeron@mathstat.yorku.ca

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

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02946-X
PII: S 0002-9947(02)02946-X
Received by editor(s): March 7, 2001
Received by editor(s) in revised form: August 6, 2001
Published electronically: February 20, 2002
Additional Notes: First author supported in part by NSERC and CRM grants.
Second author supported in part by NSERC grant OGP0170279 \ and NSF grant DMS-9022140.
Article copyright: © Copyright 2002 American Mathematical Society