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Skew Schubert functions and the Pieri formula for flag manifolds


Authors: Nantel Bergeron and Frank Sottile
Journal: Trans. Amer. Math. Soc. 354 (2002), 651-673
MSC (1991): Primary 05E15, 14M15, 05E05, 06A07, 14N10
DOI: https://doi.org/10.1090/S0002-9947-01-02845-8
Published electronically: September 21, 2001
MathSciNet review: 1862562
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Abstract: We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric functions, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtainng a combinatorial chain construction of Schubert polynomials.


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Additional Information

Nantel Bergeron
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P, 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: https://doi.org/10.1090/S0002-9947-01-02845-8
Keywords: Pieri formula, Bruhat order, Schubert polynomial, Stanley symmetric function, flag manifold, {\em jeu de taquin}, weak order
Received by editor(s): October 9, 2000
Published electronically: September 21, 2001
Additional Notes: The first author was supported in part by NSERC and CRM grants.
The second author was supported in part by NSERC grant OGP0170279 and NSF grant DMS-9022140.
Dedicated: In memory of Rodica Simion
Article copyright: © Copyright 2001 American Mathematical Society

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