Skew Schubert functions and the Pieri formula for flag manifolds

Bergeron, Nantel; Sottile, Frank

doi:10.1090/S0002-9947-01-02845-8

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ISSN 1088-6850(online) ISSN 0002-9947(print)

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
Posted: 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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