A Pieri-type formula for isotropic flag manifolds

Bergeron, Nantel; Sottile, Frank

doi:10.1090/S0002-9947-02-02946-X

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
DOI: https://doi.org/10.1090/S0002-9947-02-02946-X
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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