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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Pieri-type formula for isotropic flag manifolds
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by Nantel Bergeron and Frank Sottile PDF
Trans. Amer. Math. Soc. 354 (2002), 2659-2705 Request permission

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
  • MR Author ID: 355336
  • ORCID: 0000-0003-0087-7120
  • Email: sottile@math.umass.edu
  • 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.
  • © Copyright 2002 American Mathematical Society
  • 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
  • MathSciNet review: 1895198