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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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A Pieri-type formula for the ${K}$-theory of a flag manifold
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by Cristian Lenart and Frank Sottile PDF
Trans. Amer. Math. Soc. 359 (2007), 2317-2342 Request permission

Abstract:

We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert classes are indexed by a cycle which has either the form $(k{-}p{+}1,k{-}p{+}2,\ldots ,k{+}1)$ or the form $(k{+}p,k{+}p{-}1,\ldots ,k)$, and are pulled back from a Grassmannian projection. Our formulas are in terms of certain labeled chains in the $k$-Bruhat order on the symmetric group and are combinatorial in that they involve no cancellations. We also show that the multiplicities in the Pieri formula are naturally certain binomial coefficients.
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Additional Information
  • Cristian Lenart
  • Affiliation: Department of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222
  • MR Author ID: 259436
  • Email: lenart@albany.edu
  • Frank Sottile
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
  • MR Author ID: 355336
  • ORCID: 0000-0003-0087-7120
  • Email: sottile@math.tamu.edu
  • Received by editor(s): November 22, 2004
  • Received by editor(s) in revised form: March 31, 2005
  • Published electronically: December 19, 2006
  • Additional Notes: The research of the first author was supported by SUNY Albany Faculty Research Award 1039703
    The research of the second author was supported in part by the Clay Mathematical Institute, the MSRI, and NSF CAREER grant DMS-0134860
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 2317-2342
  • MSC (2000): Primary 14M15, 05E99, 19L64
  • DOI: https://doi.org/10.1090/S0002-9947-06-04043-8
  • MathSciNet review: 2276622