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A Pieri-type formula for the $ {K}$-theory of a flag manifold

Authors: Cristian Lenart and Frank Sottile
Journal: Trans. Amer. Math. Soc. 359 (2007), 2317-2342
MSC (2000): Primary 14M15, 05E99, 19L64
Published electronically: December 19, 2006
MathSciNet review: 2276622
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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

Frank Sottile
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Grothendieck polynomial, Schubert variety, Pieri's formula, Bruhat order
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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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