A Pieri-type formula for the ${K}$-theory of a flag manifold
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- by Cristian Lenart and Frank Sottile PDF
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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.References
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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