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ISSN 1088-6850(e) ISSN 0002-9947(p)

A Pieri-type formula for the -theory of a flag manifold

Author(s): Cristian Lenart; Frank Sottile
Journal: Trans. Amer. Math. Soc. 359 (2007), 2317-2342.
MSC (2000): Primary 14M15, 05E99, 19L64
Posted: 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 -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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