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A Pieri-Chevalley formula in the K-theory of a G/B-bundle
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## A Pieri-Chevalley formula in the K-theory of a G/B-bundle

### Harsh Pittie and Arun Ram

**Abstract.**
Let $G$ be a semisimple complex Lie group, $B$ a
Borel subgroup, and $T\subseteq B$ a maximal torus of $G$. The
projective variety $G/B$ is a generalization of the classical flag
variety. The structure sheaves of the Schubert subvarieties form a
basis of the K-theory $K(G/B)$ and every character of $T$ gives rise
to a line bundle on $G/B$. This note gives a formula for the product
of a dominant line bundle and a Schubert class in $K(G/B)$. This
result generalizes a formula of Chevalley which computes an analogous
product in cohomology. The new formula applies to the relative case,
the K-theory of a $G/B$-bundle over a smooth base $X$, and is
presented in this generality. In this setting the new formula is a
generalization of recent $G=GL_n({\mathbb C})$ results of Fulton and
Lascoux.

*Copyright 1999 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**05** (1999), pp. 102-107
- Publisher Identifier: S 1079-6762(99)00067-0
- 1991
*Mathematics Subject Classification*. Primary 14M15; Secondary 14C35, 19E08
*Key words and phrases*.
- Received by the editors February 9, 1999
- Posted on July 14, 1999
- Communicated by Efim Zelmanov
- Comments (When Available)

**Harsh Pittie**

Department of Mathematics, Graduate Center, City University of New York, New York, NY 10036

**Arun Ram**

Department of Mathematics, Princeton University, Princeton, NJ 08544

*E-mail address:* `rama@math.princeton.edu`

Research supported in part by National Science Foundation grant DMS-9622985.

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