A Pieri-Chevalley formula for K(G/B)

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 21 February 2013

Introduction

Algebraic combinatorics and geometry come together in a beautiful way in the study of the cohomology $H^{*}(G/B)$ of the generalized flag variety $G/B$ for a complex semisimple Lie group $G$ with Borel subgroup $B\text{.}$ The cohomology $H^{*}(G/B)$ is isomorphic (as a graded ring) to the quotient of a polynomial ring by the ideal generated by $W\text{-symmetric}$ functions without constant term and has a natural basis of “Schubert polynomials” $\left[X_w\right]H^{*}(G/B)$ where $\left[X_w\right]$ is the Poincaré dual of the fundamental class of the Schubert variety $\left[X_w\right]\subseteq G/B$ in $H_{*}(G/B)\text{.}$ One of the fundamental results in the theory of Schubert polynomials and the cohomology of the flag variety is a formula of Chevalley which gives an expansion of the product $\lambda ·\left[X_w\right]$ in terms of the Schubert class basis for an element $\lambda \in H^2(G/B)\text{.}$

A similar picture holds for the K-theory of $G/B\text{.}$ The ring $K(G/B)$ is isomorphic to a quotient of a Laurent polynomial ring by an ideal generated by certain $W\text{-symmetric}$ functions and has a basis given by classes $\left[{𝒪}_{X_w}\right]$ where ${𝒪}_{X_w}$ is the structure sheaf of the Schubert variety $X_w\subseteq G/B$ extended by 0 outside of $X_w\text{.}$ In this paper we give an analogue of Chevalley’s formula for the ring $K(G/B)$ Specifically, we give an explicit combinatorial formula for $e^{\lambda }\left[{𝒪}_{X_w}\right],$ the tensor product of a (negative) line bundle with the structure sheaf of a Schubert variety, expanded in terms of the Schubert class basis $\left\{\left[{𝒪}_{X_w}\right]\right\}\text{.}$

The Chern character is an isomorphism between $K(G/B)$ and $H^{*}(G/B)$ and Chevalley’s formula can be recovered from ours by applying the Chern character and comparing lowest degree terms. The higher order terms of our formula may yield further interesting identities in cohomology.

Fulton and Lascoux [FLa1994] have given a formula similar to ours for the $G={GL}_n\left(ℂ\right)$ case. Our formula is a generalization of their formula to general type except that we work only with $K(G/B)$ instead of the K-theory of the flag bundle. In our work the column strict tableaux used by Fulton and Lascoux are replaced by Littelmann’s path model. This has two advantages: (1) it allows us to work in general type and (2) it obviates the need for the complex combinatorics associated with the jeu de taquin and the “rectification” of tableaux. It is possible that the more general flag bundle result of Fulton and Lascoux can be obtained from our general commutation formula, Theorem 4.2, but to sort this out properly one would have to understand concretely the connection between the tableaux and the Littelmann paths.

In the first section of the paper we review notations and recall why one is able to work with $K(G/B)$ as a quotient of a Laurent polynomial ring. In the second section we derive an expression for $\left[{𝒪}_p\right]\in K(G/P),$ for a point $p\in G/P,$ in terms of familiar vector bundles. We are able to pull back this formula into $K(G/B)$ to obtain formulas for certain special Schubert classes in terms of line bundles. In section 3 we recall the operators which play the same role as the BGG operators in cohomology and show that they can be used to give explicit (inductive) expressions for the Schubert classes in $K(G/B)\text{.}$ In section 4 we prove the main theorem, which gives a commutation relation between line bundles and the Schubert classes. Our new Pieri-Chevalley formula is an immediate consequence of this relation. In the final section we explain how the K-theory relates to cohomology and how our formula implies the classical Chevalley formula.

The main results of the preliminary sections can all be considered well known. These results can be found, either explicitly or implicitly, in the work of Demazure [Dem1974], Kostant and Kumar [KKu1990], Fulton and Lascoux [FLa1994] and others. For the convenience of the reader, we have given short proofs or sketches of proofs for most of these results and, in the final section, we have given a dictionary between K-theory and $H^{*}(G/B)\text{.}$ This dictionary illustrates how our results relate to the theory of Schubert polynomials.

We would like to thank M. Green, S. Kumar, T. Shifrin, and A. Vasquez for stimulating conversations during our work on this paper.

Notes and References

This is an excerpt of the paper entitled A Pieri-Chevalley formula for K(G/B) authored by H. Pittie and Arun Ram (preprint May 19, 1998).

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

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