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

Abstract

The ring K(G/B) is isomorphic to a quotient of a Laurent polynomial ring by an ideal generated by certain W-symmetric functions and has a basis given by classes O_w, where O_w is the class of the structure sheaf of the Schubert variety X_w. In this paper we give an explicit combinatorial formula for the tensor product of a (negative) line bundle with the structure sheaf of a Schubert variety, expanded in terms of the Schubert class basis O_w. Chevalley's classical formula for the class of a hyperplane section of a Schubert variety in H^*(G/B) can be recovered from our formula 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 have given a formula similar to ours for the GL_n(C) case. Our formula is a generalization of their formula to general type. In our work the column strict tableaux used by Fulton and Lascoux are replaced by Littelmann's path model.