2004/01/23 by Harsh Pittie, Arun Ram, Pittie, Harsh +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.math/0401332
openalex publication_date 2004/01/23 · arxiv created 2004/02/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Ow, where Ow is the class of the structure sheaf of the Schubert variety Xw. 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 Ow. 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 GLn(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.