Identities of Structure Constants for Schubert polynomials and Orders on S_n

We use geometry to prove a number of new identities among the Littlewood-Richardson coefficients for Schubert polynomials (Schubert classes in a flag manifold). For many of these identities, there is a companion result about the Bruhat order which should imply the identity, were it known how to express these coefficients in terms of the Bruhat order. This analysis leads the determination of many of these constants. We conclude with an outline of geometric proofs for these identities.