Pieri's Formula for Flag Manifolds and Schubert Polynomials

We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the ag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the classical Pieri's formula for symmetric polynomials/Grassmann varieties to Schubert polynomials//ag manifolds. Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, some of which we express in terms of paths in the Bruhat order on the symmetric group, which in turn yields an enumerative result about the Bruhat order. 1. Introduction Schubert polynomials had their origins in the study of the cohomology of ag manifolds by Bernstein-Gelfand-Gelfand 3] and Demazure 7]. They were later deened by Lascoux and Sch utzenberger 17], who developed a purely combinatorial theory. For each permutation w in the symmetric group S n there is a Schubert polyno