Schubert polynomials, the Bruhat order, and the geometry of flag manifolds

We illuminate the relation between the Bruhat order and structure constants for the polynomial ring in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We establish a number of new identities among these structure constants. This leads to formulas for some of these constants and new results on the enumeration of chains in the Bruhat order. A new graded partial order on the symmetric group which contains Young’s lattice arises from these investigations. We also derive formulas for certain specializations of Schubert polynomials. To the memory of Marcel Paul Schützenberger