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 constants and new results on the
enumeration of chains in the Bruhat order. A new graded partial order (the
Grassmann-Bruhat order) on
the symmetric group which contains Young's lattice arises from these
investigations. We also derive formulas for certain specializations of
Schubert polynomials.