Schubert polynomials, pipe dreams, equivariant classes, and a co-transition formula

We give a new proof that three families of polynomials coincide: the double Schubert polynomials of Lascoux and Sch"utzenberger defined by divided difference operators, the pipe dream polynomials of Bergeron and Billey, and the equivariant cohomology classes of matrix Schubert varieties. All three families are shown to satisfy a “co-transition formula” which we explain to be some extent projectively dual to Lascoux’ transition formula. We comment on the K-theoretic extensions.