Universal Schubert Polynomials

The aim of this paper is to introduce some polynomials that specialize to all previously known Schubert polynomials: the classical Schubert polynomials of Lascoux and Schützenberger [L-S], [M], the quantum Schubert polynomials of Fomin, Gelfand, and Postnikov [F-G-P], and quantum Schubert polynomials for partial flag varieties of Ciocan-Fontanine [CF2]. There are also double versions of these universal Schubert polynomials that generalize the previously known double Schubert polynomials [L-S], [M], [K-M], [CF-F]. They describe degeneracy loci of maps of vector bundles, but in a more general setting than the previously known setting of [F2]. These universal Schubert polynomials possess many but not all algebraic properties of their classical specializations. Their extra structure make them useful for studying their specializations, as it can be easier to find patterns before variables are specialized. The main geometric setting to which these polynomials apply is the following. We have maps of vector bundles