Schubert polynomials and Quiver formulas

Fulton's Universal Schubert polynomials represent general degeneracy loci for maps of vector bundles with rank conditions coming from a permutation. The Buch-Fulton Quiver formula expresses this polynomial as an integer linear combination of products of Schur polynomials in the differences of the bundles. We present a positive combinatorial formula for the coefficients which are known to directly generalize the Littlewood-Richardson coefficients. Our formula counts sequences of semi-standard Young tableaux satisfying certain conditions. One consequence is a nonrecursive description of the Fomin-Gelfand-Postnikov quantization map for flag manifolds and the generalization to partial flag manifolds due to Ciocan-Fontanine.

This is joint work with Anders Buch, Andrew Kresch and Harry Tamvakis.