Schubert polynomials and Hessenberg varieties

Double Schubert polynomials are representatives for Schubert classes in the equivariant cohomology of a flag variety, distinguished by their pleasant combinatorial properties. Hessenberg varieties are certain subvarieties of the flag variety; it turns out that their (ordinary) cohomology classes are specializations of double Schubert polynomials. This motivates a combinatorial problem: find the expansion of a specialized double Schubert polynomial in the basis of (single) Schubert polynomials. I'll describe some results in this direction. This is joint work with Julianna Tymoczko.