The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special case, the Schubert calculus for Grassmannians, where one asks for the number of linear spaces of given dimensio...

We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary symmetric polynomial or a complete homogeneous symmetric polynomial. Thus, we generalize the classical Pieri’s rule for symmetric polynomials/Grassmann varietie...

We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for intersection numbers of the flag variety. Our argument utilizes recent developments in study Schubitopes, which are Newton polytopes polynomials. The resulting tableau test executes polynomial time.