The cohomology ring of the flag manifold has two distinguished bases: monomials in tautological Chern classes and classes of Schubert varieties. In 1982 Lascoux and Schützenberger gave formulas for these Schubert classes in terms of the monomial basis and used combinatorics to show the resulting polynomials have non negative coefficients. Geometrically, this positivity was a mystery: these monomials do not represent positive cycles, and existing combinatorial constructions had no relation to geometry.
The main goal of this talk is to outline a geometric proof of this positivity, identifying the coefficient of a monomial as an intersection number. This joint work with Nantel Bergeron begins by studying certain maps between flag manifolds, and then uses the Pieri formula for flag manifolds to compute the resulting maps on cohomology. This gives a new construction of these polynomial representatives and identifies the coefficient of a monomial as an intersection number.
We will close with a potential application of these maps to studying singularities of Schubert varieties via pattern avoidance in permutations.