The Littlewood-Richardson problem for flag varieties/Schubert polynomials is one of the outstanding problems in the Theory of Schubert polynomials. Nantel Bergeron and I have been studying this recently. In this talk, I will report on our progress to date, concentrating on the role of Pieri-type formulas and the interaction of the combinatorics and geometry of flag manifolds.
I will first discuss the situation for the classical flag manifold: The Pieri-type formula, its relation to the combinatorics of the Bruhat order, and some identities of the Littlewood-Richardson coefficients. The main porton of the talk will deal with our recent work on these same questions for the symplectic flag manifolds.
Specifically, We give the formula for the multiplication of an arbitrary Schubert class in the cohomology of a symplectic flag manifold by a special Schubert class pulled back from the Lagrangian Grassmannian. This formula is expressed in both terms of chains in the Bruhat order, and in terms of the cycle structure a certain permutations, showing it to be a common generalization of the Pieri-type formula for the Lagrangian Grassmannian and that for the ordinary flag manifold. Our proof uses results on the Bruhat order, identities of structure constants and intersections of Schubert varieties.