## A Pieri-type formulas for flag manifolds

### Frank Sottile,

University of Toronto

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.