We describe an approach to the Schubert calculus on full flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope. We also prove a formula for the Demazure characters of a given representation of GL(n) via exponential sums over integral points in faces of the Gelfand-Zetlin polytope associated with the representation. Time permitting, I will also discuss a (mostly conjectural) generalization of our approach that would allow to describe the K-theory of a full flag variety. I am planning to introduce (or recall) all the definitions on flag varieties during the talk, no prerequisites are necessary. The talk is based on a joint work with V.Kiritchenko and V.Timorin.
Thursday, November 2, 2017