Abstract:
Let G/B be a flag variety over C, where G is a simple algebraic group with a simply laced Dynkin diagram, and B is a Borel subgroup. The Bruhat decomposition of G defines subvarieties of G/B called Schubert subvarieties. The codimension 1 Schubert subvarieties are called Schubert divisors. We study the Chow ring of G/B (it is isomorphic to the cohomology ring of G/B viewed as a manifold with classical topology). This ring is generated as an abelian group by the classes of all Schubert varieties, and is "almost" generated as a ring by the classes of Schubert divisors. More precisely, an integer multiple of each element of G/B can be written as a polynomial in Schubert divisors with integer coefficients. In particular, each product of Schubert divisors is a linear combination of Schubert varieties with integer coefficients.
In the first part of my talk I am going to speak about the coefficients of these linear combinations. In particular, I am going to explain how to check if a coefficient of such a linear combination is nonzero and give an idea how to check if such a coefficient equals 1. If there is time left, then in the second part of my talk, I will say something about an application of my result, namely, how it makes it possible estimate so-called canonical dimension of flag varieties and groups over non-algebraically-closed fields.
427 Thackeray Hall