Rostislav Devyatov, Multiplicity-free Products of Schubert Divisors and an Application to Canonical Dimension

In the first part of my talk I am going to speak about Schubert calculus. Let G/B be a flag variety, where G is a linear simple algebraic group, and B is a Borel subgroup. Schubert calculus studies (in classical terms) multiplication in the cohomology ring of a flag variety over the complex numbers, or (in more algebraic terms) the Chow ring of the flag variety. This ring is generated as a group by the classes of so-called Schubert varieties (or their Poincare duals, if we speak about the classical cohomology ring), i. e. of the varieties of the form BwB/B, where w is an element of the Weyl group. As a ring, it is almost generated by the classes of Schubert varieties of codimension 1, called Schubert divisors. More precisely, the subring generated by Schubert divisors is a subgroup of finite index. These two facts lead to the following general question: how to decompose a product of Schubert divisors into a linear combination of Schubert varieties. In my talk, I am going to address (and answer if I have time) two more particular versions of this question: If G is of type A, D, or E, when does a coefficient in such a linear combination equal 0? When does it equal 1?

In the second part of my talk, I will define canonical dimension of varieties (which, roughly speaking, measures how hard it is to get a rational point in a given variety) and canonical dimension of algebraic groups (which, roughly speaking, measures how complicated the torsors of an algebraic group can be). Then I will state a theorem about an upper estimate on the canonical dimension of the group and its torsors following from the fact that a certain coefficient we obtained in the first part of my talk (i. e. the coefficient in the decomposition of a product of Schubert divisors into a linear combination of Schubert varieties) equals 1. As a result, we will get some explicit numerical estimates on canonical dimension of simply connected simple split algebraic groups of type A, D, and E.