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 am going to say how to apply these results to theory of torsors and their canonical dimensions. A torsor of an algebraic group G (over an arbitrary field, here this is important) is a scheme E with an action of G such that over a certain extension of the base field E becomes isomorphic to G, and the action becomes the action by left shifts of G on itself. The canonical dimension of a scheme X understood as a scheme is the minimal dimension of a subscheme Y of X such that there exists a rational map from X to Y. And the canonical dimension of an algebraic group G understood as a group is the maximum over all field extensions L of the base field of G of the canonical dimensions of all G_L-torsors. In my talk I am going to explain how to get estimates on canonical dimension of certain groups understood as groups using the result from the first part.
Rostislav Devyatov (Page professionnelle )
L’exposé sera retransmis via BBB (BigBlueButton).
Nom de réunion : « Sem AG 02-06-2020 : Rostislav Devyatov » ; contacter Luc Pirio pour le mot de passe.