Representation theory and mathematical physics seminar: New feature of the Schubert calculus

Seminar | April 21 | 4-5 p.m. | 939 Evans Hall

 Vasily Gorbunv, University of Aberdeen

 Department of Mathematics

In the talk we will describe a new feature of the classical Schubert calculus which holds for all types of the classical Lie groups. As the main example we will use the type A Grassmanians. The usual definition of the Schubert cycles involves a choice of a parameter, namely a choice of a full flag. Studying the dependence of the construction of the Schubert cycles on these parameters in the equivariant cohomology leads to an interesting 1 cocycle on the permutation group or a solution to the quantum Yang Baxter equation. This connects the Schubert calculus to the theory of quantum integrable systems. We show the above cocycle is the 'Baxterization' ( the term introduced by V. Jones) of the natural action of the nil Coxeter algebra of Berstein Gelfand Gelfand Demazure difference operators in the equivariant cohomology of partial flag varieties. We will outline some applications of this connection as well.