Vassily Gorbounov (University of Aberdeen), Physically Inspired Mathematics

Title: New feature of the Schubert calculus

Abstract:  In the talk we will describe a new feature of the Schubert calculus in the equivariant cohomology or K theory which holds for all types of the classical Lie groups. As the main example we will use the torus equivariant cohomology and 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 Bernstein Gelfand Gelfand Demazure difference operators in the equivariant cohomology of partial flag varieties. We will describe the operators of the appropriate quantum integrable system geometrically. In particular we show how the quantum equivariant cohomology appear in this geometric picture.