Mathematics Colloquia and Seminars

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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 Grassmannians. 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 outline some applications of this connection as well.