Combinatorial description of the cohomology of the affine flag variety

Abstract

We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We introduce Murnaghan-Nakayama elements and Dunklelements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutativealgebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, weobtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to expressk-Schur functions in terms of power sum symmetric functions. We also provide the defi-nition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of theaffine flag variety.