Combinatorial description of the cohomology of the affine flag variety

TL;DR

This paper develops an algebraic framework for affine Schubert calculus, introducing new operators and algebraic structures that connect cohomology, symmetric functions, and combinatorics in type A affine flag varieties.

Contribution

It constructs the affine FK algebra, introduces Murnaghan-Nakayama and Dunkl elements, and links cohomology of affine flag varieties to affine Schur functions and Schubert polynomials.

Findings

Affine FK algebra models affine Schubert calculus.

Affine Schubert polynomials are defined via BGG operators.

New combinatorial formulas for k-Schur functions and character tables.

Abstract

We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutative algebra generated by these operators is isomorphic to the cohomology of the affine flag variety. We show that the cohomology of the affine flag variety is product of the cohomology of an affine Grassmannian and a flag variety, which are generated by MN elements and Dunkl elements respectively. The Schubert classes in cohomology of the affine Grassmannian (resp. the flag variety) can be identified with affine Schur functions (resp. Schubert polynomials) in a quotient of the polynomial ring. Affine Schubert polynomials, polynomial representatives of the Schubert class in the cohomology of the affine flag variety, can be defined in the product of two quotient rings using the Bernstein-Gelfand-Gelfand operators interpreted as divided difference operators acting on the affine Fomin-Kirillov algebra. As for other applications, we obtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. We also define $k$-strong-ribbon tableaux from Murnaghan-Nakayama elements to provide a new formula of $k$-Schur functions. This formula gives the character table of the representation of the symmetric group whose Frobenius characteristic image is the $k$-Schur function.