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.
Highlights
Fomin and Kirillov defined a certain quadratic algebra, called the Fomin-Kirillov algebra, to better understand the combinatorics of the cohomology ring of the flag variety. They showed that the commutative subalgebra generated by Dunkl elements of degree 1 is isomorphic to the cohomology of the flag variety
We introduce the affine Fomin-Kirillov algebra, generalizing the Fomin-Kirillov algebra to affine type A, to describe the cohomology of the affine flag variety
We show that MurnaghanNakayama elements and Dunkl elements commute with each other as Bruhat operators and show that the algebra generated by these elements as Bruhat operators is isomorphic to the cohomology of the affine flag variety as subalgebras in HomQ(A, A)
Summary
Fomin and Kirillov defined a certain quadratic algebra, called the Fomin-Kirillov algebra, to better understand the combinatorics of the cohomology ring of the flag variety. They showed that the commutative subalgebra generated by Dunkl elements of degree 1 is isomorphic to the cohomology of the flag variety. 744 for K-theory, quantum, equivariant cohomology and for other finite types In this extended abstract, we introduce the affine Fomin-Kirillov algebra (affine FK algebra in short), generalizing the Fomin-Kirillov algebra to affine type A, to describe the cohomology of the affine flag variety. The identification combines algebraic, geometric and combinatorial components of affine Schubert calculus Those three operators will be considered as elements in HomQ(A, A). The coefficient ring of the cohomology is Q, and related combinatorics will be adjusted the three operators are well-defined over Z
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