Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Combinatorial description of the cohomology of the affine flag variety   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.

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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: q-alg
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The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph