Combinatorial description of the cohomology of the affine flag variety

Lee, Seung Jin

doi:10.1090/tran/7467

Combinatorial description of the cohomology of the affine flag variety
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Trans. Amer. Math. Soc. 371 (2019), 4029-4057 Request permission

Abstract:

We define a polynomial representative of the Schubert class in the cohomology of an affine flag variety associated to $SL(n)$, called an affine Schubert polynomial. Affine Schubert polynomials are defined by using divided difference operators, generalizing those operators used to define Schubert polynomials, so that Schubert polynomials are special cases of affine Schubert polynomials. Also, affine Stanley symmetric functions can be obtained from affine Schubert polynomials by setting certain variables to zero. We study affine Schubert polynomials and divided difference operators by constructing an affine analogue of the Fomin-Kirillov algebra called an affine Fomin-Kirillov algebra. We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine Fomin-Kirillov algebra to describe the cohomology of the affine flag variety and affine Schubert polynomials, and by doing so we also obtain a Murnaghan-Nakayama rule for the affine Schubert polynomials.

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