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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Combinatorial description of the cohomology of the affine flag variety


Author: Seung Jin Lee
Journal: Trans. Amer. Math. Soc. 371 (2019), 4029-4057
MSC (2010): Primary 05E05, 05E15, 14N15
DOI: https://doi.org/10.1090/tran/7467
Published electronically: December 3, 2018
MathSciNet review: 3917216
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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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Additional Information

Seung Jin Lee
Affiliation: Department of Mathematical Sciences, Seoul National University, GwanAkRo 1, Gwanak-Gu, Seoul 08826, Republic of Korea
Email: lsjin@snu.ac.kr

Received by editor(s): April 9, 2017
Received by editor(s) in revised form: October 23, 2017, and November 18, 2017
Published electronically: December 3, 2018
Article copyright: © Copyright 2018 American Mathematical Society