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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Combinatorial description of the cohomology of the affine flag variety
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by Seung Jin Lee PDF
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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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
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 4029-4057
  • MSC (2010): Primary 05E05, 05E15, 14N15
  • DOI: https://doi.org/10.1090/tran/7467
  • MathSciNet review: 3917216