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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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Constant term identities and Poincaré polynomials
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by Gyula Károlyi, Alain Lascoux and S. Ole Warnaar PDF
Trans. Amer. Math. Soc. 367 (2015), 6809-6836 Request permission

Abstract:

In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald’s constant term identities admit an extra set of free parameters, thereby linking them to Poincaré polynomials. We then exploit these extra degrees of freedom in the case of type $\mathrm {A}$ to give the first proof of Kadell’s orthogonality conjecture—a symmetric function generalisation of the $q$-Dyson conjecture or Zeilberger–Bressoud theorem.

Key ingredients in our proof of Kadell’s orthogonality conjecture are multivariable Lagrange interpolation, the scalar product for Demazure characters and $(0,1)$-matrices.

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Additional Information
  • Gyula Károlyi
  • Affiliation: School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia – and – Institute of Mathematics, Eötvös University, Budapest, Hungary
  • Alain Lascoux
  • Affiliation: CNRS, Institut Gaspard Monge, Université Paris-Est, Marne-la-Vallée, France
  • S. Ole Warnaar
  • Affiliation: School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia
  • MR Author ID: 269674
  • Received by editor(s): October 5, 2012
  • Received by editor(s) in revised form: February 27, 2013
  • Published electronically: June 24, 2015
  • Additional Notes: This work was supported by the Australian Research Council and by Hungarian National Scientific Research Funds (OTKA) Grant K100291.
    We deeply regret that Alain Lascoux passed away on October 20, 2013
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 367 (2015), 6809-6836
  • MSC (2010): Primary 05A19, 05E05, 17B22, 20F55
  • DOI: https://doi.org/10.1090/tran/6119
  • MathSciNet review: 3378815