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Constant term identities and Poincaré polynomials


Authors: Gyula Károlyi, Alain Lascoux and S. Ole Warnaar
Journal: Trans. Amer. Math. Soc. 367 (2015), 6809-6836
MSC (2010): Primary 05A19, 05E05, 17B22, 20F55
DOI: https://doi.org/10.1090/tran/6119
Published electronically: June 24, 2015
MathSciNet review: 3378815
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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

DOI: https://doi.org/10.1090/tran/6119
Keywords: Constant term identities, Kadell's conjecture, Poincar\'e polynomials, polynomial lemma, $(0, 1)$-matrices
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
Article copyright: © Copyright 2015 American Mathematical Society