Constant term identities and Poincaré polynomials

Károlyi, Gyula; Lascoux, Alain; Warnaar, S.

doi:10.1090/tran/6119

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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