Constant Term Identities and Poincaré Polynomials
نویسنده
چکیده
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 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.
منابع مشابه
A unified elementary approach to the Dyson, Morris, Aomoto, and Forrester constant term identities
We introduce an elementary method to give unified proofs of the Dyson, Morris, and Aomoto identities for constant terms of Laurent polynomials. These identities can be expressed as equalities of polynomials and thus can be proved by verifying them for sufficiently many values, usually at negative integers where they vanish. Our method also proves some special cases of the Forrester conjecture.
متن کاملKostka–foulkes Polynomials for Symmetrizable Kac–moody Algebras
We introduce a generalization of the classical Hall–Littlewood and Kostka–Foulkes polynomials to all symmetrizable Kac–Moody algebras. We prove that these Kostka–Foulkes polynomials coincide with the natural generalization of Lusztig’s t-analog of weight multiplicities, thereby extending a theorem of Kato. For g an affine Kac–Moody algebra, we define t-analogs of string functions and use Chered...
متن کاملA polynomial expression for the character of diagonal harmonics
Based on his study of the Hilbert scheme from algebraic geometry, Haiman [Invent. Math. 149 (2002), pp. 371–407] obtained a formula for the character of the space of diagonal harmonics under the diagonal action of the symmetric group, as a sum of Macdonald polynomials with rational coefficients. In this paper we show how Haiman’s formula, combined with identities involving plethystic symmetric ...
متن کاملHigher Order Degenerate Hermite-Bernoulli Polynomials Arising from $p$-Adic Integrals on $mathbb{Z}_p$
Our principal interest in this paper is to study higher order degenerate Hermite-Bernoulli polynomials arising from multivariate $p$-adic invariant integrals on $mathbb{Z}_p$. We give interesting identities and properties of these polynomials that are derived using the generating functions and $p$-adic integral equations. Several familiar and new results are shown to follow as special cases. So...
متن کاملViewing Some Ordinary Differential Equations from the Angle of Derivative Polynomials
In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.
متن کامل