Constant term identities and Poincaré polynomials

Abstract

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 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.

Constant Terms of Near-Dyson Polynomials

“…Next we give simple proofs of the master theorem and its transitive analogue from [2] using a similar technique. A similar proof in a different context can be found in [10] (see Theorems 1.2 and 3.5), though our proof is more direct.…”

A recursion for a symmetric function generalization of the $q$-Dyson constant term identity

“…where n k is a q-binomial coefficient. In fact Kadell only considered v = (r, 0 n−1 ) in his conjecture, but the more general statement given above is what was proved by Károlyi, Lascoux and Warnaar in [11,Theorem 1.3] using multivariable Lagrange interpolation and key polynomials. If for a sequence u = (u 1 , .…”

Macdonald symmetric functions of rectangular shapes

“…We like to mention that this splitting formula may be helpful in the computation of the non-constant term of F β,q [s; 0]. After this study was finished, we found that the first proof of Kadell's conjecture was given in 2012 [12]. Our proofs here are elementary.…”