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Séminaire Lotharingien de Combinatoire, B42m (1999), 45 pp.

# Adriano Garsia, Mark Haiman, Glenn Tesler

#
Explicit Plethystic Formulas for Macdonald *q*,*t*-Kostka Coefficients

**Abstract.**
In previous work Garsia and Tesler proved that the Macdonald
*q*,*t*-Kostka coefficients have a rather simple plethystic
representation. To be precise, they expressed the *q*,*t*-Kostka
coefficient indexed by the pair of partitions lambda and mu as a
symmetric polynomial *k*(*x*;*q*,*t*),
depending only on lambda, plethystically
evaluated at a polynomial *B*(*q*,*t*),
depending only on mu. Garsia and
Tesler gave an algorithm for the construction of the polynomial
*k*(*x*;*q*,*t*)
and derived from it the first proof of the Macdonald
polynomiality conjecture. Our main result here is a relatively
simple, entirely explicit formula for the polynomial
*k*(*x*;*q*,*t*). The
basic ingredient in this formula is the operator ``Nabla'' that has
emerged as an ubiquitous element in the recent representation
theoretical study of Macdonald polynomials carried out by F. & N.
Bergeron, Garsia, Haiman and Tesler. Further properties of Nabla are
developed here, along with a mini-theory of plethystic operators with
promising significant implications within the theory of symmetric
functions. One of the byproducts of these developments is a new
derivation of the symmetric function results of Sahi and Knop, which
throws a new light on their connection to Macdonald Theory.

Received: December 17, 1998; Accepted: April 30, 1999.

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