Séminaire Lotharingien de Combinatoire, B42m (1999), 45 pp.
Adriano Garsia, Mark Haiman, Glenn Tesler
Explicit Plethystic Formulas for Macdonald q,t-Kostka Coefficients
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
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
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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