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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