Explicit Plethystic Formulas for Macdonald q,t-Kostka Coefficients

Adriano Garsia ; Mark Haiman ; Glenn Tesler

Séminaire lotharingien de combinatoire, Tome 42 (1998-1999)

Résumé

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.