Multi-Macdonald polynomials

Abstract

We introduce generalizations of Macdonald polynomials indexed by n-tuples of partitions and characterized by certain orthogonality and triangularity relations. We prove that they can be explicitly given as products of ordinary Macdonald polynomials depending on special alphabets. With this factorization in hand, we establish their most basic properties, such as explicit formulas for their norm-squared, evaluation and reproducing kernel. Moreover, we show that the q,t-Kostka coefficients associated to the multi-Macdonald polynomials are positive and correspond to q,t-analogs of the dimensions of the irreducible representations of the wreath product Cn∼Sd, where Cn is the cyclic group of order n.