Multi-Macdonald polynomials

TL;DR

This paper introduces multi-Macdonald polynomials, providing explicit formulas, fundamental properties, and demonstrating their positivity and representation-theoretic significance related to wreath products.

Contribution

It defines multi-Macdonald polynomials, expresses them as products of ordinary Macdonald polynomials, and explores their properties and connections to representation theory.

Findings

Explicit formulas for norm-squared, evaluation, and reproducing kernel.

Positivity of q,t-Kostka coefficients and their interpretation as dimensions.

Connection to irreducible representations of wreath products.

Abstract

We introduce 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 $C_n \sim S_d$, the wreath product of the cyclic group $C_n$ with the symmetric group.