Abstract:

Wreath Macdonald polynomials were defined by Haiman in 2003 as generalizations of the modified Macdonald polynomials whereby the symmetric groups are replaced with their wreath product with a cyclic group of fixed order r. Just as how the representationtheory of the symmetric groups can be studied in terms of the ring of symmetric functions, the representation theory of these wreath products can be described by its rth tensor power; thus, wreath Macdonald polynomials can be considered as partially symmetric functions.

I will present joint work with M. Romero that establishes a wreath analogue of Macdonald reciprocity. While our methods are not combinatorial, the combinatorics of r-cores and r-quotients plays a key role in describing the results. As an application, I will present joint work with S. Albion Ferlinc that proves a (q,t)-analogue of the modular Nekrasov--Okounkov formula.