Macdonald polynomials and cyclic sieving

Abstract

The Garsia--Haiman module is a bigraded ${\mathfrak{S}}_n$-module whose Frobenius image is a Macdonald polynomial. The method of orbit harmonics promotes an ${\mathfrak{S}}_n$-set $X$ to a graded polynomial ring. The orbit harmonics can be applied to prove cyclic sieving phenomena which is a notion that encapsulates the fixed-point structure of finite cyclic group action on a finite set. By applying this idea to the Garsia--Haiman module, we provide cyclic sieving results regarding the enumeration of matrices that are invariant under certain cyclic row and column rotation and translation of entries.

A new result added, 13 pages