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.

arXiv commentA new result added, 13 pages