Macdonald polynomials and cyclic sieving

TL;DR

This paper connects Macdonald polynomials with cyclic sieving phenomena by using orbit harmonics on the Garsia--Haiman module to analyze fixed points under cyclic group actions on matrices.

Contribution

It introduces a novel application of orbit harmonics to establish cyclic sieving results related to Macdonald polynomials and matrix invariance under cyclic transformations.

Findings

Proves cyclic sieving phenomena for matrices invariant under cyclic rotations and translations.

Links Macdonald polynomials to fixed-point enumeration in cyclic group actions.

Provides new combinatorial interpretations of Macdonald polynomials.

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.