Cyclic sieving and orbit harmonics

TL;DR

This paper connects orbit harmonics from combinatorial representation theory with the cyclic sieving phenomenon in enumerative combinatorics, providing a new approach to prove cyclic sieving results using algebraic tools.

Contribution

It introduces a novel application of orbit harmonics to establish cyclic sieving results, bridging algebraic and combinatorial methods.

Findings

Orbit harmonics can be used to prove cyclic sieving phenomena.

The approach links group actions on sets to polynomial evaluations at roots of unity.

New proofs of cyclic sieving results are provided using algebraic techniques.

Abstract

Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point locus in $\mathbb{C}^n$. The cyclic sieving phenomenon is a notion in enumerative combinatorics which encapsulates the fixed-point structure of the action of a finite cyclic group $C$ on a finite set $X$ in terms of root-of-unity evaluations of an auxiliary polynomial $X(q)$. We apply orbit harmonics to prove cyclic sieving results.