Cyclic sieving and orbit harmonics

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.