The cyclic sieving phenomenon: a survey

TL;DR

This survey reviews the cyclic sieving phenomenon, a combinatorial and algebraic concept linking group actions, polynomials, and roots of unity, highlighting its applications and deep connections to representation theory.

Contribution

It compiles and discusses the existing literature on cyclic sieving, emphasizing its algebraic foundations and recent developments.

Findings

Numerous instances of cyclic sieving have been identified.

The phenomenon often involves deep representation-theoretic proofs.

It connects combinatorics with algebra and representation theory.

Abstract

The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and f(q) be a polynomial in q with nonnegative integer coefficients. Then the triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g, and w is a root of unity chosen to have the same order as g. It might seem improbable that substituting a root of unity into a polynomial with integer coefficients would have an enumerative meaning. But many instances of the cyclic sieving phenomenon have now been found. Furthermore, the proofs that this phenomenon hold often involve interesting and sometimes deep results from representation theory. We will survey the current literature on cyclic sieving, providing the necessary background about representations, Coxeter groups, and other algebraic aspects as needed.