The cyclic sieving phenomenon: a survey

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 ∈ C, we have #X = f(ω) where # denotes cardinality, X is the fixed point set of g, and ω 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. Acknowledgements The author would like to thank Vic Reiner, Martin Rubey, Bruce Westbury, and an anonymous referee for helpful suggestions. And he would particularly like to thank Vic Reiner for enthusiastic encouragement. This work was partially done while the author was a Program Officer at NSF. The views expressed are not necessarily those of the NSF. 1 What is the cyclic sieving phenomenon? Reiner, Stanton, and White introduced the cyclic sieving phenomenon in their seminal 2004 paper [58]. In order to define this concept, we need three ingredients. The first of these is a finite set, X. The second is a finite cyclic group, C, which acts on X. Given a group element g ∈ C, we denote its fixed point set by X = {y ∈ X : gy = y}. (1.1) We also let o(g) stand for the order of g in the group C. One group which will be especially important in what follows will be the group, Ω, of roots of unity. We let ωd stand for a primitive dth root of unity. The reader should think of g ∈ C and ωo(g) ∈ Ω as being linked because they have the same order in their respective groups. The final ingredient is a polynomial f(q) ∈ N[q], the set of polynomials in the variable q with nonnegative integer coefficients. Usually f(q) will be a generating function associated with X. Definition 1.1 The triple (X,C, f(q)) exhibits the cyclic sieving phenomenon (abbreviated CSP) if, for all g ∈ C, we have #X = f(ωo(g)) (1.2)