Cyclic sieving

In combinatorial mathematics, cyclic sieving is a phenomenon in which an integer polynomial evaluated at certain roots of unity counts the rotational symmetries of a finite set.

[1] Given a family of cyclic sieving phenomena, the polynomials give a q-analogue for the enumeration of the sets, and often arise from an underlying algebraic structure, such as a representation.

The first study of cyclic sieving was published by Reiner, Stanton and White in 2004.

[2] The phenomenon generalizes the "q = −1 phenomenon" of John Stembridge, which considers evaluations of the polynomial only at the first and second roots of unity (that is, q = 1 and q = −1).

be a finite set with an action of the cyclic group

exhibits the cyclic sieving phenomenon (or CSP) if for every positive integer

exhibits the cyclic sieving phenomenon if the number of elements in

which increases each element in the pair by one (and sends

exhibits the cyclic sieving phenomenon.

which is an integer polynomial evaluating to the usual binomial coefficient at

acting by increasing each element in the subset by one (and sending

exhibits the cyclic sieving phenomenon for every

[4] The cyclic sieving phenomenon can be naturally stated in the language of representation theory.

is linearly extended to obtain a representation, and the decomposition of this representation into irreducibles determines the required coefficients of the polynomial

be the vector space over the complex numbers with a basis indexed by a finite set

, then linearly extending each action turns

exhibits the cyclic sieving phenomenon if and only if

exhibits the cyclic sieving phenomenon if and only if

exhibits the cyclic sieving phenomenon.

be the set of standard Young tableaux with shape

Jeu de taquin promotion gives an action of

be the following q-analog of the hook length formula:

exhibits the cyclic sieving phenomenon.

is the character for the irreducible representation of the symmetric group associated to

is the set of semistandard Young tableaux of shape

, then promotion gives an action of the cyclic group

exhibits the cyclic sieving phenomenon.

be the set of semi-standard Young tableaux of shape

, where entries along each row and column are strictly increasing.

be the set of permutations of cycle type

A q -analogue of the hook length formula exhibits cyclic sieving, with evaluations at roots of unity counting the number of rectangular standard Young tableaux fixed by repeated applications of jeu de taquin promotion.