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