Cyclic sieving

In combinatorial mathematics, cyclic sieving is a phenomenon by which evaluating a generating function for a finite set at roots of unity counts symmetry classes of objects acted on by a cyclic group.^[1]

Definition

Let C be a cyclic group generated by an element c of order n. Suppose C acts on a set X. Let X(q) be a polynomial with integer coefficients. Then the triple (X, X(q), C) is said to exhibit the cyclic sieving phenomenon (CSP) if for all integers d, the value X(e^2πid/n) is the number of elements fixed by c^d. In particular X(1) is the cardinality of the set X, and for that reason X(q) is regarded as a generating function for X.

Examples

The q-binomial coefficient

$\left[{n \atop k}\right]_q$

is the polynomial in q defined by

$\left[{n \atop k}\right]_q = \frac{\prod_{i = 1}^n (1 + q + q^2 + \cdots + q^{i - 1})}{\left(\prod_{i = 1}^k (1 + q + q^2 + \cdots + q^{i - 1})\right) \cdot \left(\prod_{i = 1}^{n - k} (1 + q + q^2 + \cdots + q^{i - 1})\right)}.$

It is easily seen that its value at q = 1 is the usual binomial coefficient $\binom{n}{k}$, so it is a generating function for the subsets of {1, 2, ..., n} of size k. These subsets carry a natural action of the cyclic group C of order n which acts by adding 1 to each element of the set, modulo n. For example, when n = 4 and k = 2, the group orbits are

$\{1, 3\} \to \{2, 4\} \to \{1, 3\}$ (of size 2)

and

$\{1, 2\} \to \{2, 3\} \to \{3, 4\} \to \{1, 4\} \to \{1, 2\}$ (of size 4).

One can show^[2] that evaluating the q-binomial coefficient when q is an nth root of unity gives the number of subsets fixed by the corresponding group element.

In the example n = 4 and k = 2, the q-binomial coefficient is

$\left[{4 \atop 2}\right]_q = 1 + q + 2q^2 + q^3 + q^4;$

evaluating this polynomial at q = 1 gives 6 (as all six subsets are fixed by the identity element of the group); evaluating it at q = −1 gives 2 (the subsets {1, 3} and {2, 4} are fixed by two applications of the group generator); and evaluating it at q = ±i gives 0 (no subsets are fixed by one or three applications of the group generator).

Notes and references

1. Reiner, Victor; Stanton, Dennis; White, Dennis (February 2014), "What is... Cyclic Sieving?" (PDF), Notices of the American Mathematical Society 61 (2): 169–171, doi:10.1090/noti1084
2. V. Reiner, D. Stanton and D. White, The cyclic sieving phenomenon, Journal of Combinatorial Theory Series A, Volume 108 Issue 1, October 2004, Pages 17–50

• Sagan, Bruce. The cyclic sieving phenomenon: a survey. Surveys in combinatorics 2011, 183–233, London Math. Soc. Lecture Note Ser., 392, Cambridge Univ. Press, Cambridge, 2011.