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

${\displaystyle \left[{n \atop k}\right]_{q}}$

is the polynomial in q defined by

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

It is easily seen that its value at q = 1 is the usual binomial coefficient ${\displaystyle {\binom {n}{k}}}$, 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

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

and

${\displaystyle \{1,2\}\to \{2,3\}\to \{3,4\}\to \{1,4\}\to \{1,2\}}$ (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

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

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).

List of cyclic sieving phenomena

In the Reiner–Stanton–White paper, the following example is given:

Let α be a composition of n, and let W(α) be the set of all words of length n with α_i letters equal to i. A descent of a word w is any index j such that ${\displaystyle w_{j}>w_{j+1}}$. Define the major index ${\displaystyle \operatorname {maj} (w)}$ on words as the sum of all descents.

The triple ${\displaystyle (X_{n},C_{n-1},{\frac {1}{[n+1]_{q}}}\left[{2n \atop n}\right]_{q})}$ exhibit a cyclic sieving phenomenon, where ${\displaystyle X_{n}}$ is the set of non-crossing (1,2)-configurations of [n − 1].^[3]

Let λ be a rectangular partition of size n, and let X be the set of standard Young tableaux of shape λ. Let C = Z/nZ act on X via promotion. Then ${\displaystyle (X,C,{\frac {[n]!_{q}}{\prod _{(i,j)\in \lambda }[h_{ij}]_{q}}})}$ exhibit the cyclic sieving phenomenon. Note that the polynomial is a q-analogue of the hook length formula.

Furthermore, let λ be a rectangular partition of size n, and let X be the set of semi-standard Young tableaux of shape λ. Let C = Z/kZ act on X via k-promotion. Then ${\displaystyle (X,C,q^{-\kappa (\lambda )}s_{\lambda }(1,q,q^{2},\dotsc ,q^{k-1}))}$ exhibit the cyclic sieving phenomenon. Here, ${\displaystyle \kappa (\lambda )=\sum _{i}(i-1)\lambda _{i}}$ and s_λ is the Schur polynomial.^[4]

An increasing tableau is a semi-standard Young tableau, where both rows and columns are strictly increasing, and the set of entries is of the form ${\displaystyle 1,2,\dotsc ,\ell }$ for some ${\displaystyle \ell }$. Let ${\displaystyle Inc_{k}(2\times n)}$ denote the set of increasing tableau with two rows of length n, and maximal entry ${\displaystyle 2n-k}$. Then ${\displaystyle (\operatorname {Inc} _{k}(2\times n),C_{2n-k},q^{n+{\binom {k}{2}}}{\frac {\left[{n-1 \atop k}\right]_{q}\left[{2n-k \atop n-k-1}\right]_{q}}{[n-k]_{q}}})}$ exhibit the cyclic sieving phenomenon, where ${\displaystyle C_{2n-k}}$ act via K-promotion.^[5]

Let ${\displaystyle S_{\lambda ,j}}$ be the set of permutations of cycle type λ and exactly j exceedances. Let ${\displaystyle a_{\lambda ,j}(q)=\sum _{\sigma \in S_{\lambda ,j}}q^{\operatorname {maj} (\sigma )-\operatorname {exc} (\sigma )}}$, and let ${\displaystyle C_{n}}$ act on ${\displaystyle S_{\lambda ,j}}$ by conjugation.

Then ${\displaystyle (S_{\lambda ,j},C_{n},a_{\lambda ,j}(q))}$ exhibit the cyclic sieving phenomenon.^[6]

Notes and references