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.

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 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).

List of cyclic sieving phenomena

In the Reiner-Stanon-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 $w_{j}>w_{j+1}$. Define the major index $maj(w)$ on words as the sum of all descents.

The triple $(W(\alpha ),C_{n},\sum _{w\in W(\alpha )}q^{maj(w)})$ exhibit a cyclic sieving phenomena, where the cyclic group acts via shift. Recall that $\sum _{w\in W(\alpha )}q^{maj(w)}={\binom {n}{\alpha _{1},\alpha _{2},\dotsc ,\alpha _{\ell }}}_{q}$, the q-multinomial coefficient. This formula is due to MacMahon.

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

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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 $(X,C,{\frac {[n]!_{q}}{\prod _{(i,j)\in \lambda }[h_{ij}]_{q}}})$ exhibit the cyclic sieving phenomena. 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 $(X,C,q^{-\kappa (\lambda )}s_{\lambda }(1,q,q^{2},\dotsc ,q^{k-1}))$ exhibit the cyclic sieving phenomena. Here, $\kappa (\lambda )=\sum _{i}(i-1)\lambda _{i}$ and s_λ is the Schur polynomial.

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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 $1,2,\dotsc ,\ell$ for some $\ell$. Let $Inc_{k}(2\times n)$ denote the set of increasing tableau with two rows of length n, and maximal entry $2n-k$. Then $(Inc_{k}(2\times n),C_{2n-k},{\frac {\left[{n-1 \atop k}\right]_{q}\left[{2n-k \atop n-k-1}\right]_{q}}{[n-k]_{q}}})$ exhibit the cyclic sieving phenomena, where $C_{2n-k}$ act via K-promotion.

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

Then $(S_{\lambda ,j},C_{n},a_{\lambda ,j}(q))$ exhibit the cyclic sieving phenomena.

Notes and references

• 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.