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

${\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q$

is the polynomial in q defined by

${\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q=\frac{\prod _{i=1}^n(1+q+q^2+\cdots +q^{i-1})}{(\prod _{i=1}^k(1+q+q^2+\cdots +q^{i-1}))\cdot (\prod _{i=1}^{n-k}(1+q+q^2+\cdots +q^{i-1}))}.$

It is easily seen that its value at q = 1 is the usual binomial coefficient $(\genfrac{}{}{0ex}{}{n}{k})$, 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

${\left[\genfrac{}{}{0ex}{}{4}{2}\right]}_q=1+q+2q^2+q^3+q^4;$

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 $w_j>w_{j+1}$. Define the major index $\mathrm{maj}(w)$ on words as the sum of all descents.

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The triple $(X_n,C_{n-1},\frac{1}{[n+1{]}_q}{\left[\genfrac{}{}{0ex}{}{2n}{n}\right]}_q)$ exhibit a cyclic sieving phenomenon, 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 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 $(X,C,q^{-\kappa (\lambda )}{s}_{\lambda }(1,q,q^2,\dots ,q^{k-1}))$ exhibit the cyclic sieving phenomenon. 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 $1,2,\dots ,\ell$ for some $\ell$. Let $In{c}_k(2\times n)$ denote the set of increasing tableau with two rows of length n, and maximal entry $2n-k$. Then $({\mathrm{Inc}}_k(2\times n),C_{2n-k},q^{n+(\genfrac{}{}{0ex}{}{k}{2})}\frac{{\left[\genfrac{}{}{0ex}{}{n-1}{k}\right]}_q{\left[\genfrac{}{}{0ex}{}{2n-k}{n-k-1}\right]}_q}{[n-k{]}_q})$ exhibit the cyclic sieving phenomenon, 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 exceedances. Let $a_{\lambda ,j}(q)=\sum _{\sigma \in S_{\lambda ,j}}{q}^{\mathrm{maj}(\sigma )-\mathrm{exc}(\sigma )}$, 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 phenomenon.

Notes and references

• Sagan, Bruce (2011). "The cyclic sieving phenomenon: a survey". in Chapman, Robin. Surveys in Combinatorics 2011. London Mathematical Society Lecture Note Series. 392. Cambridge University Press. pp. 183–233. ISBN 978-1-139-50368-6. https://books.google.com/books?id=98-l4J34SFYC&pg=PA183.