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See the cyclic sieving phenomenon page for the definition and related theorems.

#Miscellaneous

#Root posets

Let \(\Phi\) be a root system and \(\Phi^+\) its poset of positive roots. Armstrong–Stump–Thomas [AST13] studied the action of the rowmotion operator, also called the Fon-Der-Flaass/Panyushev map, on the antichains of \(\Phi^+.\) Addressing conjectures of Panyushev and Bessis–Reiner, they showed that this action is in equivariant bijection with the Kreweras complementation action on the \(\Phi\)-noncrossing partition lattice. Moreover, they showed that the \(q\)-\(\Phi\)-Catalan polynomial is a cyclic sieving polynomial for the action of rowmotion on the antichains of \(\Phi^+\) (equivalently, for the Kreweras complement acting on \(\Phi\)-noncrossing partitions). Note that the Type \(A\) case of this CSP is equivalent to the noncrossing matchings CSP.

Rowmotion can be realized as a certain composition of toggles on order ideals. By making these toggles piecewise-linear, one obtains a piecewise-linear action of rowmotion on height \(m\) \(P\)-partitions of a poset \(P.\) Hopkins (see [Hop24]) conjectured a CSP for the action of piecewise-linear rowmotion on these \(P\)-partitions when \(P\) is a root poset of coincidental type, where the sieving polynomial is the so-called \(q\)-\(\Phi\)-multi-Catalan number. This conjecture extends the aforementioned result of Armstrong–Stump–Thomas, which is the case \(m=1.\)

D. Kim studies \(q\)-Kreweras numbers for coincidental Coxeter groups attached to limit symbols [Kim20]. This gives another family of \(q\)-analogues in the same root-poset and noncrossing-partition setting.

D. Grinberg and T. Roby prove a birational rowmotion periodicity theorem for rectangular posets over suitable noncommutative rings [GR23]. Their result extends the field-valued periodicity theorem for products of two chains and includes a noncommutative antipodal reciprocity formula.

#Polygonal dissections and Frieze patterns

In [AB25], the authors prove a CSP for the set \(A_\mu,\) consisting of noncrossing dissections of an \((n+2)\)-gon, where the number of \(j\)-gons in the dissection is \(\mu_j.\) The group action is rotation of the \((n+2)\)-gon. This gives \[\left( A_\mu, C_{n+2}, \frac{1}{[n+1]_q} \qbinom{n+k}{k}\qbinom{k}{\mu_1,\mu_2,\dotsc,\mu_n} \right)\] where \(k-1\) is the number of noncrossing diagonals in the dissection.

There is then a certain bijection between noncrossing dissections of \((n+2)\)-gons and classes of frieze patterns, see [HJ18].

#Minuscule posets

In [RS12], the authors consider rowmotion acting on the antichains of minuscule posets. The corresponding CSP-polynomial is the rank-generating function of the poset, which for minuscule posets enjoys some particularly nice properties, see [Ex. 170, Sta11].

For the product of two chains, this action is a special case of the plane partitions CSP.

Hopkins (see [Hop24]) conjectured a CSP for the action of piecewise-linear rowmotion on height \(m\) \(P\)-partitions when \(P\) is a minuscule poset, where the sieving polynomial is the rank-generating function of \(P\times[m],\) which again has nice properties (in particular, a product formula). This conjecture extends the aforementioned result of Rush–Shi, which is the case \(m=1.\)

For plane partitions with additional symmetries and related cyclic-sieving conjectures, see [Hop20].

#BiCSP on \(0\)–\(1\)-matrices and Hall–Littlewood

In [BRS08], a bi-cyclic sieving phenomenon on permutation matrices is described. Let \(X_n\) be the set of \(n\times n\) permutation matrices, and let \(\grpc_n \times \grpc_n\) act on \(X_n\) by cyclic shifts of rows and columns. Then \[\left(X_n, \grpc_n \times \grpc_n, \epsilon(q,t)\sum_{\lambda \vdash n} f^\lambda(q) f^\lambda(t) \right)\] exhibits the bi-cyclic sieving phenomenon. Here, \(\epsilon(q,t)\) is \((qt)^{n/2}\) if \(n\) is even, and \(1\) if \(n\) is odd. In fact, they prove a much stronger statement regarding complex reflection groups.

This is further generalized in [Rho10] to \(0\)–\(1\)-matrices, where the column sums and row sums in the matrices are given by fixed integer partitions.

B. Rhoades further generalizes this to matrices with nonnegative integer entries [Rho10], and proves a bi-cyclic phenomenon using \[\epsilon(q,t)\sum_{\lambda \vdash n} K_{\lambda,\mu}(q) K_{\lambda,\nu}(t)\] as a biCSP polynomial. Note that this is closely related to the Robinson–Schensted–Knuth algorithm.

#Tensor products

In [Wes16], cyclic sieving phenomena are created via representation theory and crystals, generalizing the promotion CSP by B. Rhoades. This is related to the principal specialization of Frobenius image of characters of the symmetric group.

#Lattice polytopes

J. Propp asks for a solution to the following problem. Let \(P \subset \setR^d\) be a lattice polytope and \(g\) be a linear map \(\setR^d \to \setR^d,\) such that \(\langle g \rangle\) is a cyclic group of order \(n\) where \(g(P) = P.\)

Can we find a linear function \(\sigma:\setZ^d \to \setZ\) such that for the \(q\)-analogue of the Ehrhart function of \(P,\) defined as \[p_m(q) \coloneqq \sum_{x \in mP \cap \setZ^d} q^{\sigma(x)},\] such that \((mP \cap \setZ^d, \langle g \rangle, p_m(q))\) is a CSP-triple for all \(m \in \setN\)? That is, we get a family of cyclic sieving phenomena on the lattice points of dilations of \(P.\)

D. Han, X. Wang, H. Zhang, and S. Zhang study zero-sum polytopes associated with finite abelian groups, including reciprocity, rigidity, and cyclic sieving phenomena [HWZZ26].

#Independent sets of graphs

J. A. White gives cyclic sieving phenomena for independent sets of fixed size in graphs equipped with cyclic group actions [Whi26]. The main examples are powers of cycle graphs, and the same work gives closed formulas for the number of independent sets of fixed size in powers of cycles and paths. Whiskering gives a way to build new CSP examples from old ones, and recursive methods handle independent sets in gear graphs, helm graphs, and book graphs.

#Strong dichotomy classes

O. A. Agust{\'i}n-Aquino studies cyclic sieving in the enumeration of strong dichotomy classes, that is, bicolour self-complementary and rigid patterns in \(\setZ/2k\setZ\) [Agu26]. The paper proves that, for odd \(k,\) evaluating the rigid pattern-inventory polynomial at \(-1\) gives the negative of the number of strong classes.

#Hamming codes

A. Mason, V. Reiner and S. Sridhar have some results on cyclic codes and (dual) Hamming codes in [MRS20].

#Dominant maximal weights

A cyclic sieving phenomenon on dominant maximal weights in affine Kac–Moody algebras is described in [KOO20].

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