Cyclic Sieving Phenomenon Research Articles

Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux

We construct an injection from the set of r-fans of Dyck paths (resp. vacillating tableaux) of length n into the set of chord diagrams on [n] that intertwines promotion and rotation. This is done in two different ways, namely as fillings of promotion matrices and in terms of Fomin growth diagrams. Our analysis uses the fact that r-fans of Dyck paths and vacillating tableaux can be viewed as highest weight elements of weight zero in crystals of type Br and Cr, respectively, which in turn can be analyzed using virtual crystals. On the level of Fomin growth diagrams, the virtualization process corresponds to the Roby–Krattenthaler blow up construction. One of the motivations for finding rotation invariant diagrammatic bases such as chord diagrams is the cyclic sieving phenomenon. Indeed, we give a cyclic sieving phenomenon on r-fans of Dyck paths and vacillating tableaux using the promotion action.

Free Action and Cyclic Sieving on Skew Semi-standard Young Tableaux

In this note, we provide a short proof of Theorem 4.3 in the paper titled Crystals, semistandard tableaux and cyclic sieving phenomenon, by Y.-T. Oh and E. Park, which concerns a cyclic sieving phenomenon on semi-standard Young tableaux. We also extend their result to skew shapes.

Skew Schur Polynomials and Cyclic Sieving Phenomenon

Let $k$ and $m$ be positive integers and $\lambda/\mu$ a skew partition. We compute the principal specialization of the skew Schur polynomials $s_{\lambda /\mu}(x_1, \ldots, x_{k})$ modulo $q^m-1$ under suitable conditions. We interpret the results thus obtained from the viewpoint of the cyclic sieving phenomenon on semistandard Young skew tableaux of shape $\lambda/\mu$. As an application, we deal with evaluations of the principal specialization of the skew Schur polynomials at roots of unity.

Refined Catalan and Narayana cyclic sieving

We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type \(A\) and type \(B\). Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.Keywords: Dyck paths, cyclic sieving, Narayana numbers, major index, q-analog.Mathematics Subject Classifications: 05E18, 05A19, 05A30

Macdonald polynomials and cyclic sieving

The Garsia–Haiman module is a bigraded Sn-module whose Frobenius image is a Macdonald polynomial. The method of orbit harmonics promotes an Sn-set X to a graded polynomial ring. The orbit harmonics can be applied to prove cyclic sieving phenomena which is a notion that encapsulates the fixed-point structure of finite cyclic group action on a finite set. By applying this idea to the Garsia–Haiman module, we provide cyclic sieving results regarding the enumeration of matrices that are invariant under certain cyclic row and column rotation and translation of entries.

Dihedral sieving on cluster complexes

The cyclic sieving phenomenon of Reiner, Stanton, and White characterizes the stabilizers of cyclic group actions on finite sets using q-analogue polynomials. Eu and Fu demonstrated a cyclic sieving phenomenon on generalized cluster complexes of every type using the q-Catalan numbers. In this paper, we exhibit the dihedral sieving phenomenon, introduced for odd n by Rao and Suk, on clusters of every type. In the type A case, we show that the Raney numbers count both reflection-symmetric k-angulations of an n-gon and a particular evaluation of the (q,t)-Fuss--Catalan numbers. We also introduce a sieving phenomenon for the symmetric group, and discuss possibilities for dihedral sieving for even n.

Cyclic sieving and orbit harmonics

Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group G on a finite set X to a graded action of G on a polynomial ring quotient by viewing X as a G-stable point locus in \({\mathbb {C}}^n\). The cyclic sieving phenomenon is a notion in enumerative combinatorics which encapsulates the fixed-point structure of the action of a finite cyclic group C on a finite set X in terms of root-of-unity evaluations of an auxiliary polynomial X(q). We apply orbit harmonics to prove cyclic sieving results.

Formula omitted]-dimensions of highest weight crystals and cyclic sieving phenomenon

In this paper, we compute explicitly the q-dimensions of highest weight crystals modulo qn−1 for a quantum group of arbitrary finite type under certain assumption, and interpret the modulo computations in terms of the cyclic sieving phenomenon. This interpretation gives an affirmative answer to the conjecture by Alexandersson and Amini. As an application, under the assumption that λ is a partition of length <m and there exists a fixed point in SSTm(λ) under the action c arising from the crystal structure, we show that the triple (SSTm(λ),〈c〉,sλ(1,q,q2,…,qm−1)) exhibits the cycle sieving phenomenon if and only if λ is of the form ((am)b), where either b=1 or m−1. Moreover, in this case, we give an explicit formula to compute the number of all orbits of size d for each divisor d of n.

Cyclic Sieving for cyclic codes

Prompted by a question of Jim Propp, this paper examines the cyclic sieving phenomenon (CSP) in certain cyclic codes. For example, it is shown that, among dual Hamming codes over Fq, the generating function for codedwords according to the major index statistic (resp. the inversion statistic) gives rise to a CSP when q=2 or q=3 (resp. when q=2). A byproduct is a curious characterization of the irreducible polynomials in F2[x] and F3[x] that are primitive.

Promotion and cyclic sieving on families of SSYT

We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion. The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial. The second family we consider consists of skew shapes, consisting of rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier result by B. Rhoades and later B. Fontaine and J. Kamnitzer. Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in form of a bicyclic sieving phenomenon. One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occur with the same frequency.

Skew characters and cyclic sieving

AbstractIn 2010, Rhoades proved that promotion on rectangular standard Young tableaux, together with the associated fake-degree polynomial, provides an instance of the cyclic sieving phenomenon. We extend this result tom-tuples of skew standard Young tableaux of the same shape, for fixedm, subject to the condition that themth power of the associated fake-degree polynomial evaluates to nonnegative integers at roots of unity. However, we are unable to specify an explicit group action. Put differently, we determine in which cases themth tensor power of a skew character of the symmetric group carries a permutation representation of the cyclic group.To do so, we use a method proposed by Amini and the first author, which amounts to establishing a bound on the number of border-strip tableaux of skew shape. Finally, we apply our results to the invariant theory of tensor powers of the adjoint representation of the general linear group. In particular, we prove the existence of a bijection between permutations and Stembridge’s alternating tableaux, which intertwines rotation and promotion.

Parity-Unimodality and a Cyclic Sieving Phenomenon for Necklaces

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 29 September 2020Accepted: 16 May 2021Published online: 13 September 2021Keywordscombinatorics, Catalan, unimodal, cyclic sievingAMS Subject Headings05E10, 16G99, 20C30Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

Cyclic Sieving for Plane Partitions and Symmetry

The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.

Cyclic sieving, skew Macdonald polynomials and Schur positivity

When λ is a partition, the specialized non-symmetric Macdonald polynomial E λ (x;q;0) is symmetric and related to a modified Hall–Littlewood polynomial. We show that whenever all parts of the integer partition λ are multiples of n, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under an n-fold cyclic shift of the columns. The corresponding CSP polynomial is given by E λ (x;q;0). In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed. This refinement is closely related to an earlier result by B. Rhoades.

Cyclic sieving phenomenon on dominant maximal weights over affine Kac-Moody algebras

We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for level ℓ highest weight modules over an affine Kac-Moody algebra with exactly one highest weight being taken for each equivalence class, in a way not depending on types, ranks and levels. In order to do that, we introduce S-evaluation on the set of dominant maximal weights for each highest modules, and generalize Sagan's action in [17] by considering the datum on each affine Kac-Moody algebra. As consequences, we obtain closed and recursive formulae for cardinality of the number of dominant maximal weights for every highest weight module and observe level-rank duality on the cardinalities.

Cyclic Sieving and Cluster Duality of Grassmannian

We introduce a decorated configuration space $\mathscr{C}\!{\rm onf}_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $\big(\mathscr{C}\!{\rm onf}_n^\times(a), \mathcal{W}\big)$ canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian $\operatorname{Gr}_a(n)$ with respect to the Pl\"ucker embedding. We prove that $\big(\mathscr{C}\!{\rm onf}_n^\times(a), \mathcal{W}\big)$ is equivalent to the mirror Landau-Ginzburg model of the Grassmannian considered by Eguchi-Hori-Xiong, Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.

Tableau stabilization and rectangular tableaux fixed by promotion powers

We introduce tableau stabilization, a new phenomenon and statistic on Young tableaux based on jeu de taquin. We investigate bounds for tableau stabilization, the shape of stabilized tableaux, and tableau stabilization as a permutation statistic. We apply tableau stabilization to construct the sufficiently large rectangular tableaux fixed by powers of promotion, which were counted by Brendon Rhoades via the cyclic sieving phenomenon [(Rhoades in J Combin Theory Ser A 117(1):38–76, 2010), Theorem 1.3].

The Cyclic Sieving Phenomenon on Circular Dyck Paths

We give a $q$-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this $q$-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to Möbius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving, and introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of combinatorial objects exhibiting the cyclic sieving phenomenon.

Q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding [Formula: see text]-supercongruence. Similar [Formula: see text]-supercongruences are established for binomial coefficients and the Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the [Formula: see text]-Lucas numbers.

The cone of cyclic sieving phenomena

We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone corresponds to a non-negative integer matrix which jointly records the statistic and cyclic order distribution associated with the set of objects realizing the CSP. In particular we consider a universal subcone onto which every CSP matrix linearly projects such that the projection realizes a CSP with the same cyclic orbit structure, but via a universal statistic that has even distribution on the orbits.Reiner et.al. showed that every cyclic action gives rise to a unique polynomial (mod qn−1) complementing the action to a CSP. We give a necessary and sufficient criterion for the converse to hold. This characterization allows one to determine if a combinatorial set with a statistic gives rise (in principle) to a CSP without having a combinatorial realization of the cyclic action. We apply the criterion to conjecture a new CSP involving stretched Schur polynomials and prove our conjecture for certain rectangular tableaux. Finally we study some geometric properties of the CSP cone. We explicitly determine its half-space description and in the prime order case we determine its extreme rays.