The Cyclic Sieving Phenomenon on Circular Dyck Paths

Alexandersson, Per; Linusson, Svante; Potka, Samu

doi:10.37236/8720

Cited by 5 publications

(7 citation statements)

References 18 publications

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“…One can verify that this definition agrees with the one given in [2]. Despite the name, the specialization E λ (x; q, 0) is in fact a symmetric polynomial (3) as we shall see below.…”

Section: Definition 211 a Burge Word Is A Two-line Array With Positsupporting

confidence: 78%

“…. , this family is a Lyndon-like family, a notion by P. Alexandersson, S. Linusson and S. Potka [3] (see also [19]) meaning that fixed points in COF(nλ, m) under φ k are in natural bijection with the elements in COF( n k λ, m) whenever k | n. When λ = (1), this phenomenon reduces to a classical cyclic sieving phenomenon on words of length n in the alphabet [m], see Example 2.10 below. A skew version of (1) is given in Theorem 5.10.…”

Section: Per Alexandersson and Joakim Uhlinmentioning

confidence: 94%

“…Definition 2.9 (Lyndon-like CSP, [3]). Let {X n } ∞ n=1 be a family of combinatorial objects with a cyclic group action C n acting on X n .…”

Section: 4mentioning

confidence: 99%

“…One can show that the group action on a Lyndon-like family X n corresponds to rotation on some set of words of length n, see [3,Prop. 34].…”

Section: 4mentioning

confidence: 99%

“…We use English notation rather than skyline diagrams. (3) There is an extension of the notion of inversion triples to diagrams indexed by weak compositions and then one has that E λ (x; q, 0) is independent of the order of entries in λ, see [2,Prop. 17].…”

Section: Definition 211 a Burge Word Is A Two-line Array With Positmentioning

confidence: 99%

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Cyclic sieving, skew Macdonald polynomials and Schur positivity

Alexandersson¹,

Uhlin²

2020

Algebraic Combinatorics

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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. We also introduce a skew version of E λ (x; q; 0). We show that these are symmetric and Schur positive via a variant of the Robinson-Schenstedt-Knuth correspondence and we also describe crystal raising and lowering operators for the underlying fillings. Moreover, we show that the skew specialized non-symmetric Macdonald polynomials are in some cases vertical-strip LLT polynomials. As a consequence, we get a combinatorial Schur expansion of a new family of LLT polynomials. 1.1. Main results. For an integer partition λ = (λ 1 ,. .. , λ), we let nλ denote the partition (nλ 1 ,. .. , nλ). We show that there is a natural action φ on the fillings COF(nλ, m) where each block of n consecutive columns is cyclically rotated one step. Consequently φ generates a C n-action on COF(nλ, m). In Theorem 3.2, we prove that for every n, m ∈ N + and integer partition λ, the triple (1)

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“…One can verify that this definition agrees with the one given in [2]. Despite the name, the specialization E λ (x; q, 0) is in fact a symmetric polynomial (3) as we shall see below.…”

Section: Definition 211 a Burge Word Is A Two-line Array With Positsupporting

confidence: 78%

Section: Per Alexandersson and Joakim Uhlinmentioning

confidence: 94%

“…Definition 2.9 (Lyndon-like CSP, [3]). Let {X n } ∞ n=1 be a family of combinatorial objects with a cyclic group action C n acting on X n .…”

Section: 4mentioning

confidence: 99%

“…One can show that the group action on a Lyndon-like family X n corresponds to rotation on some set of words of length n, see [3,Prop. 34].…”

Section: 4mentioning

confidence: 99%

Section: Definition 211 a Burge Word Is A Two-line Array With Positmentioning

confidence: 99%

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Cyclic sieving, skew Macdonald polynomials and Schur positivity

Alexandersson¹,

Uhlin²

2020

Algebraic Combinatorics

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Skew characters and cyclic sieving

Alexandersson

Pfannerer

Rubey

et al. 2021

Forum of Mathematics, Sigma

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In 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 to m-tuples of skew standard Young tableaux of the same shape, for fixed m, subject to the condition that the mth 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 the mth 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.

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Cyclic Sieving for Plane Partitions and Symmetry

Hopkins

2020

SIGMA

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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.

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The Cyclic Sieving Phenomenon on Circular Dyck Paths

Cited by 5 publications

References 18 publications

Cyclic sieving, skew Macdonald polynomials and Schur positivity

Cyclic sieving, skew Macdonald polynomials and Schur positivity

Skew characters and cyclic sieving

Cyclic Sieving for Plane Partitions and Symmetry

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