Constructions for Cyclic Sieving Phenomena

Berget, Andrew; Eu, Sen Peng; Reiner, Victor

doi:10.1137/100803596

Cited by 12 publications

(13 citation statements)

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“…Since the sets W α are precisely the S n -orbits for the natural S n action on length n words, Theorem 1.2 may be thought of as a "universal sieving result" as follows. A very similar observation appeared in [BER11,Prop. 3.1].…”

Section: Introductionsupporting

confidence: 81%

Cyclic Sieving, Necklaces, and Branching Rules Related to Thrall's Problem

Ahlbach

Swanson

2018

Electron. J. Combin.

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We show that the cyclic sieving phenomenon of Reiner-Stanton-White together with necklace generating functions arising from work of Klyachko offer a remarkably unified, direct, and largely bijective approach to a series of results due to Kraśkiewicz-Weyman, Stembridge, and Schocker related to the so-called higher Lie modules and branching rules for inclusions Ca ≀ S b ֒→ S ab . Extending the approach gives monomial expansions for certain graded Frobenius series arising from a generalization of Thrall's problem. Lemma 1.4. [AS18, Lemma 8.3] Let W be a finite set of length n words closed under the C n -action, where C n acts by cyclic rotations. Then, the triple W, C n , W flex (q) exhibits the CSP.

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Section: Introductionsupporting

confidence: 81%

Cyclic Sieving, Necklaces, and Branching Rules Related to Thrall's Problem

Ahlbach

Swanson

2018

Electron. J. Combin.

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“…The corresponding hooklength is h i,j = #H i,j . The hook of (2, 2) in λ = (5,4,4,2) is indicated by crosses is the following diagram…”

Section: Also Definementioning

confidence: 99%

“…Define (i, j) to be a corner of λ if neither (i + 1, j) nor (i, j + 1) is in λ. The corners of λ displayed in (7.1) are (1,5), (3,4), and (4, 2). Given T ∈ SYT(λ) we define its promotion, ∂T , by an algorithm.…”

Section: Also Definementioning

confidence: 99%

The cyclic sieving phenomenon: a survey

Sagan¹

2011

Surveys in Combinatorics 2011

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The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. It was partly motivated as an extension of the q = −1 phenomenon introduced earlier by Stembridge. We give a survey of the current literature on cyclic sieving. The techniques used to prove that the phenomenon holds are discussed, including ideas from representation theory such as Springer's Theorem on regular elements in complex reflection groups.

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“…Since Reiner, Stanton, and White introduced the cyclic sieving phenomenon (CSP) in 2004 [RSW04], it has become an important companion to any cyclic action on a finite set. Some remarkable examples of the CSP involve the action of a Springer regular element on Coxeter groups [RSW04, Theorem 1.6], the action of Schutzenberger's promotion on Young tableaux of fixed rectangular shape [Rho10], and the creation of new CSPs from old using multisets and plethysms with homogeneous symmetric functions [BER11,Proposition 8]. See [Sag11] for Sagan's thorough introduction to the cyclic sieving phenomenon.…”

Section: Introductionmentioning

confidence: 99%

Refined cyclic sieving on words for the major index statistic

Ahlbach

Swanson

2018

European Journal of Combinatorics

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Reiner-Stanton-White [RSW04] defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and a polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content.There is a natural notion of refinement for many CSP's. We formulate and prove a refinement, with respect to the major index statistic, of this CSP on words of fixed content by also fixing the cyclic descent type. The argument presented is completely different from Reiner-Stanton-White's representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective in a sense we make precise with a "universal" sieving statistic on words, flex.A building block of our argument involves cyclic sieving for shifted subset sums, which also appeared in Reiner-Stanton-White. We give an alternate, largely bijective proof of a refinement of this result by extending some ideas of Wagon-Wilf [WW94].

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Constructions for Cyclic Sieving Phenomena

Abstract: We show how to derive new instances of the cyclic sieving phenomenon from old ones via elementary representation theory. Examples are given involving objects such as words, parking functions, finite fields, and graphs.

Cited by 12 publications

References 13 publications

Cyclic Sieving, Necklaces, and Branching Rules Related to Thrall's Problem

Cyclic Sieving, Necklaces, and Branching Rules Related to Thrall's Problem

The cyclic sieving phenomenon: a survey

Refined cyclic sieving on words for the major index statistic

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