Cyclic sieving, skew Macdonald polynomials and Schur positivity

Alexandersson, Per; Uhlin, Joakim

doi:10.48550/arxiv.1908.00083

Cited by 6 publications

(34 citation statements)

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“…We remark that stretching shapes seems to be a fruitful way to construct cyclic sieving phenomena, as was previously shown with fillings related to Macdonald polynomials by P. Alexandersson & J. Uhlin [AU19]. Possibly an adaptation of the approach presented here can be used to settle the following conjecture:…”

Section: Cyclic Sieving For Skew Standard Tableauxsupporting

confidence: 63%

Skew characters and cyclic sieving

Alexandersson¹,

Pfannerer²,

Rubey³

et al. 2020

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In 2010, B. Rhoades proved that promotion together with the fakedegree polynomial associated with rectangular standard Young tableaux give an instance of the cyclic sieving phenomenon.We extend this result to all skew standard Young tableaux where the fakedegree polynomial evaluates to nonnegative integers at roots of unity, albeit without being able to specify an explicit group action. Put differently, we determine in which cases a skew character of the symmetric group carries a permutation representation of the cyclic group.We use a method proposed by N. 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 J. Stembridge's alternating tableaux, which intertwines rotation and promotion.

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Section: Cyclic Sieving For Skew Standard Tableauxsupporting

confidence: 63%

Skew characters and cyclic sieving

Alexandersson¹,

Pfannerer²,

Rubey³

et al. 2020

Preprint

Self Cite

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“…It should be remarked that there have been similar results discovered which relate root of unity specializations of q-Kostka polynomials and fixed point enumerations of matrices or fillings of tableaux (see [Rho10,AU19] for example). It should be mentioned that there is more resemblance between Theorem 1.2 and the results of Rhoades [Rho10] in which, using Hall-Littlewood polynomial, he showed that N-matrices with fixed column content µ and row content ν exhibits biCSP [Rho10].…”

Section: Introductionmentioning

confidence: 67%

Macdonald polynomials and cyclic sieving

Oh¹

2021

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The Garsia-Haiman module is a bigraded Sn-module whose Frobenius image is a Macdonald polynomial. The method of orbit harmonics promotes the 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.

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“…} is the set of weak descents of v. We shall call the elements of Bur n Burge words. This terminology is due to Alexandersson and Uhlin [1]. The connection to Burge is with his variant of the RSK correspondence [5].…”

Section: The Burge Transposementioning

confidence: 99%

Transport of patterns by Burge transpose

Cerbai¹,

Claesson²

2020

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We take the first steps in developing a theory of transport of patterns from Fishburn permutations to (modified) ascent sequences. Given a set of pattern avoiding Fishburn permutations, we provide an explicit construction for the basis of the corresponding set of modified ascent sequences. Our approach is in fact more general and can transport patterns between permutations and equivalence classes of so called Cayley permutations. This transport of patterns relies on a simple operation we call the Burge transpose. It operates on certain biwords called Burge words. Moreover, using mesh patterns on Cayley permutations, we present an alternative view of the transport of patterns as a Wilf-equivalence between subsets of Cayley permutations. We also highlight a connection with primitive ascent sequences.

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Cyclic sieving, skew Macdonald polynomials and Schur positivity

Cited by 6 publications

References 0 publications

Skew characters and cyclic sieving

Skew characters and cyclic sieving

Macdonald polynomials and cyclic sieving

Transport of patterns by Burge transpose

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