Refined cyclic sieving on words for the major index statistic

Ahlbach, Connor; Swanson, Joshua P.

doi:10.1016/j.ejc.2018.05.003

Cited by 7 publications

(13 citation statements)

References 18 publications

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“…In Section 2, we define the cyclic sieving phenomenon and give a brief overview of the relevant symmetric functions. In Section 3, we give a proof of the CSP in (1), and in Section 4 we prove (2). In Section 5 we introduce the skew specialized Macdonald polynomials and give the Schur expansion of these.…”

Section: Per Alexandersson and Joakim Uhlinmentioning

confidence: 99%

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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Section: Per Alexandersson and Joakim Uhlinmentioning

confidence: 99%

Cyclic sieving, skew Macdonald polynomials and Schur positivity

Alexandersson¹,

Uhlin²

2020

Algebraic Combinatorics

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“…A corollary of these universal sieving results is the following equidistribution result. A more refined statement appeared in [AS18].…”

Section: Introductionmentioning

confidence: 99%

“…In [AS18], the authors introduced a new statistic on words, flex. As an example, flex(221221) = 2 · 3 = 6 since 221221 is the concatenation of 2 copies of the primitive word 221 and 221221 is third in lexicographic order amongst its 3 cyclic rotations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Asymptotic normality of the major index on standard tableaux

Billey

Konvalinka

Swanson

2020

Advances in Applied Mathematics

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We consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We use cumulants to classify all possible limit laws for any sequence of such shapes in terms of a simple auxiliary statistic, aft, generalizing earlier results of Canfield-Janson-Zeilberger, Chen-Wang-Wang, and others. These results can be interpreted as giving a very precise description of the distribution of irreducible representations in different degrees of coinvariant algebras of certain complex reflection groups. We conclude with some conjectures concerning unimodality, log-concavity, and local limit theorems.

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Refined cyclic sieving on words for the major index statistic

Cited by 7 publications

References 18 publications

Cyclic sieving, skew Macdonald polynomials and Schur positivity

Cyclic sieving, skew Macdonald polynomials and Schur positivity

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

Asymptotic normality of the major index on standard tableaux

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