Abstract

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.

Highlights

  • The cyclic sieving phenomenon (CSP), introduced by V

  • We introduce a skew version of Eλ(x; q; 0). We show that these are symmetric and Schur positive via a variant of the Robinson–Schenstedt–Knuth correspondence and we describe crystal raising and lowering operators for the underlying fillings

  • We provide families of cyclic sieving on tableaux related to certain specializations of non-symmetric Macdonald polynomials

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Summary

Introduction

The cyclic sieving phenomenon (CSP), introduced by V. We provide families of cyclic sieving on tableaux related to certain specializations of non-symmetric Macdonald polynomials. This settles an earlier conjecture by the authors presented in [48]. We note that some of the results in this paper are based on earlier work done in the second author’s master’s thesis [48]

Preliminaries
Cyclic sieving on coinversion-free fillings
Refined CSP on stretched specialized Macdonald fillings
Skew specialized Macdonald polynomials
Crystal operators on words and SSYT
Schur expansion of certain vertical-strip LLT polynomials
Full Text
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