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
Summary
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]
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