Cyclic sieving, skew Macdonald polynomials and Schur positivity

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.