Cyclic sieving, skew Macdonald polynomials and Schur positivity

Alexandersson, Per; Uhlin, Joakim

Algebraic Combinatorics, Tome 3 (2020) no. 4, pp. 913-939.

When $\lambda$ is a partition, the specialized non-symmetric Macdonald polynomial ${\mathrm{E}}_{\lambda }(\mathbf{x};q;0)$ is symmetric and related to a modified Hall–Littlewood polynomial. We show that whenever all parts of the integer partition $\lambda$ 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 ${\mathrm{E}}_{\lambda }(\mathbf{x};q;0)$. 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 ${\mathrm{E}}_{\lambda }(\mathbf{x};q;0)$. 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.

Classification : 05E10, 05E05, 06A07
Mots clés : Cyclic sieving, Macdonald polynomials, LLT polynomials, crystals, Schur-positivity.

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