# ALGEBRAIC COMBINATORICS

Cyclic sieving, skew Macdonald polynomials and Schur positivity
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 913-939.

When $\lambda$ is a partition, the specialized non-symmetric Macdonald polynomial ${\mathrm{E}}_{\lambda }\left(\mathbf{x};q;0\right)$ 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 ${\mathrm{E}}_{\lambda }\left(\mathbf{x};q;0\right)$. 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 ${\mathrm{E}}_{\lambda }\left(\mathbf{x};q;0\right)$. 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.

Revised: 2020-03-25
Accepted: 2020-04-12
Published online: 2020-08-20
DOI: https://doi.org/10.5802/alco.123
Classification: 05E10,  05E05,  06A07
Keywords: Cyclic sieving, Macdonald polynomials, LLT polynomials, crystals, Schur-positivity.
@article{ALCO_2020__3_4_913_0,
author = {Alexandersson, Per and Uhlin, Joakim},
title = {Cyclic sieving, skew Macdonald polynomials and Schur positivity},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {3},
number = {4},
year = {2020},
pages = {913-939},
doi = {10.5802/alco.123},
language = {en},
url = {alco.centre-mersenne.org/item/ALCO_2020__3_4_913_0/}
}
Alexandersson, Per; Uhlin, Joakim. Cyclic sieving, skew Macdonald polynomials and Schur positivity. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 913-939. doi : 10.5802/alco.123. https://alco.centre-mersenne.org/item/ALCO_2020__3_4_913_0/

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