ALGEBRAIC COMBINATORICS

no. 4

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

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.

Received:
Revised:
Accepted:
Published online:
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},
     pages = {913--939},
     publisher = {MathOA foundation},
     volume = {3},
     number = {4},
     year = {2020},
     doi = {10.5802/alco.123},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.123/}
}
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/articles/10.5802/alco.123/

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