| Autor: |
Alexandersson, Per, Uhlin, Joakim |
| Rok vydání: |
2019 |
| Předmět: |
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| Zdroj: |
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 913-939 |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.5802/alco.123 |
| Popis: |
When $\lambda$ is a partition, the specialized non-symmetric Macdonald polynomial $E_{\lambda}(x;q;0)$ is symmetric and related to a modified Hall--Littlewood polynomial. We show that whenever all parts of the integer partition $\lambda$ is a multiple of $n$, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under a cyclic shift of the columns. The corresponding CSP polynomial is given by $E_{\lambda}(x;q;0)$. 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 $E_{\lambda}(x;q;0)$. 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. |
| Databáze: |
arXiv |
| Externí odkaz: |
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