In this paper, we study a new cyclic sieving phenomenon on the set $\mathsf{SST}_n(\lambda)$ of semistandard Young tableaux with the cyclic action $\mathsf{c}$ arising from its $U_q(\mathfrak{sl}_n)$-crystal structure. We prove that if $\lambda$ is a Young diagram with $\ell(\lambda) < n$ and $\gcd( n, |\lambda| )=1$, then the triple $\left( \mathsf{SST}_n(\lambda), \mathsf{C}, q^{- \kappa(\lambda)} s_\lambda(1,q, \ldots, q^{n-1}) \right) $ exhibits the cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by $\mathsf{c}$. We further investigate a connection between $\mathsf{c}$ and the promotion $\mathsf{pr}$ and show the bicyclic sieving phenomenon given by $\mathsf{c}$ and $\mathsf{pr}^n$ for hook shape.
@article{10_37236_8802,
author = {Young-Tak Oh and Euiyong Park},
title = {Crystals, semistandard tableaux and cyclic sieving phenomenon},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {4},
doi = {10.37236/8802},
zbl = {1428.05329},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8802/}
}
TY - JOUR
AU - Young-Tak Oh
AU - Euiyong Park
TI - Crystals, semistandard tableaux and cyclic sieving phenomenon
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/8802/
DO - 10.37236/8802
ID - 10_37236_8802
ER -
%0 Journal Article
%A Young-Tak Oh
%A Euiyong Park
%T Crystals, semistandard tableaux and cyclic sieving phenomenon
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/8802/
%R 10.37236/8802
%F 10_37236_8802
Young-Tak Oh; Euiyong Park. Crystals, semistandard tableaux and cyclic sieving phenomenon. The electronic journal of combinatorics, Tome 26 (2019) no. 4. doi: 10.37236/8802
