Crystals, semistandard tableaux and cyclic sieving phenomenon

Abstract: 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.