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 $\left({\mathsf{SST}}_n(\lambda ),\mathsf{C},q^{-\kappa (\lambda )}{s}_{\lambda }(1,q,\dots ,q^{n-1})\right)$ 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.

20 pages