Crystals, semistandard tableaux and cyclic sieving phenomenon

In this paper, we study a new cyclic sieving phenomenon on the set SSTn(λ) of semistandard Young tableaux with the cyclic action c arising from its Uq(sln)-crystal structure. We prove that if λ is a Young diagram with ℓ(λ)<n and gcd(n,|λ|)=1, then the triple (SSTn(λ),C,q−κ(λ)sλ(1,q,…,qn−1)) exhibits the cyclic sieving phenomenon, where C is the cyclic group generated by c. We further investigate a connection between c and the promotion pr and show the bicyclic sieving phenomenon given by c and prn for hook shape.