Jeu-de-taquin promotion and a cyclic sieving phenomenon for semistandard hook tableaux

Jeu-de-taquin promotion yields a bijection on the set of semistandard λ-tableaux with entries bounded by k. In this note, we determine the order of jeu-de-taquin promotion on the set of semistandard hook tableaux CST((n−m,1m),k), with entries bounded by k, and on the set of semistandard hook tableaux with fixed content α, CST((n−m,1m),k,α). We give a bijection between CST((n−m,1m),k,α) and a suitable set of standard hook tableaux that behaves nicely with respect to jeu-de-taquin promotion and use the bijection to give a cyclic sieving phenomenon for CST((n−m,1m),k,α).