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 C S T ( ( n − m , 1 m ) , k ) , with entries bounded by k , and on the set of semistandard hook tableaux with fixed content α , C S T ( ( n − m , 1 m ) , k , α ) . We give a bijection between C S T ( ( n − m , 1 m ) , 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 C S T ( ( n − m , 1 m ) , k , α ) .