Invariant Tensors and the Cyclic Sieving Phenomenon

We construct a large class of examples of the cyclic sieving phenomenon by exploiting the representation theory of semi-simple Lie algebras. Let M be a finite dimensional representation of a semi-simple Lie algebra and let B be the associated Kashiwara crystal. For r > 0, the triple (X, c, P ) which exhibits the cyclic sieving phenomenon is constructed as follows: the set X is the set of isolated vertices in the crystal ⊗rB; the map c : X → X is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial P is the fake degree of the Frobenius character of a representation of Sr related to the natural action of Sr on the subspace of invariant tensors in ⊗rM . Taking M to be the defining representation of SL(n) gives the cyclic sieving phenomenon for rectangular tableaux.