Invariant tensors and the cyclic sieving phenomenon

Abstract

We construct a large class of examples of the cyclic sieving phenomenon by expoiting 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 $r\ge 0$, 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 ${\otimes }^{r}B$; the map $c:X\to X$ 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 ${\mathfrak{S}}_r$ related to the natural action of ${\mathfrak{S}}_r$ on the subspace of invariant tensors in ${\otimes }^{r}M$. Taking $M$ to be the defining representation of $\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux.