The Cyclic Sieving Phenomenon for Faces of Generalized Cluster Complexes

The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridge's $q=-1$ phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization of the cluster complexes found by Fomin and Zelevinsky. In this paper, the faces of various dimensions of the generalized cluster complexes in type $A_n$, $B_n$, $D_n$, and $I_2(a)$ are shown to exhibit the cyclic sieving phenomenon under a cyclic group action. For the cluster complexes of exceptional type $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, and $H_4$, a verification for such a phenomenon on their maximal faces is given.