The Cyclic Sieving Phenomenon for Faces of Generalized Cluster Complexes

Details The Cyclic Sieving Phenomenon for Faces of Generalized Cluster Complexes The Cyclic Sieving Phenomenon for Faces of Generalized Cluster Complexes

2006-12-22

arxiv.org/abs/math/0612679v1

Mathematics

Combinatorics

26 pages, 14 figures

Scientific paper

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.

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