The Cyclic Sieving Phenomenon for Faces of Cyclic Polytopes

A cyclic polytope of dimension d with n vertices is a convex polytope combinatorially equivalent to the convex hull of n distinct points on a moment curve in R. In this paper, we prove the cyclic sieving phenomenon, introduced by Reiner-StantonWhite, for faces of an even-dimensional cyclic polytope, under a group action that cyclically translates the vertices. For odd-dimensional cyclic polytopes, we enumerate the faces that are invariant under an automorphism that reverses the order of the vertices and an automorphism that interchanges the two end vertices, according to the order on the curve. In particular, for n = d + 2, we give instances of the phenomenon under the groups that cyclically translate the odd-positioned and even-positioned vertices, respectively. Research partially supported by the National Science Council, Taiwan under grant NSC grants 982115-M-390-002-MY3 Research partially supported by NSC grants 97-2115-M-251-001-MY2 Research partially supported by NSC grants 98-2115-M-127-001 the electronic journal of combinatorics 17 (2010), #R47 1