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 ${Bbb R}^d$. In this paper, we prove the cyclic sieving phenomenon, introduced by Reiner-Stanton-White, 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.
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机译：具有$ n $个顶点的维$ d $的循环多边形是凸多边形，在组合上等效于$ { Bbb R} ^ d $弯矩曲线上$ n $个不同点的凸包。在本文中，我们证明了Reiner-Stanton-White引入的周期性筛分现象，该现象是在周期性循环转换顶点的群体作用下对偶数维环状多面体的表面进行的。对于奇维循环多边形，我们根据曲线的顺序枚举了在反转顶点顺序的自同构和交换两个末端顶点的自同构下不变的面。特别地，对于$ n = d + 2 $，我们给出在分别循环平移奇数个和偶数个顶点的组下的现象的实例。
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