The $f$-vector of a polytope consists of the numbers of its $i$-dimensional faces. An open field of study is the characterization of all possible $f$-vectors. It has been solved in three dimensions by Steinitz in the early 19th century. We state a related question, i.e., to characterize $f$-vectors of three dimensional polytopes respecting a symmetry, given by a finite group of matrices. We give a full answer for all three dimensional polytopes that are symmetric with respect to a finite rotation or rotary reflection group. We solve these cases constructively by developing tools that generalize Steinitz’s approach.

中文翻译：

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$F$旋转和旋转反射下对称的3-多位点的向量

这 $F$多面体的向量由其数组成 $\mathrm{一世}$维面孔。开放的研究领域是所有可能特征的表征$F$-向量。19世纪初，斯坦尼斯（Steinitz）已从三个角度解决了这一问题。我们提出一个相关的问题，即表征$F$有限的一组矩阵给出的关于对称性的三维多面体的向量。我们给出了关于有限旋转或旋转反射组对称的所有三维多面体的完整答案。我们通过开发可推广Steinitz方法的工具来建设性地解决这些情况。