Realizations of regular polytopes, III
Realizations of regular polytopes, III
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规则多面体的实现，III
DOI：
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发表时间：
2003
期刊：
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通讯作者：
P. McMullen
中科院分区：
文献类型：
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作者：
P. McMullen
欧氏空间中有限抽象正则多面体$$\数学{P}$$的几何实现有一个全面的理论。识别$${\数学{P}}$$的同余实现，它们的空间形成一个点凸锥，通过标量乘法和混合结合不同实现的运算。本文介绍了一种结合实现的新方法--张量积方法。正如混合对应于群的表示之和，乘积也对应于表示的通常乘积。文中给出了一系列的例子来说明新的理论。除了具有内在的趣味性，这些例子中的一些还导致了对已知实现空间的额外洞察；更重要的是，这里首次确定了常规600单元的实现锥体的结构。
There is a comprehensive theory of geometric realizations of a finite abstract regular polytope $${\mathcal{P}}$$ in euclidean spaces. Identifying congruent realizations of $${\mathcal{P}}$$, their space forms a pointed convex cone, with scalar multiplication and blending the operations which combine different realizations. In this paper, a new way to combine realizations is introduced, that of the (tensor) product. Just as blending corresponds to sums of representations of groups, so the product corresponds to the usual product of representations. A range of examples is given to illustrate the new theory. As well as being of intrinsic interest, some of these examples lead to extra insight into already known realization spaces; more importantly, the structure of the realization cone of the regular 600-cell is here determined for the first time.