Unitary Groups Generated by Reflections
Unitary Groups Generated by Reflections
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由反射生成的酉群
DOI：
10.4153/cjm-1953-042-7
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发表时间：
1953
期刊：
影响因子：
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通讯作者：
G. C. Shephard
中科院分区：
文献类型：
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作者：
G. C. Shephard
欧几里德n维空间中的反射是一种特殊类型的全等变换，其周期为2并且留下一个素数（即，超平面）不变。由许多这些反射产生的组已被广泛研究[5，pp. 187 - 212]。它们是有趣的，因为除了极少数例外，均匀多面体的对称群都是这种类型。考克斯特还证明了[4]，通过威索夫的构造，可以从任何由反射生成的群中导出许多均匀的多面体。他对这一结构的讨论通过使用图形符号[4，p.328; 5，p.84]来优雅地说明，由此可以从节点、分支和环的简单图中读出多面体的性质。
A reflection in Euclidean n-dimensional space is a particular type of congruent transformation which is of period two and leaves a prime (i.e., hyperplane) invariant. Groups generated by a number of these reflections have been extensively studied [5, pp. 187-212]. They are of interest since, with very few exceptions, the symmetry groups of uniform polytopes are of this type. Coxeter has also shown [4] that it is possible, by Wythoff's construction, to derive a number of uniform polytopes from any group generated by reflections. His discussion of this construction is elegantly illustrated by the use of a graphical notation [4, p. 328; 5, p. 84] whereby the properties of the polytopes can be read off from a simple graph of nodes, branches, and rings.