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Unitary Groups Generated by Reflections

Published online by Cambridge University Press:  20 November 2018

G. C. Shephard
G. C. Shephard*
Affiliation:
University of Birmingham
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 364 - 383
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Baker, H. F., A locus with 25,920 linear self-transformations (Cambridge, 1936).Google Scholar
2. Coxeter, H. S. M., The polytopes with regular-prismatic vertex figures, Phil. Trans. Royal Soc, Ser. A, 229 (1930), 329–425.Google Scholar
3. Coxeter, H. S. M., The polytopes with regular-prismatic vertex figures II, Proc. London Math. Soc. (2), 84 (1932), 126–189.Google Scholar
4. Coxeter, H. S. M., Wythoff's construction for uniform polytopes, Proc. London Math. Soc. (2), 38 (1935), 327–339.Google Scholar
4a. Coxeter, H. S. M., The abstract groups Rm = Sm = (RjSj)pi = 1 , Sm = T2 = (SjT)2pi = 1, and Sm = T2 = , Proc. London Math. Soc. (2), 41 (1936), 278–301.Google Scholar
5. Coxeter, H. S. M., Regular polytopes (London, 1948;New York, 1949).Google Scholar
6. Coxeter, H. S. M. Extreme forms, Can. J. Math., 3 (1951), 391–441.Google Scholar
7. Coxeter, H. S. M. and Todd, J. A., An extreme duodenary form, Can. J. Math., 5 (1953), 384–392.Google Scholar
8. Hamill, C. M., On a finite group of order 6,531,840, Proc. London Math. Soc. (2), 52 (1951), 401–454.Google Scholar
9. Hartley, E. M., A sextic primal in five dimensions, Proc. Cambridge Phil. Soc, 46 (1950), 91–105.Google Scholar
10. Hudson, R. W. H. T., Rummer's Quartic Surface (Cambridge, 1905).Google Scholar
11. Shephard, G. C., Regular Complex Polytopes, Proc. London Math. Soc. (3), 2 (1952), 82–97.Google Scholar
12. Todd, J. A., On the simple group of order 25,920, Proc. Royal Soc. London, Ser. A, 189 (1947), 326–358.Google Scholar
13. Todd, J. A., The characters of a collineation group in five dimensions, Proc. Royal Soc. London, Ser. A, 200 (1949) 320–336.Google Scholar
14. Todd, J. A., The invariants of a finite collineation group in five dimensions, Proc. Cambridge Phil. Soc, 40 (1950), 73–90.Google Scholar
15. Todd, J. A. and Coxeter, H. S. M., A practical method for enumerating cosets of a finite abstract group, Proc. Edinburgh Math. Soc (2), 5 (1936), 26–36.Google Scholar