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Surprising relationships among unitary reflection groups

Published online by Cambridge University Press:  20 January 2009

H. S. M. Coxeter
H. S. M. Coxeter
Affiliation:
Department of MathematicsUniversity of Toronto : Toronto M5S 1A1Canada
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The transpositions that generate a symmetric group can be represented as real reflections: symmetry operations of a regular simplex. Analogous unitary reflections serve to generate other factor groups of the braid group; they are symmetry operations of regular complex polytopes. Certain relationships among these groups have, as geometric counterparts, unexpected plane sections of the polytopes, beginning with the square sections of the regular tetrahedron. In Section 6, 5-dimensional coordinates will be used to exhibit pentagonal sections of the 4-dimensional regular simplex. The most spectacular instance of such “equatorial” sections occurs in the case of the Witting polytope in complex 4-space, so exquisitely drawn by Peter McMullen for the frontispiece of Regular Complex Polytopes [6]. This has a plane section 3{<5}3 which appears thare as Fig. 4.8B on page 48. Shephard [9, p. 92] called it 3(360)3. Its 120 vertices will be seen to be situated “inside” 120 of the 2160 faces 3{3}3 of the Witting polytope. These “faces” are self-inscribed octagons [7, p. 290].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 27 , Issue 2 , June 1984 , pp. 185 - 194
Copyright
Copyright © Edinburgh Mathematical Society 1984

References

REFERENCES

1.Brieskorn, E. and Saito, K., Artin-Gruppen and Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271.Google Scholar
2.Coxeter, H. S. M., The polytope 221, whose twenty-seven vertices correspond to the lines on the general cubic surface, Amer. J. Math. 62 (1940), 457–486.CrossRefGoogle Scholar
3.Coxeter, H. S. M., Factor groups of the braid group, Proc. Fourth Canadian Math. Congress, Toronto, 1959.Google Scholar
4.Coxeter, H. S. M., Non-Euclidean Geometry (5th ed., University of Toronto Press, 1978).Google Scholar
5.Coxeter, H. S. M., Regular Polytopes (3rd ed., Dover, New York, 1973).Google Scholar
6.Coxeter, H. S. M.Regular Complex Polytopes (Cambridge University Press, 1974).Google Scholar
7.Coxeter, H. S. M., The Pappus configuration and the self-inscribed octagon. Ill, Proc. K. Nederl. Akad. Wetens. Amsterdam A 80 (1977), 285–300.Google Scholar
8.Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups (4th ed., Springer, Berlin, 1980).CrossRefGoogle Scholar
9.Shephard, G. C., Regular complex polytopes, Proc. London Math. Soc. (3) 2 (1952), 82–97.CrossRefGoogle Scholar