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Groups Generated by Unitary Reflections of Period Two

Published online by Cambridge University Press:  20 November 2018

H. S. M. Coxeter
H. S. M. Coxeter*
Affiliation:
University of Toronto
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In complex affine n-space with a unitary metric, a reflection is a congruent transformation leaving invariant all the points of a hyperplane. Thus the characteristic roots of a unitary reflection of period p consist of a primitive pth root of unity and n – 1 unities. A group generated by n reflections is conveniently represented by a graph having a node for each generator and a branch for each pair of non-commutative generators.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 9 , 1957 , pp. 243 - 272
Copyright
Copyright © Canadian Mathematical Society 1957

References

1. Bagnera, G., I gruppi finiti di tYasformazioni UneaYi dello spazio che contengono ontologie, Rend. Circ. Mat. Palermo, 19 (1905), 1–56.Google Scholar
2. Baker, H. F., A Locus with 25920 LineaY Self TYansfor mations (Cambridge, 1946).Google Scholar
3. Barlow, W., Crystal symmetry: The actual basis of the thirty-two classes, Phil. Mag. (6), 1 (1901), 1–36.Google Scholar
4. Barnes, E. S., The perfect and extreme senary forms, Can. J. Math., 9 (1957), 235–242.Google Scholar
5. Barnes, E. S., The complete enumevation of extreme senary forms, Phil. Trans. Royal Soc, London, A 249 (1957), 461–506.Google Scholar
6. Burnside, W., Theory of Groups of Finite Order (2nd ed., Cambridge 1911).Google Scholar
7. Coxeter, H. S. M., The polytopes with regular-prismatic vevtex figures, Phil. Trans. Royal Soc. London, A 229 (1931), 329–425.Google Scholar
8. Coxeter, H. S. M., Discrete groups genevated by reflections, Ann. Math., 85 (1934), 588–621.Google Scholar
9. Coxeter, H. S. M., Wythoff's construction for unifovm polytopes, Proc. London Math. Soc. (2), 38 (1935), 327–339.Google Scholar
10. Coxeter, H. S. M., The abstract groups Rm = Sm = (RjSj)pj = 1, etc., Proc. London Math. Soc. (2), 41 (1936), 278–301.Google Scholar
1l. Coxeter, H. S. M., Regular skew polyhedva in three andfour dimensions, and their topological analogues, Proc. London Math. Soc. (2), 42 (1937), 33–62.Google Scholar
12. Coxeter, H. S. M., The groups Gm,n,p , Trans. Amer. Math. Soc, 45 (1939), 73–150.Google Scholar
13. Coxeter, H. S. M., A method for proving certain abstract groups to be infinite, Bull. Amer. Math. Soc, 46 (1940), 246–251.Google Scholar
14. Coxeter, H. S. M., Regular Polytopes (London, 1948; New York, 1949).Google Scholar
15. Coxeter, H. S. M., Self-dual configurations and regular graphs, Bull. Amer. Math. Soc, 56 (1950), 413–455.Google Scholar
16. Coxeter, H. S. M., The product of the generators of a finite group generated by reflections, Duke Math. J., 18 (1951), 765–782.Google Scholar
17. Coxeter, H. S. M., Extreme forms, Can. J. Math., 3 (1951), 391–441.Google Scholar
18. Coxeter, H. S. M. and Todd, J. A., An extreme duodenary form, Can. J. Math., 5 (1953), 384–392.Google Scholar
19. Dickson, L. E., Linear Groups, with an Exposition of the Galois Field Theory (Leipzig, 1901).Google Scholar
20. Dieudonné, J., La géométrie des groupes classiques, Ergeb. Math. (Neue Folge), 5 (1955).Google Scholar
20a. Edge, W. L., The Kummer quartic and the tetrahedroid based on the Maschke forms, Proc. Cambridge Philos. Soc, 45 (1949), 519–535.Google Scholar
21. Edington, W. E., Abstract group definitions and applications, Trans. Amer. Math. Soc, 25 (1923), 193–210.Google Scholar
22. Enriques, F. and Chisini, O., Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche, 3 (Bologna, 1924).Google Scholar
23. Hamill, C. M., On a finite group of order 6,531,840, Proc. London Math. Soc (2), 52 (1951), 401–454.Google Scholar
24. Hudson, R. W. H. T., Rummer's Quartic Surface (Cambridge, 1905).Google Scholar
25. Killing, W., Die Zusammensetzung der stetigen endlichen Transformations gruppen, II, Math. Ann. 33 (1888), 1–48.Google Scholar
26. Klein, F., Ueber die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann., 14 (1879), 428–471.Google Scholar
26a. Klein, F., Lectures on the Icosahedron (London, 1913).Google Scholar
27. Maschke, H., Ueber die quaternare, endliche, lineare Substitutionsgruppe der Borchard'schen Moduln, Math. Ann., 30 (1887), 496–515.Google Scholar
28. Shephard, G. C., Regular complex polytopes, Proc. London Math. Soc. (3), 2 (1952), 82–97.Google Scholar
29. Shephard, G. C., Unitary groups generated by reflections, Can. J. Math. 5 (1953), 364–383.Google Scholar
30. Shephard, G. C., Some problems on finite reflection groups, L'Enseignement Math. (2), 2 (1956), 42–48.Google Scholar
31. Shephard, G. C. and Todd, J. A., Finite unitary reflection groups, Can. J. Math., 6 (1954), 274–304.Google Scholar
32. Threlfall, W., Gruppenbilder, Abh. Sächs. Akad. Wiss. Math.-phys. KL, 41 (1932), 1–59.Google Scholar
33. Todd, J. A., On the simple group of order 25920, Proc. Royal Soc. London, A 189 (1947), 326–358.Google Scholar
34. Todd, J. A., The invariants of a finite collineation group in 5 dimensions, Proc Cambridge Philos. Soc. 46 (1950), 73–90.Google Scholar
35. Valentiner, H., Théorie des groupes de transformations finis, K. danske vidensk. selsk. (6), 5 (1889), 205–235.Google Scholar
36. Veblen, O. and Young, J. W., Projective Geometry, 2 (Boston, 1918).Google Scholar
37. Wiman, A., Ueber eine einfache Gruppe von 360 ebenen Collineationen, Math. Ann. 47 (1896), 531–556.Google Scholar