Hostname: page-component-7b7567c8c-mnklg Total loading time: 0 Render date: 2026-11-13T01:25:07.724Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Unitary Reflection Groups

Published online by Cambridge University Press:  20 November 2018

G. C. Shephard and
J. A. Todd
G. C. Shephard
Affiliation:
University of Birmingham
J. A. Todd
Affiliation:
University of Cambridge
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of unitary transformations (5, p. 257). Such a group may be thought of as a group of congruent transformations, keeping the origin fixed, in a unitary space Un of n dimensions, in which the points are specified by complex vectors with n components, and the distance between two points is the norm of the difference between their corresponding vectors.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954, pp. 274 - 304
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Bagnera, G., I gruppi finiti di trasformazioni lineari dello spazio che contengono ontologie, Rend. Cire. Mat. Palermo, 19 (1905), 1– 56.Google Scholar
2. Baker, H. F., A locus with 25920 linear self transformations (Cambridge, 1946).Google Scholar
3. Blichfeldt, H. F., The finite discontinuous primitive groups of collineations in four variables, Math. Annalen, 60 (1905), 204–231.Google Scholar
4. Burkhardt, H., Untersuchungen auf dem Gebiete der hyperelliptischen Modulfunctionen (Zweiter Teil), Math. Annalen, 38 (1891), 161–224.Google Scholar
5. Burnside, W., Theory of groups of finite order (second edition, Cambridge, 1911).Google Scholar
6. Coxeter, H. S. M., Discrete groups generated by reflections, Ann. Math., 35 (1934), 588–621.Google Scholar
7. Coxeter, H. S. M., The abstract groups Gm,n,p , Trans. Amer. Math. Soc. 45 (1939), 73–150.Google Scholar
8. 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.Google Scholar
9. Coxeter, H. S. M., Regular polytopes (London, 1948; New York, 1949).Google Scholar
10. 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
10a. Frame, J. S., The classes and representations of the groups of 27 lines and 28 bitangents, Annali di Math., 32 (1951), 83–119.Google Scholar
11. Hamill, C. M., The finite primitive collineation groups which contain homologies of period two (Thesis, University of Cambridge, 1950).Google Scholar
12. Hamill, C. M., On a finite group of order 6,531,840, Proc. London Math. Soc. (2), 52 (1951), 401–454.Google Scholar
13. Hamill, C. M., A collineation group of order 213.35.52.7, Proc. London Math. Soc. (3), 3 (1953), 54–79.Google Scholar
14. Klein, F., Ueber die Transformationen siebenter Ordnung der elliptischen Funktionen, Math. Annalen, 14 (1879), 428–471.Google Scholar
15. Klein, F., Lectures on the icosahedron (trans. Morrice), (London, 1913).Google Scholar
16. Maschke, H., Ueber die quaternäre, endliche, linear e Substitutions gruppe der Borchardt's chen Moduln, Math. Annalen, 30 (1887), 496–515.Google Scholar
17. Maschke, H., Aufstellung des vollen Formensy stems einer quaternären Gruppe von 51840 linear en substitutionen, Math. Annalen, 33 (1889), 317–344.Google Scholar
18. Miller, G. A., Blichfeldt, H. F. and Dickson, L. E., Theory and applications of finite groups (New York, 1916).Google Scholar
19. Mitchell, H. H., Determination of the ordinary and modular ternary linear groups, Trans. Amer. Math. Soc, 12 (1911), 207–242.Google Scholar
20. Mitchell, H. H., Determination of the finite quaternary linear groups, Trans. Amer. Math. Soc, 14(1913), 23–142.Google Scholar
21. Mitchell, H. H., Determination of all primitive collineation groups in more than four variables which contain homologies, Amer. J. of Math., 36 (1914), 1–12.Google Scholar
22. Molien, T., Ueber die Invarianten der linear en Substitutions gruppe, Berliner Sitzungsber., (1898), 1152–1156.Google Scholar
23. Shephard, G. C., Regular complex polytopes, Proc London Math. Soc (3), 2 (1952), 82–97.Google Scholar
24. Shephard, G. C., Unitary groups generated by reflections, Can. J. Math., 5 (1953), 364–383.Google Scholar
25. Todd, J. A., On the simple group of order 25920, Proc Royal Soc. (A), 189 (1947), 326–358.Google Scholar
26. Todd, J. A., The invariants of a finite collineation group in five dimensions, Proc. Cambridge Phil. Soc, 46 (1950), 73–90.Google Scholar
27. 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), 26–34.Google Scholar
28. Valentiner, H., De endelige Transformations-Gruppers Theori, K. danske vidensk, selsk. (Copenhagen) (6), 5 (1889), 64–235.Google Scholar
29. Wiman, A., Ueber eine einfache Gruppe von 360 ebenen Collineationen, Math. Annalen, 47 (1896), 531–556.Google Scholar