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Finite Unitary Reflection Groups

  • G. C. Shephard (a1) and J. A. Todd (a2)
Extract

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.

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References
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1. Bagnera, G., I gruppi finiti di trasformazioni lineari dello spazio che contengono ontologie, Rend. Cire. Mat. Palermo, 19 (1905), 1– 56.
2. Baker, H. F., A locus with 25920 linear self transformations (Cambridge, 1946).
3. Blichfeldt, H. F., The finite discontinuous primitive groups of collineations in four variables, Math. Annalen, 60 (1905), 204–231.
4. Burkhardt, H., Untersuchungen auf dem Gebiete der hyperelliptischen Modulfunctionen (Zweiter Teil), Math. Annalen, 38 (1891), 161–224.
5. Burnside, W., Theory of groups of finite order (second edition, Cambridge, 1911).
6. Coxeter, H. S. M., Discrete groups generated by reflections, Ann. Math., 35 (1934), 588–621.
7. Coxeter, H. S. M., The abstract groups Gm,n,p , Trans. Amer. Math. Soc. 45 (1939), 73–150.
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.
9. Coxeter, H. S. M., Regular polytopes (London, 1948; New York, 1949).
10. Coxeter, H. S. M., The product of the generators of a finite group generated by reflections, Duke Math. J., 18 (1951), 765–782.
10a. Frame, J. S., The classes and representations of the groups of 27 lines and 28 bitangents, Annali di Math., 32 (1951), 83–119.
11. Hamill, C. M., The finite primitive collineation groups which contain homologies of period two (Thesis, University of Cambridge, 1950).
12. Hamill, C. M., On a finite group of order 6,531,840, Proc. London Math. Soc. (2), 52 (1951), 401–454.
13. Hamill, C. M., A collineation group of order 213.35.52.7, Proc. London Math. Soc. (3), 3 (1953), 54–79.
14. Klein, F., Ueber die Transformationen siebenter Ordnung der elliptischen Funktionen, Math. Annalen, 14 (1879), 428–471.
15. Klein, F., Lectures on the icosahedron (trans. Morrice), (London, 1913).
16. Maschke, H., Ueber die quaternäre, endliche, linear e Substitutions gruppe der Borchardt's chen Moduln, Math. Annalen, 30 (1887), 496–515.
17. Maschke, H., Aufstellung des vollen Formensy stems einer quaternären Gruppe von 51840 linear en substitutionen, Math. Annalen, 33 (1889), 317–344.
18. Miller, G. A., Blichfeldt, H. F. and Dickson, L. E., Theory and applications of finite groups (New York, 1916).
19. Mitchell, H. H., Determination of the ordinary and modular ternary linear groups, Trans. Amer. Math. Soc, 12 (1911), 207–242.
20. Mitchell, H. H., Determination of the finite quaternary linear groups, Trans. Amer. Math. Soc, 14(1913), 23–142.
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.
22. Molien, T., Ueber die Invarianten der linear en Substitutions gruppe, Berliner Sitzungsber., (1898), 1152–1156.
23. Shephard, G. C., Regular complex polytopes, Proc London Math. Soc (3), 2 (1952), 82–97.
24. Shephard, G. C., Unitary groups generated by reflections, Can. J. Math., 5 (1953), 364–383.
25. Todd, J. A., On the simple group of order 25920, Proc Royal Soc. (A), 189 (1947), 326–358.
26. Todd, J. A., The invariants of a finite collineation group in five dimensions, Proc. Cambridge Phil. Soc, 46 (1950), 73–90.
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.
28. Valentiner, H., De endelige Transformations-Gruppers Theori, K. danske vidensk, selsk. (Copenhagen) (6), 5 (1889), 64–235.
29. Wiman, A., Ueber eine einfache Gruppe von 360 ebenen Collineationen, Math. Annalen, 47 (1896), 531–556.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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