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On the crystallography of infinite Coxeter groups

Published online by Cambridge University Press:  24 October 2008

George Maxwell
George Maxwell
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver 8, B.C., Canada
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Extract

Let V be the vector space of translations of a finite dimensional real affine space. The principal aim of this paper is to study (generally non-Euclidean) space groups whose point groups K are ‘linear’ Coxeter groups in the sense of Vinberg (4). This involves the investigation of lattices Λ in V left invariant by K and the calculation of cohomology groups H1(K, V/Λ) (3). The first problem is solved by generalizing classical concepts of ‘bases’ of root systems and their ‘weights’, while the second is carried out completely in the case when the Coxeter graph Γ of K contains only edges marked by 3. An important part in the calculation of H1(K, V/Λ) is then played by certain subgraphs of Γ which are complete multipartite graphs. The only subgraphs of this kind which correspond to finite Coxeter groups are of type Al× … × A1, A2, A3 or D4. This may help to explain why, in our earlier work on space groups with finite Coxeter point groups (3), (2), components of r belonging to these types played a rather mysterious exceptional role.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 1 , July 1977 , pp. 13 - 24
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Bourbaki, N.Groupes et algèbres de Lie, chap. IV–VI (Paris, Hermann, 1968).Google Scholar
(2)Broderick, N. and Maxwell, G.The crystallography of Coxeter groups: II. J. Algebra 44 (1977), 290–318.CrossRefGoogle Scholar
(3)Maxwell, G.The crystallography of Coxeter groups, J. Algebra 35 (1975), 159–177.Google Scholar
(4)Vinberg, E. B.Discrete linear groups generated by reflections. [In Russian.] Izv. Akad. Nauk SSSR 35 (1971), 1072–1112.Google Scholar