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The Betti Numbers of the Simple Lie Groups

Published online by Cambridge University Press:  20 November 2018

A. J. Coleman
A. J. Coleman*
Affiliation:
University of Toronto
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The purpose of the present paper1 is to simplify the calculation of the Betti numbers of the simple compact Lie groups.

For the unimodular group and the orthogonal group on a space of odd dimension the form of the Poincaré polynomial was correctly guessed by E. Cartan in 1929 (5, p. 183). The proof of his conjecture and its extension to the four classes of classical groups was given by L. Pontrjagin (13) using topological arguments and then by R. Brauer (2) using algebraic methods.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 349 - 356
Copyright
Copyright © Canadian Mathematical Society 1958

References

1. Borel, A. and Chevalley, C., The Betti numbers of the exceptional Lie groups, Mem. Amer. Math. Soc, 14 (1955), 1-9.Google Scholar
2. Brauer, R., Sur les invariants intégraux des variétés representatives des groupes de Lie simples clos, C. R. Acad. Sci. Paris, 201 (1935), 419-421.Google Scholar
3. Cartan, E., Sur la structure des groupes de transformations finis et continus, thesis (Paris, 1894).Google Scholar
4. Cartan, E., Sur la reduction à sa forme canonique de la structure d'un groupe de transformations fini et continu, Amer. J. Math., 18 (1896), 1-61.Google Scholar
5. Cartan, E., Sur les invariants intégraux de certains espaces homogènes clos et les propriétés topologiques de ces espaces, Ann. Soc. Pol. Math., 8 (1929), 181–225.Google Scholar
6. Chevalley, C., The Betti numbers of the exceptional simple Lie groups, Proc. Int. Math. Cong. II (1950), 21-24.Google Scholar
7. Chevalley, C., Invariants of finite groups generated by reflections, Amer. J. Math., 77 (1955), 778-782.Google Scholar
8. Coxeter, H. S. M., Regular Poly topes (London, 1948).Google Scholar
9. Coxeter, H. S. M., The product of the generators of a finite group generated by reflections, Duke Math. J., 18 (1952), 765-782.Google Scholar
10. Hopf, H., Ueber die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen, Ann. Math., 42 (1941), 22-52.Google Scholar
11. Hopf, H., Maximale Toroide und singuläre Elemente in geschlossenen Lieschen Gruppen, Comm. Math. Helv., 15 (1942), 59-70.Google Scholar
12. Killing, W., Die Zusammensetzung der stetigen endlichen Transformations gruppen, Math. Ann. 33 (1889), 1-48.Google Scholar
13. Pontrjagin, L., Sur les nombres de Betti des groupes de Lie, C. R. Acad. Sci. Paris, 200 (1935), 1277-1280.Google Scholar
14. Racah, G., Sulla caratterizzazione délia rappresentazioni irriducibili dei gruppi semisimplici di Lie, Rend. Acad. Naz. Lincei, 8 (1950), 108-112.Google Scholar
15. Samelson, H., Beiträge zur Topologie der Gruppen-Mannigfaltigkeiten, Ann. Math., 42 (1941), 1091-1136.Google Scholar
16. Shephard, G. C. and Todd, J. A., Finite unitary reflection groups, Can. J. Math. 6 (1954), 274-304.Google Scholar
17. Shephard, G. C., Some problems on finite reflection groups, L'enseignement mathématique, II (1956), 42-48.Google Scholar
18. Stiefel, E., Ueber eine Beziehung zwischen geschlossenen Liesche Gruppen und diskontinuierlichen Bewegungsgruppen, Comm. Math. Helv., 14 (1942), 350-380.Google Scholar
19. Weyl, H., Theorie der Darstellung kontinuierlicher halb-einfach Gruppen durch lineare transformationen, Math. Zeit., 24 (1926), 377-395).Google Scholar
20. Witt, E., Spiegelungsgruppen und Aufzählung halbeinfacher Liescher Ringe, Abhand. Math. Sem., Hamburg, 14 (1941), 289-322.Google Scholar
21. Yen, C. T., Sur les polynomes de Poincaré des groupes simples exceptionnels, C. R. Acad. Sci. Paris, 228 (1949), 628-634.Google Scholar