Hostname: page-component-6bb9c88b65-kzqxb Total loading time: 0 Render date: 2025-07-19T21:56:37.539Z Has data issue: false hasContentIssue false
  • English
  • Français

Metabelian Lie powers of group representations

Part of: Representation theory of groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

R. M. Bryant ,
R. Stöhr  and
R. Zerck
R. M. Bryant
Affiliation:
University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester, M60 1QD, United Kingdom
R. Stöhr
Affiliation:
University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester, M60 1QD, United Kingdom
R. Zerck
Affiliation:
Karl-Weierstrass-Institut für Mathematik, Mohrenstrasse 39, Postfach 1304, O-1086 Berlin, Germany
Rights & Permissions [Opens in a new window]

Abstract

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 representation of a group G on a vector space V extends uniquely to a representation of G on the free metabelian Lie algebra on V. In this paper we study such representations and make some group-theoretic applications.

MSC classification

Secondary: 20C15: Ordinary representations and characters 20F40: Associated Lie structures

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 56 , Issue 2 , April 1994 , pp. 145 - 168
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Alperin, J. L., Local representation theory (Cambridge Univ. Press, Cambridge, 1986).CrossRefGoogle Scholar
[2]Amayo, R. K. and Stewart, I., Infinite-dimensional Lie algebras (Noordhoff International, Leyden, 1974).CrossRefGoogle Scholar
[3]Auslander, M. and Lyndon, R. C., ‘Commutator subgroups of free groups’, Amer. J. Math. 77 (1955), 929–931.CrossRefGoogle Scholar
[4]Bourbaki, N., Groupes et algébres de Lie, Ch. I (Hermann, Paris, 1960).Google Scholar
[5]Bourbaki, N., Groupes et algébres de Lie, Ch. II (Hermann, Paris, 1972).Google Scholar
[6]Brandt, A., ‘The free Lie ring and Lie representations of the full linear group’, Trans. Amer. Math. Soc. 56 (1944), 528–536.CrossRefGoogle Scholar
[7]Bryant, R. M. and Kovács, L. G., ‘Tensor products of representations of finite groups’, Bull. London Math. Soc. 4 (1972), 133–135.CrossRefGoogle Scholar
[8]Bryant, R. M. and Kovács, L. G., ‘Lie representations and groups of prime power order’, J. London Math. Soc. (2) 17 (1978), 415–21.CrossRefGoogle Scholar
[9]Chen, K.-T., ‘Integration in free groups’, Ann. of Math. (2) 54 (1951), 147–162.CrossRefGoogle Scholar
[10]Feit, W., Characters of finite groups (Benjamin, New York, 1967).Google Scholar
[11]Gaschütz, W., ‘Über modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden’, Math. Z. 60 (1954), 274–286.CrossRefGoogle Scholar
[12]Green, J. A., Polynomial representations of GLn, Lecture Notes in Math. 830 (Springer, Berlin, 1980).Google Scholar
[13]Gupta, N. D., Laffey, T. J. and Thomson, M. W., ‘On the higher relation modules of a finite group’, J. Algebra 59 (1979), 172–187.CrossRefGoogle Scholar
[14]Hannebauer, T. and Stöhr, R., ‘Homology of groups with coefficients in free metabelian Lie powers and exterior powers of relation modules and applications to group theory’, in: Proc. Second Internal. Group Theory Conference (Bressanone, 1989), Rend. Circ. Mat., Palermo, (2) Suppl. No 23 (1990) pp. 77–113.Google Scholar
[15]Huppert, B. and Blackburn, N., Finite groups II (Springer, Berlin, 1982).CrossRefGoogle Scholar
[16]Lazard, M., ‘Sur les groupes nilpotents et les anneaux de Lie’, Ann. Sci. École Norm. Sup. (3) 71 (1954), 101–190.CrossRefGoogle Scholar
[17]Macdonald, I. G., Symmetric functions and Hall polynomials (Oxford Univ. Press, Oxford, 1979).Google Scholar
[18]Neumann, H., Varieties of groups (Springer, Berlin, 1967).CrossRefGoogle Scholar
[19]Shmel'kin, A. L., ‘Free polynilpotent groups’, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 91–122 (in Russian).Google Scholar