Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-30T18:57:30.652Z Has data issue: false hasContentIssue false
  • English
  • Français

Irreducible Modules for Polycyclic Group Algebras

Published online by Cambridge University Press:  20 November 2018

I. M. Musson
I. M. Musson*
Affiliation:
The University of Wisconsin-Madison, Madison, Wisconsin
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.

If G is a polycyclic group and k an absolute field then every irreducible kG-module is finite dimensional [10], while if k is nonabsolute every irreducible module is finite dimensional if and only if G is abelian-by-finite [3]. However something more can be said about the infinite dimensional irreducible modules. For example P. Hall showed that if G is a finitely generated nilpotent group and V an irreducible kG-module, then the image of kZ in EndkGV is algebraic over k [3]. Here Z = Z(G) denotes the centre of G. It follows that the restriction Vz of V to Z is generated by finite dimensional kZ-modules. In this paper we prove a generalization of this result to polycyclic group algebras.

We introduce some terminology.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 4 , 01 August 1981 , pp. 901 - 914
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Donkin, S., Locally finite representations oj poly cyclic groups, to appear.Google Scholar
2. Farkas, D. R. and Snider, R. L., Induced representations of polycyclic groups, Proc. London Math. Soc. (3) 39 (1979), 193–207.Google Scholar
3. Hall, P., On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595–622.Google Scholar
4. Harper, D. L., Primitive irreducible representations of nilpotent groups, Proc. Camb. Philos. Soc. 82 (1977), 241–247.Google Scholar
5. Harper, D. L., Thesis (Queen's College, Cambridge, 1977).Google Scholar
6. Harper, D. L., Primitivity in representations of polycyclic groups, Proc. Camb. Philos. Soc. 88 (1980), 15–31.Google Scholar
7. Musson, I. M., Injective modules for group rings of polycyclic groups I, Quarterly J. Math. 31 (1980), 429–448.Google Scholar
8. Musson, I. M., Injective modules for group rings of poly cyclic groups II, Quarterly J. Math. 31 (1980), 449–466.Google Scholar
9. Passman, D. S., The algebraic structure of group rings (Wiley-Interscience, New York, 1977).Google Scholar
10. Roseblade, J. E., Group rings of polycyclic groups, J. Pure and Applied Algebra 3 (1973), 307–328.Google Scholar
11. Roseblade, J. E., Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. (3) 36 (1978), 385–447.Google Scholar
12. Segal, D., Irreducible representations of finitely generated nilpotent groups, Proc. Camb. Philos. Soc. 81 (1977), 201–208.Google Scholar
13. Zaitsev, D. I., On soluble groups of finite rank, Algebra i Logika 16 (1977), 300–312. English translation Algebra and Logic 16 (1977), 199-207.Google Scholar