Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T19:10:18.016Z Has data issue: false hasContentIssue false
  • English
  • Français

Lie powers of free modules for certain groups of prime power order

Part of: Lie algebras and Lie superalgebras Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

R. M. Bryant  and
I. C. Michos
R. M. Bryant
Affiliation:
UMIST, PO Box 88, Manchester M60 1QDEngland e-mail: [email protected]
I. C. Michos
Affiliation:
LIAFA, Université Paris 775251 Paris, France e-mail: [email protected]
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.

Let G be a finite group of order pk, where p is a prime and k ≥ 1, such that G is either cyclic, quaternion or generalised quaternion. Let V be a finite-dimensional free KG-module where K is a field of characteristic p. The Lie powers Ln(V) are naturally KG-modules and the main result identifies these modules up to isomorphism. There are only two isomorphism types of indecomposables occurring as direct summands of these modules, namely the regular KG-module and the indecomposable of dimension pk – pk−1 induced from the indecomposable K H-module of dimension p − 1, where H is the unique subgroup of G of order p. Formulae are given for the multiplicities of these indecomposables in Ln(V). This extends and utilises work of the first author and R. Stöhr concerned with the case where G has order p.

MSC classification

Secondary: 17B01: Identities, free Lie (super)algebras 20C20: Modular representations and characters
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 2 , October 2001 , pp. 149 - 158
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Benson, D. J., Representations and cohomology II (Cambridge University Press, Cambridge, 1991).Google Scholar
[2]Bourbaki, N., Lie groups and Lie algebras, Part I: Chapters 1–3 (Hermann, Paris, 1975).Google Scholar
[3]Brandt, A., ‘The free Lie ring and Lie representations of the full linear group’, Trans. Amer. Math. Soc. 56 (1944), 528–536.Google Scholar
[4]Bryant, R. M., Kovács, L. G. and Stöhr, R., ‘Lie powers of modules for groups of prime order’, Proc. London Math. Soc. (3), to appear.Google Scholar
[5]Bryant, R. M. and Stöhr, R., ‘On the module structure of free Lie algebras’, Trans. Amer. Math. Soc. 352 (2000), 901–934.CrossRefGoogle Scholar
[6]Huppert, B. and Blackburn, N., Finite groups II (Springer, Berlin, 1982).CrossRefGoogle Scholar
[7]Michos, I. C., Finite groups acting on free Lie algebras (Ph.D. Thesis, Manchester, 1998).Google Scholar
[8]Reutenauer, C., Free Lie algebras (Clarendon Press, Oxford, 1993).CrossRefGoogle Scholar