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Modular Lie representations of finite groups

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

Published online by Cambridge University Press:  09 April 2009

R. M. Bryant
R. M. Bryant
Affiliation:
School of Mathematics, University of Manchester, PO Box 88, Manchester M60 1QD, England e-mail: [email protected]
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Abstract

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Let K be a field of prime characteristic p and let G be a finite group with a Sylow p-subgroup of order p. For any finite-dimensional K G-module V and any positive integer n, let Ln (V) denote the nth homogeneous component of the free Lie K-algebra generated by (a basis of) V. Then Ln(V) can be considered as a K G-module, called the nth Lie power of V. The main result of the paper is a formula which describes the module structure of Ln(V) up to isomorphism.

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 77 , Issue 3 , December 2004 , pp. 401 - 424
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Almkvist, G., ‘Representations of Z/pZ in characteristic p and reciprocity theorems’, J. Algebra 68 (1981), 1–27.Google Scholar
[2]Alperin, J. L., Local representation theory, Cambridge Stud. Adv. Math. 11 (Cambridge University Press, Cambridge, 1986).Google Scholar
[3]Benson, D. J., Representations and cohomology I (Cambridge University Press, Cambridge, 1995).Google Scholar
[4]Brandt, A. J., ‘The free Lie ring and Lie representations of the full linear group’, Trans. Amer. Math. Soc. 56 (1944), 528–536.Google Scholar
[5]Bryant, R. M., ‘Free Lie algebras and Adams operations’, J. London Math. Soc. (2) 68 (2003), 355–370.Google Scholar
[6]Bryant, R. M., ‘Modular Lie representations of groups of prime order’, Math. Z. 246 (2004), 603–617.CrossRefGoogle Scholar
[7]Bryant, R. M., ‘Free Lie algebras and formal power series’, J. Algebra 253 (2002), 167–188.CrossRefGoogle Scholar
[8]Bryant, R. M., Kovács, L. G. and Stöhr, R., ‘Free Lie algebras as modules for symmetric groups’, J. Austral. Math. Soc. Ser. A 67 (1999), 143–156.CrossRefGoogle Scholar
[9]Bryant, R. M., ‘Lie powers of modules for groups of prime order’, Proc. London Math. Soc. (3) 84 (2002), 343–374.Google Scholar
[10]Bryant, R. M., ‘Lie powers of modules for GL(2, p)’, J. Algebra 260 (2003), 617–630.CrossRefGoogle Scholar
[11]Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Wiley Interscience, New York, 1962).Google Scholar
[12]Curtis, C. W. and Reiner, I., Methods in representation theory, II (J. Wiley and Sons, New York, 1987).Google Scholar
[13]Donkin, S. and Erdmann, K., ‘Tilting modules, symmetric functions, and the module structure of the free Lie algebra’, J. Algebra 203 (1998), 69–90.CrossRefGoogle Scholar
[14]Hughes, I. and Kemper, G., ‘Symmetric powers of modular representations for groups with a Sylow subgroup of prime order’, J. Algebra 241 (2001), 759–788.CrossRefGoogle Scholar
[15]Kouwenhoven, F. M., ‘The λ-structure of the Green ring of GL(2, Fp) in characteristic p, III’, Comm. Algebra 18 (1990), 1701–1728.Google Scholar
[16]Kouwenhoven, F. M., ‘The λ-structure of the Green ring of GL(2, Fp) in characteristic p, IV’, Comm. Algebra 18 (1990), 1729–1747.Google Scholar
[17]Kovács, L. G. and Stöhr, R., ‘Lie powers of the natural module for GL(2)’, J. Algebra 229 (2000), 435–462.Google Scholar