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The Complexity of the Lie Module

Published online by Cambridge University Press:  05 September 2013

Karin Erdmann ,
Kay Jin Lim  and
Kai Meng Tan
Karin Erdmann
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK, ([email protected])
Kay Jin Lim
Affiliation:
Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076, Singapore, ([email protected]; [email protected])
Kai Meng Tan
Affiliation:
Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076, Singapore, ([email protected]; [email protected])
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Abstract

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We show that the complexity of the Lie module Lie(n) in characteristic p is bounded above by m, where pm is the largest p-power dividing n, and, if n is not a p-power, is equal to the maximum of the complexities of Lie(pi) for 1≤i≤m.

Keywords

Lie modulep-modular representationsymmetric groupwreath productcomplexityPrimary 20C30
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 393 - 404
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Adem, A. and Milgram, R. J., Cohomology of finite groups (2nd edn), Grundlehren der Mathematischen Wissenschaften, Volume 309 (Springer, 2004).Google Scholar
2.Benson, D. J., Representations and cohomology, II, Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics, Volume 31 (Cambridge University Press, 1991).Google Scholar
3.Bryant, R. M. and Johnson, M., Lie powers and Witt vectors, J. Algebraic Combin. 28 (2008), 169–187.Google Scholar
4.Bryant, R. M. and Johnson, M., A modular version of Klyachko's theorem on Lie representations of the general linear group, Math. Proc. Camb. Phil. Soc. 153 (2012), 79–98.CrossRefGoogle Scholar
5.Bryant, R. M. and Schocker, M., The decomposition of Lie powers, Proc. Lond. Math. Soc. 93 (2006), 175–196.Google Scholar
6.Bryant, R. M., Erdmann, K., Danz, S. and Müller, J., Vertices of Lie modules, preprint (2013).Google Scholar
7.Donkin, S. and Erdmann, K., Tilting modules, symmetric functions, and the module structure of the free Lie algebra, J. Alg. 203 (1998), 69–90.CrossRefGoogle Scholar
8.Erdmann, K. and Schocker, M., Modular Lie powers and the Solomon descent algebra, Math. Z. 253 (2006), 295–313.Google Scholar
9.Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, Volume 52 (Springer, 1977).CrossRefGoogle Scholar
10.Lim, K. J. and Tan, K. M., The Schur functor on tensor powers, Arch. Math. 96 (2012), 513–518.Google Scholar
11.Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory (2nd edn) (Dover, New York, 1976).Google Scholar
12.Reutenauer, C., Free Lie algebras (Clarendon Press, Oxford, 1993).Google Scholar
13.Selick, P. and Wu, J., On natural coalgebra decomposition of tensor algebras and loop suspensions, Memoirs of the American Mathematical Society, Volume 701 (American Mathematical Society, Providence, RI, 2000).Google Scholar