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A modular version of Klyachko's theorem on Lie representations of the general linear group

Published online by Cambridge University Press:  28 February 2012

R. M. BRYANT  and
MARIANNE JOHNSON
R. M. BRYANT
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL. e-mail: roger.bryant@manchester.ac.uk, marianne.johnson@manchester.ac.uk
MARIANNE JOHNSON
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL. e-mail: roger.bryant@manchester.ac.uk, marianne.johnson@manchester.ac.uk
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Abstract

Klyachko, in 1974, considered the tensor and Lie powers of the natural module for the general linear group over a field of characteristic 0 and showed that nearly all of the irreducible submodules of the rth tensor power also occur up to isomorphism as submodules of the rth Lie power. Here we prove an analogue for infinite fields of prime characteristic by showing, with some restrictions on r, that nearly all of the indecomposable direct summands of the rth tensor power also occur up to isomorphism as summands of the rth Lie power.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 1 , July 2012 , pp. 79 - 98
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

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