Mathematical Proceedings of the Cambridge Philosophical Society

Table of Contents - July 2012 - Volume 153, Issue 01  

Research Article

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

R. M. BRYANTa1 and MARIANNE JOHNSONa1

a1 School of Mathematics, University of Manchester, Manchester M13 9PL. e-mail: roger.bryant@manchester.ac.uk, marianne.johnson@manchester.ac.uk

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.

(Received February 15 2011)

(Revised October 26 2011)

(Online publication February 28 2012)