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

Abstract

AbstractKlyachko, 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.