Asymptotic behaviour of Lie powers and Lie modules

Abstract

Let V be a finite-dimensional FG-module, where F is a field of prime characteristic p and G is a group. We show that, when r is not a power of p, the Lie power L r(V) has a direct summand B r(V), which is a direct summand of the tensor power V ⊗r such that dim B r (V)/ dim L r (V) → 1 as r→∞. Similarly, for the same values of r, we obtain a projective submodule C(r) of the Lie module Lie (r) over F such that dim C(r)/dim Lie (r)→1 as r→∞. © 2011. Published by Oxford University Press. All rights reserved.