Asymptotic behaviour of Lie powers and Lie modules

Roger M. Bryant, Kay Jin Lim and Kai Meng Tan

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^{\otimes r} and which satisfies
dim B^r(V)/dim L^r(V) \to 1 as r \to \infty. 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) \to 1 as r \to \infty.