Asymptotic Behaviour of Lie Powers and Lie Modules

@inproceedings{BRYANT2010AsymptoticBO,
  title={Asymptotic Behaviour of Lie Powers and Lie Modules},
  author={ROGER M. BRYANT and Kay Jin Lim and Kai Meng Tan},
  year={2010}
}
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(V ) has a direct summand B(V ) which is a direct summand of the tensor power V ⊗r and which satisfies dim B(V )/ dim L(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 →∞.