Asymptotic behaviour of Lie powers and Lie modules

TL;DR

This paper investigates the asymptotic structure of Lie powers and Lie modules over fields of prime characteristic, showing that certain submodules dominate their dimensions as the degree grows large.

Contribution

It establishes that for degrees not a power of the characteristic, Lie powers and modules contain large submodules whose dimensions approach those of the entire modules as the degree increases.

Findings

Existence of a large direct summand in Lie powers for degrees not a power of p.

Construction of a projective submodule in Lie modules with dimension ratio approaching 1.

Asymptotic dominance of these submodules in the module structure as r tends to infinity.

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 $r \to \infty$.