The Lie module and its complexity

Abstract

The complexity of a module is an important homological invariant that measures the polynomial rate of growth of its minimal projective resolution. For the symmetric group $Σ_n$, the Lie module $\mathsf{Lie}(n)$ has attracted a great deal of interest in recent years. We prove here that the complexity of $\mathsf{Lie}(n)$ in characteristic $p$ is $t$ where $p^t$ is the largest power of $p$ dividing $n$, thus proving a conjecture of Erdmann, Lim and Tan. The proof uses work of Arone and Kankaanrinta which describes the homology $\operatorname{H}_\bullet(Σ_n, \mathsf{Lie}(n))$ and earlier work of Hemmer and Nakano on complexity for modules over $Σ_n$ that involves restriction to Young subgroups.