The Lie module and its complexity

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 Σn, the Lie module Lie(n) has attracted a great deal of interest in recent years. We prove here that the complexity of Lie(n) in characteristic p is t, where p 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 H•(Σn, Lie(n)) and earlier work of Hemmer and Nakano on complexity for modules over Σn that involves restriction to Young subgroups.