Vertices of Lie modules

Let LieF(n)LieF(n) be the Lie module of the symmetric group SnSn over a field F   of characteristic p>0p>0, that is, LieF(n)LieF(n) is the left ideal of FSnFSn generated by the Dynkin–Specht–Wever element ωnωn. We study the problem of parametrizing non-projective indecomposable summands of LieF(n)LieF(n), via describing their vertices and sources. Our main result shows that this can be reduced to the case when n is a power of p  . When n=9n=9 and p=3p=3, and when n=8n=8 and p=2p=2, we present a precise answer. This suggests a possible parametrization for arbitrary prime powers.