Vertices of Lie Modules

Abstract

Let Lie(n) be the Lie module of the symmetric group S_n over a field F of characteristic p>0, that is, Lie(n) is the left ideal of FS_n generated by the Dynkin-Specht-Wever element. We study the problem of parametrizing non-projective indecomposable summands of Lie(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=9 and p=3, and when n=8 and p=2, we present a precise answer. This suggests a possible parametrization for arbitrary prime powers.