Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2007 / B / 8 / d / Solution

In characteristic three with $n=5$, a constant vector $(c,c,c,c,c)$ belongs to the augmentation submodule only if $5c=0$, which forces $c=0$. Thus $S^{(4,1)}$ has no trivial submodule. Part (b) proves that it is simple, of dimension four, and therefore equals $D^{(4,1)}$.