Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2009 / B / 5 / d / Solution

The four constructed submodules have the distinct shapes $(n-3,3)$, $(n-2,2)$, $(n-1,1)$ and $(n)$, so they are pairwise nonisomorphic irreducible characteristic-zero modules. Complete reducibility therefore makes their sum direct. The given decomposition of $M^{(n-2,2)}$ and the dimension identity in (a) show their dimensions sum to $\dim M^{(n-3,3)}$. Their direct sum is consequently the whole module: