= 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:
$$
\boxed{M^{(n-3,3)}=S^{(n-3,3)}\oplus S^{(n-2,2)}\oplus S^{(n-1,1)}\oplus S^{(n)}.}
$$
Back to article page
