Let $V_m$ be the $m$dimensional complex vector space with basis $\{e_1, \dots, e_m\}$ and let $i\leq m$. Consider the element ${v}_0^i \in S^i(S^m(V_m))$, where ${v}_0$ is the element $e_1\dots e_m \in S^m(V_m)$. Then, is the $GL(V_m)$submodule of $S^i(S^m(V_m))$ generated by the element ${v}_0^i$ is the whole of $S^i(S^m(V_m))$?

$$S^2( S^2 ) = S^4 \oplus S^{2,2}.$$ In this case, the first space is five dimensional and the second is one dimensional. Checking my work again this morning, I made a mistake. In fact, $i=m=2$ is not a counterexample. If $V_2$ is spanned by $x$ and $y$, the correct components are $$(xy)^2=\left(\frac{1}{3}(x^2)(y^2) + \frac{2}{3}(xy)^2\right) + \left(\frac{1}{3}(x^2)(y^2) + \frac{1}{3}(xy)^2\right).$$ 

