Let $V_{2}$ denote the irreducible representation $\operatorname{Sym}^{2}\left(\mathbb{C}^{2}\right)$ of $S U(2)$; thus $V_{2}$ has dimension 3. Compute the character of the representation $\operatorname{Sym}^{n}\left(V_{2}\right)$ of $S U(2)$ for any $n \geqslant 0$. Compute the dimension of the invariants $\operatorname{Sym}^{n}\left(V_{2}\right)^{S U(2)}$, meaning the subspace of $\operatorname{Sym}^{n}\left(V_{2}\right)$ where $S U(2)$ acts trivially.

Hence, or otherwise, show that the ring of complex polynomials in three variables $x, y, z$ which are invariant under the action of $S O(3)$ is a polynomial ring. Find a generator for this polynomial ring.