Let $G=\mathrm{SU}(2)$ and let $V_{n}$ be the vector space of complex homogeneous polynomials of degree $n$ in two variables.

(a) Prove that $V_{n}$ has the structure of an irreducible representation for $G$.

(b) State and prove the Clebsch-Gordan theorem.

(c) Quoting without proof any properties of symmetric and exterior powers which you need, decompose $\mathrm{S}^{2} V_{n}$ and $\Lambda^{2} V_{n}(n \geqslant 1)$ into irreducible $G$-spaces.