= Solution
The <special unitary group> in two dimensions is
$$
G=SU(2)=\{A\in M_2(\mathbb C):A^*A=I,\ \det A=1\}.
$$
Let $V_1=\mathbb C^2$ be its defining <group representation>, and for $n\geq0$ let
$$
V_n=S^nV_1,
$$
the <homogeneous polynomial representation of SU2> on homogeneous polynomials of degree $n$ in two variables. It has <degree of a representation> $n+1$. The representations $V_0,V_1,V_2,\ldots$ are pairwise nonisomorphic, continuous, finite-dimensional <irreducible representations>, and the <classification of finite-dimensional representations of SU2> says that this is the complete list.
Every $A\in SU(2)$ is conjugate to $\operatorname{diag}(z,z^{-1})$ with $|z|=1$. The monomial basis of $V_n$ gives the <character of the homogeneous polynomial representation of SU2>
$$
\boxed{\chi_{V_n}(A)=z^n+z^{n-2}+\cdots+z^{-n}}.
$$
Writing $z=e^{i\theta}$, this is $\sin((n+1)\theta)/\sin\theta$, interpreted by continuity when $\sin\theta=0$. The <Clebsch-Gordan decomposition for SU2> is
$$
\boxed{V_m\otimes V_n\cong\bigoplus_{j=0}^{\min(m,n)}V_{m+n-2j}}.
$$
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