= Solution
Write
$$
g=\begin{pmatrix}a&b\\-\bar b&\bar a\end{pmatrix}\in SU_2,
\qquad |a|^2+|b|^2=1.
$$
The <homogeneous polynomial representation of SU2> is the symmetric-power action determined by
$$
g\cdot x=ax-\bar b y,\qquad
g\cdot y=bx+\bar a y,
$$
and extended multiplicatively to $V_n$.
In the ordered basis $(x^3,x^2y,xy^2,y^3)$, the matrix for $g$ is
$$
\begin{pmatrix}
a^3&a^2b&ab^2&b^3\\
-3a^2\bar b&a^2\bar a-2ab\bar b&2ab\bar a-b^2\bar b&3b^2\bar a\\
3a\bar b^2&b\bar b^2-2a\bar a\bar b&a\bar a^2-2b\bar a\bar b&3b\bar a^2\\
-\bar b^3&\bar a\bar b^2&-\bar b\bar a^2&\bar a^3
\end{pmatrix}.
$$
The character of a representation $(V,\rho)$ is $\chi_V(g)=\operatorname{tr}\rho(g)$. Every $g\in SU_2$ is conjugate to $\operatorname{diag}(z,z^{-1})$ with $|z|=1$, and hence the <character of the homogeneous polynomial representation of SU2> is
$$
\chi_n(g)=z^n+z^{n-2}+\cdots+z^{-n}.
$$
For $z=e^{i\theta}$ this is $\sin((n+1)\theta)/\sin\theta$, interpreted by continuity when $\sin\theta=0$.
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