= Solution
The left <group action> on the homogeneous <polynomials> is $\rho_n(g)P(z)=P(g^{-1}z)$, since $\rho_n(g)\rho_n(h)=\rho_n(gh)$. Every <SU(2) matrix> is conjugate to $\operatorname{diag}(e^{i\theta},e^{-i\theta})$ with $0\le\theta\le\pi$. In the monomial basis $z_1^{n-j}z_2^j$, its eigenvalues are $e^{-i(n-2j)\theta}$. Consequently the <character of the homogeneous polynomial representation of SU2> is
$$
\boxed{\chi_n(\theta)=\sum_{j=0}^n e^{i(n-2j)\theta}=\frac{\sin((n+1)\theta)}{\sin\theta}.}
$$
The quotient is interpreted by continuity at the endpoints: $\chi_n(0)=n+1$ and $\chi_n(\pi)=(-1)^n(n+1)$. The introductory action and character requests are independent of parts (i)–(iii), so this root Solution slot has been restored.
Back to article page
