= Solution
Every element of $SU(2)$ is unitarily conjugate to an element of the maximal torus
$$
T=\{\operatorname{diag}(z,z^{-1}):z\in S^1\}.
$$
Since characters are class <functions>, $\chi_V$ is determined on $T$. The restriction decomposes into finitely many torus weights, so
$$
\chi_V(z)=\sum_{n\in\mathbb Z}a_nz^n,
\qquad a_n\in\mathbb Z_{\geq0},
\qquad\sum_na_n=\dim V.
$$
A Weyl-group element conjugates $\operatorname{diag}(z,z^{-1})$ to $\operatorname{diag}(z^{-1},z)$, so $\chi_V(z)=\chi_V(z^{-1})$ and $a_n=a_{-n}$.
Back to article page
