= Solution
For $g$ conjugate to $\operatorname{diag}(z,z^{-1})$, multiplication of the weight sums gives
$$
\chi_n(g)^2
=\sum_{j=0}^n\chi_{2n-2j}(g).
$$
Indeed, the coefficient of $z^{2n-2m}$ on the left counts pairs $(r,s)$ with $0\leq r,s\leq n$ and $r+s=m$; the nested weight strings on the right give the same multiplicity. Characters determine finite-dimensional representations of the compact group $SU_2$, so
$$
\boxed{V_n\otimes V_n
\cong\bigoplus_{j=0}^nV_{2n-2j}}.
$$
This weight-counting argument proves the required case of the <Clebsch-Gordan decomposition for SU2> rather than merely quoting it.
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