Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 19J / d / Solution

For $g$ conjugate to $\mathrm{diag}(z,z^{-1})$, multiplication of the weight sums givesIndeed, the coefficient of $z^{2n-2m}$ on the left counts pairs $(r,s)$ with $0\le r,s\le n$ and $r+s=m$; the nested weight strings on the right give the same multiplicity. Characters determine finite-dimensional representations of the compact group $S{U}_2$, soThis weight-counting argument proves the required case of the Clebsch-Gordan decomposition for SU2 rather than merely quoting it.