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

WriteThe homogeneous polynomial representation of SU2 is the symmetric-power action determined byand extended multiplicatively to $V_n$.

In the ordered basis $(x^3,x^{2}y,x{y}^2,y^3)$, the matrix for $g$ is

The character of a representation $(V,\rho )$ is ${\chi }_V(g)=\mathrm{tr}\rho (g)$. Every $g\in S{U}_2$ is conjugate to $\mathrm{diag}(z,z^{-1})$ with $|z|=1$, and hence the character of the homogeneous polynomial representation of SU2 isFor $z=e^{i\theta }$ this is $\sin ((n+1)\theta )/\sin \theta$, interpreted by continuity when $\sin \theta =0$.