Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 19F / 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 $\mathrm{diag}(e^{i\theta },e^{-i\theta })$ 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 isThe 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.