Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 15F / b / i / Solution

The character of a representation is ${\chi }_V(g)=\mathrm{tr}{\rho }_V(g)$. Average any positive inner product over the compact group using Haar measure to obtain a unitary representation. On the maximal torus $\mathrm{diag}(z,z^{-1})$, the commuting unitary operators admit a simultaneous eigenbasis. Each eigenline carries a circle character of a representation $z^n$ by part (a). Hencewith finitely many nonzero coefficients. Every element of special unitary group is conjugate to an element of this torus, so this describes the full character of a representation via its torus restriction. Here the coefficient notation includes zero multiplicities.