Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 19I / ii

Let $\chi$ be the character of a representation of $\rho$. The character orthogonality identity for an irreducible character is ${\sum }_{g\in G}|\chi (g){|}^2=|G|=n$. For $g\in Z$, the scalar formula from part (i) gives $|\chi (g)|=d$. Keeping these $m$ nonnegative summands yields