Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 19J / a / Solution

Over $\mathbb{C}$, the operator representing $r$ has an eigenvector $v$ with eigenvalue $\lambda \ne 0$. The relation $tr{t}^{-1}=r^{-1}$ givesTherefore $\mathrm{span}\{v,tv\}$ is a nonzero $G$-invariant subspace of dimension at most two. If $V$ is irreducible it equals this subspace, so $\dim V\le 2$.