Past exam of the groups and representations course of the University of Oxford / 2023 / 1 / c / Solution

Let $\zeta =e^{2\pi i/n}$. The group homomorphism ${\mathbb{Z}}_n\to U(1)$ given by $[k]\mapsto {\zeta }^k$ is injective, identifying the finite cyclic group with the subgroup of $n$th roots of unity in the circle group.

The continuous irreducible representations of the circle group are the one-dimensional representations $R_Q(e^{i\theta })=e^{iQ\theta }$ with $Q\in \mathbb{Z}$. One way to see this is to make the representation unitary by averaging, simultaneously diagonalize its commuting unitary matrices, and impose $R_Q(e^{i(\theta +2\pi )})=R_Q(e^{i\theta })$. Its restriction of a representation isThus each irreducible representation remains one-dimensional, and representations whose integer labels differ by a multiple of $n$ become equivalent. Every irreducible representation of the finite cyclic group occurs this way.