Irreducible complex representations of a finite dihedral group
= Irreducible complex representations of a finite dihedral group
For $D_{2n}=\langle r,s:r^n=s^2=1,\ srs=r^{-1}\rangle$, every irreducible complex representation has degree at most two. If $v$ is an eigenvector of $r$ with eigenvalue $\lambda$, then $sv$ has eigenvalue $\lambda^{-1}$, so irreducibility makes the representation equal to $\operatorname{span}\{v,sv\}$.