Irreducible complex representations of a finite dihedral group

For $D_{2n}=⟨r,s:r^n=s^2=1,srs=r^{-1}⟩$, 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 $\mathrm{span}\{v,sv\}$.

For the dihedral group of order $2m$ with odd $m$, let $D_k(r)=\mathrm{diag}({\zeta }^k,{\zeta }^{-k})$ and $D_k(s)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, where $\zeta =e^{2\pi i/m}$ and $1\le k\le (m-1)/2$. An invariant tensor defining a bilinear form satisfies $D_k(g{)}^{T}L{D}_k(g)=L$. The rotation forces $L_{11}=L_{22}=0$ and the reflection forces $L_{12}=L_{21}$. Thus the invariant tensor is a multiple of $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, and a singlet Yukawa interaction gives two equal physical masses by singular value decomposition. Distinct inequivalent doublets admit no such bilinear form, by Schur lemma and their self-duality.

For $\omega =e^{2\pi i/n}$ and $1\le j<n/2$, defineThese representations are irreducible and pairwise nonisomorphic. They give all two-dimensional irreducibles: $(n-1)/2$ of them for odd $n$, and $n/2-1$ for even $n$.

For $D_8=⟨r,s:r^4=s^2=1,srs=r^{-1}⟩$, the matricesdefine a faithful irreducible two-dimensional complex representation.