This post concerns the proof of the Artin conjecture in the case of degree 2 representations.
It is known that the finite subgroups of $\textrm{GL}(2,\mathbb{C})$ can be classified according to their images in $\textrm{PGL}(2,\mathbb{C})$, being isomorphic to one of: $C_n$, $D_n$, $A_4$, $S_4$, $A_5$, in which case they are called cyclic, dihedral, tetrahedral, octahedral, and icosahedral respectively.
Any degree 2 representation $(V,\rho)$ of a finite group $G$ can hence be classified by the image of $\rho(G) < \textrm{GL}(2,\mathbb{C})$ under the canonical projection $\pi: \textrm{GL}(2,\mathbb{C}) \to \textrm{PGL}(2,\mathbb{C})$. We say that $(V,\rho)$ is cyclic if $\pi(\rho(G)) \cong C_n$, dihedral if $\pi(\rho(G)) \cong D_n$, et cetera.
The work of Langlands and Tunnell resolved the Artin conjecture for tetra- and octahedral representations. I will not be going into that here. But in the cases of cyclic and dihedral representations, it is said that the following two are equivalent:
(1) $(V,\rho)$ is a cyclic or dihedral degree 2 representation.
(2) $(V,\rho)$ is a monomial representation.
This would resolve the Artin conjecture for cyclic and dihedral representations.
I have been trying to prove the equivalence of (1) and (2), but all my attempts so far have only led to a cul-de-sac.
All help or input would be highly appreciated.