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A representation of a group \(G\) is a homomorphism \(\phi: G\to GL(V)\), for some (finite-dimensional) non-zero vector space \(V\). The dimension of \(V\) is called the degree of \(\phi\).
Usually, \(\phi(g)\) is denoted by \(\phi_g\) for \(g\in G\).
Two representations \(\phi: G\to GL(V)\) and \(\psi: G\to GL(W)\) are equivalent if there exists an isomorphism \(T: V\to W\) such that \(\psi_g = T\phi_g T^{-1}\) for all \(g\in G\). In this case, we write \(\phi \sim \psi\).
Define \(\phi: \Bbb Z_n \to GL(\Bbb C^2)\) by
\[\phi(m) = \begin{pmatrix} e^{2\pi mi/n} & 0 \\ 0 & e^{-2\pi mi/n} \end{pmatrix}.\]Also, define \(\psi: \Bbb Z_n \to GL(\Bbb C^2)\) by
\[\psi(m) = \begin{pmatrix} \cos(2\pi m/n) & -\sin(2\pi m/n) \\ \sin(2\pi m/n) & \cos(2\pi m/n). \end{pmatrix}\]We can show that \(\phi \sim \psi\) with
\[T = \begin{pmatrix} i & -i \\ 1 & 1 \end{pmatrix}.\]