Linear algebra/representation theory - an example needed

Question

I am reading Linear representations of finite groups by Serre, the following makes sense but can I see this with a concrete example since I cannot think of one?

Let $\rho$ and $\rho\,'$ be two representations of the same group $G$ in vector spaces $V$ and $V\,'$. These representations are said to be isomorphic if there exits a linear homomorphism $\kappa\colon V \rightarrow V\,'$ which ''converts'' $\rho$ to $\rho\,'$, if the following equality holds $$ \kappa \circ \rho(s) = \rho(s)\,'\circ \kappa \quad \forall s \in G.$$

If $\rho$ and $\rho\,'$ are given in matrix form by $\Gamma_s$ and $\Gamma_s'$, this implies, there exists an invertible matrix $K$ such that the following is holds: $$ \Gamma_s = K^{-1} \cdot \Gamma_s' \cdot K \quad \forall s \in G. $$

Answer 1

Let the base field be $\mathbb{C}$, and let $V=V'=\mathbb{V}_2(\mathbb{C})$.

Let $G=\mathbb{Z}_3=\langle d \rangle$.

In matrix terms let $$ \rho:d\mapsto \begin{pmatrix} 0 & 1\\-1 & -1\end{pmatrix} $$ and $$ \rho' :d\mapsto \begin{pmatrix}\omega & 0\\ 0 & \omega^2 \end{pmatrix} $$ where $\omega$ is the cube root of unity.

You can find the matrix of $\kappa$ for yourself, it's the usual matrix of eigenvectors.

Answer 2

Take two representations of the cyclic group of order two $C_2=\langle c\mid c^2=1\rangle$ in $\mathbb{R}^2$ given by $$ \rho(c)=\begin{pmatrix}-1&0\\0&1\end{pmatrix} $$ and $$ \rho'(c)=\begin{pmatrix}\frac{3}{5}&-\frac{4}{5}\\-\frac{4}{5}&-\frac{3}{5}\end{pmatrix}. $$ The map $\kappa:\mathbb{R}^2\to\mathbb{R}^2$ is the map $$ \kappa\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}1\\2\end{pmatrix},\;\;\;\mbox{and}\;\;\;\kappa\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}-2\\1\end{pmatrix}. $$ Geometrically, $\rho$ represents $c$ as a reflection over the $y$-axis, and $\rho'$ represents $c$ as a reflection over the line $y=2x$.