Understanding Representation theory and group actions

Question

This might be a silly question, but I need help understanding an example of a representation of a group ,the following example is from Yvette Kosmann-Scwartbach's book called Groups and Symmetries:

''Let $t\in S_3 $ be the transposition $123\to 132$ and $c$ the cyclic permutation $123\to 231$ that generates $S_3$.We set $j=e^{2i\pi/3}$ ,so that $j^2+j+1=0$.We can represent $S_3$ on $\mathbb{C}^2 $ by defining $ρ(e)=I_2$,$ρ(t)=\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}$ and $ρ(c)=\begin{pmatrix} j & 0 \\ 0 & j^2 \\ \end{pmatrix}$''.

My understanding is that $ρ_g , (ρ_g=ρ(g))$ is defined by the action of $S_3$ on $\mathbb{C^2}$ and so for $ρ(e)=I_2$ we get $ρ(e)=[a_{e(1)},a_{e(2)}]$,where $e$ the ''neutral'' permutation and $a=\{a_1,a_2\}=\{(1,0),(0,1)\}$ a basis of $\mathbb{C^2}$ and thus $ρ(e)=[a_1,a_2]=I_2 $ ,since $e:123 \to 123$.
That kind of logic doens't seem to work on the rest.What am I missing and how does the author introduce $j$ on the matrices?
Thank you in advance !

Answer 1

Here is one way to understand this. For each $\sigma\in S_3$ you have $\rho(\sigma): \mathbb{C}^2 \to \mathbb{C}^2$ and this is simply given as matrix multiplication. So, for example $$ \rho(t)\begin{pmatrix}x \\ y \end{pmatrix} = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix} = \begin{pmatrix}y \\ x \end{pmatrix}. $$

The point is that since $S_3$ is generated by $t$ and $c$, the three (really only need $2$ equations you have is enough to fully define $\rho$. You can't just select any matrices for the value of $\rho$ at $t$ and $c$. So, there are things to check.