# Understanding Representation theory and group actions

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?

• I'm not sure I understand your question. You need to show that you have a homomorphism from $S_3$ to $GL_2(\mathbb C)$ compatible with those two datapoints $\rho(c)$ and $\rho(t)$. The $j$ is coming from the fact that since $t^3=e$, $\rho(t)^3=I_2$, and so the only possible eigenvalues of $\rho(t)$ are 1, j, and $j^2$. All the matrices in a reputation are diagonalizable, and there is no loss in generality picking a basis such that one particular matrix ends up diagonal. If $\rho(t)$ is the matrix you make diagonal, there aren't many choices. Oct 9, 2022 at 15:14
• I,m sry $ρ(c)=\begin{pmatrix} j & 0 \\ 0 & j^2 \\ \end{pmatrix}$,not ρ(t),i just edited.
– GGG
Oct 9, 2022 at 15:39
• Regardless, any relation that holds in the original group holds in the representation, and any collection of matrices that satisfies all the relations between all of the generators will be a representation. The j is coming in because you have a generator with order 3. However, there are many other choices. If you want a real 2 dimensional representation, take a reflection and a rotation by 120 degrees. Oct 9, 2022 at 15:43
• Ah I see ,I see .Thank you Aaron !
– GGG
Oct 9, 2022 at 15:47

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.
• And so if every element of $S_3$ is a ''power'' of c or t ,then every ρ is a power of ρ(t) and (or ) ρ(c)
• @GGG It isn't that every element in $S_3$ is a power of $x$ and $t$, you can also have products. For example, to find $\rho$ at $(1\;3)$, you can use $(1\;3) = tc$ (assuming I understand your notation). So $\rho(1\;3) = \rho(tc) = \rho(t)\rho(c)$. Oct 9, 2022 at 16:21