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The following is from Representation Theory of Finite Groups, by B. Steinberg; Example 3.1.14, p.16:

Let $\rho:S_3\to GL_2(\mathbb{C})$ be specified on the generators $(1 \ 2)$ and $(1 \ 2\ 3)$ by

$$\rho_{(1 \ 2)}=\begin{bmatrix}-1 & -1 \\ 0 & 1\end{bmatrix}, \ \rho_{(1 \ 2 \ 3)}=\begin{bmatrix}-1 & -1 \\ 1 & 0\end{bmatrix} $$

I'm trying to verify that $\rho:S_3\to GL_2(\mathbb{C})$ is a representation. By definition a representation is homomorphism, thus I want to show that $\rho_{gh}=\rho_g\rho_h$ for every $g,h\in S_3$. Starting with $$\rho_{(1 \ 2)}\rho_{(1 \ 2 \ 3)}=\begin{bmatrix}-1 & -1 \\ 0 & 1\end{bmatrix}\begin{bmatrix}-1 & -1 \\ 1 & 0\end{bmatrix}=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$$ we need to verify that $\rho_{(1 \ 2)(1 \ 2 \ 3)}=\rho_{(2 \ 3)}=\begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix}$, but I can't see how to do that. Moreover, I have doubts on checking the homomorphism property only for generators, that is, why checking the homomorphism property for the generators is enough to establish that $\rho$ is a representation?

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    $\begingroup$ You are supposed to define $\rho_{(23)}$ to be that. $\endgroup$ – Randall Jul 29 '18 at 2:17
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Hint: See this answer .

So, a presentation for $S_3$ is $\langle a,b\mid a^2=b^3=(ab)^2=1\rangle$.

Take $a=(12)$ and $b=(123)$.

You just need to check that $\rho (r_i)=1$ for each of the relations.

Once you establish that $\rho$ is a homomorphism, you will have that it's a representation of $S_3$ on $\mathbb C^2$ by definition.

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    $\begingroup$ Of course, you need to still show it's a homomorphism. Defining it on generators is not enough. $\endgroup$ – Steve D Jul 29 '18 at 4:35
  • $\begingroup$ Right you are... I was trying to correct it when I noticed your comment. $\endgroup$ – Chris Custer Jul 29 '18 at 5:35

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