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Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group $S_3$ on {$V_1, V_2, V_3$} where $\pi \in S_3$ takes each vertex $V_i $ to $V_{\pi(i)}$. This extends to a unique (linear) action $S_3$ by $X:S_3 → GL_2{\mathbb{R}}$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Let $X : S_3 → GL_2(\mathbb{R})$ be a matrix representation.

attemtp: I already found the six matrices. And I just want help on showing the matrices faithfully represent $S_3$.

I was trying to relate that a homomorphism $X$ is injective if and only if the kernel of $X$ is the identity . And by definition, a linear or matrix representation is faithful if it is injective.

Suppose $X$ is injective , then $X(e) = \begin{bmatrix} 1 \ 0 \\ 0 \ 1 \end{bmatrix}$. Where $e$ is the identity permutation , and $ \begin{bmatrix} 1 \ 0 \\ 0 \ 1 \end{bmatrix}$. is the identity in $GL_2(\mathbb{R})$.

Then $kerX = X^{-1}( \begin{bmatrix} 1 \ 0 \\ 0 \ 1 \end{bmatrix})$ = {$e$}.

And conversely, suppose $ker X$ ={$e$} and $X(\pi_1) = X(\pi_2)$.

Then $X(\pi_1\pi_2^{-1}) = X(\pi_1)X(\pi_2)^{-1} = \begin{bmatrix} 1 \ 0 \\ 0 \ 1 \end{bmatrix}$. Then $\pi_1\pi_2^{-1} \in kerX$ and so $\pi_1\pi_2^{-1} = e$, so $\pi_1 = \pi_2$. SO $X$ is injective, hence is a faithful representation of $S_3$.

Can someone please verify this? Any suggestion or feedback would be really appreciated it. Thank you!

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  • $\begingroup$ Faithfulness is just another name for injectivity for representations; there's nothing to prove there. (Presumably it's to avoid confusing the injectivity of an action $\rho:G \to \operatorname{End}(V)$ with the injectivity of each element $\rho(g)$.) What part of the proof are you stuck on? If you explicitly have $X(g)$ for all $g\in S_3$, then there's nothing more to do. $\endgroup$ – anomaly Feb 5 '16 at 21:05
  • $\begingroup$ I thought I had to prove it was injective. So by a formal conclusion, I could say $X(\pi)$ for all $\pi \in S_3$ implies the function is injective because each elements gets map to a unique element in $GL_2$? $\endgroup$ – Mahidevran Feb 5 '16 at 21:09

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