TITLE: Representation of Triangle vertices permutations group in $\Bbb{R}^2$
QUESTION [1 upvotes]: Let $S_3$ be the symmetry group of vertices permutations of equilateral triangle in $\Bbb{R}^2$ with origin in $0$.

Show that the group operation of $S_3$ could be extended to a linear map $\rho (\sigma), \sigma\in S_3$ over the plane.

Show that $\rho : S_3 \to GL(\Bbb{R}^2)$ is a group representation.


Our course definition of a representation of a (finite group) is a pair $(V,\rho)$ where $V$ is a vector space over a field $\Bbb{F}$ and $\rho: \Bbb{G}\to GL(V)$ is a homomorphism between a group $G$ and a space of invertible linear transformations.
This is a Group Representation Theory question, and I am a little confused about the different elements in $S_3$. How can I define element of $S_3$ so that they can be mapped into reflection and rotation matrices in $GL(\Bbb{R}^2)$? I could use some help. I am really not sure what I am doing here. Do the matrices of reflections and rotations have anything to do with it?
So I guess my group is a group of permutations, which I hope can be represented as a set of $2\times 3$ matrices where the first row is $(A B C)$ for each vertex and the second is any of the 6 rearrangements of "ABC".
So my $\rho$ takes a such permutation and maps it to a rotation matrix or a reflection rotation matrix in $\Bbb{R}^2$, but other than writing permutation and saying that $
    \begin{pmatrix}
    A & B & C \\
    C & A & B \\
    \end{pmatrix} 
$ fits
\begin{pmatrix}
{\cos({2\pi\over 3})} & {-\sin({2\pi\over 3})}  \\
{\cos({2\pi\over 3})} &  {\cos({2\pi\over 3})}  \\
\end{pmatrix}
is there anything more specific I am required to?

REPLY [1 votes]: In 1 you need to show that for every permutation $\sigma\in S_3$, there is a linear map $\rho(\sigma): \mathbb R^2 \to \mathbb R^2$ (i.e. a matrix) such that applying $\rho(\sigma)$ to the vertices of the equilateral triangle gives the same result as applying the permutation $\sigma$ to the vertices. This is basically what you have done for a specific permutation at the end of your question.
For 2 you need to show that $\rho$ actually is a representation of $S_3$. I.e. that
$
\rho(\sigma\circ\tau) = \rho(\sigma)\rho(\tau).
$
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