Two dimensional representation of $S_3$

Question

I am reading some notes and I’m trying to find out about this two-dimensional representation of $S_3$ and I can’t really find much anywhere that actually explains what it is. So I thought I’d ask here.

I found something that says the transpositions $(1 2)$ and $(2 3)$ generate $S_3$ and then define $\phi : S_3 \rightarrow \text{GL}_2({\mathbb{C}}) $ by $$ \phi ((1 2))= \begin{pmatrix} 1 &0\\ -1 & -1 \end{pmatrix}$$ and $$ \phi ((2 3))= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} .$$

Is there any insight into how they’ve come up with these images of these transpositions?

Answer 1

$S_3$ can be regarded as the group of symmetries of an equilateral triangle. There are $6$ symmetry operations (the identity, three reflections and two rotations), and they each realize a distinct permutation of the vertices (the identity, three transpositions and two $3$-cycles). The points of an equilateral triangle can conveniently be described by coordinates that respect the symmetry, for instance by the three distances from the centre along the medians. These add up to $0$, so they’re linearly dependent, and we can choose any two of them, for instance $\pmatrix{d_3\\d_2}$, to represent points. These are interchanged by the reflection that exchanges these two vertices (corresponding to the transposition $(23)$), and the reflection that exchanges the vertices $1$ and $2$ (corresponding to the transposition $(12)$) leaves $d_3$ unchanged and replaces $d_2$ by $d_1=-(d_2+d_3)$. That yields the two matrices you quote.