Linear representation theory of symmetric groups: $S_{2}$

Question

I don't really get how i do a group representations. I know the representations of $S_{2}$ are the same of $Z_{2}$, i know, for this group that we have, the trivial and sign representations, but i don't really get how i put this on matrix form, i would be very thankfull if someone elaborate in how to make those representations.

Answer 1

Both these representations are 1 dimensional, so they're given by maps $\rho:S_{2} \rightarrow GL(\mathbb{C})=\mathbb{C}\setminus\{0\}$, as follows;

$\rho_{triv}(g)=1\text{, }\forall g\in S_{2}$ for the trivial representation and $\rho_{sgn}((1)(2))=1$, $\rho_{sgn}((1\quad2))=-1$ for the sign representation.

You can think of these maps as $1\times 1$ invertible matrices.

Answer 2

The trivial (complex) representation is the map $\phi:\Bbb Z_2\to \Bbb C^*$ that sends everything to the $(1\times1)$ matrix $1$. The sign representation is the map $\psi:\Bbb Z_2\to \Bbb C^*$ that sends $n+2\Bbb Z\mapsto (-1)^n$.