Sign representation of $S_n$
Let $G=S_n$ the $n$ -th symmetric group, and consider $X(\sigma) = \mathrm{sign}(\sigma)$ where $\sigma$ is any permutation in $S_n$ . That is, $\mathrm{sign}(\sigma)=1$ when $\sigma$ is an even permutation, and $\mathrm{sign}(\sigma)=-1$ when $\sigma$ is an odd permutation.

The function $X$ is a group homomorphism between $S_n$ and (that is invertible matrices of size $1\times1$ , which is the set of non-zero complex numbers). And thus we say that carries a representation of the symmetric group.

Defining representation of $S_n$
For each $\sigma \in S_n$ , let the function given by $X(\sigma)=(a_{ij})_{n\times n}$ where $(a_{ij})$ is the permutation matrix given by