example of matrix representations

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 $GL(ℂ)=ℂ\setminus \{0\}$ (that is invertible matrices of size $1\times 1$, which is the set of non-zero complex numbers). And thus we say that $ℂ\setminus \{0\}$ carries a representation of the symmetric group.

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

Such matrices are called permutation matrices because they are obtained permuting the colums of the identity matrix. The function so defined is then a group homomorphism, and thus $G{L}_n(ℂ)$ carries a representation of the symmetric group.