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

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

 $a_{ij}=\begin{cases}1&\text{if }\sigma(i)=j\\ 0&\text{if }\sigma(i)\neq j\\ \end{cases}$

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 $GL_{n}(\mathbbmss{C})$ carries a representation of the symmetric group.

Title example of matrix representations ExampleOfMatrixRepresentations 2013-03-22 14:53:31 2013-03-22 14:53:31 drini (3) drini (3) 6 drini (3) Example msc 20C99