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
Canonical name	ExampleOfMatrixRepresentations
Date of creation	2013-03-22 14:53:31
Last modified on	2013-03-22 14:53:31
Owner	drini (3)
Last modified by	drini (3)
Numerical id	6
Author	drini (3)
Entry type	Example
Classification	msc 20C99