# 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 $GL(\u2102)=\u2102\setminus \{0\}$ (that is invertible matrices of size $1\times 1$, which is the set of non-zero complex numbers^{}). And thus we say that $\u2102\setminus \{0\}$ carries a representation of the symmetric group.

Defining representation of ${S}_{n}$

For each $\sigma \in {S}_{n}$, let $X:{S}_{n}\to G{L}_{n}(\u2102)$ the function given by $X(\sigma )={({a}_{ij})}_{n\times n}$ where $({a}_{ij})$ is the *permutation matrix ^{}* given by

$${a}_{ij}=\{\begin{array}{cc}1\hfill & \text{if}\sigma (i)=j\hfill \\ 0\hfill & \text{if}\sigma (i)\ne j\hfill \end{array}$$ |

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 $G{L}_{n}(\u2102)$ 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 |