# quotient representations

We assume that all representations ($G$-modules) are finite-dimensional.

###### Definition 1

If ${N}_{\mathrm{1}}$ and ${N}_{\mathrm{2}}$ are $G$-modules over a field $k$ (i.e. representations of $G$ in ${N}_{\mathrm{1}}$ and ${N}_{\mathrm{2}}$), then a map $\phi \mathrm{:}{N}_{\mathrm{1}}\mathrm{\to}{N}_{\mathrm{2}}$ is a *$G$-map* if $\phi $ is $k$-linear and preserves the $G$-action, i.e. if

$$\phi (\sigma \cdot x)=\sigma \cdot \phi (x)$$ |

$G$-maps have subrepresentations, also called $G$-submodules, as their kernel and image. To see this, let $\phi :{N}_{1}\to {N}_{2}$ be a $G$-map; let ${M}_{1}\subset {N}_{1}$ and ${M}_{2}\subset {N}_{2}$ be the kernel and image respectively of $\phi $. ${M}_{1}$ is a submodule of ${N}_{1}$ if it is stable under the action of $G$, but

$$x\in {M}_{1}\Rightarrow \phi (\sigma \cdot x)=\sigma \cdot \phi (x)=0\Rightarrow \sigma \cdot x\in {M}_{1}$$ |

${M}_{2}$ is a submodule of ${N}_{2}$ if it is stable under the action of $G$, but

$$y=\phi (x)\in {M}_{2}\Rightarrow \sigma \cdot y=\sigma \cdot \phi (x)=\phi (\sigma \cdot x)\Rightarrow \sigma \cdot y\in {M}_{2}$$ |

Finally, we define the intuitive concept of a quotient $G$-module. Suppose ${N}^{\prime}\subset N$ is a $G$-submodule. Then $N/{N}^{\prime}$ is a finite-dimensional vector space^{}. We can define an action of $G$ on $N/{N}^{\prime}$ via $\sigma (n+{N}^{\prime})=\sigma (n)+\sigma ({N}^{\prime})=\sigma (n)+{N}^{\prime}$, so that $n+{N}^{\prime}$ is well-defined under the action and $N/{N}^{\prime}$ is a $G$-module.

Title | quotient representations |
---|---|

Canonical name | QuotientRepresentations |

Date of creation | 2013-03-22 16:37:59 |

Last modified on | 2013-03-22 16:37:59 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 6 |

Author | rm50 (10146) |

Entry type | Definition |

Classification | msc 20C99 |