# quotient representations

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

###### Definition 1

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

 $\varphi(\sigma\cdot x)=\sigma\cdot\varphi(x)$

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

 $x\in M_{1}\Rightarrow\varphi(\sigma\cdot x)=\sigma\cdot\varphi(x)=0\Rightarrow% \sigma\cdot x\in M_{1}$

$M_{2}$ is a submodule of $N_{2}$ if it is stable under the action of $G$, but

 $y=\varphi(x)\in M_{2}\Rightarrow\sigma\cdot y=\sigma\cdot\varphi(x)=\varphi(% \sigma\cdot x)\Rightarrow\sigma\cdot y\in M_{2}$

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 QuotientRepresentations 2013-03-22 16:37:59 2013-03-22 16:37:59 rm50 (10146) rm50 (10146) 6 rm50 (10146) Definition msc 20C99