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.