Does the dihedral group $D_4$ have any sub-representations using the representation provided? if so, please can I have an example?
so we have:
$$D_4 = \lbrace e, r, r^2, r^3, c, d, x, y\rbrace$$
where $r$ is the rotation about the origin through $\pi/2$ (sending A to B).
while $c, d, x, y$ are reflections in $AC,BD, O_x$ and $O_y$
we have the map
$$A(g):\mathbb{R}^2\rightarrow \mathbb{R}^2$$
the map $$g \mapsto A(g) \in GL(2,\mathbb{R})$$
defines a 2-dimensional representation $\rho$ over $\mathbb{R}$
the representation $\rho$ is given in matrix terms by
$e \mapsto \begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix} $, $r \mapsto \begin{bmatrix} 0 & -1\\ 1 & 0\end{bmatrix}$, $r^2 \mapsto \begin{bmatrix} -1 & 0\\ 0 &-1\end{bmatrix}$, $r^3 \mapsto \begin{bmatrix} 0 & 1\\ -1 & 0\end{bmatrix}$,
$c \mapsto \begin{bmatrix} 0 & 1\\ 1 & 0\end{bmatrix}$, $d \mapsto \begin{bmatrix} 0 & -1\\ -1 & 0\end{bmatrix}$, $x \mapsto \begin{bmatrix} 1 & 0\\ 0 & -1\end{bmatrix}$, $y \mapsto \begin{bmatrix} -1 & 0\\ 0 & 1\end{bmatrix}$