# Show that the given representation of the group $G$ is reducible

Let $$a$$ be the reflection of the plane $$\mathbb{R}^2$$ over the bisector of the odd quadrants (line with equation $$y = x$$), and let $$b$$ be the reflection of the same plane over the bisector of the even quadrants (line with equation $$y = -x$$).

Show that the given representation of the group $$G$$ is reducible: write the bases of the invariant subspaces $$U, V \subseteq \mathbb{R}^2$$, for which $$U \oplus V = \mathbb{R}^2$$ holds.

I ask these question here: Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants and as I stated any idea on how to even start this task?

Attempt: To show that the representation of the group $$G$$ is reducible, we need to analyze the actions of the reflections $$a$$ and $$b$$ on $$\mathbb{R}^2$$. The reflection $$a$$ over the line $$y = x$$ swaps the coordinates of any vector, transforming $$(x, y)$$ to $$(y, x)$$, represented by the matrix $$A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$. The reflection $$b$$ over the line $$y = -x$$ transforms $$(x, y)$$ to $$(-y, -x)$$, represented by the matrix $$B = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$. Have I started correctly? How to continue?

Here, you might want to review what "reducible" means in that context. In general, saying that the representation of $$G$$ is reducible means that you should find two nonzero subspaces $$U$$ and $$V$$ stable by the action of $$G$$, which sum to the whole space. In this special case, the whole space is $$\Bbb R^2$$: what does this say about the dimension of $$U$$ and $$V$$ ? In view of this value for the dimension, how can you reformulate what you are looking for ? Sketching the situation might help you.
• Strictly speaking, what you wrote down is the definition of "decomposable". Reducible means merely that there is a proper non-zero stable subspace. These two notions happen to be equivalent for representations of a finite group over a field of characteristic $0$, but this is a nontrivial fact (it is the content of Maschke's Theorem), and is not true for arbitrary group representations. Commented Jul 13 at 15:57