# Indecomposable but not irreducible representation and direct sums

I have two questions concerning representation theory of groups:

1) Consider the group $G=(\mathbb{R},+)$ and the representation $\rho: G\to\text{GL}(\mathbb{R}^2), r\mapsto\begin{pmatrix}1&r\\0&1\end{pmatrix}$.

Why is this representation indecomposable but not irreducible?

It is not irreducible as the space $\{(r,0), r\in \mathbb{R}\}$ is invariant under this action.

But why is this indecomposable?

2) If I have an exercise like 'Show that $V=V_1\oplus V_2$ as $G$-representations: What do I have to show? The first thing is that this is a direct sum of vector spaces, i.e. $V=V_1 + V_2$ and trivial intersection. But what else?

For the first question, indecomposable means that you have to show that the representation is not equivalent to a direct sum of non-zero representations of $G$. This is clear in the first example, so it is indecomposable. In the second part, you have to verify the definition of direct sum of representations, see
• Thank you! Why is this clear in the first example? And for the second part I need to show that the representation $G\to \text{GL}(V)$ is equal to the map $G\to \text{GL}(V_1\oplus V_2)$, right? But this is equivalent to show that $V_i$ are invariant, isn't it? – user337073 Oct 22 '17 at 19:55
1) Suppose on the contrary that $\rho$ is decomposable, i.e. exist $G$-invariant and linearly independent vectors $v = (v_{1}, v_{2} )^{t}, w = (w_{1}, w_{2} )^{t} \in \mathbb{R}^{2}$. Then, the following must be true: \begin{align} \rho_{r} v = \lambda v \tag{1} \\ \rho_{r} w = \gamma w \tag{2} \end{align} $\forall r \in \mathbb{R}$ y and some $\lambda, \gamma \in \mathbb{R}$. Equation (1) implies \begin{align} v_{1} + v_{2} r = \lambda v_{1} \\ v_{2} = \lambda v_{2} \end{align} so $\lambda = 1$ and $v_{2} r = 0$ for all $r \in \mathbb{R}$. Then $v_{2} = 0$, and $\mathbb{R}v = \mathbb{R} e_{1}$. From equation (2) we reach to the same conclusion, i.e. $\mathbb{R}w = \mathbb{R} e_{1}$. But this is absurd because we supposed $v$ and $w$ are linearly independent. So $\rho$ is indecomposable.
2) For your second question, you need to show that $V_{i}$ is $G$-invariant and that $\left. \rho\right|_{V_{i}} : G \to GL(V_{i})$ is a representation.