Can there be any representation of a finite group which is reducible but indecomposable?
I know that it is true that irreducibility implies indecomposability. But the converse is not necessarily true. For instance we could take $\Bbb Z$ and the representation $\varphi_n = \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}$ which is not irreducible but indecomposable. Is there any analogous example for the representation of finite groups? Any help in this regard will be highly appreciated.
Thanks in advance.
There are analogues in finite characteristic -- the same as your example for $n=1$ if the group is cyclic of order $m$, and $m \bmod p=0$. In characteristic zero, irreducibility for finite group representations is equivalent to indecomposability. This is due to Maschke's theorem, see for example Understanding Maschke's Theorem