I'm currently reading Serre's Linear Representations of Finite Groups, and I'm kind of confused regarding the concepts of irreducibility and indecomposability.
If I'm understanding it correctly, an irreducible representation is a representation $(\rho, V)$ of a group $G$ which does not has a non-trivial subrepresentation (i.e. a representation $(\rho|_W, W)$ where $W\subseteq V$ is a $G$-stable subspace). While an indecomposable representation is a representation that is not isomorphic to any direct sum of other representations.
Wikipedia and other sources say that irreducibility implies indecomposability, which seems logical (not 100 % sure why though), while Serre says the following:
Let $\rho:G \rightarrow GL(V) $ be a linear representation of $G$. We say that it is irreducible or simple if $V$ is not $0$ and if no vector subspace of $V$ is stable under $G$, except of course $0$ and $V$. By theorem I [which says that there exists a $G$-stable complement $W^0$ of a $G$-stable subspace $W\subseteq V$], this second condition is equivalent to saying $V$ is not the direct sum of two representations.
Does this not mean that irreducibility is equivalent indecomposability, and thereby going against what Wikipedia says?
Thanks!
P.S. I'm getting kind of annoyed at this book for being "to concise" and skipping a lot of details. Does anyone have any other recommendations on introductory books on representation theory?
Best Answer
Let $G=(\mathbb{R},+)$ and let $V=\mathbb{R}^2$. Consider $\rho\colon G\longrightarrow GL(V)$ defined by$$\rho(\lambda)(x,y)=(x+\lambda y,y).$$Then $\rho$ is a representation of $G$ which is indecomposable (the only non-trivial subspace is $\mathbb{R}\times\{0\}$) but not irreducible.
However, Serre is dealing with finite-dimensional complex representations of finite groups, and in that case, yes, every indecomposable representation is irreducible.