I came across the definition of a completely reducible representation of a group. The way I understand the definition there is no real difference between a reducible representation and a completely reducible representation. The book I'm currently reading states that "This means that a completely reducible representation can be written, with a suitable choice of basis, as the direct sum of irreducible representations.". But, as far as I know, any representation is either reducible or irreducible so, eventually, any reducible representation can be written as a direct sum of irreducible representations.
What am I missing? What is the difference between reducible and completely reducible?
Thanks.
Best Answer
For a field of characteristic zero we have the Maschke theorem : if $W \subset V$ is a subrepresentation, there is another subrepresentation $U \subset V$ such that $U \oplus W \cong V$. (Here I am assuming that $G$ is finite, as if $G$ is infinite there are already counter-example as given in the comments).
This become false if the field is not of characteristic zero, and a very cute example is $S_2$ acting on the plane over the field $\mathbb F_2$. There is exactly one line which is invariant (namely $L = \text{span}(e_1 + e_2)$) so our representation is reducible but not completely reducible.