About the decomposable vector space 
I want to ask you whether the direct sum mentioned in the definition is unique, or the vector space $V$ could be written as a direct sum of another two $G$-invariant subspaces? 
I ask about the existence and unique cases. thanks!
 A: It is not necessarily unique. If
$$V_1 = V_3 \oplus V_4$$
with $V_3$ and $V_4$ being non-zero invariant subspaces of $V_1$, then both $V_3$ and $V_4 \oplus V_2$ are non-zero invariant subspaces of $V$ and additionally,
$$V = V_3 \oplus (V_4 \oplus V_2)$$
A: There may be many ways to decompose a representation into nonzero invariant subspaces. One may use an analogy with arithmetic: in general, composite numbers can be written as the product of two numbers (neither equal to $1$) in many different ways. For instance, $24$ may be written as any of the products $2\times 12$, $3\times 8$, $4\times 6$. In general, however, we can refine the decomposition of $V$, for instance if $V=V_1\oplus W$ is an internal direct sum and $W=V_2\oplus V_3$ is an internal direct sum, then $V=V_1\oplus W=V_1\oplus V_2\oplus V_3$ is an internal direct sum.
We may proceed in this way to continue refining a decomposition until we can't anymore; then we have a decomposition of $V$ into a direct sum of indecomposable subspaces. In general, even if a representation is indecomposable, it may still have subrepresentations, it's just that those subreps have no complementary reps as part of a decomposition. However, for complex representations of finite groups, indecomposability and irreducibility are equivalent.
The decomposition into indecomposables/irreps (meaning, the set of subspaces used in the internal direct sum) is generally not unique. For instance, suppose $G$ acts trivially on the two-dimensional real vector space $\mathbb{R}^2$. The irreps are precisely the one-dimensional subspaces, so decomposing $\mathbb{R}^2$ into an internal direct sum of irreps is equivalent to picking any pair of distinct lines - but there are many pairs of distinct lines to use!
However, the multiset (set with repeated elements allowed) of isomorphism classes of ireps used in the decomposition into irreps is uniquely determined. Moreover, there given any isomorphism class $(\ast)$ of irreducible representation of $G$, there is a unique subrep $W\le V$ with the following properties: all irreducible subreps of $W$ are isomorphic to $(\ast)$, and all subreps of $V$ isomorphic to $(\ast)$ are contained in $W$. In other words, the subspace $W$ which sums all of the irreducible subreps of a certain type is uniquely determined. It is called an isotypical component.
