Find a basis of Hom$_{\mathbb{C}G}(V,W)$. Let $G=C_4$ and let $V,W$ be $\mathbb{C}G-$modules with
$[g]_V=\left(\begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix}\right)$ and $[g]_W=\left(\begin{matrix} 17 & 29 & 0\\-10 & -17 & 0 \\ 0 & 0 & -1 \end{matrix}\right)$
Find a basis of Hom$_{\mathbb{C}G}(V,W)$ and determine if $V$ and $W$ share any common composition factors.
Attempt:
$[g]_V$ has eigenvalues $\{i, -i\}$ and eigenspaces $V_i= \left\langle \left( \begin{matrix} 1 \\ -i\end{matrix} \right) \right\rangle $, $V_{-i}= \left\langle \left( \begin{matrix} 1 \\ i\end{matrix}  \right) \right\rangle $.
$[g]_W$ has eigenvalues $\{i, -i, -1\}$ and eigenspaces $W_i= \left\langle \left( \begin{matrix} \frac{-17-i}{10} \\ 1 \\ 0\end{matrix} \right) \right\rangle $, $W_{-i}= \left\langle \left( \begin{matrix} \frac{-17+i}{10} \\ 1 \\ 0\end{matrix}  \right) \right\rangle , W_{-1}= \left\langle \left( \begin{matrix} 0 \\ 0 \\ 1\end{matrix} \right) \right\rangle $.
We can write $V=V_i \bigoplus V_{-i} $ and $W=W_i\bigoplus W_{-i} \bigoplus W_{-1}$.
This is where I am not sure how to proceed. I think we want to see which of the submodules are isomorphic, but I'm not sure. Any help is appreciated!
 A: Recall the following Lemma:

Let G a finite abelian group, then all the irreducible representations over $\mathbb{C}$ have dimesion $1$. 
(Sketch: Let $V$ be an irreducible representation, define the map $L_g:V\rightarrow V$, $v\mapsto [g]_Vv$. Thanks to the hypothesis is a $\mathbb{C}G$-map. Apply Shur's Lemma to obtain $L_g = \lambda_g Id$. Finally $<v>$ is a non trivial submodule, so $<v>=V$ for irreducibility)

In our case we have the cyclic group $C_4$ so it's easy to find all the irreducible representation. Let $g$ be the generator of $C_4$, and $V=\mathbb{C}$ we have $[g]_V^4 = 1$ so $[g]_V$ can be only $1,-1,i,-i$. So there are only $4$ distinct irreducible representation. We denote them $U_1,U_{-1},U_i,U_{-i}$.
Now you can show that $V_i\cong W_i \cong U_i$, $V_{-i} \cong W_{-i} \cong U_{-i}$ and $W_{-1} \cong U_{-1}$. And now you have a decomposition of $V$ and $W$ in irreducible representations:
\begin{gather}
V\cong U_i \oplus U_{-i}\\
W \cong U_{i} \oplus U_{-i} \oplus U_{-1}
\end{gather}
So $U_i$ and $U_{-i} are common decomposition factor.
Now try to compute $Hom_{\mathbb{C}G}(V,W)$. We have the following lemma:

Let $A,B,C$ be $G$-representations, then
  \begin{gather}
Hom(A\oplus B, C) \cong Hom(A,C) \oplus Hom(B,C)\\
Hom(A,B\oplus C) \cong Hom(A,B) \oplus Hom(A,C)
\end{gather}
(Sketch: write down the easiest map and it will work)

Thanks to the lemma we have:
\begin{equation}
Hom_{\mathbb{C}G}(V,W) \cong Hom_{\mathbb{C}G}(U_i,U_i) \oplus Hom_{\mathbb{C}G}(U_i,U_{-i}) \oplus Hom_{\mathbb{C}G}(U_i,U_{-1})\oplus Hom_{\mathbb{C}G}(U_{-i},U_i)\oplus Hom_{\mathbb{C}G}(U_{-i},U_{-i}) \oplus Hom_{\mathbb{C}G}(U_{-i},U_{-1})
\end{equation}
If two representations are non isomorphic, then the $Hom$ space is zero; if they are isomorphic then the $Hom$ space is isomorphic to $\mathbb{C}$ (Shur's Lemma). We obtain
\begin{equation}
Hom_{\mathbb{C}G}(V,W) \cong Hom_{\mathbb{C}G}(U_i,U_i)\oplus Hom_{\mathbb{C}G}(U_{-i},U_{-i}) \cong U_1 \oplus U_1
\end{equation}
Where the last isomorphism comes to the fact that the action of $G$ over the $Hom$ space is given by
\begin{equation}
([g]_{Hom(V,W)}f)(v) = [g]_{W}f([g]_V^{-1}(v))
\end{equation}
and if you do the computation you obtain the result.
If you want an explicit base of $Hom_{\mathbb{C}G}(V,W)$ it's sufficient to read what we have done in terms of $V_i,V_{-i}$ and $W_i,W_{-i},W_{-1}$; your generator of $V_i$ has to go into an element of $W_i$, and your generator of $V_{-i}$ has to go into an element of $W_{-i}$.
