Define the $Hom(V,W)$ representation by $\phi: V → W$
by $g\phi(v) = g\phi(g^{-1}v)$
where
$\phi \in Hom(V,W) = {\{f: V → W}\}$ and $g\in G$.
Show that $Hom(V,W)$ is indeed a representation/homomorphism.
attempt: Let $\varphi_1 : G → GL(V)$ and $\varphi_2: G → GL(W)$ be representations.
Suppose for every $g_1,g_2 \in G, v \in V$, we have
$(g_1g_2)\phi(v) = (g_1g_2)\phi[(g_1g_2)^{-1}v] = (g_1g_2)\phi[(g_2^{-1}g_1^{-1})v] $
I am stuck. Can someone please help me?I need to show it's a homomorphism. Thank you!
Define a map $$\rho: G \to GL(Hom(V,W))$$ by $g \cdot \varphi(v)=g\varphi(g^{-1}v)$. Let $\varphi, \psi \in Hom(V,W)$. Then for any scalar $c$ and vector $v \in V$, we have \begin{align*} g \cdot (c\varphi+\psi)(v)&=g(c\varphi+\psi)(g^{-1}v) \\ &=g(c\varphi(g^{-1}v)+\psi(g^{-1}v))\\ &=cg\varphi(g^{-1}v)+g\psi(g^{-1}v) \\ &=(gc\varphi+g\psi)(v) \end{align*}
The last two lines follow from the fact that $G$ acts linearly on both $V$ and $W$. Hence we have $g(c\varphi+\psi)=c(g\varphi)+g\psi$, implying $G$ acts linearly on $Hom(V,W)$.
For the next condition, we check $\forall g, h \in G$ we have $(gh)\varphi=g(h\varphi)$. Let $v \in V$. Then \begin{align*} (gh)\varphi(v)&=(gh)\varphi((gh)^{-1}v) \\ &=(gh)\varphi((h^{-1}g^{-1})v)) \\ &= g(h \varphi(h^{-1}(g^{-1}v))) \end{align*} which is what we wanted.
If $e \in G$ is the identity, we clearly have $e \varphi=\varphi$, and so $\rho$ is a representation.