Suppose $\theta$ is a one-dimensional representation of a group $G$, and $\rho : G \to \mathrm{GL}(V)$ is another representation. Define $\theta \otimes \rho : G \to \mathrm{GL}(V)$ given by $\theta \otimes \rho (g) = \theta(g)\cdot\rho(g)$.
What exactly does the "$\cdot$" mean in the definition of this new map?
Let the vector spaces be over a field $k$. If I interpet $\theta$ as a map $\theta: G \to k^\times$, $\theta(g)$ would be an element of $k$ and so the "$\cdot$" would make sense as scalar multiplication in the vector space $V$. But if $\theta : G \to \mathrm{GL}(W)$ more generally for a one-dimensional vector space $W$, all I can say is that I may write \theta = \phi \theta' \phi^{-1} where $\phi : k \to W$ is an isomorphism and \theta' : G \to k^\times. So then \theta \otimes \rho(g) = \phi \theta'(g) \phi^{-1} \cdot \rho(g), and I don't know how to make sense of the "$\cdot$".
I've a feeling I'm being stupid here - any explanation would be most appreciated.