This is a crosspost from [math.SE](https://math.stackexchange.com/questions/413888/when-is-rg-times-h-rg-otimes-rh). Suppose $G$ and $H$ are discrete groups. Is it always the case that any finite dimensional complex representation of $G\times H$ is of the form
$$
\bigoplus\_i V\_i \otimes W\_i,
$$
where $V\_i, W\_i$ are reps of $G$ and $H$, respectively?


I know this is true when $G$ and $H$ are finite and when the representation of $G\times H$ is completely reducible, but is there a simple counterexample to the general case? 


I'm also curious if it is ``usually true," in some sense, that any rep of $G\times H$ has the above form.



Are all representations of $G\times H$ induced from representations of $G$ and $H$?
Let $G = H = \mathbb{Z}$. Now a $G \times H$ representation is a pair of commuting invertible matrices. Let's try
$$\left(
\begin{array}{cc}
 1 & 1 \\
 0 & 1 \\
\end{array}
\right)\mbox{ and } \left(
\begin{array}{cc}
 -1 & 1 \\
 0 & -1 \\
\end{array}
\right).$$
Certainly this rep is indecomposable since it already is after restricting to either group. Why can't it be a tensor product of two representations of $\mathbb{Z}$? One representation would have to be one-dimensional, given by a scalar multiplication. But this is impossible since neither matrix is a scalar.


I think it is true that there is an open set on the representation variety where things decompose into a sum of tensor products as you say but I do not see why exactly.


