Tensor products of $FG$-modules and tensor products of representations

Question

Let $\rho_1$ and $\rho_2$ be two representations of $G$, $V$ and $W$ be the correspondence $FG$-modules, respectively.

Recall the tensor product $\rho_1\otimes \rho_2$ of $\rho_1$ and $\rho_2$ is defined by $\rho_1\otimes \rho_2:G\to GL(V\otimes_F W)$, $(\rho_1\otimes \rho_2)(g)(v\otimes w) =(\rho_1(g)\otimes \rho_2(g))(v\otimes w) =(\rho_1(g)(v),\rho_2(g)(w))$.

On the other hand, suppose that the $FG$-module $V\otimes_{FG} W$ afford the representation $\rho_3$.

My Question: Is $\rho_3\sim \rho_1\otimes \rho_2$?

$~~~~~~~~~~~~~~$ $FG$-module $V$ $~~~~~~~~~~~~~~$ $\to$ $FG$-module $V\otimes_{FG} W$ $\leftarrow$ $~~~~~~~~~~~~$ $FG$-module $W$
$~~~~~~~~~~~~~~~~~~~~~~~~~~$ $\updownarrow$ $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $\updownarrow$? $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $\updownarrow$
representation $\rho_1:G\to GL(V)$ $\to$ $~~~~~~~~~~$ $\rho_1\otimes \rho_2$ $~~~~~~~~~~~~~$ $\leftarrow$ representation $\rho_2:G\to GL(W)$

Answer 1

The tensor product of the $FG$ modules is the tensor product over the field $F$. This might seem strange, as you were expecting a tensor product of bimodules and hence wrote a tensor product over $FG$.

But consider two algebras $A$ and $B$ with modules $V$ and $W$. Then $V \otimes_F W$ is an $A \otimes_F B$ module in the evident way. We are going to take $A = B = F G$. But how does $V \otimes_F W$ become an $A$ module when it should be an $A \otimes A$ module? There is an algebra map $A \to A \otimes A$ extending $g \mapsto g \otimes g$, and one composes this with the representation of $A \otimes A$ on $V \otimes W$, ending up with the representation $\rho_1 \otimes \rho_2$ as you have described it.

Answer 2

No. For a very simple example, note $V=FG$ (with the regular representation) is the unit of the tensor product $\otimes_{FG}$, while the trivial representation $W=F$ is the unit of the tensor product of representations. So, for instance, $FG\otimes_{FG} F=F$ but the tensor product of representations would be $FG$ instead. What is going on here is that the tensor product of representations is just a tensor product over the field $F$, not over the ring $FG$.

In fact, there is a larger issue: it is not even clear how to define $V\otimes_{FG} W$ in general. To define this tensor product over the noncommutative ring $FG$, you would need $V$ to be a right $FG$-module, and the tensor product would just have the structure of an $F$-vector space, not an $FG$-module. Only if $V$ happens to be an $FG$-bimodule would the tensor product have a natural structure of an $FG$-module.

In fact, any left $FG$-module $V$ can be made into a right $FG$-module by defining $v\cdot g=g^{-1}\cdot v$ for $g\in G$, $v\in V$ (where $g^{-1}\cdot v$ is using the left module structure). So given two representations, you can always naturally define a vector space (but not a representation!) $V\otimes_{FG} W$. The relationship with the ordinary tensor product of representation is that $V\otimes_{FG} W$ is the vector space of coinvariants of the ordinary tensor product.