Induced representations and tensor products I want to prove the following formula:
$${\rm Ind}(W)\otimes E \cong {\rm Ind}(W \otimes{\rm Res}(E)),$$
where $H$ is a subgroup of a finite group $G$, $W$ is an irreducible representation of $H$, and $E$ is an irreducible representation of $G$.
I think the proof shouldn't use much more than Frobenius reciprocity. I tried breaking the ${\rm Ind}_H^G$ into tensor products:
$$(W \otimes_{\mathbb{C}[H]} \mathbb{C}[G]) \otimes_{\mathbb{C}[G]} E$$ and
$$\mathbb{C}[G] \otimes_{\mathbb{C}[H]} (W \otimes_{\mathbb{C}[H]} E)$$
but I'm not quite sure what to do when we have tensor products that are linear over two different rings in the same expression.
 A: If you want to prove the result with tensor products; use bimodules:

*

*$\mathbb{C}[G]$ is a $(\mathbb{C}[G],\mathbb{C}[H])$-bimodule.

*$U$ is a $(\mathbb{C}[H], \mathbb{C}[H])$-bimodule.

*$V$ is a $(\mathbb{C}[H], \mathbb{C}[G])$-bimodule.

Here the left action of $H$ on $U$ is the one you are given, the right action of $H$ on $U$ is $v \cdot h = h^{-1}v$. Similarly for the $(\mathbb{C}[H], \mathbb{C}[G])$-bimodule structure on $V$.
Then by associativity of tensor products of bimodules, we have an isomorphism $$(\mathbb{C}[G] \otimes U) \otimes V \cong \mathbb{C}[G] \otimes (U \otimes V)$$
of $(\mathbb{C}[G], \mathbb{C}[G])$-bimodules.
As a left $\mathbb{C}[G]$-module, the module on the left hand is isomorphic to $\operatorname{Ind}(U) \otimes V$. As a left $\mathbb{C}[G]$-module, the module on the right hand side is isomorphic to $\operatorname{Ind}(U \otimes V)$. Hence the desired isomorphism $$\operatorname{Ind}(U) \otimes V \cong \operatorname{Ind}(U \otimes V)$$ of left $\mathbb{C}[G]$-modules.
Of course, all of this works over any field, not just $\mathbb{C}$.
