This is a pure curiosity question and may turn out completely devoid of substance.


Let $G$ be a finite group and $H$ a subgroup, and let $V$ and $W$ be two representations of $H$ (representations are finite-dimensional per definitionem, at least per my definitions). With $\otimes$ denoting inner tensor product, how are the two representations $\mathrm{Ind}^G\_H\left(V\otimes W\right)$ and $\mathrm{Ind}^G\_HV\otimes \mathrm{Ind}^G\_HW$ are related to each other? There is a fairly obvious map of representations from the latter to the former, but it is neither injective nor surjective in general. I am wondering whether we can say anything about the decompositions of the two representations into irreducibles.



Induction of tensor product vs. tensor product of inductions
Surely you mean "former to latter"?


I think the natural map is injective. Let $V$ and $W$ have
bases $v\_1,\ldots,v\_r$ and $w\_1,\ldots,w\_s$ respectively.
Let $g\_1,\ldots,g\_t$ be coset representatives for $H$ in $G$.
Then a basis for $\mathrm{Ind}\_H^G V\otimes \mathrm{Ind}\_H^G W$
consists of the $(v\_i g\_k)\otimes(w\_j g\_l)$. The image of
the natural injection from $\mathrm{Ind}\_H^G(V\otimes W)$
is spanned by the $(v\_i g\_k)\otimes(w\_j g\_l)$ with $k=l$.
There are exactly the right number of these.


