Isomorphic Definitions of Induced Representation From Serre's book on Representation theory:
Given groups $H\le G$ and a representation of $H$, call it $W$, there are two (or three) ways to define the induced representation:
$$
V=\bigoplus_{\sigma\in G/H}W_\sigma,
$$
where $W_\sigma=\rho_sW$ for any $s\in\sigma$. The other way is to define:
$$
V=\mathbb{C}[G]\otimes_{\mathbb C[H]}W.
$$
These two, of course are isomorphic. But what is the explicit isomorphism (map) between them?
Moreover, there's a third way to define the induced representation, i.e.
$$
V=\operatorname{Hom}_{\mathbb C[H]}(\mathbb C[G],W).
$$
It seems even harder to show that $\operatorname{Hom}_{\mathbb C[H]}(\mathbb C[G],W)$ is isomorphic to $\mathbb{C}[G]\otimes_{\mathbb C[H]}W$. I have been thinking about this for a long while but failed to come up with a proof.
Any help is appreciated! Thanks in advance!
 A: I suppose by the "elementary property of tensor products" you mean the "universal property of tensor products".  If so,
this is the main tool to produce homomorphisms whose domain is a tensor product space and I believe it is precisely what you need here!
Here is a brief definition of that property in a very special case, suitable for the present situation.  Suppose you have
a ring $R$  (e.g. $R={\mathbb C}[G]$, as  in the question), a subring $S\subseteq R$ (e.g. $S={\mathbb C}[H]$), and $W$, a left $S$-module.  Then the
tensor product $R\otimes _SW$ is a left $R$-module satisfying the following "universal" property:

*

*There is a map
$$
  \beta :R\times W\to R\otimes _SW,
  $$
(think $(r,w)\mapsto r\otimes w$) which is (a) $R$-linear in the first variable,  (b) additive in the
second variable, and (3) balanced,  meaning that  $\beta (rs,w)= \beta (r,sw)$, for all $r\in  R$, $s\in  S$, and $w\in W$.


*For every $R$-module $Z$, and for every map
$$
  \gamma :R\times W\to Z
  $$
satisfying (a), (b) and (c), as above, there exists a unique
$R$-linear map $\tilde \gamma :R\otimes _SW\to Z$, such that $\gamma =\tilde \gamma \circ \beta $.
(In a more general situation, namely when, in addition, $U$ is an $R$-$S$-bimodule, the above generalizes to a similar universal
property regarding the tensor product $U⊗_SW$.)
This applies to give a concrete map
$$
  \tilde \gamma : \mathbb{C}[G]\otimes_{\mathbb C[H]}W \to  \bigoplus_{\sigma\in G/H}W_\sigma,
  $$
starting with
$$
  \gamma : \mathbb{C}[G]\times W \to  \bigoplus_{\sigma\in G/H}W_\sigma,
  $$
defined by $\gamma (g,w)=\rho _sw$, viewed as lying in the direct summand $W_\sigma$.
Perhaps you can take it from here?

$\newcommand\sb[1]{_{#1}}$ EDIT:
Here is the isomorphism between the second and third definitions of induced representation.  Of course this will in
turn lead to an isomorphism between these and the first form (I must say that I try to avoid the first form given its
weird dependence on  left coset  representatives).
So $H$ is a subgroup of the finite group $G$,  we are given  a representation $W$ of $H$, and we want to find a $G$-covariant isomorphism
$$
  \Lambda :\text{Hom}\sb{{\mathbb C}[H]}({\mathbb C}[G],W) \to  {\mathbb C}[G]\otimes \sb{{\mathbb C}[H]}W.
  $$
We now need to choose a system of representatives of left cosets, say
$$
  R=\{g_1,\ldots , g_n\}
  $$
(despite what I said above).  This means that, for every $g$ in $G$, there exists a unique $r$ in $R$ such that $gH=rH$.
For each $\varphi $ in $\text{Hom}\sb{{\mathbb C}[H]}({\mathbb C}[G],W)$ we then define
$$
  \Lambda (\varphi ) = \sum_{r\in R}r\otimes \varphi (r^{-1}).
  $$
It is important to notice that the map $\Lambda $ itself does not depend on the choice of $R$.  In fact,
if $f,g\in  G$, and $fH=gH$, then
$$
  f\otimes \varphi (f^{-1})=g\otimes \varphi (g^{-1})
  $$
because we may write $f=gh$, for some $h\in  H$, and then
$$
  f\otimes \varphi (f^{-1})=  gh\otimes \varphi (h^{-1}g^{-1})=g\otimes h\varphi (h^{-1}g^{-1})=g\otimes \varphi (g^{-1}),
  $$
where we have used that the tensor product is balanced in the second step, and the fact that $\varphi $ is ${\mathbb C}[H]$-linear in the
last one.
Let us now show that $\Lambda $ is $G$-covariant (meaning that it is ${\mathbb C}[G)$-linear).  Given $\varphi $ in
$\text{Hom}\sb{{\mathbb C}[H]}({\mathbb C}[G],W)$, recall that for every $a\in  {\mathbb C}[G]$, according to the definition of the ${\mathbb C}[G)$-module
structure, $a\cdot \varphi $ is given by
$$
  (a\cdot \varphi )(x) = \varphi (xa), \quad \forall x\in  {\mathbb C}[G].
  $$
In the special case that $a=g\in G$, we have
$$
  \Lambda (g\cdot \varphi ) =
  \sum_{r\in R}r\otimes (g\cdot \varphi )(r^{-1}) =
  \sum_{r\in R}r\otimes \varphi (r^{-1}g) = \cdots
  $$
As noted above, the definition of $\Lambda $ doesn't depend on the choice  of representatives, so we opt for a new set, namely
$$
  R'=gR=\{gg_1,\ldots , gg_n\}.
  $$
The above then equals
$$
  \cdots =
  \sum_{r\in R'}r\otimes \varphi (r^{-1}g) =
  \sum_{r\in R}gr\otimes \varphi ((gr)^{-1}g) =
  g\sum_{r\in R}r\otimes \varphi (r^{-1}) = g\Lambda (\varphi ),
  $$
as desired.  Since $G$ spans ${\mathbb C}[G]$, this proves that $\Lambda $ is ${\mathbb C}[G]$-linear.
In order to prove that $\Lambda $ is bijective, we explicitly exhibit  its inverse.  For this it is convenient to introduce the
so called conditional expectation
$$
  E: {\mathbb C}[G] \to  {\mathbb C}[H],
  $$
namely the canonical projection given by
$$
  E\Big (\sum_{g\in  G}\lambda _g \ g\Big )=\sum_{g\in  H}\lambda _g\ g.
  $$
Two relevant properties of $E$ which we will need in what follows are:

*

*$E$ is a ${\mathbb C}[H]$-${\mathbb C}[H]$-bimodule map,


*If $R$ is a set of representatives, as above, then
$$
  a=\sum_{r\in  R} rE(r^{-1}a) = \sum_{r\in R}E(ar)r^{-1},\quad \forall a\in  {\mathbb C}[G].
  $$
Since the domain of the inverse of $\Lambda $ is a tensor product module, we make use of the universal property and hence
we first define
$$
  \beta : {\mathbb C}[G]\times W \to  \text{Hom}\sb{{\mathbb C}[H]}({\mathbb C}[G],W)
  $$
by
$$
  \beta (a,v) : x\in  {\mathbb C}[G] \mapsto  E(xa)v\in  W,
  $$
for all $(a,v)\in  {\mathbb C}[G]\times W$.  Letting $\gamma $ be the corresponding map defined on ${\mathbb C}[G]\otimes \sb{{\mathbb C}[H]}W$, we have for all $a\otimes v$ in
${\mathbb C}[G]\otimes \sb{{\mathbb C}[H]}W$ that
$$
  \Lambda (\gamma (a\otimes v))=
  \sum_{r\in R}r\otimes \gamma (a\otimes v)|_{r^{-1}} = $$$$ =
  \sum_{r\in R}r\otimes E(r^{-1}a)v =
  \sum_{r\in R}rE(r^{-1}a)\otimes v = a\otimes v.
  $$
This shows that   $\Lambda \circ \gamma $ coincides with the identity on ${\mathbb C}[G]\otimes \sb{{\mathbb C}[H]}W$.  Regarding the composition
$\gamma \circ \Lambda $, pick $\varphi $ in $\text{Hom}\sb{{\mathbb C}[H]}({\mathbb C}[G],W)$ and $x$ in ${\mathbb C}[G]$.  then
$$
  \gamma (\Lambda (\varphi ))|_x =
  \sum_{r\in R}\gamma (r\otimes \varphi (r^{-1}))|_x =
  \sum_{r\in R}E(xr)\varphi (r^{-1}) =
  \varphi \Big (\sum_{r\in R}E(xr)r^{-1}\Big ) = \varphi (x),
  $$
so $\gamma \circ \Lambda $ is also the identity.
