TITLE: Equivalence of induced representation
QUESTION [2 upvotes]: Let $H$ be a subgroup of $G$.
In Wiki, it gives an algebraic construction of induced representation.
And it is equivalent to the vector space $Hom_{H}(\mathbb{C}[G],V)$ i.e.   $ \{ f:G \rightarrow V   |  f(hg)=hf(g), \forall h \in H, g \in G \} $, and the $g$-action on $Hom_{H}(\mathbb{C}[G],V)$ is defined by $(g'f)(g) = f(gg')$.
Why are they equivalent? What is  the isomorphism between them that preserves the $g$-action? i.e  $\phi$: $Hom_{H}(\mathbb{C}[G],V)\rightarrow W=\bigoplus_{i=1}^n x_i V.$ st. $\phi(gf)=g\phi(f)$.

REPLY [2 votes]: Let me assume $G$ is finite to match the situation in the wikipedia page. 
Take $x_1,\dots,x_n$ to be representatives for the cosets in $H\backslash G$. The wiki page uses left-cosets but it's equivalent and would require slightly more messy notation.
A given element $f\in Hom_H(\mathbb{C}[G],V)$, is completely determined by its values on $x_i$. Every element of $g$ can be written uniquely as $hx_i$ for some $h\in H$, and we have $f(g)=f(hx_i)=hf(x_i)$. So in fact the data of each function $f$ is simply $n$ vectors in $V$, one for each coset of $H$.
The isomorphism is then
$$Hom_H(\mathbb{C}[G],V) \rightarrow \oplus_{i=1}^n x_i V, \ \  f \mapsto (f(x_1),f(x_2),\dots,f(x_n)),$$ 
where each $f(x_i)$ is considered in the copy of $V$ labelled $x_iV$.
For a fixed $g\in G$, and for each $x_i$, we have $x_i g = h x_j$ for a unique $j=j(i)$ and $h=h_i\in H$. Then $(gf)(x_i)=f(x_ig)=f(h_ix_{j(i)})=h_i f(x_{j(i)})$, so under the above map $gf$ goes to $h_if(x_{j(i)})_{1\leq i \leq n}$ which shows the map is $G$-equivariant. 
The idea in both descriptions is to extend the representation $V$ of $H<G$ to all of $G$ by taking several copies of $V$, one for each coset of $H$, and having $G$ act by permuting these copies according to how it permutes the cosets, and by acting on each one simultaneously via the action of $H$.
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