Facts about induced representations

can i have help with induced representations? I find the definition of $Ind_H^GV$ difficult but i can think about it formaly. But then i want to proof the follwoing facts:

  • $\dim(Ind_H^GV)=|G:H|\cdot\dim(V)$
  • $H\subset K\subset G$ then $Ind_K^GInd_H^K=Ind_H^G$
  • $Ind_H^GV$ is naturally isomorphic to $Hom_H(\mathbb{C}[G],V)$
  • the space of $W$-valued maps on $G$ is isomorphic to $C[G]\otimes W$

Another question from me: In the definition we don’t use right-cosets, but then we begin to work with them. Why?! I hope someone can help me 🙂 Thanks

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So, looking at the given definition:
$$Ind_H^GV=\left\{f:G\rightarrow V : f(hx)=\rho(h)f(x)\ \forall x\in G\ \forall h\in H\right\}\,,$$
we can see that if an $f:G\to V$ is in $Ind_H^GV$, then the value $f(x)$ for an $x\in G$ already determines $f$ on the whole coset $Hx$ (namely $f(hx)=\rho(h)f(x)$). On the other side, fixing one $x_i$ of each coset and letting $f_1(x_i)\in V$ arbitrary, then as
$G=\bigcup_i^*Hx_i$ (disjoint union), by the given rule we can uniquely extend $f_1$ to $G$, and then $f_1\in Ind_H^G(V)$. This proves 1.

$\,$ 2. should be a simple verification.

For 3., observe that any function $f:G\to V$, using the linear operations on $V$, naturally (and uniquely) extends to a linear map $\bar f:\Bbb C[G]\to V$. Show that such a map $F$ is in addition a $H$-homorphism (with the action of $H$ on $V$ given by $\rho|_H$), if and only if $F(hs)=\rho(h)F(s)$ for all $s\in\Bbb C[G]$.

Finally, for 4., I guess $G$ is (throughout) assumed to be finite, so $\Bbb C[G]$ is finite dimensional, then use the previous hint and that $\hom_{\Bbb C}(U,W)\cong U\otimes_{\Bbb C}W$ if $\dim U<\infty$.