I am trying to come to terms with the notion of an induced representation.
The following is an excerpt from section 3.3 of $\textit{Représentations linéaires des groupes finis}$ by J.-P. Serre.
Let $\rho:G \to \textrm{GL}(V)$ be a linear representation of $G$, and let $\rho_H$ be its restriction to $H$. Let $W$ be a subrepresentation of $\rho_H$, that is, a vector subspace of $V$ stable under the $\rho_t$, $t \in H$. Denote by $\theta: H \to \textrm{GL}(W)$ the representation of $H$ in $W$ thus defined. Let $s \in G$; the vector space $\rho_s W$ depends only on the left coset $sH$ of $s$; indeed, if we replace $s$ by $st$, with $t \in H$, we have $\rho_{st}W=\rho_s \rho_t W = \rho_s W$, since $\rho_t W = W$. If $\sigma$ is a left coset of $H$, we can thus define a subspace $W_{\sigma}$ of $V$ to be $\rho_s W$ for any $s \in \sigma$. It is clear that the $W_{\sigma}$ are permuted among themselves by the $\rho_s$, $s \in G$. Their sum $\sum_{\sigma\in G/H}W_{\sigma}$ is thus a subrepresentation of $V$.
$\textbf{Definition.}$ We say that the representation $\rho$ of $G$ in $V$ is $\textit{induced}$ by the representation $\theta$ of $H$ in $W$ if $V$ is equal to the sum of the $W_{\sigma}$ ($\sigma \in G/H$) and if this sum is direct (that is, if $V = \oplus_{\sigma \in G/H}W_{\sigma}$).
My confusion arises primarily from the definition of the representation $\theta$. How does the following paragraph
Denote by $\theta: H \to \textrm{GL}(W)$ the representation of $H$ in $W$ thus defined. Let $s \in G$; the vector space $\rho_s W$ depends only on the left coset $sH$ of $s$; indeed, if we replace $s$ by $st$, with $t \in H$, we have $\rho_{st}W=\rho_s \rho_t W = \rho_s W$, since $\rho_t W = W$. If $\sigma$ is a left coset of $H$, we can thus define a subspace $W_{\sigma}$ of $V$ to be $\rho_s W$ for any $s \in \sigma$. It is clear that the $W_{\sigma}$ are permuted among themselves by the $\rho_s$, $s \in G$. Their sum $\sum_{\sigma\in G/H}W_{\sigma}$ is thus a subrepresentation of $V$.
constitute a definition of a representation of the group $H$?
I am of course aware that the entirety of the cited paragraph is not a definition of $\theta$, but it is not even clear to me exactly where definition of $\theta$ begins and where it ends.
I have tried consulting other resources, and I have found the definition given by Serre to be the least unintelligible one.
If anyone could reformulate – or make explicit – the definition of $\theta$, or provide a reference to a resource that does so, I would be most grateful.
Best Answer
You only need the following excerpt:
Thus, the construction is as follows: you pick an arbitratry subspace $W$ of $V$ which is $H$-invariant and define a representation $\theta\colon H\to\mathrm{GL}(W)$ by restricting $\rho_t$, $t\in H$, to $W$. This is well-defined because $W$ was chosen $H$-invariant.
What comes next in the text is studying the properties of $\theta$ which arises in this way.