# Induced representations, as defined by Serre

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.

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.
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.
• I completely understand now. The "thus defined" refers to the preceding sentence, as opposed to the succeeding one (in the latter case, one would of course have expected the sentence to have ended with a colon). I thought he was going to define an entirely different representation $\theta$. Thank you. Commented Dec 26, 2019 at 12:47