Character of restricted representation.

Question

My lecture notes state that the restriction of an irreducible character is in general not irreducible and gives the following example:

"Restricting any non-linear irreducible character to the trivial subgroup will produce a reducible character".

I want to be sure that I understand this. For example, if we have a representation of degree $2$, then the representation of the trivial subgroup $\{e\}$ is just $\rho(e)=\begin{pmatrix}1&0 \\ 0&1\end{pmatrix}$ which is composed of a direct sum of the irreducible representation $\rho'(e)=(1)$ with multiplicity $2$, and therefore $\rho$ is reducible. Is this the correct interpretation?

Answer 1

Your interpretation seems correct to me. It also easily follows using the inner product.
If $\rho$ is a non-linear character then $[$$\rho_{\{e\}}$,$\rho_{\{e\}}$$]$$=\rho(e)^2>1$ and hence $\rho_{\{e\}}$ is reducible.

Answer 2

By the way, in addition to the previous answer, there are situations (in finite groups) in which the restriction is irreducible: for example let $N \unlhd G$, $\chi \in Irr(G)$, with $gcd(\chi(1), |G:N|)=1$. Then $\chi_N \in Irr(N)$.