I apologize in advance if my question sounds naive to those who are experts in representation theory.
Let $H$ be a subgroup of some finite group $G$. Let $\pi: G \rightarrow GL_n(C)$ be some (finite-dimensional) complex irreducible representation of $G$. Now, can we say the following for the restriction of $\pi$ to the subgroup $H$
- $\pi|_H$ is a irreducible representation of H.
- or $\pi|_H$ can be expressed directly as direct sum of irreducible representations of $H$. For example, $\pi|_H = \phi_1 \bigoplus \phi_2$, where, $\phi_1$ and $\phi_2$ be two irreducible representations of $H$.
According to my understanding, we can say so if $H$ is a normal subgroup. But what about any subgroup $H$?
Edit: For point 2, I am not asking about the equivalence to the direct sum of irreducible representation. I am asking if the restriction $\pi|_H$ can be expressed directly.
Best Answer
Consider a 2-dimensional irreducible representation of $S_3$ with $$(1,2,3) \mapsto \left(\begin{array}{rr}-1&1\\-1&0\end{array}\right),\ \ \ \ (1,2) \mapsto \left(\begin{array}{rr}0&1\\1&0\end{array}\right).$$ The restriction to $H := \langle (1,2,3) \rangle$ is equivalent to the sum of two $1$-dimensional complex representation. But it is irreducible over ${\mathbb R}$ and it is not equal to such a sum.
Note that $H$ is a normal subgroup of $G=S_3$, so your belief that this property holds for normal subgroups is wrong.