If you have an irreducible representation of a subgroup, is the induced representation of the whole group still irreducible?
Solution
No.
1. Let \(G=C_2\) and let \(H=\{e\}\).
The trivial subgroup has a unique irreducible representation over \(\CC\), namely the one-dimensional trivial representation \(\mathbf 1_H\).
2. Its induced representation is the regular representation of \(C_2\).
By definition, \[ \operatorname{Ind}_H^G(\mathbf 1_H)\cong \CC[G] \] as a \(G\)-representation because \(H\) is trivial.
3. This induced representation is reducible.
The regular representation of \(C_2=\{1,s\}\) has basis \(e_1,e_s\). The lines \[ \CC(e_1+e_s),\qquad \CC(e_1-e_s) \] are both \(G\)-stable, giving the decomposition into the trivial and sign representations. Thus the induced representation is not irreducible.
Hence induction does not in general preserve irreducibility.