Proposition 15.4.1 (Dimension of induced representations).. This is immediate from the construction .
Note. An induced representation is almost never irreducible. For instance, (the regular representation), which decomposes into all irreducibles of . Frobenius reciprocity gives a criterion: is irreducible iff , which by Frobenius equals . Analyzing leads to Mackey’s formula, which we state without proof.
Proposition 15.4.2 (Mackey’s irreducibility criterion). Let be an irreducible representation of . Then is irreducible if and only if for every , the representations and of have no common irreducible constituent, where denotes the conjugate representation restricted to .
Example 15.4.1 (Induction from normal subgroups). If and is an irreducible representation of , then is irreducible iff for all , the conjugate is not isomorphic to as an -representation. In other words, is not fixed by any nontrivial element of acting on the set of irreducible representations of by conjugation.
Example 15.4.2 (Inducing from index-2 subgroups). Let and an irreducible representation of . Since is normal of index 2, there is a single nontrivial coset representative . Then is irreducible iff , and splits as a direct sum of two irreducibles (each of dimension ) when .
For example, let and . The nontrivial irreducibles of are the characters and (where ). Conjugation by swaps , so and is irreducible. Since , this is the standard representation of .