Let \(H\) be a subgroup of a finite group \(G\) and \(A\colon H\to GL(n,\mathbb{C})\) be a representation of \(H\).
Prove that for any character \(\phi\) of \(G\), \((\chi^{A\uparrow_H^G},\phi)_G=(\chi^A,\phi\downarrow^G_H)_H\).
Given an example to show that it is not always the case that if \(A\) is irreducible, then \(A\uparrow_H^G\) is irreducible.
Prove that if \(K\) is a subgroup of \(H\) and \(B\colon K\to GL(m,\mathbb{C})\) is a representation of \(K\), then the representation \(B\uparrow_K^G\) is similar to the representation \((B\uparrow_K^H)\uparrow_H^G\).
Solution
1. For any character \(\phi\) of \(G\),\[ (\chi^{A\uparrow_H^G},\phi)_G =(\chi^A,\phi\downarrow_H^G)_H. \]
The induced-character formula is \[ \chi^{A\uparrow_H^G}(g) =\frac1{|H|}\sum_{\substack{x\in G\\x^{-1}gx\in H}} \chi^A(x^{-1}gx). \] Therefore \[ \begin{aligned} (\chi^{A\uparrow_H^G},\phi)_G &=\frac1{|G|}\sum_{g\in G} \chi^{A\uparrow_H^G}(g)\overline{\phi(g)}\\ &=\frac1{|G||H|} \sum_{x\in G}\sum_{h\in H} \chi^A(h)\overline{\phi(xhx^{-1})}. \end{aligned} \] Because characters are class functions, \[ \phi(xhx^{-1})=\phi(h). \] Thus the inner sum is independent of \(x\), and \[ \begin{aligned} (\chi^{A\uparrow_H^G},\phi)_G &=\frac1{|H|}\sum_{h\in H} \chi^A(h)\overline{\phi(h)}\\ &=(\chi^A,\phi\downarrow_H^G)_H. \end{aligned} \] This is Frobenius reciprocity.
2. An irreducible representation can induce to a reducible representation.
Take \[ G=C_2=\{1,s\},\qquad H=\{1\}. \] The unique one-dimensional representation \(A\) of the trivial group \(H\) is irreducible. Its induction to \(G\) is the regular representation \[ A\uparrow_H^G\cong\mathbb C[G], \] which decomposes as \[ \mathbb C[G]\cong \mathbf1\oplus\operatorname{sgn}. \] Hence \(A\uparrow_H^G\) is reducible.
3. If \(K\le H\le G\) and \(B\) is a representation of \(K\), then\[ B\uparrow_K^G \cong (B\uparrow_K^H)\uparrow_H^G. \]
Using the group-algebra construction of induction, \[ B\uparrow_K^H =\mathbb C[H]\otimes_{\mathbb C[K]}B \] and hence \[ (B\uparrow_K^H)\uparrow_H^G =\mathbb C[G]\otimes_{\mathbb C[H]} \bigl(\mathbb C[H]\otimes_{\mathbb C[K]}B\bigr). \] Define \[ \Phi:\mathbb C[G]\otimes_{\mathbb C[H]} \bigl(\mathbb C[H]\otimes_{\mathbb C[K]}B\bigr) \longrightarrow \mathbb C[G]\otimes_{\mathbb C[K]}B \] by \[ \Phi(g\otimes(h\otimes v))=gh\otimes v. \] This is well-defined because the balancing relations over \(\mathbb C[H]\) and \(\mathbb C[K]\) map to the balancing relation in the target.
Conversely define \[ \Psi(g\otimes v)=g\otimes(1\otimes v). \] This is well-defined over \(\mathbb C[K]\): for \(k\in K\), \[ \Psi(gk\otimes v) =gk\otimes(1\otimes v) =g\otimes(k\otimes v) =g\otimes(1\otimes kv) =\Psi(g\otimes kv). \] The maps \(\Phi\) and \(\Psi\) are inverse to one another and commute with the left \(G\)-action. Therefore they give an isomorphism of \(G\)-representations, proving transitivity of induction.