Let \(G\) and \(H\) be finite groups and let \(A:G\to\mathrm{GL}_n(\mathbb{C})\) and \(B:H\to\mathrm{GL}_m(\mathbb{C})\) be representations of \(G\) and \(H\) respectively.
Show that \(A\times B:G\times H\to\mathrm{GL}_{nm}(\mathbb{C})\) is a representation where for \((\sigma,\tau)\in G\times H\), \[ (A\times B)((\sigma,\tau))=A(\sigma)\otimes B(\tau) \] and for matrices \(M\) and \(N\), \(M\otimes N\) is the Kronecker product of \(M\) and \(N\).
Show that if \(A\) is an irreducible representation of \(G\) and \(B\) is an irreducible representation of \(H\), then \(A\times B\) is an irreducible representation of \(G\times H\).
Show that every irreducible representation of \(G\times H\) is of the form \(A\times B\) where \(A\) is an irreducible representation of \(G\) and \(B\) is an irreducible representation of \(H\).
Solution
1. The map\[ A\times B:G\times H\to \mathrm{GL}_{nm}(\mathbb C),\qquad (A\times B)(g,h)=A(g)\otimes B(h), \]is a representation.
For compatible matrices, \[ (M_1\otimes N_1)(M_2\otimes N_2)=(M_1M_2)\otimes(N_1N_2). \] Hence multiplication in \(G\times H\) is respected, and \((A\times B)(e_G,e_H)=I_{nm}\).
2. Its character is\[ \chi_{A\times B}(g,h)=\chi_A(g)\chi_B(h). \]
This follows from \(\operatorname{tr}(M\otimes N)=\operatorname{tr}(M)\operatorname{tr}(N)\).
3. If \(A\) and \(B\) are irreducible, then \(A\times B\) is irreducible.
Using 2, \[ \langle\chi_{A\times B},\chi_{A\times B}\rangle_{G\times H} =\langle\chi_A,\chi_A\rangle_G\,\langle\chi_B,\chi_B\rangle_H=1. \] Thus the character has norm \(1\), hence is irreducible.
4. More generally, for irreducibles \(A_i,A_j\) of \(G\) and \(B_r,B_s\) of \(H\),\[ \langle\chi_{A_i\times B_r},\chi_{A_j\times B_s}\rangle_{G\times H} =\langle\chi_{A_i},\chi_{A_j}\rangle_G\,\langle\chi_{B_r},\chi_{B_s}\rangle_H. \]
Expand the inner product as a double sum over \(G\times H\) and separate the \(g\)- and \(h\)-sums.
5. If \(G\) has \(r\) irreducible characters and \(H\) has \(s\), then the \(rs\) characters \(\chi_{A_i\times B_j}\) are pairwise orthonormal irreducible characters of \(G\times H\).
Irreducibility is 3, and pairwise orthogonality is 4 together with orthogonality of irreducible characters on each factor.
6. The group \(G\times H\) has exactly \(rs\) conjugacy classes.
Two pairs \((g,h)\) and \((g',h')\) are conjugate in \(G\times H\) exactly when \(g,g'\) are conjugate in \(G\) and \(h,h'\) are conjugate in \(H\). Hence conjugacy classes are precisely products \(C\times D\) of conjugacy classes from the two factors. Since a finite group has as many irreducible complex characters as conjugacy classes, \(G\) and \(H\) have \(r\) and \(s\) classes respectively, so \(G\times H\) has \(rs\).
7. Every irreducible representation of \(G\times H\) is of the form \(A_i\times B_j\).
By 5 there are already \(rs\) pairwise nonisomorphic irreducible representations of this form, while 6 shows that \(G\times H\) has exactly \(rs\) irreducible representations up to isomorphism. Therefore the list is complete.