Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 19F (Representation Theory)

If a group representation of $G$ has character $\chi$, its restriction of a representation to $H$ has character $\chi {|}_H$. For an $H$-representation $V$ with character $\psi$, the induced representation is $\mathbb{C}[G]{\otimes }_{\mathbb{C}[H]}V$, with $G$ acting on the first factor. Decomposing into coset copies of $V$ shows that its character isIndeed only cosets fixed by $g$ contribute to the trace, and each coset has $|H|$ representatives.

The Frobenius reciprocity says $⟨{\mathrm{Ind}}_H^G\psi ,\chi {⟩}_G=⟨\psi ,{\mathrm{Res}}_H^G\chi {⟩}_H$. Directly, with the usual character inner product,The first equality substitutes $g=xh{x}^{-1}$ in the finite sum, and the second uses conjugacy invariance. This proves the theorem.

In the induced character formula above, there are no $x\in G$ with $x^{-1}gx\in H$ when the conjugacy class of $g$ misses $H$. The sum is empty, so $({\mathrm{Ind}}_H^G\psi )(g)=0$.

Each $H$-conjugacy class represented by $x_i$ has $|H|/|C_H(x_i)|$ elements. For every element of that class, the number of conjugators from $g$ to it is $|C_G(g)|$. Grouping the induced character formula by these classes therefore gives

In the example, $a$ has order four, $b$ has order two, and $bab=a^{-1}$, so $H\cong D_8$. Its five conjugacy classes are $1$, $a^2$, $\{a,a^3\}$, $\{b,a^{2}b\}$ and $\{ab,a^{3}b\}$, of sizes $1,1,2,2,2$. The four one-dimensional irreducible characters are determined by $\psi (a)=\epsilon$ and $\psi (b)=\eta$, each sign independently chosen. The remaining two-dimensional irreducible character comes from the plane action of the square:Their squared degrees sum to eight, so this is the full character table.

In $S_4$, use class representatives $1,(12),(12)(34),(123),(1234)$. Their intersections with $H$ are respectively $\{1\}$, $\{b,a^{2}b\}$, $\{a^2\}\cup \{ab,a^{3}b\}$, the empty set, and $\{a,a^3\}$. The corresponding $S_4$ centralizer orders are $24,4,8,3,4$, while the noncentral $H$ centralizers have order four. The induced values are consequentlyIn the same $S_4$ class order, all five induced irreducible characters are