Induced class functions and restricted class functions

Let $G$ be a finite group, let $H\le G$ be a subgroup, and let $\theta \in \mathrm{c}\mathrm{f}(H)$ be a complex class function on $H$ (Class functions and the complex vector space $\mathrm{c}\mathrm{f}(G)$, Subgroup).

Induction. The induced class function ${\mathrm{Ind}}_H^G\theta \in \mathrm{c}\mathrm{f}(G)$ is defined by the Frobenius formula ${\mathrm{Ind}}_H^G\theta (g):=\frac{1}{\mathrm{\mid }H\mathrm{\mid }}{\sum }_{\begin{array}{c}x\in G\\ x^{-1}gx\in H\end{array}}\theta \left(x^{-1}gx\right),g\in G\mathrm{.}$ The sum is over the finite set of $x\in G$ with $x^{-1}gx\in H$, and each summand is a complex number, so the formula is well defined; the normalising factor $1\mathrm{/}\mathrm{\mid }H\mathrm{\mid }$ is the one used for honest induced characters in Frobenius' formula for the character of an induced representation.

Restriction. For $\psi \in \mathrm{c}\mathrm{f}(G)$ the restricted class function ${\mathrm{Res}}_H^G\psi \in \mathrm{c}\mathrm{f}(H)$ is the restriction $\psi {\mathrm{\mid }}_H$. It is a class function because the conjugating elements in the equation $f(hk{h}^{-1})=f(k)$ for $k,h\in H$ are also elements of $G$.

Elementary properties. The definition is arranged so that the following three statements hold; each is a direct computation from the displayed sum.

1. ${\mathrm{Ind}}_H^G\theta$ is a class function on $G$: for $t,g\in G$ the substitution $y=tx$ is a bijection from $\{x\in G:x^{-1}gx\in H\}$ onto $\{y\in G:y^{-1}(tg{t}^{-1})y\in H\}$, because $(tx{)}^{-1}(tg{t}^{-1})(tx)=x^{-1}gx$, and then ${\mathrm{Ind}}_H^G\theta (tg{t}^{-1})={\mathrm{Ind}}_H^G\theta (g)$.
2. Induction is $\mathbb{C}$-linear in $\theta$: for ${\theta }_1,{\theta }_2\in \mathrm{c}\mathrm{f}(H)$ and $\lambda \in \mathbb{C}$ one has ${\mathrm{Ind}}_H^G({\theta }_1+{\theta }_2)={\mathrm{Ind}}_H^G{\theta }_1+{\mathrm{Ind}}_H^G{\theta }_2$ and ${\mathrm{Ind}}_H^G(\lambda \theta )=\lambda {\mathrm{Ind}}_H^G\theta$, because both sides are the same finite sum of values of ${\theta }_1+{\theta }_2$, respectively $\lambda \theta$.
3. If $\theta =\chi$ is the character of a finite-dimensional complex representation of $H$, then ${\mathrm{Ind}}_H^G\chi$ as defined here is the honest induced character of The induced character ${\mathrm{Ind}}_H^G\chi$ of a complex character: this is exactly the content of the Frobenius formula Frobenius' formula for the character of an induced representation.

The map $\theta \mapsto {\mathrm{Ind}}_H^G\theta$ is thus a $\mathbb{C}$-linear map $\mathrm{c}\mathrm{f}(H)\to \mathrm{c}\mathrm{f}(G)$ extending the honest induction of characters; no claim of positivity, integrality or dependence only on a character is made for a general $\theta$.

• Why class functions are needed. The Frobenius kernel argument of this page applies induction to the virtual character $\theta =\phi -\phi (1)1_H$ of Frobenius character extension construction, which is not an honest character, and the library defines ${\mathrm{Ind}}_H^G\chi$ only for honest characters (The induced character ${\mathrm{Ind}}_H^G\chi$ of a complex character). The display above is the unique $\mathbb{C}$-linear extension of that construction, so Zero at identity induction restriction for a frobenius complement and Frobenius character extension construction may apply it to $\theta$. For an honest character $\chi$ of $H$ the two notions agree, by property 3.

• Value at the identity. At $g=1$ every $x\in G$ contributes to the defining sum, and $x^{-1}1x=1$, so ${\mathrm{Ind}}_H^G\theta (1)=\frac{1}{\mathrm{\mid }H\mathrm{\mid }}{\sum }_{x\in G}\theta (1)=[G:H]\theta (1)$. The normalising factor $1\mathrm{/}\mathrm{\mid }H\mathrm{\mid }$ is therefore what makes induction of the trivial character return $[G:H]$; for the virtual character $\theta =\phi -\phi (1)1_H$ of the kernel argument one has $\theta (1)=0$, so ${\mathrm{Ind}}_H^G\theta (1)=0$.

• Reciprocity. The $\mathbb{C}$-bilinear pairing between induction and restriction is recorded in Frobenius reciprocity for class functions.