induced representation

《Introduction to Mathematical Logic, Sixth Edition》 · 印刷第 216 页 Section 4.4.1「Induced Representations」，节首定义段

Let $G$ be a finite group and $H$ a subgroup of $G$. Given any finite-dimensional representation $(\rho,F)$ of $G$, we obtain a representation $(\operatorname{Res}_H^G(\rho),F)$ of $H$ by restriction. Conversely, a representation $(\mu,E)$ of $H$ induces a representation of $G$ by the following construction: We take the space $\mathbb C[G;E]$ of all functions from $G$ to $E$, and let $G$ act by left translations: $L(g)f(x)=f(g^{-1}x)$. Let $\mathcal I_\mu\subset\mathbb C[G;E]$ be the subspace of functions such that $f(gh)=\mu(h)^{-1}f(g)$, for all $h\in H$ and $g\in G$. The space $\mathcal I_\mu$ is invariant under left translation by $G$. We define the induced representation $\operatorname{Ind}_H^G(\mu)$ of $G$ to be the left translation action of $G$ on the space $\mathcal I_\mu$.

《Cohomologie Locale Des Faisceaux Coherents (Sga 2): Seminaire De Geometrie Algebrique Du Bois Marie 1962 (Documents Mathematiques)》 · 印刷第 60 页 Section 4.3「Induction and Restriction」，节首群环张量积构造与 induced module 命名（省略记号旁注）

We now consider how to construct representations of a group from representations of its subgroups. Let $H$ be a subgroup of $G$ and $V$ an $RH$-module where $R$ is a commutative ring with $1$. We define an $RG$-module $V\uparrow_H^G=RG\otimes_{RH}V$ with the action of $G$ coming from the left module action on $RG$: $x\cdot\left(\sum_{g\in G}a_gg\otimes v\right)=\left(x\sum_{g\in G}a_gg\right)\otimes v$, where $x,g\in G$, $a_g\in R$, and $v\in V$. We refer to this module as $V$ induced from $H$ to $G$, and say that $V\uparrow_H^G$ is an induced module. […] The operation $\uparrow_H^G$ is called induction.