Frobenius reciprocity theorem

《Introduction to Mathematical Logic, Sixth Edition》 · 印刷第 216 页 Section 4.4.1「Induced Representations」的群、表示与诱导记号约定及 Theorem 4.4.1 完整陈述

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 $\mathcal I_\mu$. Theorem 4.4.1 (Frobenius Reciprocity). There is a vector space isomorphism $\operatorname{Hom}_G\left(\rho,\operatorname{Ind}_H^G(\mu)\right)\cong\operatorname{Hom}_H\left(\operatorname{Res}_H^G(\rho),\mu\right)$.