# Frobenius reciprocity

Let $V$ be a finite-dimensional representation of a finite group $G$, and let $W$ be a representation of a subgroup $H\subset G$. Then the characters of $V$ and $W$ satisfy the inner product relation

 $(\chi_{\operatorname{Ind}(W)},\chi_{V})=(\chi_{W},\chi_{\operatorname{Res}(V)})$

where $\operatorname{Ind}$ and $\operatorname{Res}$ denote the induced representation $\operatorname{Ind}_{H}^{G}$ and the restriction representation $\operatorname{Res}_{H}^{G}$.

The Frobenius reciprocity theorem is often given in the stronger form which states that $\operatorname{Res}$ and $\operatorname{Ind}$ are adjoint functors between the category of $G$–modules and the category of $H$–modules:

 $\operatorname{Hom}_{H}(W,\operatorname{Res}(V))=\operatorname{Hom}_{G}(% \operatorname{Ind}(W),V),$

or, equivalently

 $V\otimes\operatorname{Ind}(W)=\operatorname{Ind}(\operatorname{Res}(V)\otimes W).$
Title Frobenius reciprocity FrobeniusReciprocity 2013-03-22 12:17:51 2013-03-22 12:17:51 djao (24) djao (24) 7 djao (24) Theorem msc 20C99