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
Canonical name	FrobeniusReciprocity
Date of creation	2013-03-22 12:17:51
Last modified on	2013-03-22 12:17:51
Owner	djao (24)
Last modified by	djao (24)
Numerical id	7
Author	djao (24)
Entry type	Theorem
Classification	msc 20C99