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).