Frobenius Reciprocity

An adjunction statement in representation theory asserting a canonical isomorphism between Hom-spaces: for a subgroup H of G and suitable representations V of H and W of G, Hom_G(Ind_H^G V, W) ≅ Hom_H(V, Res_H^G W); equivalently, induction is left adjoint to restriction.

A natural adjunction between induction and restriction functors in representation theory: for a subgroup H of a group G (or more generally for a pair of algebras and a module restriction/extension situation) there is a canonical isomorphism Hom_G(Ind_H^G V, W) ≅ Hom_H(V, Res^G_H W) identifying G-maps from an induced representation with H-maps into the restricted representation.