Frobenius reciprocity

Frobenius reciprocity is a precise balance between the two basic operations on representations: induction (building a representation of the whole group from one of a subgroup) and restriction (forgetting down to a subgroup). It says these operations are adjoint — counting how a piece sits inside an induced representation is exactly the same as counting how it sits inside the restriction, just computed on the other side.

Let H be a subgroup of G, let W be a representation of H, and let U be a representation of G. Frobenius reciprocity is the natural isomorphism Hom_G(Ind_H^G W, U) isomorphic to Hom_H(W, Res^G_H U), where Res restricts U to H. In categorical language, induction is left adjoint to restriction. Taking dimensions over C and translating to characters gives the inner-product form: (Ind_H^G chi_W, psi_U)_G = (chi_W, Res psi_U)_H.

In practice this is the workhorse for decomposing induced representations. To find the multiplicity of an irreducible U of G inside Ind_H^G W, you instead restrict U to H and count the multiplicity of W there — usually a far easier computation in the smaller group. The symmetry also runs the other way through the right-adjoint (coinduction), which coincides with induction for finite index, so reciprocity holds with restriction adjoint to induction on both sides in that case.