Induced Representation - Alternate Formulations

Alternate Formulations

The central theorem in the finite group case is the Frobenius reciprocity theorem. It is stated in terms of another construction of representations, the restriction map (which is a functor): any linear representation of G, as a K-module where K is the group ring of G over a field K, is also a K-module. The theorem states that, given representations ρ of G and σ of H, the space of G-equivariant linear maps from ρ to Ind(σ) has the same dimension as that of the H-equivariant linear maps from Res(ρ) to σ. (Here Res stands for restricted representation, and Ind for induced representation.) It is useful (in the typical case of non-modular representations, anyway - say with K = C) for computing the decomposition of the induced representation: we can do calculations on the side of H, which is the 'small' group.

The Frobenius formula states that if χ is the character of the representation σ, given by χ(h) = Tr σ(h), then the character ψ of the induced representation is given by

where is defined to be χ on H and 0 off H.

Frobenius reciprocity shows that Res and Ind are adjoint functors. More precisely, Ind is the left adjoint to Res. But in the finite group case, it is also a right adjoint, so (Res, Ind) is a Frobenius pair. The content of that statement is more than the dimensions: it requires that the isomorphism of vector spaces of intertwining maps be natural, in the sense of category theory. It actually suggests that induced representation can in this case be defined by means of the adjunction. That's not the only way to do it - and perhaps not the only helpful way - but it means that the theory will not be ad hoc in its start.

One can therefore make the reciprocity theorem the way to define the induced representation. There is another way, suggested by the permutation examples of the introductory paragraph. The induced representation Ind(σ) should be realized as a space of functions on G transforming under H according to the representation σ. Therefore if σ acts on the vector space V, we should look at V-valued functions on G on which H acts via σ (this must be said carefully with explicit talk about left- and right-actions; see below). This approach allows the induced representation to be a kind of free module construction.

The two approaches outlined above can be reconciled in the case of finite groups, by using the tensor product with K as a K-module. There is a third and classical approach, of simply writing down the character (trace) of the induced representation, in terms of conjugation in G of elements g into H.

The reciprocity formula can sometimes be generalized to more general topological groups; for example, the Selberg trace formula and the Arthur-Selberg trace formula are generalizations of Frobenius reciprocity to discrete cofinite subgroups of certain locally compact groups.