Induced representation
Let G be a finite group and H any subgroup of G. Furthermore let (π, V) be a representation of H. Let n = [G : H] be the index of H in G and let g1, ..., gn be a full set of representatives in G of the left cosets in G/H.In the case of finite groups and finite-dimensional representations, the Frobenius reciprocity theorem states that, given representations σ of H and ρ of G, the space of H-equivariant linear maps from σ to Res(ρ) has the same dimension over K as that of G-equivariant linear maps from Ind(σ) to ρ.[2] The universal property of the induced representation, which is also valid for infinite groups, is equivalent to the adjunction asserted in the reciprocity theorem.with the following property: given any representation (ρ,W) of G and H-equivariant linear mapThe 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 the sum is taken over a system of representatives of the left cosets of H in G and If G is a locally compact topological group (possibly infinite) and H is a closed subgroup then there is a common analytic construction of the induced representation.Let (π, V) be a continuous unitary representation of H into a Hilbert space V. We can then let: Here φ∈L2(G/H) means: the space G/H carries a suitable invariant measure, and since the norm of φ(g) is constant on each left coset of H, we can integrate the square of these norms over G/H and obtain a finite result.The group G acts on the induced representation space by translation, that is, (g.φ)(x)=φ(g−1x) for g,x∈G and φ∈IndGH π.This construction is often modified in various ways to fit the applications needed.The definition of the representation space is as follows: Here ΔG, ΔH are the modular functions of G and H respectively.This is just standard induction restricted to functions with compact support.Suppose G is a topological group and H is a closed subgroup of G. Also, suppose π is a representation of H over the vector space V. Then G acts on the product G × V as follows: where g and g′ are elements of G and x is an element of V. Define on G × V the equivalence relation Denote the equivalence class ofNote that this equivalence relation is invariant under the action of G; consequently, G acts on (G × V)/~ .The latter is a vector bundle over the quotient space G/H with H as the structure group and V as the fiber.This is the vector space underlying the induced representation IndGH π.as follows: In the case of unitary representations of locally compact groups, the induction construction can be formulated in terms of systems of imprimitivity.