Induced Representations

Core Idea

Given a subgroup H ≤ G and a representation σ of H, the induced representation Ind_H^G(σ) is a representation of G constructed by "extending" σ from H to all of G. It acts on a space of dimension [G:H]·dim(σ), built by taking one copy of σ for each coset of H in G. Induction is the primary method for constructing representations of a group from representations of its subgroups, and its character can be computed by an explicit formula.

Explainer

Induction is the process of building a representation of a group G from a representation of a subgroup H. Given a representation σ: H → GL(W), we construct the induced representation Ind_H^G(σ) on a larger space. Choose left coset representatives g₁, …, g_m for H in G (where m = [G:H]). The induced space is V = g₁W ⊕ g₂W ⊕ ··· ⊕ g_mW, a direct sum of m copies of W. An element g ∈ G acts by permuting these copies (since g maps one coset to another) and applying elements of H within each copy.

More formally, V = ℂ[G] ⊗_{ℂ[H]} W, where the tensor product is over the group algebra of H. The G-action is g · (x ⊗ w) = (gx) ⊗ w. This construction is basis-independent and functorial. The dimension is [G:H] · dim(W), which makes sense: you need one copy of W for each coset.

The character of the induced representation has an explicit formula: χ_{Ind}(g) = (1/|H|) Σ_{x∈G} χ̃_σ(x⁻¹gx), where χ̃_σ extends χ_σ by zero outside H. Equivalently, summing over coset representatives: χ_{Ind}(g) = Σᵢ χ̃_σ(gᵢ⁻¹g gᵢ). Only conjugates of g that land in H contribute. This formula is computable and connects the induced character to the conjugacy class structure of G relative to H.

Induction has a natural partner: restriction. Given a representation ρ of G, restricting it to H (just forgetting the G-action on elements outside H) gives Res_H^G(ρ). These two operations are adjoint in a precise sense captured by Frobenius reciprocity: ⟨Ind_H^G(σ), ρ⟩_G = ⟨σ, Res_H^G(ρ)⟩_H. This adjunction is one of the most powerful tools in representation theory, relating the representation theories of a group and its subgroups. Many important representations — including the irreducible representations of symmetric groups — are most naturally constructed via induction from carefully chosen subgroups.