Induction: from a subgroup up to the group
Let H ≤ G and let W be a representation of H. The induced representation Ind_H^G W builds a representation of the bigger group by, roughly, placing a copy of W on each coset of H and letting G permute the cosets while acting within each by H. Its dimension is [G:H]·dim W. Concretely Ind_H^G W = C[G] ⊗_{C[H]} W, an extension of scalars along the inclusion C[H] ↪ C[G] — the cleanest definition once you have tensor products of modules.
The tool that makes induction usable is Frobenius reciprocity: ⟨Ind_H^G W, V⟩_G = ⟨W, Res_H^G V⟩_H, where Res is restriction back down to H. In words, the multiplicity of an irreducible V of G inside the induced representation equals the multiplicity of W inside V restricted to H. It turns a hard upstairs computation into an easy downstairs one — and it is the adjunction between two functors, induction left-adjoint to restriction.
Inducing from H = A_3 = {e,(123),(132)} (index 2) up to G = S_3.
A_3 = Z/3Z is abelian, three 1-dim characters psi_0, psi_1, psi_2
with psi_j((123)) = omega^j, omega = e^{2 pi i /3}.
Induce the trivial psi_0 of A_3: dim = [S_3:A_3]*1 = 2.
Its character on classes (e,(12),(123)) is
Ind psi_0 (e) = 2
Ind psi_0 ((12)) = 0 (transpositions have no A_3-conjugate fixed)
Ind psi_0 ((123)) = psi_0((123)) + psi_0((132)) = 1 + 1 = 2
=> character (2, 0, 2) = trivial (+) sign of S_3. (reducible)
Induce psi_1 instead: character (2, 0, omega + omega^2) = (2, 0, -1)
=> exactly the STANDARD irreducible of S_3.
Check by Frobenius reciprocity: <Ind psi_1, standard>_{S_3}
= <psi_1, Res standard>_{A_3} = 1. OKCharacters as the shadow of an algebra
Step back to the group algebra. Because C[G] is a semisimple ring, the Artin–Wedderburn theorem says it splits as a product of matrix algebras: C[G] ≅ ∏ᵢ Mat_{dᵢ}(C), one factor per irreducible, of size its degree dᵢ. Comparing dimensions on both sides recovers Σ dᵢ² = |G| instantly. Everything about characters — orthogonality, the count of irreducibles, the regular representation — is the structure of this product of matrix rings, viewed through the trace.
Letting the group be infinite
What survives when G is a continuous group like the circle U(1) or the rotation group SO(3)? The key move was averaging (1/|G|) Σ_g. For a compact group there is a unique invariant probability measure — Haar measure — so averaging becomes integration ∫_G ⋯ dg. With that single substitution, Maschke, Schur, and orthogonality all go through verbatim: every finite-dimensional representation is completely reducible and characters are still orthonormal class functions.
The finite-group statement that C[G] decomposes into matrix blocks containing every irreducible becomes the Peter–Weyl theorem: the matrix coefficients of the irreducible representations form an orthonormal basis of L²(G). For U(1) this is precisely the classical Fourier series — its irreducibles are the characters z ↦ zⁿ, and Peter–Weyl is the completeness of {eⁱⁿᵗ}. Representation theory thus contains harmonic analysis as the abelian special case.