Representations of Abelian Groups

Core Idea

Every irreducible complex representation of a finite abelian group is one-dimensional, a consequence of Schur's lemma (every group element commutes with the entire representation, hence acts as a scalar). Since every finite abelian group is a product of cyclic groups ℤ/n₁ℤ × ··· × ℤ/nₖℤ, its irreducible representations are products of irreducible representations of the cyclic factors — each specified by a tuple of roots of unity. The set of all irreducible representations forms the dual group Ĝ, which is (non-canonically) isomorphic to G itself.

Explainer

The representation theory of finite abelian groups is completely determined by a single fact: Schur's lemma forces all irreducible representations to be 1-dimensional. Here is why. If G is abelian, then for any representation ρ: G → GL(V), every ρ(g) commutes with every ρ(h). So every ρ(g) is a G-equivariant endomorphism of V. By Schur's lemma (over ℂ), if V is irreducible, each ρ(g) must be a scalar λ_g · I. But then every subspace of V is invariant, so irreducibility forces dim(V) = 1.

Since every irreducible representation is a homomorphism χ: G → ℂ*, the set of irreducible representations forms a group under pointwise multiplication: (χ₁ · χ₂)(g) = χ₁(g) · χ₂(g). This group is the dual group (or character group) Ĝ = Hom(G, ℂ*). For G = ℤ/n₁ℤ × ··· × ℤ/nₖℤ, the fundamental theorem of finite abelian groups gives Ĝ ≅ ℤ/n₁ℤ × ··· × ℤ/nₖℤ ≅ G. The isomorphism Ĝ ≅ G is non-canonical (it depends on choices of generators), but the double dual Ĝ̂ ≅ G has a canonical isomorphism g ↦ (χ ↦ χ(g)). This is the finite-group version of Pontryagin duality.

The Fourier analysis on a finite abelian group decomposes functions f: G → ℂ into irreducible components. Every function can be written as f = Σ_{χ∈Ĝ} f̂(χ)·χ, where f̂(χ) = (1/|G|) Σ_{g∈G} f(g)·conjugate(χ(g)) are the Fourier coefficients. The Plancherel formula Σ_{g∈G} |f(g)|² = |G| Σ_{χ∈Ĝ} |f̂(χ)|² is a consequence of the orthogonality relations. For G = ℤ/nℤ, this is the classical discrete Fourier transform. For general abelian groups, the Fourier transform factors according to the product decomposition of G, recovering the multidimensional FFT.

The representation ring R(G) of a finite abelian group is particularly simple: it is isomorphic to the group ring ℤ[Ĝ] ≅ ℤ[x₁, …, xₖ]/(x₁^{n₁} − 1, …, xₖ^{nₖ} − 1). The tensor product of representations corresponds to multiplication of characters in Ĝ, and direct sum corresponds to addition. This ring structure encodes all the decomposition rules for representations of G and connects naturally to algebraic number theory through the cyclotomic fields generated by the roots of unity involved.