Representations of Cyclic Groups

Core Idea

Every irreducible complex representation of a cyclic group ℤ/nℤ is one-dimensional, given by sending a generator g to an nth root of unity ζ = e^2πik/n for k = 0, 1, …, n−1. This gives exactly n irreducible representations, matching the n conjugacy classes (each element is its own conjugacy class since the group is abelian). The character table is the DFT matrix (1/√n)(ζʲᵏ), and the representation theory of cyclic groups is equivalent to the discrete Fourier transform — a fact that connects algebra to signal processing.

Explainer

The cyclic group ℤ/nℤ = ⟨g | gⁿ = e⟩ has the simplest representation theory of any group family. Since the group is abelian, Schur's lemma implies that every irreducible complex representation is one-dimensional. A 1-dimensional representation is just a group homomorphism ρ: ℤ/nℤ → ℂ*, which is determined by ρ(g) since g generates the group. The constraint ρ(g)ⁿ = ρ(gⁿ) = ρ(e) = 1 means ρ(g) must be an nth root of unity. There are exactly n choices: ρ_k(g) = e^2πik/n for k = 0, 1, …, n−1. These n representations are pairwise non-isomorphic and exhaust all irreducibles.

The character table of ℤ/nℤ is the n×n matrix with entry (j,k) equal to ω^{jk}, where ω = e^2πi/n. This is exactly the discrete Fourier transform (DFT) matrix. The orthogonality relations for characters — Σ_{g∈G} χᵢ(g)·conjugate(χⱼ(g)) = |G|·δᵢⱼ — become the statement that the DFT matrix (scaled by 1/√n) is unitary. This is not a coincidence: the DFT decomposes functions on ℤ/nℤ into irreducible components, and the Fourier inversion formula is the character orthogonality relation. Representation theory of cyclic groups is Fourier analysis on finite cyclic groups.

Over the real numbers, the picture changes. The representations ρ_k and ρ_{n−k} are complex conjugates, and when k ≠ 0, n/2, they cannot be individually realized over ℝ. Instead, they combine into a 2-dimensional real irreducible representation where g acts as the rotation matrix [[cos(2πk/n), −sin(2πk/n)], [sin(2πk/n), cos(2πk/n)]]. So the real irreducible representations of ℤ/nℤ consist of some 1-dimensional ones (corresponding to real roots of unity: ±1 when they exist) and some 2-dimensional ones (corresponding to conjugate pairs of complex roots).

The group algebra ℂ[ℤ/nℤ] ≅ ℂ[x]/(xⁿ − 1) ≅ ℂ ⊕ ℂ ⊕ ··· ⊕ ℂ (n copies), where the isomorphism uses the Chinese Remainder Theorem and the factorization xⁿ − 1 = ∏(x − ω^k). Each factor ℂ corresponds to one irreducible representation. This is the simplest instance of the Artin-Wedderburn decomposition. For cyclic groups, the representation ring R(ℤ/nℤ) ≅ ℤ[x]/(xⁿ − 1), with the tensor product of representations corresponding to multiplication of characters (pointwise product of roots of unity).