Regular Representation

Core Idea

The (left) regular representation of a finite group G acts on the vector space ℂ[G] with basis {eᵍ : g ∈ G} by left multiplication: ρ(g)(eₕ) = e_{gh}. It has dimension |G| and contains every irreducible representation Vᵢ with multiplicity equal to dim(Vᵢ). The regular representation is the "universal" representation from which all others can be extracted, and its decomposition yields the fundamental formula |G| = Σ dᵢ².

Explainer

The regular representation is constructed from the group itself. Form a vector space ℂ[G] with one basis vector eᵍ for each element g ∈ G, so dim(ℂ[G]) = |G|. Define the left action of G by ρ(g)(eₕ) = e_{gh} — each group element permutes the basis vectors by left multiplication. This is always a faithful representation (distinct group elements give distinct permutations), and it carries maximal information about the group's structure.

The character of the regular representation has a striking form: χ_reg(e) = |G| and χ_reg(g) = 0 for all g ≠ e. The identity fixes every basis vector (trace = |G|), while any non-identity element moves every basis vector (no diagonal entries, trace = 0). Using this character to compute multiplicities: nᵢ = ⟨χ_reg, χᵢ⟩ = (1/|G|) Σ_{g∈G} χ_reg(g) conjugate(χᵢ(g)) = (1/|G|) · |G| · χᵢ(e) = dᵢ. Each irreducible representation Vᵢ appears with multiplicity exactly equal to its dimension.

This decomposition ℂ[G] ≅ ⊕ᵢ Vᵢ^{⊕dᵢ} has profound consequences. Comparing dimensions: |G| = Σ dᵢ². This sum-of-squares formula is one of the most basic constraints in finite group representation theory — it limits which sets of dimensions can arise as irreducible degrees. For example, a group of order 12 might have irreducibles of dimensions 1, 1, 1, 3 (since 1+1+1+9 = 12) or 1, 1, 1, 1, 2, 2 (since 1+1+1+1+4+4 = 12), but never 1, 1, 1, 1, 1, 1, 1, 5 (since 5² = 25 > 12).

The regular representation also reveals the connection between representation theory and the group algebra ℂ[G]. The group algebra is ℂ[G] with multiplication defined by extending the group multiplication linearly: (Σ aᵍeᵍ)(Σ bₕeₕ) = Σ aᵍbₕe_{gh}. The decomposition ℂ[G] ≅ ⊕ Mₐᵢ(ℂ) (as an algebra, by the Artin-Wedderburn theorem) connects the representation theory of G to the structure theory of semisimple algebras.