Group Algebras

Core Idea

The group algebra k[G] of a finite group G over a field k is the vector space with basis {eᵍ : g ∈ G} equipped with multiplication extending the group operation linearly: (Σ aᵍeᵍ)(Σ bₕeₕ) = Σ aᵍbₕe_{gh}. This construction translates group representation theory into module theory: a representation of G over k is precisely the same thing as a left k[G]-module. This equivalence is not merely a convenience — it unlocks the full machinery of ring theory (ideals, radicals, semisimplicity) for studying representations.

Explainer

The group algebra k[G] is the algebraic structure that bridges group theory and ring theory. As a vector space, k[G] has dimension |G| with the group elements as a basis. Multiplication is defined by extending the group operation linearly: if a = Σ aᵍeᵍ and b = Σ bₕeₕ, then ab = Σ_{g,h} aᵍbₕe_{gh}. This makes k[G] an associative algebra with identity e_e (the identity element of G). For non-abelian G, the algebra is non-commutative. The group algebra can also be thought of as the algebra of k-valued functions on G with convolution as multiplication, connecting it to harmonic analysis.

The fundamental theorem of this subject is that representations of G over k are equivalent to left k[G]-modules. Given a representation ρ: G → GL(V), define the module action by (Σ aᵍeᵍ)·v = Σ aᵍρ(g)v — this extends the G-action on V linearly to all of k[G]. Conversely, any left k[G]-module V gives a representation by restricting the action to the basis elements eᵍ. Under this correspondence, subrepresentations are submodules, intertwining operators are module homomorphisms, direct sums are direct sums, and irreducible representations are simple modules. Every theorem about representations has a module-theoretic counterpart.

This perspective reveals why Maschke's theorem is really a statement about semisimplicity. When char(k) does not divide |G|, the group algebra k[G] is a semisimple ring — every module is a direct sum of simple modules. The Artin-Wedderburn theorem then gives the structure: k[G] ≅ M_{d₁}(D₁) ⊕ ··· ⊕ M_{dₖ}(Dₖ), a direct sum of matrix algebras over division rings. Over ℂ, each Dᵢ = ℂ (by Schur's lemma), so ℂ[G] ≅ M_{d₁}(ℂ) ⊕ ··· ⊕ M_{dₖ}(ℂ), where d₁, …, dₖ are the dimensions of the irreducible representations. This isomorphism is the deepest structural result in finite group representation theory.

The center Z(k[G]) plays a special role. Its basis consists of the class sums — the formal sums of all elements in each conjugacy class. Since dim(Z(ℂ[G])) equals the number of conjugacy classes, which equals the number of irreducible representations, the center encodes the character theory. The primitive central idempotents eᵢ = (dᵢ/|G|) Σ_{g∈G} χᵢ(g⁻¹)eᵍ project k[G] onto its simple components, providing an algebraic realization of the decomposition into irreducible representations.