Representation Ring

Core Idea

The representation ring R(G) (also called the Green ring or character ring) is the Grothendieck group of finite-dimensional representations of G, with addition from direct sum and multiplication from tensor product. Its elements are formal differences of isomorphism classes of representations — "virtual representations" — and it is a commutative ring where the irreducible representations form a ℤ-basis. The character map χ: R(G) → Class(G, ℂ) embeds R(G) as a subring of class functions, making R(G) a bridge between representation theory, K-theory, and algebraic number theory.

Explainer

The representation ring R(G) organizes all representations of G into an algebraic structure. Start with the free abelian group on isomorphism classes of finite-dimensional representations, then impose the relation [V ⊕ W] = [V] + [W]. This is the Grothendieck group construction, which formally adds additive inverses to get "virtual representations." The tensor product of representations defines a multiplication [V]·[W] = [V ⊗ W], making R(G) a commutative ring. The isomorphism classes of irreducible representations form a ℤ-basis, so every element is a unique integer linear combination of irreducibles.

The character map χ: R(G) → Class(G, ℂ) sends each representation to its character. Since characters are additive (χ_{V⊕W} = χ_V + χ_W) and multiplicative (χ_{V⊗W} = χ_V · χ_W), this is a ring homomorphism. Over ℂ, it is injective (characters determine representations up to isomorphism), so R(G) embeds as a subring of the ring of class functions. The image consists of the virtual characters — ℤ-linear combinations of irreducible characters. The full ring of class functions is R(G) ⊗_ℤ ℂ.

The Adams operations ψᵏ: R(G) → R(G) are ring endomorphisms defined by ψᵏ(V) being the virtual representation whose character at g is χ_V(gᵏ). These operations satisfy ψᵏ ∘ ψˡ = ψᵏˡ and encode the interplay between the ring structure and the group structure. For 1-dimensional representations, ψᵏ(ρ) = ρᵏ (the kth tensor power). For general representations, ψᵏ is related to exterior and symmetric powers by Newton's identities. Adams operations make R(G) a λ-ring, connecting it to algebraic K-theory.

The representation ring has deep connections to number theory. For a cyclic group ℤ/nℤ, R(G) ≅ ℤ[ζₙ], the ring of integers in the cyclotomic field (after tensoring appropriately). For general finite groups, R(G) captures the "representation-theoretic arithmetic" of G. The rank of R(G) as a ℤ-module equals the number of irreducible representations (= number of conjugacy classes). The representation ring functor G ↦ R(G) is contravariant in G via restriction and covariant via induction, and these operations satisfy Frobenius reciprocity at the level of rings, providing the algebraic backbone for the Mackey machine and equivariant K-theory.