Representation Rings — Mathematics Knowledge System

Overview

The representation ring R(G) (also called the Grothendieck ring of representations) of a group G is the free abelian group generated by isomorphism classes of irreducible representations, with multiplication given by tensor product. As a ring, R(G) encodes the additive structure (direct sum) and multiplicative structure (tensor product) of G-representations. For a finite group over ℂ, R(G) ≅ ℤ^r where r is the number of conjugacy classes.

Intuition

R(G) is like a 'shadow' of the category of representations. You take the Grothendieck group — freely generated by isoclasses, with direct sum as addition — and tensor product gives the ring structure. Characters embed R(G) into the ring of class functions on G, making the irreducible characters a ℤ-basis.

Formal Definition

Definition

The representation ring R(G) of G is the Grothendieck group of the category Rep(G) of finite-dimensional k-representations of G (for char k = 0), made into a ring by tensor product.

R(G) = \mathbb{Z}\text{-span}\{[V] : V \text{ irred. rep of } G\} \cong \mathbb{Z}^{\#\mathrm{Irr}(G)}

As an abelian group (over ℂ or char 0 field)

[V] + [W] = [V \oplus W], \quad [V] \cdot [W] = [V \otimes_k W]

Ring operations: addition and multiplication

\chi: R(G) \hookrightarrow \mathrm{CF}(G, \mathbb{Z}) \quad ([V] \mapsto \chi_V), \quad \text{injective ring map into class functions}

Character map: embedding into class functions

R(G) \otimes_{\mathbb{Z}} \mathbb{C} \cong \mathrm{CF}(G, \mathbb{C}) \quad (\text{after tensoring with } \mathbb{C})

Rationalisation: class functions over ℂ

Notation

Notation	Meaning
$R(G)$	Representation ring (Green ring) of G
$[V]$	Isomorphism class of representation V in R(G)
$\mathrm{CF}(G, k)$	Ring of class functions G → k
$K_0$	Grothendieck group K₀(Rep(G))

Properties

Commutative ring

R(G) is a commutative ring (since V⊗W ≅ W⊗V as G-reps), unital with unit [triv] = the trivial representation.

Lambda ring structure

R(G) carries a λ-ring structure: λᵏ([V]) = [Λᵏ V] (exterior powers), satisfying the λ-ring axioms.

Theorems

Theorem 1: Structure over ℂ

For a finite group G over ℂ, R(G) is a commutative ring freely generated as a ℤ-module by the classes of irreducible representations. R(G) ⊗ ℂ is isomorphic to the ring of complex-valued class functions on G.

Theorem 2: Adams Operations

For each n ≥ 1, there is a ring homomorphism ψⁿ: R(G) → R(G) (the nth Adams operation) satisfying ψⁿ([V]) = [Sym^{(n)}V] - corrections. On characters: ψⁿ(χ)(g) = χ(gⁿ).

Worked Examples

1
G = ℤ/nℤ has n conjugacy classes (it's abelian) so n irreducible representations. Over ℂ, the irreps are χₖ: g ↦ ζᵏ for k = 0,…,n−1, where ζ = e^{2πi/n}.
$\mathrm{Irr}(\mathbb{Z}/n\mathbb{Z}) = \{\chi_0, \chi_1, \ldots, \chi_{n-1}\}, \quad \chi_k(g) = \zeta^k$
2
As an abelian group R(G) ≅ ℤⁿ. The ring structure: [χⱼ]·[χₖ] = [χⱼ ⊗ χₖ] = [χⱼ₊ₖ mod n].
$[\chi_j] \cdot [\chi_k] = [\chi_{(j+k) \bmod n}]$
3
So R(ℤ/nℤ) ≅ ℤ[x]/(xⁿ−1) as rings, where x = [χ₁]. The ring is isomorphic to the group ring ℤ[ℤ/nℤ].
$R(\mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}[x]/(x^n - 1)$

✓ Answer

R(ℤ/nℤ) ≅ ℤ[x]/(xⁿ−1), generated by the fundamental character χ₁.

Practice Problems

Mediumfree response

Show that R(G) ⊗_ℤ ℂ ≅ CF(G, ℂ) as rings, where CF(G,ℂ) is the ring of complex class functions.

Hardfree response

Define the Adams operation ψ² on R(G) in terms of characters and verify it is a ring homomorphism.

Quiz

Summary

R(G) is the Grothendieck ring of finite-dimensional G-representations: free ℤ-module on Irr(G), with tensor product as multiplication.
R(G) ≅ ℤ^r as an abelian group, where r = number of irreducibles = number of conjugacy classes.
Characters embed R(G) into CF(G,ℂ); after tensoring with ℂ, R(G)⊗ℂ ≅ CF(G,ℂ).
R(ℤ/nℤ) ≅ ℤ[x]/(xⁿ−1).
Adams operations ψⁿ: R(G) → R(G) defined by (ψⁿχ)(g) = χ(gⁿ) are ring homomorphisms.

References

BookAtiyah, M.F. — K-Theory (1967), Chapter 2
BookSerre, J.-P. — Linear Representations of Finite Groups (1977), Chapter 4