Mathematics.

algebraic representations

Burnside Ring

Representation Theory65 minDifficulty8 out of 10

You should know: group representations

Overview

The Burnside ring Ω(G) of a finite group G is the Grothendieck ring of finite G-sets: the free abelian group on isomorphism classes of transitive G-sets (i.e., orbits), with multiplication given by Cartesian product of G-sets. It sits naturally alongside the representation ring R(G), and there is a linearisation map Ω(G) → R(G) sending a G-set X to its permutation representation ℂ[X]. The Burnside ring encodes the combinatorics of G-actions and is related to Burnside's lemma.

Intuition

Just as R(G) is the Grothendieck ring of G-vector spaces, Ω(G) is the Grothendieck ring of G-sets. The basic transitive G-sets are G/H for subgroups H ≤ G, and these give the ℤ-basis. Multiplication corresponds to Cartesian product: (G/H) × (G/K) decomposes as a sum of orbits (G/L) according to double cosets.

Formal Definition

Definition

Let G be a finite group. A finite G-set is a finite set X with a left G-action. Every finite G-set is a disjoint union of transitive G-sets, and the transitive G-sets are the coset spaces G/H for subgroups H ≤ G (up to conjugacy in G).

Ω(G)=Z-span{[G/H]:H≤G, up to conjugacy}\Omega(G) = \mathbb{Z}\text{-span}\{[G/H] : H \leq G, \text{ up to conjugacy}\}
Burnside ring: free ℤ-module on conjugacy classes of subgroups
[X]+[Y]=[X⊔Y],[X]⋅[Y]=[X×Y][X] + [Y] = [X \sqcup Y], \quad [X] \cdot [Y] = [X \times Y]
Ring operations: disjoint union and Cartesian product
[G/H]×[G/K]=∑HxK∈H\G/K[G/(H∩xKx−1)][G/H] \times [G/K] = \sum_{HxK \in H\backslash G/K} [G/(H \cap xKx^{-1})]
Product formula via double cosets
Lin:Ω(G)→R(G),[X]↦[C[X]](linearisation map)\mathrm{Lin}: \Omega(G) \to R(G), \quad [X] \mapsto [\mathbb{C}[X]] \quad (\text{linearisation map})
Linearisation: G-set to permutation representation

Notation

NotationMeaning
Ω(G)\Omega(G)Burnside ring of G
G/HG/HTransitive G-set of left cosets of H
C[X]\mathbb{C}[X]Permutation representation: free ℂ-vector space on X
Lin\mathrm{Lin}Linearisation map Ω(G) → R(G)

Properties

Unit element

TheunitofΩ(G)istheG−setptwithtrivialaction=G/G(one−pointset).The unit of Ω(G) is the G-set {pt} with trivial action = G/G (one-point set).

Augmentation

Thereisaringhomomorphismε:Ω(G)→Zsending[G/H]to1(countingorbits),withkerneltheaugmentationideal.There is a ring homomorphism ε: Ω(G) → ℤ sending [G/H] to 1 (counting orbits), with kernel the augmentation ideal.

Theorems

Theorem 1: Burnside's Lemma (via the ring)
ThenumberofG−orbitsonaG−setXequals(1/∣G∣)Σg∈G∣Xg∣,whereXg=x∈X:gx=x.IntermsofΩ(G):[X]=ΣHaH[G/H]whereaH=∣orbitsisomorphictoG/H∣.The number of G-orbits on a G-set X equals (1/|G|) Σ_{g∈G} |X^g|, where X^g = {x ∈ X : gx = x}. In terms of Ω(G): [X] = Σ_H a_H [G/H] where a_H = |{orbits isomorphic to G/H}|.
Theorem 2: Ghost Map
Thereisaninjectiveringhomomorphismφ:Ω(G)→∏H≤G,conjZsending[X]to(∣XH∣)H.Itsimagecanbecharacterisedbycongruenceconditions(tomDieck′stheorem).There is an injective ring homomorphism φ: Ω(G) → ∏_{H ≤ G, conj} ℤ sending [X] to (|X^H|)_H. Its image can be characterised by congruence conditions (tom Dieck's theorem).
Theorem 3: Rank of Ω(G)
Ω(G) is a free abelian group of rank equal to the number of conjugacy classes of subgroups of G.

Worked Examples

  1. 1

    Subgroups of ℤ/2ℤ up to conjugacy: {e} and G = ℤ/2ℤ. So Ω(G) has ℤ-basis {[G/{e}], [G/G]} = {[G], [{pt}]}.

    Ω(Z/2Z)=Z[G/{e}]⊕Z[G/G]\Omega(\mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}[G/\{e\}] \oplus \mathbb{Z}[G/G]
  2. 2

    G/{e} = G (the 2-element set with G acting by left multiplication); G/G = {pt}. Dimensions: |G/{e}| = 2, |G/G| = 1.

    [G/{e}]↔2-element G-set,[G/G]↔1-point set[G/\{e\}] \leftrightarrow \text{2-element G-set}, \quad [G/G] \leftrightarrow \text{1-point set}
  3. 3

    Products: [pt]·[pt]=[pt], [pt]·[G]=[G], [G]·[G]=[G×G]. G×G has 4 elements; G acts diagonally. Fixed points: (g,g) fixed by all, so G×G ≅ [G] ⊕ [G] ⊕ [pt] ... wait: G×G decomposes as G-set. Diagonal {(e,e),(s,s)} is a G-orbit isomorphic to {pt}? No: s·(e,e)=(s,s), so {(e,e),(s,s)} is one G-orbit of size 2, isomorphic to G. {(e,s),(s,e)}: s·(e,s)=(s,e), so another orbit of size 2. So G×G = [G] ⊕ [G] = 2[G].

    [G/{e}]⋅[G/{e}]=2[G/{e}][G/\{e\}] \cdot [G/\{e\}] = 2[G/\{e\}]

✓ Answer

Ω(ℤ/2ℤ) ≅ ℤ² with basis [G] and [pt]. Multiplication: [G]² = 2[G], [G]·[pt] = [G], [pt]² = [pt].

Practice Problems

Mediumfree response

How many conjugacy classes of subgroups does S₃ have? What is the rank of Ω(S₃)?

Hardfree response

Explain why the linearisation map Lin: Ω(G) → R(G) is not in general an isomorphism, giving an example.

Quiz

The Burnside ring Ω(G) is the Grothendieck ring of:
As a ℤ-module, the rank of Ω(G) equals:
The linearisation map Lin: Ω(G) → R(G) sends a G-set X to:

Summary

  • Ω(G) = Grothendieck ring of finite G-sets: free ℤ-module on {[G/H] : H ≤ G, up to conjugacy}.
  • Ring structure: [X]+[Y]=[X⊔Y], [X]·[Y]=[X×Y].
  • rank Ω(G) = number of conjugacy classes of subgroups of G.
  • Linearisation map Lin: Ω(G) → R(G), [X] ↦ [ℂ[X]], is a ring homomorphism but not in general an isomorphism.
  • The ghost map φ: Ω(G) → ∏_H ℤ, [X] ↦ (|X^H|)_H, is injective.

References

  1. Booktom Dieck, T. — Transformation Groups and Representation Theory (1979), Chapter 5
  2. WebsiteWikipedia — Burnside ring