# nLab Burnside ring

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

The Burnside ring $A(G)$ of a finite group $G$ is the analogue of the representation ring in the category of finite sets, as opposed to the category of finite dimensional vector spaces over a field $F$.

Elements of the Burnside ring are thus formal differences of G-sets (with respect to disjoint union).

$A$ is a contravariant functor $\text{FinGrp} \xrightarrow{A} \text{AbRing}$.

## Definition

For any group $G$, the Burnside rig of $G$ is the set of isomorphism classes of the topos $FinSet^G$, the category of permutation representations of $G$ on finite sets, equipped with the addition operation descended from coproducts in $FinSet^G$ and the multiplication operation descended from products in $FinSet^G$. In fact the Burnside rig $B(G)$ is an exponential rig?, where exponentiation is derived from the cartesian closed structure of the topos.

The Burnside ring $A(G)$ is then the (additive) group completion of the Burnside rig, $A(G) = \mathbb{Z} \otimes_{\mathbb{N}} B(G)$. (This tensor product in commutative monoids is the coproduct of $\mathbb{Z}$ and $B(G)$ in the category of commutative rigs, and $\mathbb{Z} \otimes_{\mathbb{N}} -$ is left adjoint to the forgetful functor from commutative rings to commutative rigs.)

More generally, any distributive category determines a Burnside rig (Schanuel91).

Explicitly, the Burnside ring can be seen to be the free abelian group on the set $G / H_1 , G / H_2, \ldots, G / H_t$ of $G$, where $H_{1}, \ldots, H_{t}$ are representatives of the distinct conjugacy classes of $G$, equipped with the product described in Definition .

## Properties

### Span by the coset spaces $G/H$

Since every finite G-set is a direct sum of the basic coset spaces $G/H$, for $H \hookrightarrow G$ subgroups of $G$, and since $G/H_1$ and $G/H_2$ are isomorphic G-sets of $H_1$ and $H_2$ are conjugate to each other, the Burnside ring is spanned, as an abelian group by the $[G/H]$ for $H$ ranging over conjugacy classes of subgroups.

$A(G) \;\simeq_{\mathbb{Z}}\; \underset{[H \subset G]}{\oplus} [G/H] \,.$

Notice that the permutation representations $k[G/H]$ corresponding to these generators are precisely the induced representations of trivial representations: $k[G/H] \simeq ind_H^G(\mathbf{1})$.

### In terms of the table of marks

We discuss how the product in the Burnside ring is encoded in the table of marks of the given finite group.

###### Definition

(Burnside multiplicities)

Given a choice of linear order on the conjugacy classes of subgroups of $G$ (for instance as in this Lemma), we say that the corresponding structure constants of the Burnside ring (or Burnside multiplicities) are the natural numbers

$n_{i j}^\ell \;\in\; \mathbb{N}$

uniquely defined by the equation

(1)$[G/H_i] \times [G/H_j] \;=\; \underset{ \ell }{\sum} n_{i j}^\ell [G/H_\ell] \,.$
###### Proposition

(Burnside ring product in terms of table of marks)

The Burnside ring structure constants $\left( n_{i j}^\ell\right)$ (Def. ) are equal to the following algebraic expression in the table of marks $M$ and its inverse matrix $M^{-1}$ (which exists by this Prop.):

$n_{i j}^l \;=\; \underset{1 \leq m \leq t}{\sum} M_{i m} \cdot M_{j m} \cdot (M^{-1})_{m l}$

where $t$ is the dimension of $M$, i.e. $M$ is a $t \times t$ matrix.

For proof see at table of marks, this Prop.

### As the equivariant stable cohomotopy of the point

###### Proposition

(Burnside ring is equivariant stable cohomotopy of the point)

Let $G$ be a finite group, then its Burnside ring $A(G)$ is isomorphic to the equivariant stable cohomotopy cohomology ring $\mathbb{S}_G(\ast)$ of the point in degree 0.

$A(G) \overset{\simeq}{\longrightarrow} \mathbb{S}_G(\ast) \,.$

This is originally due to Segal 71, a detailed proof is given by tom Dieck 79, theorem 7.6.7, 8.5.1. See also Lück 05, theorem 1.13, tom Dieck-Petrie 78.

From a broader perspective, this statement is a special case of tom Dieck splitting of equivariant suspension spectra (e.g. Schwede 15, theorem 6.14), see there.

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point $\ast$
cohomology
of classifying space $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation ring
$KU_G(\ast) \simeq R_{\mathbb{C}}(G)$
Atiyah-Segal completion theorem
$R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant)
complex cobordism cohomology
MU$MU_G(\ast)$completion theorem for complex cobordism cohomology
$MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant)
algebraic K-theory
$K \mathbb{F}_p$representation ring
$(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$
Rector completion theorem
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
$K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

More explicitly, this means that the Burnside ring of a group $G$ is isomorphic to the colimit

$A(G) \simeq \underset{\longrightarrow}{\lim}_V [S^V,S^V]_G$

over $G$-representations in a complete G-universe, of $G$-homotopy classes of $G$-equivariant based continuous functions from the representation sphere $S^V$ to itself (Greenlees-May 95, p. 8).

### As the endomorphism ring of the additive Burnside category

###### Example

(Burnside ring is endomorphism ring of additive Burnside category)

The endomorphism ring of the terminal G-set (the point $\ast$ equipped with the, necessarily, trivial action) in the additive Burnside category $G Burn_{ad}$ is the Burnside ring $A(G)$:

$End_{G Burn_{ad}}(\ast, \ast) \;\simeq\; A(G) \,.$

### Relation to representation ring

Let $G$ be a finite group. Consider

1. the Burnside ring $A(G)$, which is the Grothendieck group of the monoidal category $G Set$ of finite G-sets;

2. the representation ring $R(G)$, which is the Grothendieck group of the monoidal category $G Rep$ of finite dimensional $G$-linear representations.

Then then map that sends a G-set to the corresponding linear permutation representation is a strong monoidal functor

$G Set \overset{\mathbb{C}[-]}{\longrightarrow} G Rep$

and hence induces a ring homomorphism

$A(G) \overset{ \mathbb{C}[-] }{\longrightarrow} R(G)$

Under the identitification

1. of the Burnside ring with the equivariant stable cohomotopy of the point

$A(G) \;\simeq\; \mathbb{S}_G(\ast)$

(as above)

2. of the representation ring with the equivariant K-theory of the point

$R(G) \;\simeq\; K_G(\ast)$

(see there)

this should be the image of the initial morphism of E-infinity ring spectra

$\mathbb{S} \longrightarrow KU$

from the sphere spectrum to KU.

For details on this comparison map see at permutation representation, this section.

The Burnside product seems to first appear as equation (i) in:

• William Burnside, On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups, Proceedings of the American Mathematical Society 1901 (doi:10.1112/plms/s1-34.1.159)

(beware the terminology: a G-set is called a “permutation group $G$” in that article, a subset is called a “compound” and the Cartesian product of $G$-sets is called their “compounding”).

It is then included (not in the first but) in the second edition (Sections 184-185) of:

The term “Burnside ring” as well as “Burnside algebra” is then due to (see NMT 04, Vol. 1 p. 60 for historical comments)

Modern textbook accounts and lecture notes:

• John Greenlees, Peter May, section 2 of Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

• Stephen Schanuel, 1991, Negative sets have Euler characteristic and dimension, in: Proc. Como 1990. Lecture Notes in Mathematics, vol. 1488. Springer-Verlag, Berlin, pp. 379–385.

Discussion in relation to equivariant stable cohomotopy and the Segal-Carlsson completion theorem is in

• Graeme Segal, Equivariant stable homotopy theory, In Actes du Congrès International des Math ématiciens (Nice, 1970), Tome 2 , pages 59–63. Gauthier-Villars, Paris, 1971 (pdf)

• Tammo tom Dieck, T. Petrie, Geometric modules over the Burnside ring, Invent. Math. 47 (1978) 273-287;

chapter 10 in: Transformation Groups and Representation Theory (pdf)

• Erkki Laitinen, On the Burnside ring and stable cohomotopy of a finite group, Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 (jstor:24491306, pdf)

• Gunnar Carlsson, Equivariant Stable Homotopy and Segal’s Burnside Ring Conjecture, Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 (jstor:2006940, pdf)

• C. D. Gay, G. C. Morris, and I. Morris, Computing Adams operations on the Burnside ring of a finite group, J. Reine Angew. Math., 341 (1983), pp. 87–97.

• Wolfgang Lück, The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups (arXiv:math/0504051)

Computational aspects are discussed in

• Martin Kreuzer, Computational aspects of Burnside rings, part I: the ring structure, D.P. Beitr Algebra Geom (2017) 58: 427 (doi:10.1007/s13366-016-0324-4)