nLab Burnside ring

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 $FinSet^G$ of finite G-sets, 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 ring 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 if $H_1$ 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, Segal 78, p. 2, 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) cohomology	representing spectrum	equivariant cohomology of the point $\ast$	cohomology of classifying space $B G$
(equivariant) ordinary cohomology	HZ	Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant) complex K-theory	KU	representation 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 $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant) stable cohomotopy	$K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S	Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$	Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

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.

Related concepts

• orbit category

• Segal conjecture

• equivariant stable cohomotopy

• beta-ring

• objective number theory

References

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:

• William Burnside, Theory of Groups of Finite Order, Second edition Cambridge 1911 (pdf)

  reprinted by Cambridge University Press 2012 (doi:10.1017/CBO9781139237253)

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

• Louis Solomon, The Burnside algebra of a finite group, Journal of Combinatorial Theory Volume 2, Issue 4, June 1967, Pages 603-615 (doi:10.1016/S0021-9800(67)80064-4)

Modern textbook accounts and lecture notes:

• Charles Curtis, Irving Reiner, chapter XI of Representation theory of finite groups and associative algebras, AMS 1962

• Tammo tom Dieck, Transformation Groups and Representation Theory, Springer 1979

• Tammo tom Dieck, Chapter IV of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

• Serge Bouc, Burnside rings, in Handbook of Algebra Volume 2, 2000, Pages 739-804 (doi:10.1016/S1570-7954(00)80043-1)

• Tammo tom Dieck, Representation theory, 2009 (pdf)

• Stefan Schwede, section 6 of Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)

See also

• 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:

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

• Graeme Segal, p. -2- of: Some results in equivariant homotopy theory (1978) [scan: web, pdf]

• Tammo tom Dieck, T. Petrie, Geometric modules over the Burnside ring, Invent. Math. 47 ;

  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:, 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)