nLab
Burnside ring

Contents

Contents

Idea

The Burnside ring A(G)A(G) of a finite group GG 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 FF.

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

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

Definition

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

The Burnside ring A(G)A(G) is then the (additive) group completion of the Burnside rig, A(G)= B(G)A(G) = \mathbb{Z} \otimes_{\mathbb{N}} B(G). (This tensor product in commutative monoids is the coproduct of \mathbb{Z} and B(G)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,,G/H tG / H_1 , G / H_2, \ldots, G / H_t of GG, where H 1,,H tH_{1}, \ldots, H_{t} are representatives of the distinct conjugacy classes of GG, equipped with the product described in Definition .

Properties

Span by the coset spaces G/HG/H

Since every finite G-set is a direct sum of the basic coset spaces G/HG/H, for HGH \hookrightarrow G subgroups of GG, and since G/H 1G/H_1 and G/H 2G/H_2 are isomorphic G-sets of H 1H_1 and H 2H_2 are conjugate to each other, the Burnside ring is spanned, as an abelian group by the [G/H][G/H] for HH ranging over conjugacy classes of subgroups.

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

Notice that the permutation representations k[G/H]k[G/H] corresponding to these generators are precisely the induced representations of trivial representations: k[G/H]ind H G(1)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 GG (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 ij n_{i j}^\ell \;\in\; \mathbb{N}

uniquely defined by the equation

(1)[G/H i]×[G/H j]=n ij [G/H ]. [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 (n ij )\left( n_{i j}^\ell\right) (Def. ) are equal to the following algebraic expression in the table of marks MM and its inverse matrix M 1M^{-1} (which exists by this Prop.):

n ij l=1mtM imM jm(M 1) ml n_{i j}^l \;=\; \underset{1 \leq m \leq t}{\sum} M_{i m} \cdot M_{j m} \cdot (M^{-1})_{m l}

where tt is the dimension of MM, i.e. MM is a t×tt \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 GG be a finite group, then its Burnside ring A(G)A(G) is isomorphic to the equivariant stable cohomotopy cohomology ring 𝕊 G(*)\mathbb{S}_G(\ast) of the point in degree 0.

A(G)𝕊 G(*). 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 BGB G
(equivariant)
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
(equivariant)
complex K-theory
KUrepresentation ring
KU G(*)R (G)KU_G(\ast) \simeq R_{\mathbb{C}}(G)
Atiyah-Segal completion theorem
R(G)KU G(*)compl.KU G(*)^KU(BG)R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
(equivariant)
complex cobordism cohomology
MUMU G(*)MU_G(\ast)completion theorem for complex cobordism cohomology
MU G(*)compl.MU G(*)^MU(BG)MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)
(equivariant)
algebraic K-theory
K𝔽 pK \mathbb{F}_prepresentation ring
(K𝔽 p) G(*)R p(G)(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)
Rector completion theorem
R 𝔽 p(G)K(𝔽 p) G(*)compl.(K𝔽 p) G(*)^Rector 73K𝔽 p(BG)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
K𝔽 1Segal 74K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} SBurnside ring
𝕊 G(*)A(G)\mathbb{S}_G(\ast) \simeq A(G)
Segal-Carlsson completion theorem
A(G)Segal 71𝕊 G(*)compl.𝕊 G(*)^Carlsson 84𝕊(BG)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 GG is isomorphic to the colimit

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

over GG-representations in a complete G-universe, of GG-homotopy classes of GG-equivariant based continuous functions from the representation sphere S VS^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 GBurn adG Burn_{ad} is the Burnside ring A(G)A(G):

End GBurn ad(*,*)A(G). End_{G Burn_{ad}}(\ast, \ast) \;\simeq\; A(G) \,.

Relation to representation ring

Let GG be a finite group. Consider

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

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

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

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

and hence induces a ring homomorphism

A(G)[]R(G) 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)𝕊 G(*) A(G) \;\simeq\; \mathbb{S}_G(\ast)

    (as above)

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

    R(G)K G(*) R(G) \;\simeq\; K_G(\ast)

    (see there)

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

𝕊KU \mathbb{S} \longrightarrow KU

from the sphere spectrum to KU.

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

References

The concept was named by Dress, following

Textbook accounts and lecture notes include

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 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)

Last revised on September 19, 2019 at 05:06:28. See the history of this page for a list of all contributions to it.