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\section*{Burnside ring}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{span_by_the_coset_spaces_}{Span by the coset spaces $G/H$}\dotfill \pageref*{span_by_the_coset_spaces_} \linebreak
\noindent\hyperlink{InTermsOfTheTableOfMarks}{In terms of the table of marks}\dotfill \pageref*{InTermsOfTheTableOfMarks} \linebreak
\noindent\hyperlink{AsTheEquivariantStableCohomotopyOfThePoint}{As the equivariant stable cohomotopy of the point}\dotfill \pageref*{AsTheEquivariantStableCohomotopyOfThePoint} \linebreak
\noindent\hyperlink{as_the_endomorphism_ring_of_the_additive_burnside_category}{As the endomorphism ring of the additive Burnside category}\dotfill \pageref*{as_the_endomorphism_ring_of_the_additive_burnside_category} \linebreak
\noindent\hyperlink{relation_to_representation_ring}{Relation to representation ring}\dotfill \pageref*{relation_to_representation_ring} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 $\text{FinGrp} \xrightarrow{A} \text{AbRing}$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For \emph{any} group $G$, the \emph{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 category|cartesian closed structure]] of the topos.

The \emph{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 (\hyperlink{Schanuel91}{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 \ref{BurnsideMultiplicities}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{span_by_the_coset_spaces_}{}\subsubsection*{{Span by the coset spaces $G/H$}}\label{span_by_the_coset_spaces_}

Since every [[finite set|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 [[isomorphism|isomorphic]] [[G-sets]] of $H_1$ and $H_2$ are [[conjugation action|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.

\begin{displaymath}
A(G)
  \;\simeq_{\mathbb{Z}}\;
  \underset{[H \subset G]}{\oplus}
  [G/H]
  \,.
\end{displaymath}
Notice that the [[permutation representations]] $k[G/H]$ corresponding to these generators are precisely the [[induced representations of trivial representations]]: $k[G/H] \simeq ind_H^G(\mathbf{1})$.

\hypertarget{InTermsOfTheTableOfMarks}{}\subsubsection*{{In terms of the table of marks}}\label{InTermsOfTheTableOfMarks}

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

\begin{defn}
\label{BurnsideMultiplicities}\hypertarget{BurnsideMultiplicities}{}
\textbf{(Burnside multiplicities)}

Given a choice of [[linear order]] on the [[conjugacy classes]] of [[subgroups]] of $G$ (for instance as in \href{table+of+marks#LinearOrderOnConjugacyClassesOfSubgroups}{this Lemma}), we say that the corresponding \emph{structure constants} of the [[Burnside ring]] (or \emph{Burnside multiplicities}) are the [[natural numbers]]

\begin{displaymath}
n_{i j}^\ell \;\in\; \mathbb{N}
\end{displaymath}
uniquely defined by the [[equation]]

\begin{equation}
[G/H_i] \times [G/H_j]
  \;=\;
  \underset{ \ell }{\sum}
  n_{i j}^\ell [G/H_\ell]
  \,.
\label{BurnsideStructureConstants}\end{equation}
\end{defn}
\begin{prop}
\label{BurnsideProductInTermsOfTableOfMarks}\hypertarget{BurnsideProductInTermsOfTableOfMarks}{}
\textbf{([[Burnside ring]] product in terms of [[table of marks]])}

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

\begin{displaymath}
n_{i j}^l
  \;=\;
  \underset{1 \leq m \leq t}{\sum}
  M_{i m} \cdot M_{j m}
  \cdot
  (M^{-1})_{m l}
\end{displaymath}
where $t$ is the dimension of $M$, i.e. $M$ is a $t \times t$ matrix.

\end{prop}
For [[proof]] see at \emph{[[table of marks]]}, \href{table+of+marks#BurnsideProductInTermsOfTableOfMarks}{this Prop.}



\hypertarget{AsTheEquivariantStableCohomotopyOfThePoint}{}\subsubsection*{{As the equivariant stable cohomotopy of the point}}\label{AsTheEquivariantStableCohomotopyOfThePoint}

\begin{prop}
\label{BurnsideRingIsEquivariantStableCohomotopyOfPoint}\hypertarget{BurnsideRingIsEquivariantStableCohomotopyOfPoint}{}
\textbf{([[Burnside ring is equivariant stable cohomotopy of the point]])}

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

\begin{displaymath}
A(G) \overset{\simeq}{\longrightarrow} \mathbb{S}_G(\ast)
  \,.
\end{displaymath}
\end{prop}
This is originally due to \hyperlink{Segal71}{Segal 71}, a detailed proof is given by \hyperlink{tomDieck79}{tom Dieck 79, theorem 7.6.7, 8.5.1}. See also \hyperlink{Lueck05}{Lück 05, theorem 1.13}, \hyperlink{tomDieckPetrie78}{tom Dieck-Petrie 78}.

From a broader perspective, this statement is a special case of [[tom Dieck splitting]] of [[equivariant suspension spectra]] (e.g. \hyperlink{Schwede15}{Schwede 15, theorem 6.14}), see \href{tom+Dieck+splitting#OfFixedPointSpectraOfEquivariantSuspensionSpectra}{there}.

[[!include Segal completion -- table]]

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

\begin{displaymath}
A(G) \simeq \underset{\longrightarrow}{\lim}_V [S^V,S^V]_G
\end{displaymath}
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 (\hyperlink{GreenleesMay95}{Greenlees-May 95, p. 8}).

\hypertarget{as_the_endomorphism_ring_of_the_additive_burnside_category}{}\subsubsection*{{As the endomorphism ring of the additive Burnside category}}\label{as_the_endomorphism_ring_of_the_additive_burnside_category}

\begin{example}
\label{BurnsideRingIsEndomorphismRingInBurnsideCategory}\hypertarget{BurnsideRingIsEndomorphismRingInBurnsideCategory}{}
\textbf{([[Burnside ring]] is [[endomorphism ring]] of [[additive Burnside category]])}

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

\begin{displaymath}
End_{G Burn_{ad}}(\ast, \ast)
  \;\simeq\;
  A(G)
  \,.
\end{displaymath}
\end{example}
\hypertarget{relation_to_representation_ring}{}\subsubsection*{{Relation to representation ring}}\label{relation_to_representation_ring}

Let $G$ be a [[finite group]]. Consider

\begin{enumerate}%
\item the [[Burnside ring]] $A(G)$, which is the [[Grothendieck group]] of the [[monoidal category]] $G Set$ of [[finite set|finite]] [[G-sets]];


\item the [[representation ring]] $R(G)$, which is the [[Grothendieck group]] of the monoidal category $G Rep$ of [[finite dimensional vector space|finite dimensional]] $G$-[[linear representations]].



\end{enumerate}
Then then map that sends a G-set to the corresponding linear [[permutation representation]] is a [[strong monoidal functor]]

\begin{displaymath}
G Set \overset{\mathbb{C}[-]}{\longrightarrow} G Rep
\end{displaymath}
and hence induces a [[ring homomorphism]]

\begin{displaymath}
A(G) \overset{ \mathbb{C}[-] }{\longrightarrow} R(G)
\end{displaymath}
Under the identitification

\begin{enumerate}%
\item of the [[Burnside ring]] with the [[equivariant stable cohomotopy]] of the point

\begin{displaymath}
A(G) \;\simeq\; \mathbb{S}_G(\ast)
\end{displaymath}
(as \hyperlink{AsTheEquivariantStableCohomotopyOfThePoint}{above})


\item of the [[representation ring]] with the [[equivariant K-theory]] of the point

\begin{displaymath}
R(G) \;\simeq\; K_G(\ast)
\end{displaymath}
(see \href{representation+ring#AsEquivariantKTheoryOfThePoint}{there})



\end{enumerate}
this should be the image of the initial morphism of [[E-infinity ring spectra]]

\begin{displaymath}
\mathbb{S} \longrightarrow KU
\end{displaymath}
from the [[sphere spectrum]] to [[KU]].

For details on this comparison map see at \emph{[[permutation representation]]}, \href{permutation+representation#ExamplesVirtualPermutationRepresentations}{this section}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[orbit category]]


\item [[Segal conjecture]]


\item [[equivariant stable cohomotopy]]


\item [[beta-ring]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The concept was named by Dress, following

\begin{itemize}%
\item [[William Burnside]], \emph{Theory of Groups of Finite Order}, 1897 (\href{http://www.gutenberg.org/files/40395/40395-pdf.pdf}{pdf})

\end{itemize}
Textbook accounts and lecture notes include

\begin{itemize}%
\item Charles Curtis, Irving Reiner, chapter XI of \emph{Representation theory of finite groups and associative algebras}, AMS 1962


\item [[Tammo tom Dieck]], \emph{[[Transformation Groups and Representation Theory]]}, Springer 1979


\item Serge Bouc, \emph{Burnside rings}, in \emph{Handbook of Algebra} Volume 2, 2000, Pages 739-804 ()


\item [[Tammo tom Dieck]], \emph{Representation theory}, 2009 (\href{http://www.uni-math.gwdg.de/tammo/rep.pdf}{pdf})


\item [[Stefan Schwede]], section 6 of \emph{[[Lectures on Equivariant Stable Homotopy Theory]]}, 2015 (\href{http://www.math.uni-bonn.de/people/schwede/equivariant.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item [[John Greenlees]], [[Peter May]], section 2 of \emph{Equivariant stable homotopy theory}, in I.M. James (ed.), \emph{Handbook of Algebraic Topology} , pp. 279-325. 1995. (\href{http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf}{pdf})


\item [[Stephen Schanuel]], 1991, \emph{Negative sets have Euler characteristic and dimension}, in: Proc. Como 1990. Lecture Notes in Mathematics, vol. 1488. Springer-Verlag, Berlin, pp. 379--385.



\end{itemize}
Discussion in relation to [[equivariant stable cohomotopy]] and the [[Segal-Carlsson completion theorem]] is in

\begin{itemize}%
\item [[Graeme Segal]], \emph{Equivariant stable homotopy theory}, In Actes du Congr\`e{}s International des Math \'e{}maticiens (Nice, 1970), Tome 2 , pages 59–63. Gauthier-Villars, Paris, 1971 ([[SegalEquivariantStableHomotopyTheory.pdf:file]])


\item [[Tammo tom Dieck]], T. Petrie, \emph{Geometric modules over the Burnside ring}, Invent. Math. 47 (1978) 273-287; chapter 10 in: \emph{[[Transformation Groups and Representation Theory]]} (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-geometric.pdf}{pdf})


\item Erkki Laitinen, \emph{On the Burnside ring and stable cohomotopy of a finite group}, Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 (\href{https://www.jstor.org/stable/24491306}{jstor:24491306}, [[Laitinen79.pdf:file]])


\item [[Gunnar Carlsson]], \emph{Equivariant Stable Homotopy and Segal's Burnside Ring Conjecture}, Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 (\href{https://www.jstor.org/stable/2006940}{jstor:2006940}, \href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/carlsson.pdf}{pdf})


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


\item [[Wolfgang Lück]], \emph{The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups} (\href{https://arxiv.org/abs/math/0504051}{arXiv:math/0504051})



\end{itemize}
Computational aspects are discussed in

\begin{itemize}%
\item Martin Kreuzer, \emph{Computational aspects of Burnside rings, part I: the ring structure}, D.P. Beitr Algebra Geom (2017) 58: 427 (\href{https://doi.org/10.1007/s13366-016-0324-4}{doi:10.1007/s13366-016-0324-4})

\end{itemize}
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