Ibrahima Tounkara

Simplicial Burnside ring

This paper develops links between the Burnside ring of a finite group $G$ and the slice Burnside ring. The goal is to gain a better understanding of ghost maps,  idempotents, prime spectrum of these Burnside rings and connections between them.

MSC(2010): 19A22, 18G30, 06A11, 20J15

Keywords: Burnside ring, simplicial sets, Poset, biset functor.

§ INTRODUCTION

Starting from questions in representation theory and homotopy theory, the investigation of  biset functors received considerable attention over the last decades.

We refer to  [2], which covered this subject in detail. We recall some key definitions relevant for the current paper, the presentation follows  recent  articles very closely.
Let $G$ be a finite group. The Grothendieck ring constructed from the category of $G$-sets is denoted by $B(G)$ and is called the Burnside ring of $G$. If $X$ is a finite $G$-set, let $[X]$ be its image in $B(G)$. Additively, $B(G)$ is  the free abelian group on isomorphism classes of transitive $G$-sets. Equivalently, an additive $\ZZ$-basis is given by the $[G/H]$ where $H$ runs through a set  $[C(G)]$ of representatives of conjugacy classes of subgroups of $G$. The multiplication comes from the decomposition of $G/H \times G/K$ into orbits. The ring $B(G)$ is commutative with unit $[G/G]$.  If $H$ is a subgroup  of $G$, then there is a unique linear form $\phi_H: B(G )\ra \ZZ$ such that $\phi_H([X])= \vert X^H \vert $ for any $G$-set  $X$. It is clear moreover that $\phi_H$ is a  ring homomorphism, and  Burnside's theorem ( [4] Chap. XII Theorem I) is equivalent to the ghost map
$\Phi = \prod_{H \in [C(G)]} \phi^G_H : B(G) \ra \prod_{H \in [C(G)]} \ZZ$
being injective.
The cokernel of the ghost map is finite, and has been explicitly described by Dress [5]. In particular,  the ghost map $\QQ \Phi:  \QQ B(G) \ra \prod_{H \in [C(G)]} \QQ$ is an algebra isomorphism, where $\QQ B(G)= \QQ \otimes_{\ZZ} B(G)$. This shows that $\QQ B(G)$ is a split semisimple commutative $\QQ$-algebra.  Explicit formulas for its primitive idempotents have been given by Gluck [7] and independently by Yoshida [14].
Andreas Dress proved in [5] that  if $p$ is  $0$ or a prime and $I_{H,p}= \{ X \in B(G)~\vert ~ \phi_H( X)  \in p\ZZ\}$, then any prime ideal in $B(G)$ is of the form $I_{H,p}$ for some $H,p$. Moreover, a finite group is solvable if and only if the spectrum of its Burnside ring is connected (in the sense of Zariski's topology), i.e.  if and only if $0$ and $1$ are the only idempotents in $B(G)$.
If $X$ is a finite $G$-set, the $\QQ$-vector space $\QQ X$ with basis $X$ has a natural $\QQ G$-module structure, induced by the action of $G$ on $X$. The construction $X \mapsto \QQ X$ maps disjoint unions of $G$-sets to direct sums of $\QQ G$-modules, and so it induces a map $Ch: B(G) \ra R_\QQ(G)$. There are important connections between  the Burnside ring and the permutation representations.
This latter map, leads to an associated map $Spec:  Spec (R_\QQ(G)) \ra Spec B(G)$ which is always injective (See [6]).The slice Burnside ring $\Xi(G)$ introduced by Serge Bouc,  is built as the Grothendieck ring of the category of morphisms of finite $G$-sets, instead of the category of finite $G$-sets used to build the usual Burnside ring. It  shares  almost  all   properties of the Burnside ring. In particular, as already shown (see [3] for a more  complete description), the slice Burnside ring  is a commutative ring, which is free of finite rank as a $\ZZ$-module.
The investigation of the slices, that is the pairs of groups $(T,S)$ such that $S$ is a subgroup of $T$, is a central subject in the study of the slice Burnside ring. One reason for considering slices is that if $f: X \ra Y$ is a morphism of finite $G$-sets  then in $\Xi(G)$

$$[ \xymatrix{X \ar[r]^{f} &  Y} ]= \sum_{x \in [G\backslash X]} [ \xymatrix{G/G_x \ar[r]^{p} &  G/G_{f(x)}} ],$$

where square brackets to denote here the image of the isomorphism class of $f$ in $\Xi(G)$ and $G_x$ denotes the stabilizer of $x$.
Thus, the group $\Xi(G)$ is generated by the elements $[ \xymatrix{G/S_0 \ar[r]^{p} &  G/S_1} ]$ where $(S_1,S_0) $ runs through a set $[\Pi(G)]$ of representatives of conjugacy classes of slices of $G$.   One  can show that this generating set is actually a basis of the slice Burnside group.
There is an analogue of Burnside's theorem. After  tensoring with $\QQ$, the slice Burnside ring becomes a split semisimple $\QQ$-algebra, and an explicit formula for his primitive idempotents  can be stated. The prime spectrum of this ring has been described, and Dress's characterization of solvable groups in terms of the connectedness of the spectrum of the Burnside ring can be generalized as well.In this paper, we introduce a ring $B_n(G)$, for $n \in \NN$ such that $B_0(G)=B(G)$ and $B_1(G)=\Xi(G)$. We extend to $B_n(G)$ most of the properties recalled above in the case $n=0$ or $n=1$.Recall that any poset $\Pi$ can be treated as a category in which the objects are the elements of $\Pi$ and in which there is exactly one morphism $x \ra y$ if $x \le y$ and there are no other morphisms. The nerve of $\Pi$ is then the same as the ordered  simplicial complex associated ( that is the vertices of $\Pi_n$  are the objects of $\Pi$ and the   $n$-simplices are the chains of objects of $\Pi$ of length $n$, with face maps given by $d_i(S_0, \dots, S_n)= (S_0, \dots, \hat S_i , \dots, S_n)$, where as usual, the term $ \hat {} $ denotes a term that is being omitted and the degeneracy maps given by   $s_j(S_0, \dots, S_n)= (S_0, \dots, S_j, S_j , \dots, S_n)$ ). In particular, if we consider the collection of all subgroups of $G$ ordered by inclusion, we get the nerve category  $\Pi_\bullet(G)$ where elements in $\Pi_n(G)$, whose are just chains

$$\bar \cS:S_0 \subseteq S_{1} \subseteq  \dots \subseteq  S_n$$

will be called  $n$-slice, this is the simplicial complex considered in [11] for $p$-groups. Our  ring $B_n(G)$ has basis the set of conjugacy classes of $n$-slices, with multiplication given by

$$(S_0, \dots, S_n)\cdot (T_0, \dots, T_n)=  \sum\limits_{\substack{g \in [S_0\backslash G/ T_0]}}  (S_0 \cap {}^{g}T_0, \dots,   S_n \cap {}^{g}T_n ).$$

This paper is organized as follows:

In Sect. 2, we examine the Grothendieck group of the nerve of the skeleton   of the category  of finite $G$-sets (we abuse for using  the term nerve of  ${}_GSet$). It turns out that the obtained group $B_n(G)$, called the $n$-simplicial group,  is very similar to the classical Burnside group. It is worthwhile to discover wether these well-known properties of the Burnside ring  characterize $B_n(G)$, since then will known regard this ring as a "geometric realization" of some simplicial $G$-set.
Hence in this part we  deepen the links between the classical Burnside rings and the slice Burnside rings. In Sect. 3, we establish that the $n$-simplicial  Burnside ring embeds in a product of copies of the integers, via a ghost map, and this map has a finite cokernel. It turns out that $B_n(G)$ is a commutative  semisimple  algebra afer tensoring with $\QQ$, isomorphic to a direct sum indexed by  $[\Pi_n(G)]$ of copies of $\QQ$. In sect. 4. we give an explicit formula for the  primitive idempotents of $\QQ B_n(G)$. Sect.5 is devoted to the study of the prime spectrum of $B_n(G)$ by extending  Dress's characterization of solvable groups for $B(G)$. The last section examines the Green biset functor structure of $B_n$.

§ SIMPLICIAL BURNSIDE GROUP

For $G$  a finite group, let $\cC_\bullet^G$ be the nerve  of the category ${}_GSet$ of finite $G$-sets (see [9] for more details, [8] P.177), that is,  the simplicial set whose $n$-simplices are diagrams

$$\cC_n^G= \{\cX_n^f:= \xymatrix{X_0 \ar[r]^{f_1} &  X_{1}\ar[r]^-{f_{2}} & \dots X_{n-1} \ar[r]^-{f_{n}} &  X_{n}} \}$$

where the $X_i$ are $G-sets$ and the $f_i$ are morphisms of $G$-sets, and the degeneracy $s_i$ and face $d_i$ [Normally, we might be careful to label the face maps from $\cC_n^G$ to $\cC_{n-1}^G$ as $d^n_0, \dots, d^n_n$, similarly for the degeneracy maps, but this is rarely done in practice.] maps are defined by including an identity $\xymatrix{X_i \ar[r]^{id} & X_i}$, and leaving out $X_0$ if $i=0$, contracting
$\xymatrix{X_{i-1} \ar[r]^{f_i} & X_{i}\ar[r]^{f_{i+1}} & X_{i+1}}$ to  $\xymatrix{X_{i-1} \ar[r]^{f_{i+1} \circ f_i} & X_{i+1}}$ if $0<i<n$, leaving $X_n$ if $i=n$, respectively.
Then the following identities may be verified directly

\begin{equation}\label{1}
d_j \circ d_i  = d_{i}  \circ d_{j+1} ~~~ \mbox{for} ~~  i \le j \\
\end{equation}

\begin{equation}\label{2}
s_i \circ s_j  = s_{j+1} \circ  s_{i}  ~~~ \mbox{for} ~~  i \le j\\
\end{equation}

\begin{equation}\label{3}
d_i \circ  s_{j} = \left\{ \begin{array}{rcl}
s_{j-1}  \circ  d_{i}& \mbox{if}&  i < j,\\
id_{[n]} & \mbox{if}& i= j, j+1,\\
s_{j} \circ  d_{i-1}  & \mbox{otherwise }&.
\end{array}\right.
\end{equation}

Let $\cC_n^G$ be the category defined as follows:

* The objects of $\cC_n^G$ are  sequences $\cX^f_n: \xymatrix{X_0 \ar[r]^{f_1} &  X_{1}\ar[r]^-{f_{2}} & \dots X_{n-1} \ar[r]^-{f_{n}} &  X_{n}}$ of morphisms of $G$-sets called $(n,G)$-simplices.

* If $\cX^f_n$ and $\cY^g_n$ are objects of  $\cC_n^G$, a morphism from $\cX^f_n$ to $\cY^g_n$ is  a family $(\mu_i: X_i \ra Y_i)_{0 \le i \le n}$ of morphisms of $G$-sets such that $\mu_{i} \circ f_i = g_i \circ \mu_{i-1}$, for $i=1, \dots, n$.

* Morphisms of $(n,G)$-simplices   compose in the obvious way (coordinate-wise).

For a non-negative integer $n$, the category  $\cC_n^G$  has finite products $\times$ and  coproducts  $\sqcup$  induced by those of the category of finite  $G$-sets, respectively. It has also an  initial object $\emptyset=  (\xymatrix{\emptyset \ar[r]^{} &  \emptyset \ar[r]^-{} & \dots \emptyset \ar[r]^-{} &  \emptyset })$.

This is straightforward.

Let $G$ be a finite group.
A $n$-slice  of $G$ is a $(n+1)$-tuple $(S_0, S_{1}, \dots, S_n)$ of subgroups of $G$, with $S_{i-1} \le S_i$,  $\forall i   \in \{1, \dots, n\}$. It is helpful to refer the tuple as $\bar \cS$, in which each composite $ \cS_i$ as representing the group $S_i$. The set of all $n$-slices of $G$ will be denoted by $\Pi_n(G)$.

For any nerve $\cC_\bullet$, define a pre-ordering $\preceq$ of $Ob(\cC_n)$ by $A \preceq B$ if $\Hom_{\cC_n} (A, B) \neq \emptyset$ and an equivalence $\cong$ on $Ob(\cC_n)$ by $A \cong B$ if and only if $A \preceq B$ and $B \preceq A$. So, on $\Pi_n(G)$, we have  the following relation

$$\bar \cT:= (\cT_0, \dots, \cT_n) \preceq \bar \cS:= (\cS_0, \dots, \cS_n) \Longleftrightarrow \cT_i \le \cS_i,~~~~~~ \forall i=0,\dots, n$$

Recall that in ${}_GSet$, $X$ is indecomposable if and only if $X$ is simple and any simple $G$-set is isomorphic to $G/H$ for some subgroup $H$ of $G$.

* Any  $n$-slice $\bar \cS$ of $G$ gives rise to an $(n,G)$-simplex

$$(G/\bar \cS)_n := (\xymatrix{G/S_0 \ar[r]^{p_1} &  G/S_1\ar[r]^{p_2} & \dots G/S_{n-1} \ar[r]^{p_{n}} &  G/S_n}),$$

where $p_i$ are the projection morphisms.

The $(n,G)$-simplices   $(G/\bar \cS)_n$ are indecomposable in the sense that  $(G/\bar \cS)_n=\cY_n^g \sqcup ~\cZ_n^h \Longrightarrow \cY_n^g=\emptyset$ or $\cZ_n^h=\emptyset$.

* For  a $(n,G)$-simplex $\cX^f_n$, we set

$$\phi_{ \bar \cS} \big(\cX^f_n \big):= \big\vert \Hom_{\cC^G_n} \big((G/ \bar \cS)_n,  \cX^f_n \big) \big\vert,$$
the number of elements in the set $\Hom_{\cC^G_n} \big((G/ \bar \cS)_n,  \cX^f_n \big)$.

Let  $\cX^f_n$, $\cY_n^g$ be $(n,G)$-simplices

$$\phi_{\bar \cS} (\cX^f_n)=\Bigg\vert f_1^{-1}\Bigg(  f_2^{-1}\Bigg( ... f_i^{-1}\bigg(...f_{n-1}^{-1}\big( f_{n}^{-1}(X_n^{S_n}) ^{S_{n-1}}\big)^{S_{n-2}} ...\bigg)^{S_i} ...  \Bigg)^{S_1} \Bigg)^{S_0}\Bigg\vert, $$

in particular, for any $n$-slice $\bar \cT$, one has

$$\phi_{\bar \cS}((G/ \bar \cT)_n)= \vert \{ g \in G/\cT_0 ~\vert~ {\bar \cS}^g \preceq \bar \cT\} \vert.$$

Observe that for any $i= 0, \dots, n$, there is a bijection between $\Hom(G/\cS_i, X_i)$ and the set $X_i^{\cS_i}:=\{x \in X_i ~\vert~ gx=x~~\text{for all} ~g \in \cS_i\}$. Indeed, each $f$  in $ \Hom(G/\cS_i, X_i)$ maps the element $\cS_i \in G/\cS_i$ onto an element $x_i \in X_i$ which is invariant  by $\cS_i$. Further, $f$ is completely  determined by $x_i$, since $f(g \cS_i)=gx_i$ for all $g \in G$. The correspondence $f\mapsto x_i$ gives the desired bijection.
Therefore, the set $\Hom_{} \big((G/ \bar \cS)_n,  \cX^f_n \big)$  is in bijection with de set
$$\{(x_0,x_1, \!\dots,\! x_n)\! \in X_0^{S_0}\! \times \!X_1^{S_1} \!\times\! \dots \!\times X_n^{S_n} \vert ~  f_1(x_0)\!=\!x_1;  \dots; f_{n}(x_{n-1})\!=\!x_n \},$$

where the last equalities follow from the commutativity of the diagram

\[
\xymatrix{
G/S_0 \ar[d]_{\mu_0} \ar[r]^{p_1}&G/S_{1} \ar[d]_{\mu_{1}} \ar[r]^{p_{2}}&G/S_{2}  \ar[d]_{\mu_{2}} \ar[r]& \dots \dots &G/S_{n-1}\ar[d]_{\mu_{n-1}} \ar[r]^{p_{n}}&G/S_n\ar[d]^{\mu_n}\\
X_0\ar[r]^{f_{1}}&X_{1}\ar[r]^{f_{2}}&X_{n-2} \ar[r]&  \dots \dots& X_{n-1}\ar[r]^{f_{n}}&X_n
\]

If we take $X_i=G/\cT_i$ for $i=0,\dots, n$, we note that the left coset $g\cT_i$ is $\cS_i$-invariant if and only if $\cS_i.g \cT_i=g\cT_i$, that is, $g^{-1}\cS_ig \le \cT_i$. This establishes that
$$(G/ \cT_i)^{\cS_i}=\{g \cT_i~\vert~ g\in G, g^{-1}S_ig \le \cT_i \}$$
and therefore  $(G/\cT_i)^{S_i}= \emptyset $ unless $\cS_i \le_G \cT_i$ (  by which we mean  that   $S_i$ is conjugated to a subgroup of $\cT_i$ ). In particular,

\begin{eqnarray*}
\phi_{\bar \cS}((G/ \bar \cT)_n) &=& \vert \{(g\cT_0, \dots, g\cT_n) ~\vert~ g \in G, S_0^g\le \cT_0, \dots, S_n^g\le \cT_n ~\text{and}~g\cT_{i-1}= g\cT_{i}  \} \vert\\
&=&\vert \{ g \in G/\cT_0 ~\vert~ {\bar \cS}^g \preceq \bar \cT\} \vert.
\end{eqnarray*}

${}$  Let  $\cX^f_n$, $\cY_n^g$ be $(n,G)$-simplices

* If $\cX^f_n$ and  $\cY_n^g$ are isomorphic then $\phi_{\bar \cS}(\cX_n^f ) =\phi_{\bar \cS}(\cY_n^g )$.

Let $p$ be a prime and let $\bar \cS$ be a $n$-slice of $G$. If $P$ is a $p$-subgroup of $N_G(\bar \cS)=\cap_{i=0}^n N_G(S_i)$ and $P\bar \cS$ denotes the $n$-slice $(PS_0, \dots, PS_n)$, then

$$\phi_{\bar \cS}(\cX^f_n) \equiv \phi_{P\bar \cS}(\cX^f_n)~ (\mod.~p),$$

for any $(n,G)$-simplex $\cX^f_n$.

* Two indecomposable $(n,G)$-simplices $ (G/ \bar \cS)_n$ and  $(G/ \bar \cT)_n$ are isomorphic if and only if the $n$-slices $ \bar \cS$ and $\bar \cT$ are conjugate (we set $\bar \cS=_G \bar \cT$).

* If $\bar \cS$ and $\bar \cT$ are two $n$-slices, then $\phi_{\bar \cS}(\cX_n^f ) \le \phi_{\bar \cT}(\cX_n^f )$ for any $(n,G)$-simplex $\cX_n^f$ if and only if $\bar \cT \preceq_G \bar \cS$.
In particular, $\phi_{\bar \cS}(\cX_n^f ) = \phi_{\bar \cT}(\cX_n^f )$ for all $(n,G)$-simplices $\cX_n^f$ if and only if $\bar \cT=_G~ \bar \cS$.

* For any two indecomposable $(n,G)$-simplices $(G/\bar \cS)_n$ and  $(G/\bar \cT)_n$, one has    $\phi_{\bar \cS}((G/ \bar \cS)_n)$ divides $\phi_{\bar \cT}((G/ \bar \cS)_n)$.

* It is clear, since any  isomorphism  of $G$-sets $\mu_i: X_i \ra Y_i$ induces a bijection $X_i^S\ra Y_i^S$ on the sets of fixed points by any subgroup $S$ of $G$.

* For any $(n,G)$-simplex $\cX_n^f$, the set
$$ \Inv_{\bar \cS} (\cX_n^f):=  f_1^{-1}\Bigg(  f_2^{-1}\Bigg( ... f_i^{-1}\bigg(...f_{n-1}^{-1}\big( f_{n}^{-1}(X_n^{S_n}) ^{S_{n-1}}\big)^{S_{n-2}} ...\bigg)^{S_i} ...  \Bigg)^{S_1} \Bigg)^{S_0}$$

is invariant by $N_G(\bar \cS )$, and so

$$\vert  \Inv_{\bar \cS} (\cX_n^f) \vert  \equiv \Bigg\vert f_1^{-1}\Bigg(   ... f_i^{-1}\bigg(... f_{n}^{-1}(X_n^{S_n}) ^{S_{n-1}} ...\bigg)^{S_i} ...  \Bigg)^{PS_0}  \Bigg\vert~(\mod~p),$$

and moreover

$$f_1^{-1}\Bigg(   ... f_i^{-1}\bigg(... f_{n}^{-1}(X_n^{S_n}) ^{S_{n-1}} ...\bigg)^{S_i} ...  \Bigg)^{PS_0} \equiv f_1^{-1}\Bigg(   ... f_i^{-1}\bigg(... f_{n}^{-1}(X_n^{PS_n}) ^{PS_{n-1}} ...\bigg)^{PS_i} ...  \Bigg)^{PS_0}.$$

* If $(G/\bar \cS)_n \cong (G/\bar \cT)_n$ then $\Hom(G/\bar \cS, G/\bar \cT) \neq \emptyset $ and so $\bar \cS \le_G  \bar \cT$. Therefore $ \bar \cS =_G \bar \cT$ by symmetry. Conversely  if $\bar \cS =_G \bar \cT$, for example $\bar \cS^{g} = \bar \cT$ with $g \in G$, then there exists an isomorphism of $(n,G)$-simplices $(\mu_i: G/\cT_i \ra G/\cS_i)_{0 \le i \le n}$, given by

$$\mu_i (g'T_i) =\mu_i (g'g^{-1}S_ig)=g'g^{-1}S_i, ~~\text{for}~ g' \in G.$$

So $(G/\bar \cS)_n \cong (G/\bar \cT)_n$ as $(n,G)$-simplices if and only $\bar \cS=_G\bar \cT$.

* If $\phi_{\bar \cS}(\cX_n^f ) \le \phi_{\bar \cT}(\cX_n^f )$ for any $(n,G)$-simplex, then in particular $ \phi_{\bar \cT}((G/ \bar \cS)_n ) \neq 0$ since $ \phi_{\bar \cS}(G/ \bar \cS ) \neq 0$, and so $\bar \cT \preceq_G \bar \cS$. On the other hand, if $ \bar \cT \preceq {}^g  \bar \cS= \bar \cK $, then $\phi_{\bar \cS}(\cX_n^f )= \vert  \Inv_{\bar \cS} (\cX_n^f) \vert =     \vert  \Inv_{\bar \cK} (\cX_n^f)  \vert \le \vert  \Inv_{\bar \cT} (\cX_n^f)   \vert = \phi_{\bar \cT}(\cX_n^f )$.

* Consider the action of $N_G(\bar \cS)$ on $G/S_0$ defined by
$$x.gS_0= gx^{-1}S_0$$

for $x\in N_G(\bar \cS)$ and $gS_0 \in G/S_0$. Then $S_0$ acts trivially on $G/S_0$ and so it becomes a left $N_G(\bar \cS)/S_0$. Moreover, $N_G(\bar \cS)/S_0$ acts freely on $G/S_0$. Note that for any $(n,G)$-slice  $\bar \cT$,  the set

$$\Inv_{\bar \cT} ((G/\bar \cS)_n):=\{gS_0 ~\vert ~ \bar \cT\cdot g \bar \cS= g\bar \cS \} $$
is an $N_G(\bar \cS)/S_0$-subset of $G/S_0$. So $N_G(\bar \cS)/S_0$ acts freely on  $\Inv_{\bar \cT} ((G/\bar \cS)_n)$, and so $\vert N_G(\bar \cS)/S_0 \vert $ divides $\vert  \Inv_{\bar \cT} ((G/\bar \cS)_n) \vert$.

A $(n,G)$-simplex  $\cX^f_n$ is called $i$-sliceable  with $i$ in $\{0,\dots, n\}$ if $X_i= A_i \sqcup B_i$ as disjoint union of two non-empty $G$-sets.

Note that for any i-sliceable $(n,G)$-simplex  $\cX^f_n$,  one has $(n,G)$-simplices

* $\cA_n^f:\xymatrix{A_0 \ar[r]^{f_1} &  A_{1}\ar[r]^{f_{2}} &  \ar[r] \dots A_{i} \ar[r]^{f_{i+1}} & X_{i+1} \dots \ar[r]^{f_{n-1}}   &X_{n-1} \ar[r]^{f_{n}} &  X_{n}} $,

* $\cB_n^f: \xymatrix{B_0 \ar[r]^{f_1} &  B_{1}\ar[r]^{f_{1}} &  \ar[r] \dots B_i \ar[r]^{f_{i+1}} & X_{i+1} \dots \ar[r]^{f_{n-1}}  &X_{n-1} \ar[r]^{f_{n}} &  X_{n}}$,

defined inductively by

$$A_{j-1}:= f_{j }^{-1}(A_j) ~~\text{and}~~ B_{j-1}:= f_{j }^{-1}(B_j),$$
for any $1 \le j\le i$.

We set $[\cX^f_n, \cA^f_n, \cB^f_n]_i$ to denote the corresponding triple. If $\Gamma$ denotes the class of such triple $(\cX^f_n, \cA^f_n, \cB^f_n)$, then $\Gamma$ is closed by isomorphism.

For any $(\cX_n^f, \cA_n^f, \cB_n^f) \in \Gamma$ and for a $n$-slice $\bar \cS$ fixed, we  have

\begin{equation}
\phi_{\bar \cS} (\cX^f_n)= \phi_{\bar \cS} (\cA^f_n) + \phi_{\bar \cS} (\cB^f_n).
\end{equation}

Let $x_n$ be an element of $X_{n}^{S_{n}}$ and $i$ be an integer such that $X_i= A_i \sqcup B_i$.Then, either $f_{i+1}^{-1} \circ \dots \circ f_{n}^{-1}(x_n) \in A_i$ and $f_{j+1}^{-1} \circ \dots \circ f_{n}^{-1}(x_n) \in A_j$ for any $j<i$,

or $f_{i+1}^{-1} \circ \dots \circ f_{n}^{-1}(x_n) \in B_i$ and $f_{j+1}^{-1} \circ \dots \circ f_{n}^{-1}(x_n) \in B_j$ for any $j<i$.
Hence $\phi_{\bar \cS} (\cX^f_n)= \vert \big(A_0\cap f^{-1}(\cX_n^f)^{\bar \cS} \big)\sqcup \big( B_0\cap f^{-1}(\cX_n^f)^{\bar \cS}\big) \vert = \phi_{\bar \cS} (\cA^f_n) + \phi_{\bar \cS} (\cB^f_n)$.

Let $G$ be a finite group.

We denote $B_n(G)$ the Grothendieck group  of $\cC^G_n$ with respect to the relations $\Gamma$ that is, the quotient
$$B_n(G):= \Omega_n(G)/\tilde \Omega_n(G) $$
of the free abelian group $\Omega_n(G)$ on the set of isomorphism classes of $(n,G)$-simplices by the subgroup $\tilde \Omega_n(G)$ generated by the formal differences $[\cX^f_n]- [\cA^f_n]-[\cB^f_n]$.

* The construction satisfies the following property: if $\phi: \cC^G_n \ra A $ is a map from  $\cC^G_n$  to an abelian group $A$ given that $\phi (\cX^f_n )$ depends only on the isomorphism  class of $\cC^G_n$ and $\phi (\cX^f_n )= \phi (\cA^f_n )+\phi (\cB^f_n )$ for any element $(\cX^f_n , \cA^f_n ,\cB^f_n )$ in $\Gamma$, then there exists a unique $\bar \phi: B_n(G) \ra A$ such that $\phi (\cX^f_n)= \bar \phi ([\cX^f_n])$ for any $(n,G)$-simplex $\cX^f_n$.
Let now  $(G/ \bar \cS)_n$ be a fixed $(n,G)$-simplex.
The function $\cX^f_n \mapsto   \big\vert  \Hom_{\cC_n^G} \big((G/ \bar \cS)_n,  \cX^f_n \big) \big\vert$ defined on the class of $(n,G)$-simplices (up to isomorphism), with values in $\ZZ$ extends to a group homomorphism $B_n(G) \ra \ZZ$.
So, to see that $B_n(G)$ is non-trivial, it suffices to find $\cX_n^f$ with $\big \vert  \Hom_{} \big((G/ \bar \cS)_n,  \cX^f_n \big) \big \vert \neq 0$. Since

$$\big \vert  \Hom_{} \big((G/ \bar \cS)_n,  (G/ \bar \cS)_n \big) \big\vert \neq 0 ,$$

we have that $B_n (G) \neq 0$.

* In the special where $n=0$, one recovers the classical Burnside group $B(G)$ (see [1]), and for $n=1$, we have $B_1(G)= \Xi(G)$ the slice Burnside group introduced by Bouc in [3].

The functor $d_j $, $j=0,\dots, n$   (resp.  $s_i $, $i=0,\dots, n-1$)
induces a group  homomorphism
$$d_j : B_n(G) \ra B_{n-1}(G) ~~~~\big(\text{resp.}~   s_i: B_{n-1}(G) \ra B_{n}(G) \big) \footnote{Note that the maps $d_j $ and  $s_i $ should more appropriately be labeled, but we stick common practice and use $d_j $ and $s_i $ for face and degeneracy wherever we find them.}$$
such that the identities (<ref>), (<ref>), (<ref>) hold.

Indeed,  for any $(n,G)$-simplices $\cX^f_n$ with decomposition $[\cX^f_n, \cA^f_n, \cB^f_n]_k$,  i.e.
$\xymatrix{A_0 \sqcup B_0 \ar[r]^{f_1} &  A_{1} \sqcup B_1 \ar[r]^{f_{2}} &  \ar[r] \dots A_{k} \sqcup B_k\ar[r]^{f_{k+1}} & X_{k+1} \dots \ar[r]^{f_{n-1}}   &X_{n-1} \ar[r]^{f_{n}} &  X_{n}} $

then there is a decomposition of $d_j( \cX^f_n)$ as $[d_j( \cX^f_n),d_j(\cA^f_n), d_j( \cB^f_n )]_k$ if $k< j$ and
$[d_j( \cX^f_n), d_j(\cA^f_n), d_j( \cB^f_n )]_{k-1}$ if $k\ge j$. So the map which assigns an $(n,G)$-simplex $\cX^f_n$  to the image of  $d_j( \cX^f_n)$ in $B_{n-1}(G)$ induces, by the universal property, a well-defined map from $B_n(G)$ to $B_{n-1}(G)$.

Similarly, there is a decomposition of $s_j( \cX^f_n)$ as $[s_j( \cX^f_n), s_j(\cA^f_n) , s_j( \cB^f_n )]_k$ if $k< j$ and  $[s_j( \cX^f_n), s_j(\cA^f_n), s_j( \cB^f_n )]_{k+1}$ if $k\ge j$.
So  $s_j$  maps  the defining relations   $B_{n-1}(G)$ to those of  $B_n(G)$) and therefore gives a well-defined map (which is a group homomorphism) from $B_{n-1}(G)$ to $B_{n}(G)$. The equalities (<ref>), (<ref>), (<ref>) may be verified easily.

* Let $\pi(\cX^f_n)$ denote the image in $B_n(G)$ of the isomorphism class of $\cX^f_n$.

* We set $\langle \bar \cG \rangle_G:= \pi((G/\bar \cG)_n)$, for any $n$-slice $\bar \cG=(G_0, G_{1}, \dots, G_n)$.

* For any $(n,G)$-simplex $\cX_n^f$, we denote by $f(\cX^f_n)$ the $(n,G)$-simplex

$\xymatrix{X_0 \ar[r]^{f_1} & f_1( X_{0})\ar[r]^-{f_{2}}&\dots \ar[r]^-{f_{n}} &   f_{n}(f_{n-1}(f_{n-2}(...f_1( X_{0})...)) }.$

Let $\cX_n^f$ be a $(n,G)$-simplex. Then in the group $B_n(G)$, we have

$$\pi(\cX^f_n)= \sum\limits_{\substack{x \in [G\backslash X_0]}}  \langle \bar \cG_f^x \rangle_G,$$

where $\bar \cG_f^x$ denotes the $n$-slice $(G_x, G_{f_1(x)},\dots ,G_{f_{n}...f_{2}f_1(x)}  )$ and $G_{\bullet}$ denotes the stabilizer of the element $\bullet$.

Note first  that in the group $B_n(G)$ we have that $\pi (\cX^f_n)= \pi(f(\cX^f_n))$. Indeed
writing $X_n= X_n \sqcup \emptyset$, we get  by the defining relations of $B_n(G)$ that

\begin{eqnarray*}
\pi (\cX^f_n)&=& \pi(\emptyset) +  \pi(\xymatrix{f_{1}^{-1}(X_1) \ar[r]^{f_1} & f_{2}^{-1}( X_{2})\ar[r]^-{f_{2}}&\dots  f_{n}^{-1}( X_{n}) \ar[r]^-{f_{n}} &   X_{n}})\\
&=& \pi(\xymatrix{f_{1}^{-1}(X_1) \ar[r]^{f_1} & f_{2}^{-1}( X_{2})\ar[r]^-{f_{2}}&\dots  f_{n}^{-1}( X_{n}) \ar[r]^-{f_{n}} &   X_{n}})\\
&=& \pi(f(\cX^f_n)).
\end{eqnarray*}

\begin{eqnarray*}
\pi(f(\cX^f_n)) &=&  \pi(\xymatrix{\coprod\limits_{\substack{x \in [G\backslash X_0]}} \cO_x \ar[r]^{f_1} & \coprod\limits_{\substack{x \in [G\backslash X_0]}} \cO_{f_1(x)}\ar[r]^-{f_{2}}&\dots \coprod\limits_{\substack{x \in [G\backslash X_0]}} \cO_{f_{n}(f_{n-1}(...f_1( x)...)) }})\\
&=& \sum\limits_{\substack{x \in [G\backslash X_0]}}  \pi( \emph{\textunderbrace{ \xymatrix{ \cO_x \ar[r]^{f_1} &  \cO_{f_1(x)}\ar[r]^-{f_{2}}&\dots \cO_{f_{n}(f_{n-1}(...f_1( x)...)) }} }{$\cX^{f\vert \cO}_n $}} )\\
\end{eqnarray*}

where the last equality follows from the defining relations of $B_n(G)$. Now, the $(n,G)$-simplex $\cX^{f\vert \cO}_n$ is obviously isomorphic to the indecomposable $(n,G)$-simplex
$$\!\xymatrix{G/G_x\ar[r]^-{p_1} &  G/G_{f_1(x)}\ar[r]^-{p_{2}} &\dots   G/G_{f_{n}(f_{n-1}(...f_1( x)...)) }\!}.$$

Since two indecomposable $(n,G)$-simplices $ (G/ \bar \cS)_n$ and  $(G/ \bar \cT)_n$ are isomorphic if and only if the $n$-slices $ \bar \cS$ and $\bar \cT$ are conjugate we have the following  corollary:

The group $B_n(G)$ is generated by the elements $\langle \bar \cS \rangle_G$ where $\bar \cS $ runs through a set $[\Pi_n(G)]$ of representatives of conjugacy classes of $n$-slices of $G$.

The product of $(n,G)$-simplices induces a commutative  ring structure on $B_n(G)$ with identity  $\textbf e_n:= [\!\xymatrix{ \bullet \ar[r]^-{} &\dots \bullet \ar[r]^-{} &  \bullet \!}]$, where $\bullet$ is a $G$-set of cardinality $1$. This ring is called the $n$-simplicial ring of the finite group $G$.

Moreover,  the morphisms $d_j$ ($j=1, \dots, n$) and $s_i$ ($i=0, \dots, n-1$) in Proposition <ref> define morphisms of rings.

We have to show that the product of $(n,G)$-simplices induces a well-defined bilinear product $B_n(G) \times B_n(G) \ra B_n(G)$.
Let $\cX_n^f$ be a  $(n,G)$-simplex such that there is a decomposition $[\cX^f_n, \cA^f_n, \cB^f_n]_i$, and let $\cY^g_n$ be any $(n,G)$-simplex. We set $\cZ_n^h:= \cX_n^f \times \cY^g_n$. Then  the $(n,G)$-simplex $\cZ_n^h$ can be visualized  as follows

$$\!\xymatrix{(\!A_0 \times Y_0) \sqcup (B_0 \times Y_0) \ar[r]^-{f_0 \times g_0} & \dots (A_i \times Y_i) \!\sqcup \! (B_i \! \times \!Y_i) \!\ar[r]^-{f_{i} \! \times \! g_i} & X_{i+1}  \!  \times  \! Y_{i+1}\dots  \ar[r]^-{h_{n-1}} & \!   X_n \!  \times  \! Y_n\!},$$

since $(A_i \sqcup B_i) \times Y_i = \underbrace{(A_i \times Y_i)}_{C_i} \sqcup \underbrace{ (B_i \! \times \!Y_i)}_{D_i}  $.

\begin{eqnarray*}
\pi(\cZ_n^h)&=& \pi(\!\xymatrix{C_0 \sqcup D_0\ar[r]^-{h_0} & \dots C_i \sqcup D_i \!\ar[r]^-{h_{i}} & X_{i+1}  \!  \times  \! Y_{i+1}\dots  \ar[r]^-{h_{n-1}} & \!   X_n \!  \times  \! Y_n\!}))\\
&=&  \pi(\!\xymatrix{C_0\ar[r]^-{h_0} & \dots C_i \!\ar[r]^-{h_{i}} & X_{i+1}  \!  \times  \! Y_{i+1}\dots  \ar[r]^-{h_{n-1}} & \!   X_n \!  \times  \! Y_n\!}))\\
&+&  \pi(\!\xymatrix{D_0\ar[r]^-{h_0} & \dots D_i \!\ar[r]^-{h_{i}} & X_{i+1}  \!  \times  \! Y_{i+1}\dots  \ar[r]^-{h_{n-1}} & \!   X_n \!  \times  \! Y_n\!}))\\
&=&  \pi(\!\xymatrix{ f_1^{-1}(A_1) \times g_1^{-1}(Y_1) \ar[r]^-{h_0}&f_2^{-1}(A_2) \times g_2^{-1}(Y_2) \dots \!\ar[r]^-{h_{i}} & A_{i}  \!  \times  \! Y_{i}\dots  \ar[r]^-{h_{n-1}} & \!   X_n \!  \times  \! Y_n\!}))\\
&+&  \pi(\!\xymatrix{ f_1^{-1}(B_1) \times g_1^{-1}(Y_1) \ar[r]^-{h_0}&f_2^{-1}(B_2) \times g_2^{-1}(Y_2) \dots \!\ar[r]^-{h_{i}} & A_{i}  \!  \times  \! Y_{i}\dots  \ar[r]^-{h_{n-1}} & \!   X_n \!  \times  \! Y_n\!}))\\
%&=&\pi(\cA_n^f \times \cY^g_n ) + \pi(\cB_n^f \times \cY^g_n )
\end{eqnarray*}

Clearly, we have (it suffices to set $Y_i=Y_i \sqcup \emptyset$)

$$  \pi(\!\xymatrix{ g_1^{-1}(Y_1) \ar[r]^-{g_0}& g_2^{-1}(Y_2) \dots \!\ar[r]^-{g_{i}} &   \! Y_{i}\dots  \ar[r]^-{g_{n-1}} & \!   \! Y_n\!})= \pi(\cY^g_n).$$

So $\pi(\cZ_n^h)= \pi(\cA_n^f \times \cY^g_n ) + \pi(\cB_n^f \times \cY^g_n )$,
this shows that the product preserves the defining relations of $B_n(G)$. Now it is obvious to see that the product is associative, commutative and admits $\textbf e_n$ as an identity element.

Let $\cY^g_n$ be $(n,G)$-simplex. Since $(f_{j+1}\circ f_j) \times (g_{j+1}\circ g_j)=(f_{j+1} \times g_{j+1})\circ (f_{j} \times g_{j})$, we have that

$$d_j\big( \cX^f_n  \times \cY^g_n ) = d_j(\cX^f_n  ) \times d_j( \cY^g_n ) , ~~ s_j\big( \cX^f_n  \times \cY^g_n ) = s_j(\cX^f_n  ) \times s_j( \cY^g_n ) $$

and so  the induced maps $d_j$ and $s_j$ are ring homomorphisms.
Furthermore $d_j (\textbf e_n)= \textbf e_{n-1}$ and $s_j (\textbf e_n)= \textbf e_{n+1}$.

Letting $\Delta$  the simplicial category that is $\Delta$ has as objects all finite ordinal numbers $[n]=\{0,1, \dots, n\}$
and as morphisms $f: [n] \ra [m]$ all monotone maps; that is, the maps $f$ such that  $f(i)\le f(j)$ if $i<j$.

Then an induced functor $\cC_n^G \ra \cC_m^G$  for any finite $G$ and an induced $G$-equivariant map $\Pi_n(G) \ra \Pi_{m}(G) $  and a group homomorphism $B_n(G) \ra B_{m}(G)$ given by,
$$\cX^f_n  \mapsto  s^{n-1}_{j_k} \dots s^{n-k}_{j_1}d^{n-k}_{i_1}\dots d^m_{i_h} \cX^f_n$$
where $k$ and $h$ satisfy the following conditions:

* $n+k-h=m$,

* $0 \le i_1 < \dots <i_k \le m$,

* $0 \le j_1 < \dots j_h \le n$,

* $i_1, \dots,i_k $ are the elements of $[m]$ not in the image of $f$,

* $j_1, \dots ,i_h $ are the elements of $[n]$ at which $f$ does not increase.

Using the generators of $B_n(G)$, the multiplication is given by

$$\langle \bar \cS \rangle_G . \langle \bar \cT \rangle_G= \sum\limits_{\substack{g \in [S_0\backslash G/ T_0]}}  \langle \bar \cS \cap {}^{g}\bar \cT   \rangle_G$$

Note that each $G$-orbit of the $G$-set $(G/\cS_0) \times (G/ \cT_0)$ determines a double coset $\cS_0g\cT_0$, in the following way:

$$\cO_{(xS_0, yT_0)} \ra S_0x^{-1}yT_0.$$

the $G$-orbit of $(G/\cS_0) \times (G/\cT_0)$ corresponding to $\cS_0a\cT_0$ consists of all distinct pairs in the collection $\{(x \cS_0, y\cT_0)~\vert ~x^{-1}y \in \cS_0a \cT_0\}$. The stabilizer of the pair  $(\cS_0, g \cT_0)$ is precisely $\cS_0\cap {}^g \cT_0$. Therefore, the orbit of $(G/\cS_0) \times (G/\cT_0)$ corresponding to $\cS_0g \cT_0$ is isomorphic (as $G$-set) to $G/(\cS_0\cap {}^g \cT_0)$, and therefore

$$(G/\cS_0) \times (G/\cT_0) \cong \coprod\limits_{\substack{g \in  [\cS_0\backslash G/ \cT_0]}} G/(\cS_0 \cap {}^g\cT_0)$$

On the other hand, the image of $(\cS_i, g\cT_i)$ by the map
$$(G/\cS_i) \! \times \! (G/\cT_i)  \ra  (G/\cS_{i+1})\! \times \! \!(G/\cT_{i+1})$$
is the pair $(\cS_{i+1}, g\cT_{i+1})$, whose stabilizer is $\cS_{i+1} \cap {}^g \cT_{i+1}$. Hence inductively the result follows from Lemma <ref>.

§ GHOST MAPS

In this section, we examine a map $\Phi^G_n$ that this intrinsically related with the $n$-simplicial ring  $B_n(G$ in the sense that $\Phi^G_n$ may be discovered from the ring structure of $B_n(G)$.

* For a $n$-slice $\bar \cS$ fixed,
the correspondence $\cX_n^f \mapsto \phi_{\bar \cS} \big(\cX^f_n\big)$  extends to a ring homomorphism still denoted by

$$  \phi_{\bar \cS}: B_n(G) \longrightarrow  \ZZ.$$

* (An analogue of Burnside's Theorem) We let

$$\Phi^G_n= ( \phi_{\bar \cS}): B_n(G) \longrightarrow  \prod_{\bar \cS \in [\Pi_n(G)]} \ZZ=C_n(G)$$

be the product of the $\phi_{\bar \cS}$.
Then $\Phi^G_n$ is an injective ring homomorphism, with finite cokernel as morphism of abelian groups.  The set
$$\{\langle \bar \cS \rangle_G  ~\vert ~  \bar \cS  \in [\Pi_n(G)]\}$$
form a basis of $B_n(G)$.

* Let $R$ be an integral domain, $\phi: B_n(G) \ra R$ any ring homomorphism, and
$$T(\phi) := \{ \bar \cS \in \Pi_n(G)~\vert ~ \phi(\langle \bar \cS \rangle_G) \neq 0  \}.$$
Then, there exists exactly one element $\bar \cK \in [\Pi_n(G)]$ that is minimal with respect to $\preceq$ in $T(\phi)$. Moreover, one has $\phi(x)= \phi_{\bar \cK}(x)\cdot 1_R$ for all $x \in B_n(G)$ and this minimal $\bar \cK$ in  $T(\phi)$.

* Corollary <ref> shows that $\phi_{\bar \cS}$ the defining relations of $B_n(G)$ are mapped to $0$ by $\phi_{\bar \cS}$. So it induces a well defined map from $B_n(G)$ to $\ZZ$. Now, since the product of $(n,G)$-simplices is the product of the category of $(n,G)$-simplices, it follows that

$$\phi_{\bar \cS}\big( \cX_n^f . \cY_n^g\big) = \phi_{\bar \cS}(\cX_n^f ) \phi_{\bar \cS}(\cY_n^g).$$
The image of the identity $\textbf e_n$ by $\phi_{\bar \cS}$ is obviously $1$.

* By definition $\Phi^G_n$ is a ring homomorphism. Suppose that $u \neq 0$ is in the kernel of $\Phi^G_n$. We write $u$ in terms of the generators
$$u=  \sum\limits_{\substack{ \bar \cS \in [\Pi_n(G)]}} a_{\bar \cS} \langle \bar \cS \rangle_G.$$

We have a partial ordering on the $ \langle S_0 \dots S_n \rangle_G$ induced by $\preceq$.
Let $ \langle \bar \cS \rangle_G$ be maximal among the generators with $a_{\bar \cS}  \neq 0$. Then $\phi_{\bar \cS}( \langle \bar \cT \rangle_G ) \neq 0$  implies that $ \langle \bar \cS \rangle_G \preceq  \langle \bar \cT \rangle_G$. Hence
$$0= \phi_{\bar \cS} (u)=a_{\bar \cS} \phi_{\bar \cS}( \langle \bar \cS \rangle_G )\neq 0,$$

a contradiction.

Now by the first part of the proof, $B_n(G)$ has $\ZZ$-rank $\vert [\Pi_n(G)] \vert =rank_{\ZZ} C_n(G)$, and since $\Phi^G_n$ is injective, then $\Im(\Phi^G_n)$ and $C_n(G)$ have the same rank, and so,

the cokernel of $\Phi^G_n$ is finite.

* The set $T(\phi)$ is not empty because $\textbf e_n \notin \Ker \phi$.
Let $\bar \cS$ be minimal in $T(\phi)$ with respect to the relation $\preceq$. Then by Proposition <ref>,  for any $n$-slice $\bar \cT$

\begin{eqnarray*}
\langle \bar \cS \rangle_G.\langle \bar \cT \rangle_G&=&\sum\limits_{\substack{g \in [S_0\backslash G/ T_0]}}  \langle S_0 \cap {}^g T_0,\dots ,  S_n \cap {}^g T_n \rangle_G\\
%&\equiv&\sum\limits_{\substack{g \in  G/ T_0 \\ S_n \le {}^g T_n \\ \dots \dots \\ S_0 \le {}^g T_0}}   \langle \bar \cS \rangle_G~~~(\mod ~Ker \phi)\\
%&=& \phi_{\bar \cS}^G \big(\langle \bar \cT \rangle_G \big)\langle \bar \cS \rangle_G
\end{eqnarray*}

Since $\phi(\langle \bar \cS \rangle_G.\langle \bar \cT \rangle_G)= \phi(\langle \bar \cS \rangle_G).\phi(\langle \bar \cT \rangle_G) \neq 0$ in $R$, there exists $ g \in S_0\backslash G/ T_0$ such that $\phi(\bar \cS \cap {}^g \bar \cT) \neq 0$. Hence, by minimality of $\bar \cS$, we have that those $g$ ranges over the set $\{g \in  G/ T_0 ~\vert ~ S_n \le {}^g T_n  \dots \dots S_0 \le {}^g T_0 \}$. So $\phi(\langle \bar \cS \rangle_G).\phi(\langle \bar \cT \rangle_G )= \phi_{\bar \cS}\big(\langle \bar \cT \rangle_G \big)\phi(\langle \bar \cS \rangle_G$. Since $R$ is an integral domain, we can divide both sides by $\phi(\langle \bar \cS \rangle_G )$ and we then have $\phi(\langle \bar \cT \rangle_G)= \phi_{\bar \cS}\big(\langle \bar \cT \rangle_G \big)1_R$.
So by linearity, $\phi(x)= \phi_{\bar \cS}(x)\cdot 1_R$ for all $x \in B_n(G)$. Now, if $\bar \cK$ is another minimal element, then $\phi(\langle \bar \cK \rangle_G)= \phi_{\bar \cS}(\langle \bar \cK \rangle_G)$ and $\phi( \langle \bar \cK \rangle_G)= \phi_{\bar \cK}(\langle \bar \cK \rangle_G)$. So $\bar \cS \preceq_G \bar \cK$ and by symmetry $\bar \cS =_G \bar \cK$.

* If $R$ is an integral domain, then two homomorphisms $\phi, \phi': B_n(G) \ra R$ coincide if and only if they have the same kernel. Moreover, any homomorphism $\phi: B_n(G) \ra R$ factors through some $\phi_{\bar \cS}: B_n(G) \ra \ZZ$ and the unique homomorphism $\ZZ \ra R$ given by $n \mapsto n.1_R$.

* The ring $B_n(G)$ is finitely generated as  $\ZZ$-module, hence is a noetherian ring.
For each $n$-slice $\bar \cS$, the kernel of $\phi_{\bar \cS}$ is a prime ideal, since $\ZZ$ is an integral domain, and the intersection of all those kernels for $n$-slices $\bar \cS$ of $G$ is zero. In particular, the ring $B_n(G)$ is reduced (the intersection of all the prime ideals of $B_n(G)$  is zero).

* Since $C_n(G)$ is  an integral over $B_n(G)$, the Theorem of Cohen-Seidenberg implies that their Krull dimensions coincide, and every prime ideal of $B_n(G)$ comes from $C_n(G)$.

* The point 2) of the proposition shows in particular that the set

$$\{ \langle \bar \cS \rangle_G ~ \vert ~  \bar \cS \in [\Pi_n(G)] \}$$
form a basis of $B_n(G)$
and a  $(1,G)$-simplicial Burnside  can also be seen as the lattice Burnside ring of some lattice introduced by Oda, Takegahara and Yoshida in [10].

We have the following corollary,

The map
$$ \QQ \Phi^G_n: \QQ B_n(G) \ra \prod_{\bar \cS \in [\Pi_n(G)]} \QQ,$$

where $\QQ \Phi^G_n$ is the rational extension of $\Phi^G_n$, is an isomorphism of $ \QQ$-algebras.

Note that every $\QQ$-algebra homomorphism $\QQ B_n (G) \ra \QQ$ is of the form $\QQ \phi_{\bar \cS}$ for some $\bar \cS \in \Pi_n(G)$. For $\bar \cS , \bar \cT  \in \Pi_n(G)$, we have  $\QQ \phi_{\bar \cS}= \QQ \phi_{\bar \cT}$ if and only $\bar \cS=_G \bar \cT$.

More generally,
Put $W_G(\bar \cS)= N_G(\bar \cS)/S_0$.
Let $p$ be a prime, and $\infty$ be just a symbol. For each $\ZZ$-module $M$, we shall set $M_{(p)} = \ZZ_{(p)} \otimes_\ZZ M$, where $ \ZZ_{(p)}$ is the localisation of $\ZZ$ at $p$, and $M_{(\infty)}=M$. For a $n$-slice $\bar \cS$, we denote $W_G(\bar \cS)_p$ a Sylow $p$-subgroup of $W_G(\bar \cS)$, and set $W_G(\bar \cS)_{\infty} =W_G(\bar \cS)$. Let $ \big( \Phi_{n}^G \big)_p$ or simply $\Phi_{(p)}^G$ (if there is no risk of confusion)   denote the  homomorphism of $ \ZZ_{(p)}$-modules from ${B_n(G)}_{(p)}$  to ${C_n(G)}_{(p)}$ induced by $\Phi_n^G$.

* We consider     ${B_n(G)}_{(p)}$ and ${C_n(G)}_{(p)}$ as  subrings of   $\prod_{\bar \cS \in [\Pi_n(G)]} \QQ$.

The set $J' := \big\{ \frac{1}{\phi_{\bar \cS} (\bar \cS)}\bar \cS:= \big(\frac{\phi_{\bar \cT} (\bar \cS)}{\phi_{\bar \cS} (\bar \cS)} \big)_{\bar \cT \in [\Pi_n(G)]} ~\vert~  \bar \cS \in [\Pi_n(G)] \big\}$ is a basis of $C_n(G)$.

* If we define $Obs(G)$ as $ \underset{\bar \cS \in [\Pi_n(G)]} {\bigoplus} \ZZ \big/\vert W_G(\bar \cS)  \vert\ZZ$, then
$$C_n(G)_{(p)}/ \Im \Phi^G_{(p)} \cong Obs(G)_{(p)}$$

* Define  a homomorphism of $ \ZZ_{(p)}$-modules           $\Psi_{(p)}^G: C_n(G)_{(p)} \ra   Obs(G)_{(p)}$ by

$$ (x_{\bar \cS})_{\bar \cS \in [\Pi_n(G)]} \mapsto \big( \sum_{gT_0 \in W_G(\bar \cT)_{(p)}}  x_{<g>\bar \cT} ~~\mod ~ \vert W_G(\bar \cT)_{(p)} \vert   \big)_{\bar \cT \in [\Pi_n(G)]}$$
Then $\Psi_{(p)}^G$ is surjective.

* The sequence

$$\xymatrix{ 0 \ar[r]&  {B_n(G)}_{(p)}  \ar[r]^{ \Phi_{(p)}^G} &{C_n(G)}_{(p)} \ar[r]^-{\Psi_{(p)}^G} &  Obs(G)_{(p)} \ar[r]&0 }$$
of $ \ZZ_{(p)}$-modules is exact.

* By Lemma <ref>  $4)$, we have that $\phi_{\bar \cS} (\bar \cS)$ divides $\phi_{\bar \cT} (\bar \cS)$, and so, $\big(\frac{\phi_{\bar \cT} (\bar \cS)}{\phi_{\bar \cS} (\bar \cS)} \big)_{\bar \cT \in [\Pi_n(G)]}$ is an element of ${C_n(G)}_{(p)}$, for any $ \bar \cS \in [\Pi_n(G)]$. Now, compare the set $J'$ with the canonical basis

$$J:= \big\{ i_{\cS}= \big( \delta(\bar \cS ,\bar \cT  ) \big)_{\bar \cT \in [\Pi_n]} ~~\vert~~  \bar \cS \in [\Pi_n]  \big\}$$
of ${C_n(G)}_{(p)}$ where $\delta$ is the Kronecker's symbol, i.e.

$ \delta(\bar \cS ,\bar \cT  )=  \left\{
\begin{array}{ccc}
1& \mbox{for}  &  \bar \cS =\bar \cT \\
0&  \mbox{for}   & \bar \cS \neq \bar \cT
\end{array}
\right.$

Since $\vert J\vert=\vert J' \vert=\vert \Pi_n(G)\vert$, it suffices to prove that each $ i_{\bar \cS}$ can be written as an integral  combination of the  $\frac{1}{\phi_{\bar \cS} (\bar \cS)}\bar \cS$. This is done by induction with respect to $\preceq$.
If $\bar \cS= (1, \dots, 1)$ then $\frac{\phi_{\bar \cT} (\bar \cS)}{\phi_{\bar \cS} (\bar \cS)}= \delta(\bar \cS ,\bar \cT  )$, for any $\bar \cT$, and so, $ i_{\bar \cS} \in J'$. For arbitrary $\bar \cS$, we have $\frac{\phi_{\bar \cS} (\bar \cS)}{\phi_{\bar \cS} (\bar \cS)}=1$ and $\frac{\phi_{\bar \cT} (\bar \cS)}{\phi_{\bar \cS} (\bar \cS)}=0$ for $\bar \cT \npreceq \bar \cS$, and so, $\frac{1}{\phi_{\bar \cS} (\bar \cS)}\bar \cS=  i_{\bar \cS} + \sum\limits_{\substack{\bar \cT \in [\Pi_n(G)]\\ \bar \cT \prec \bar \cS }} n_{\bar \cT,\bar \cS}  i_{\bar \cT}$ with $ n_{\bar \cT,\bar \cS} \in \ZZ$.

Now by induction hypothesis, any $i_{\bar \cT} $ with $\bar \cT \prec \bar \cS$ is an integral linear combination of the elements of $J'$. So, the same is true for $ i_{\bar \cS}.$

* By Proposition <ref> 2), we have $\Im  \Phi_{(p)}^G = \bigoplus_{\bar \cS  \in [\Pi_n(G)] } \Phi_{(p)}^G(\langle \bar \cS \rangle_G ) $ and by Assertion 1)
$${C_n(G)}_{(p)}=  \bigoplus_{\bar \cS  \in [\Pi_n(G)] } \frac{1}{\vert W_G(\bar \cS)_p \vert }  \Phi_{(p)}^G(\langle \bar \cS \rangle_G) \ZZ.$$
Hence $C_n(G)_{(p)}/ \Im \Phi^G_{(p)} \cong Obs(G)_{(p)}$.

* Let $\bar i_{\bar \cS}= \big( \delta(\bar \cS ,\bar \cT  ) \mod \vert W_G(\bar \cS)_p \vert \big)_{\bar \cT \in [\Pi_n]}$. Obviously, the elements $\bar i_{\bar \cS}$  for $\bar \cS \in [\Pi_n]$ form a $\ZZ_{(p)}$-basis of $Obs(G)_{(p)}$. Now set

$$R_0=\{  \bar \cS \in [\Pi_n] ~\vert~  \bar i_{\bar \cS} \notin \Im \Psi_{(p)}^G    \}$$

Suppose that $R_0 \neq \emptyset$, and let $\bar \cS$ be a minimal element of $R_0$ with respect to $\preceq_G$. Then no element $\bar \cT$ of $R_0- \{ \bar \cS\}$ satisfies $\bar \cT \preceq_G \bar \cS$, and so $\Psi_{(p)}^G  ((i_{\bar \cT})_{\bar \cT \in [\Pi_n] })=(y_{\bar \cT})_{\bar \cT \in [\Pi_n] } $, where

\[
y_{\bar \cT}=  \left\{
\begin{array}{ccc}
1 ~\mod ~ \vert W_G(\bar \cS)_p \vert  & \mbox{for}  &  \bar \cT =\bar \cS \\
0~ \mod~  \vert W_G(\bar \cS)_p \vert &  \mbox{for}   &  \bar \cT  \in R_0- \{ \bar \cS\}
\end{array}
\right.\\
\]

But, $\bar i_{\bar \cT} \in \Im \Psi_{(p)}^G $ for any $\bar \cT \notin R_0$, which yields $\bar i_{\bar \cS} \in \Im \Psi^G_n$. This is a contradiction. Consequently, we have $R_0 =\emptyset$, and so $\Psi^G_{(p)}$ is surjective.

* Let $\bar \cS \in [\Pi_n(G)]$. Then
$$ \Psi_{(p)}^G( \Phi_{(p)}^G (\langle \bar \cS \rangle_G))= \big( \sum_{rT_0 \in W_G(\bar \cT)_{(p)}}  \vert \Inv_{<r>\bar \cT}(\bar \cS)\vert  ~~\mod ~ \vert W_G(\bar \cT)_{(p)} \vert   \big)_{\bar \cT \in [\Pi_n(G)]}$$

where $ \Inv_{<r>\bar \cT}(\bar \cS) = \{gS_0 \in G/S_0~\vert~ <r> \bar \cT \preceq {}^g\bar \cS \} := I_{<r>\bar \cT}$. Set $W= W_G(\bar \cS)_p $. Then $\Inv_{\bar \cT}(\bar \cS) $  can be view as a left $W$-set by the action given by $rT_0. gS_0=rgS_0$. With this action, one has
$ \Inv_{<r>\bar \cT}(\bar \cS) = \{gS_0 \in \Inv_{\bar \cT}(\bar \cS)~\vert~ rT_0 \cdot gS_0=gS_0 \}$. So

\begin{eqnarray*}
\sum_{rT_0 \in W}  \vert \Inv_{<r>\bar \cT}(\bar \cS)\vert&=& \sum_{gS_0 \in I_{\bar \cT}} \{rT_0 \in W~\vert~  rT_0 \cdot gS_0=gS_0 \}\\
&=& \sum_{gS_0 \in I_{\bar \cT}} \vert W_{gS_0}\vert \\
&=&  \sum_{gS_0 \in [W \backslash I_{\bar \cT}]} \vert O_{gK_0}\vert \vert W_{gS_0}\vert\\
&~~\equiv& 0 ~\mod ~\vert W \vert,
\end{eqnarray*}

and so $\Im  \Phi_{(p)}^G\subseteq \Ker \Psi_{(p)}^G$. It remains to prove that $  \Ker \Psi_{(p)}^G \subseteq \Im  \Phi_{(p)}^G$. Now, since $\Psi_{(p)}^G$ is surjective and $\Psi_{(p)}^G \circ \Phi_{(p)}^G =0$, we have that $\Psi_{(p)}^G$ factorizes through the cokernel of $\Phi_{(p)}^G$, which is isomorphic by $Obs_{(p)}$ by $2)$. So, we obtain a surjective map $\Coker \Phi_{(p)}^G \ra Obs_{(p)}$ between two isomorphic groups. This map is then an isomorphism, and so $\Ker \Psi_{(p)}^G $ is equal to $\Im  \Phi_{(p)}^G$.

§ IDEMPOTENTS ELEMENTS

By the isomorphism $\QQ \Phi$, there is an element $e^G_{\bar \cT} \in \QQ B_n(G)$ for each $n$-slice  $\bar \cT$ of $G$ such that

$$\QQ\phi_{\bar \cS}^G (e^G_{\bar \cT} )= \left\{ \begin{array}{rcl}
1& \mbox{if}
& \bar \cS=_G \bar \cT \\ 0 & \mbox{otherwise} &
\end{array}\right.$$

Clearly, the set $\{e^G_{\bar \cT}  ~ \vert ~  \bar \cT \in [\Pi_n(G)] \}$ is the set of primitive idempotents of $\QQ B_n(G)$.In order to give the explicit formula of the primitive idempotent $e^G_{\bar \cT}$, we need  the Möbius function of  a  finite poset $(\cX, \le)$.
The Möbius function $\mu_\cX: \cX \times \cX \ra \ZZ$ of a finite poset is defined  inductively as follows (see [13]):

$$\mu_\cX(x,x)=1, ~~~ \mu_\cX(x,y)=0 ~\text{if}~ x \nleq y, ~~ \sum\limits_{\substack{t \le y}} \mu_\cX(x,t)=0  ~\text{if}~ x < y.$$

Let $ \bar \cS$ be a $n$-slice of $G$. Then  the explicit formula of the primitive idempotent $e^G_{ \bar \cS}$ is given by

$$e^G_{ \bar \cS}= \frac{1}{\vert N_G(\bar \cS)\vert} \sum_{\bar \cT \preceq \bar \cS} \vert {\cT}_0 \vert \mu_{\Pi} \big( \bar \cT, \bar \cS \big)\langle \bar \cT \rangle_G,$$

where  $\mu_{\Pi}$ is the Möbius function of the poset $(\Pi_n(G), \preceq)$.

Let $\bar \cK$ be any $n$-slice of $G$. Then

\begin{eqnarray*}
\QQ\phi^G_{\bar \cK}( \vert G \vert e^G_{\bar \cS}) &=& \frac{\vert G \vert}{\vert N_G(\bar \cS)\vert}\! \sum_{\bar \cT \preceq \bar \cS} \! \vert \bar \cT_0 \vert \mu_{\Pi} \big( \bar \cT, \bar \cS \big) \QQ\phi^G_{\bar \cK} (\langle \bar \cT \rangle_G)\\
&=&  \frac{\vert G \vert}{\vert N_G(\bar \cS)\vert}\! \sum_{\bar \cT \preceq \bar \cS} \! \vert \bar \cT_0 \vert \mu_{\Pi} \big( \bar \cT, \bar \cS \big) \vert \{ g \in G/\bar \cT_0~\vert ~ \bar \cK \preceq {}^g\bar \cT\} \vert\\
&=&\frac{\vert G \vert}{\vert N_G(\bar \cS)\vert}\! \sum_{g\in G} \sum\limits_{\substack{\bar \cK \preceq {}^g\bar \cT \preceq {}^g\bar \cS}}\! \mu_{\Pi} \big( \bar \cT, \bar \cS \big)\\
&=&\vert G \vert \delta(\bar \cS,  \bar \cT)
\end{eqnarray*}

where the second equality follows from Lemma <ref>.
By the property of the Möbius function, we have that the sum $\sum\limits_{\substack{\bar \cK \preceq {}^g\bar \cT \preceq {}^g \bar \cS}}\! \mu_{\Pi} \big( \bar \cT, \bar \cS \big)$ is zero unless $\bar \cK={}^g\bar \cS$, for any $g \in G$. Therefore,

$$\phi_{\bar \cK}^G (e^G_{\bar \cS})= \left\{ \begin{array}{rcl}
1& \mbox{if}
&\bar \cK=_G \bar \cS \\ 0 & \mbox{otherwise} &
\end{array}\right.$$

In the case $n=0$ the formula of the primitive idempotent was given by Gluck in [7] and independently by Yoshida in [14]. The formula in the case $n=1$ was stated by Bouc in [3]. The idempotent $e^G_{\bar \cS}$ is the only idempotent of $\QQ B_n(G)$  with the following property:

[Characterization of $e^G_{\bar \cS}$]

* Let $\bar \cS$ be  a $n$-slice of $G$. Then  $\mathcal X . e^G_{\bar \cS}=  \QQ\phi_{\bar \cS}^G(\mathcal X)  e^G_{\bar \cS},$ for any $\cX \in \QQ B_n(G)$.

Conversely, if $\mathcal Y \in \QQ B_n(G)$ is such that $\mathcal X \cdot \mathcal Y=  \QQ\phi_{\bar \cS}^G(\mathcal X) \mathcal Y,$
then $\mathcal Y \in \QQ e^G_{\bar \cS}$ (that is $\mathcal Y $ is a rational multiple of $ e^G_{\bar \cS}$).

* Note that a $\QQ$-basis of $\QQ B_n(G)$ is given by the $e^G_{\bar \cT}$ where $\bar \cT$ runs through the set $\Pi_n(G)$ of conjugacy classes of $n$-slices of $G$. So for any $\mathcal X \in \QQ B_n(G)$, we have that

$$\cX= \sum_{\bar \cT \in [\Pi_n(G)]} \lambda_{\bar \cT} e^G_{\bar \cT},$$

where $\lambda_{\bar \cT}$ are rational numbers.
Since the  elements $e^G_{\bar \cT}$ are orthogonal, it follows that for any $n$-slice  $\bar \cS \in [\Pi_n(G)]$ we have,

$$\phi_{\bar \cS}^G(\mathcal X)= \lambda_{\bar \cS}~~ \text{and} ~~\cX.e^G_{\bar \cS}= \phi_{\bar \cS}^G(\mathcal X)e^G_{\bar \cS}.$$

Conversely, let $\cY$ be an element of $\QQ B_n(G)$ verifying $\cX . \cY = \phi_{\bar \cS}^G(\mathcal X) \mathcal Y$ for any $\cX \in  \QQ B_n(G)$.
Then  in particular $e^G_{\bar \cT}. \cY=0$ if $\bar \cS \neq_G \bar \cT$, thus $\cY= \phi^G_{\bar \cS} (\cY) e^G_{\bar \cS}$  is a rational multiple of $e^G_{\bar \cS}$.

Let $(\cX, \le )$ be a finite poset. Let $\Pi_n(\cX)$ denote the set of $n$-tuples $(x_0, \dots, x_n)$ of elements of $\cX$ such that $x_0 \le \dots \le x_n$. Define a partial order $\preceq$ on $\Pi_n(\cX)$ by

$$ (x_0, \dots, x_n) \preceq (y_0, \dots, y_n) \Leftrightarrow x_i \le y_i, \forall i=0, \dots, n$$

Then the Möbius function $\mu_\Pi$ of the poset $(\Pi_n(\cX), \preceq)$ can be computed as follows, for any $\bar x:=(x_0, \dots, x_n) , \bar y:=(y_0, \dots, y_n)  \in \Pi_n(\cX)$:

\begin{equation}\label{cinq}
\mu_{\Pi} \big( \bar x , \bar y \big)= \left\{ \begin{array}{rcl}
\prod\limits_{\substack{ i=0}}^n\mu_{\cX}(x_i,y_i)& \mbox{if}
&x_0 \le y_0 \le x_1 \le y_1 \dots \le x_n \le y_n \\ 0 & \mbox{otherwise} &
\end{array}\right.
\end{equation}

where $\mu_\cX$ is the Möbius function of the poset $(\cX, \le)$.

Let $m\big( \bar x , \bar y \big)$ denote the expression  defined by the right hand side of (<ref>). Then if $\bar x \preceq \bar y$

\begin{eqnarray*}
\sum\limits_{\substack{ \bar t \in \Pi_n(\cX) \\  \bar x \preceq \bar t \preceq \bar y}} m \big(\bar x,  \bar t  \big)&=&  \sum\limits_{\substack{ (t_0, \dots, t_n) \in \cP_n}} \mu_\cX (x_0, t_0) \dots \mu_\cX (x_n, t_n)\\
\end{eqnarray*}

where $\cP_n := \{ \bar t \in \Pi_n(\cX) ~\vert~ \left\{ \begin{array}{rcl}
x_i \le t_i \le y_i& \mbox{for all $i=0, \dots, n$}\\
x_0 \le \dots \le x_n & \\
y_0 \le \dots \le y_n & \\
t_i \le x_{i+1} & \mbox{for all $i=0, \dots, n-1$}
\end{array}\right.\}$.

$$\sum\limits_{\substack{ (t_0, \dots, t_n) \in \cP_n}} \mu_\cX (x_0, t_0) \dots \mu_\cX (x_n, t_n) =$$

$$ \big( \sum_{x_n \le t_n \le y_n} \mu_\cX (x_n, t_n)\big)\big( \sum_{(t_0, \dots, t_{n-1}) \in \cP'_{n-1}} \mu_\cX (x_0, t_0) \dots \mu_\cX (x_{n-1}, t_{n-1}) \big) ~~~~(\star)$$

where $\cP'_{n-1} := \{ (t_0, \dots, t_{n-1}) \in \Pi_{n-1}(\cX) ~\vert~ \left\{ \begin{array}{rcl}
x_i \le t_i \le y_i& \mbox{for all $i=0, \dots, {n-1}$}\\
t_i \le x_{i+1} & \mbox{for all $i=0, \dots,  n-1$}
\end{array}\right.\}$.

The first factor of $(\star)$ is equal to $0$ if $x_n \neq y_n$, and $1$ if $x_n = y_n$. Hence, if $x_n = y_n$, then the second factor is equal to

$$ \sum_{(t_{0}, \dots, t_{n-1}) \in \cP_{n-1}} \mu_\cX (x_0, t_0) \dots \mu_\cX (x_{n-1}, t_{n-1}) $$

Hence inductively, we have

\begin{eqnarray*}
\sum\limits_{\substack{ \bar t \in \Pi_n(\cX) \\  \bar x \preceq \bar t \preceq \bar y}} m \big(\bar x,  \bar t \big)&=&  \prod\limits_{\substack{ i=0}}^n \delta(x_i, y_i)\\
\end{eqnarray*}

and the proposition follows.

Let $ \bar \cS= (S_0, \dots, S_n)$ and $\bar \cT=(T_0, \dots, T_n)$ be $n$-slices of $G$. Then

$$\mu_{\Pi} \big( \bar \cT, \bar \cS\big)= \left\{ \begin{array}{rcl}
\prod\limits_{\substack{ i=0}}^n\mu(T_i,S_i)& \mbox{if}
&T_0 \le S_0 \le T_1 \le S_1 \dots \le T_n \le S_n \\ 0 & \mbox{otherwise} &
\end{array}\right.$$

where $\mu$ is the Möbius function of the poset of subgroups of $G$.

In particular

$$e^G_{\bar \cS}= \frac{1}{\vert N_G(\bar \cS)\vert} \sum_{T_0 \le S_0 \le \dots \le T_n \le S_n} \vert T_0 \vert \mu  (T_0, S_0) \dots \mu( T_n, S_n)\langle \bar \cT \rangle_G,$$

§ PRIME IDEALS

The aim of this part is the proof that a finite group $G$ is solvable if and only if the prime ideal spectrum of $B_n(G)$ is connected, i.e., if and only if $0$ and $1$  are the only idempotents in $B_n(G)$, like established by A.Dress in [5] for $B_0(G)$ and S.Bouc in [3].
Note that $C_n(G)$ is integral over $B_n(G)$,  because it  is generated by idempotent elements which are integral over any subring.   Hence by the going-up theorem, every prime ideal of $B_n(G)$ comes from  $C_n(G)$.

Let $p$ denote either $0$ or a prime number.
If $ \bar \cS \in \Pi_n(G)$, let $I_{\bar \cS,p}$ be the prime ideal of $B_n(G)$ defined as the kernel of the ring homomorphism

$$\xymatrix{ B_n(G) \ar[r]^-{\phi^G_{\bar \cS}} & \ZZ \!\ar[r]^-{p} & \ZZ/p\ZZ}.$$

Then every prime ideal $I$ of $B_n(G)$  has the form $I_{\bar \cS,p}$ for a suitable $\bar \cS \in \Pi_n(G)$.
Moreover,  given a prime ideal $I$ there exists a unique $\bar \cK \in [\Pi_n(G)]$ with $I= I_{\cK,p}$ and  $\phi_{\bar \cK}^G(\langle \bar \cK \rangle_G) \neq 0 ~(\mod ~p) $ where $p$ is the characteristic of the ring $R:= B_n(G)/I$.

Consider the natural map $\phi: B_n(G) \ra B_n(G)/I$. Then  $\phi(x)= \phi_{\bar \cS}(x)\cdot 1_R$ for some $n$-slice $\bar \cS$ by Proposition <ref>.

\begin{eqnarray*}
I&=&\{x \in B_n(G) ~\vert~ \phi(x)=0\}\\
&=&\{x \in B_n(G) ~\vert~ \phi_{\bar \cS}(x)\cdot 1_R=0\}\\
&=&\{x \in B_n(G) ~\vert~ \phi_{\bar \cS}(x) \equiv 0 (\mod ~p)\}= I_{ \bar \cS,p}
\end{eqnarray*}

where $p$ is the characteristic of the ring $B_n(G)/I$.

Let $\cI_n:=\{\bar \cS \in \Pi_n(G)~\vert~ \langle \bar \cS \rangle_G \notin I\}$. If $\bar \cT$ and $\bar \cK$ are both minimal in $\cI$, then

\begin{eqnarray*}
\langle \bar \cT \rangle_G.\langle \bar \cK \rangle_G&=& \sum_{g \in [T_0\backslash G/K_0]}\langle \bar \cT \cap {}^g\bar \cK   \rangle_G\\
&=& \phi_{\bar \cT}(\langle \bar \cK \rangle_G)\langle \bar \cT \rangle_G ~(\mod~I)
\end{eqnarray*}

(This relation must hold for any $\langle \bar \cK \rangle_G \notin I$)

Since $\langle \bar \cT \rangle_G.\langle \bar \cK \rangle_G \notin I$, we have $ \phi_{\bar \cT}(\langle \bar \cK \rangle_G) \neq 0$ and $ \phi_{\bar \cK}(\langle \bar \cT \rangle_G) \neq 0$ by symmetry, and so $\bar \cK=_G \bar \cT$. So the set $\cI_n$ has a unique minimal element, up to isomorphism.  On the other hand, the  quotient ring $B_n(G)/I$ is an integral domain, we have that either $p=0$ or  $p$ is a prime number. If $p=0$, then the projection $\phi: B_n(G) \ra B_n(G)/I$ is equal to $\phi_{\bar \cK}$ and   $I= I_{ \cK,p}$. If $p \neq 0$, then $\phi$ is equal to the reduction  of $\phi_{\bar \cK}$ modulo $p$.

If $\bar \cG$ is any $n$-slice of $G$ with $I=  I_{\cG,p}$ and $\phi_{\bar \cG} (\langle \bar \cG \rangle_G)  \neq 0~~(\mod ~p)$  then for a $\bar \cK$ minimal  in $\cI_n$

$$\phi_{\bar \cK} (\langle \bar \cK \rangle_G) \equiv \phi_{ \bar \cG} (\langle \bar \cK \rangle_G) \neq 0~~(\mod~p).$$
In particular $\phi_{\bar \cK} (\langle \bar \cG \rangle_G) $ is non zero; and similarly $\phi_{\bar \cG} (\langle \bar \cK \rangle_G) $ is non zero. This can only occur if $\bar \cK =_G \bar \cG$.

Let $p$ be a prime number.

* Let $\big(\Pi_n(G) \big)_{(p)}$ denote the subset of $\Pi_n(G)$ consisting of the $n$-slices $\bar \cS$ such that $N_G(\bar \cS)/S_0$ is a $p'$-group.

* For any $\bar \cS \in \Pi_n(G)$, let $\bar \cS^{\small\to}_p$ be the unique element $\bar \cK$ of $\big(\Pi_n(G) \big)_{(p)}$, up to conjugation, such that $I_{\bar \cS,p}= I_{\bar \cK,p}$.

* If $\bar \cS$  is a $n$-slice of $G$, let $\bar \cS^{+}_p$ denote a slice of the form $P\bar \cS:= (PS_0, \dots, PS_n)$ of $G$, where $P$ is a Sylow $p$-subgroup of $N_G(\bar \cS)$.

Define inductively an increasing sequence $(\bar \cS^i )_i$ in $(\Pi_n(G), \preceq)$ by
$\bar \cS^0= \bar \cS$ and $\bar \cS^{i+1}=(\bar \cS^i )^{+}_p $, for $i \in \NN$. We set $\bar \cS^{\infty}$ for  the largest term  of the sequence $(\bar \cS^i )_i$.

* By Proposition <ref>, the prime ideal of $B_n(G)$  are parametrized by pairs  $(\bar \cS, p)$ where $p$ is equal to $0$ or a prime number and  $\bar \cS \in \Pi_n(G)$ is such that
$\vert N_G(\bar \cS)/S_0 \vert \not \equiv 0 ~(\mod~p)$. We set  $\Theta_n (G)$  the set of  $(\bar \cS, p)$ where $p$ is equal to $0$ or a prime number and  $\bar \cS \in \Pi_n(G)$ is such that
$\vert N_G(\bar \cS)/S_0 \vert \not \equiv 0 ~(\mod~p)$.

Note that if we define a relation $  ``\overset{p}{\sim} "$ on $\Pi_n(G)$ by $\bar \cS \overset{p}{\sim} \bar \cS'$ if $\bar \cS^{\to}_p  =_G \bar \cS'^{\to}_p$ then $  ``\overset{p}{\sim} "$  is an equivalence relation.

The $n$-slice  $\bar \cS^{\to}_p$ is conjugate to $\bar \cS^{\infty}$.

By definition, the $n$-slice $\bar \cS^{\to}_p$ is a minimal element $\bar \cK$ of the $(\Pi_n(G), \preceq~)$ such that

$$\phi_{\bar \cS }(\langle \bar \cK  \rangle_G):= \vert \{ g \in G/K_0~\vert~\bar \cS^g \preceq  \bar \cK \} \vert \not\equiv 0~(\mod~p).$$

Thus one can assume that $\bar \cS \preceq \bar \cK$. Since $\phi_{\bar \cS} \equiv \phi_{P\bar \cS} ~(\mod~p)$  for any $p$-subgroup $P$ of $N_G(\bar \cS)$ by Corollary <ref>, one can also assume that $\bar \cS^{+}_p  \preceq \bar \cK$, and inductively, that $\bar \cS^{\infty}  \preceq \bar \cK$. Moreover $\phi_{\bar \cS^{\infty}} \equiv \phi_{\bar \cK } ~(\mod~p)$. As $N_G(\bar \cS^{\infty}) / S_0^{\infty}$ is a $p'$-group, it follows that $\bar \cS^{\infty} =_G \bar \cK$.

Let $(\bar \cS, p)$, $(\bar \cS', p')$ be elements of  $\Theta_n (G)$.
Then $I_{\bar \cS', p'} \subseteq I_{\bar \cS, p}$ if and only if

* either $p'=p$ and the $n$-slices $\bar \cS'$ and $\bar \cS$ are conjugate in $G$.

* or  $p'=0$ and $p>0$, and the $n$-slices $\bar \cS'^{\to}_p$ and $\bar \cS$ are conjugate in $G$.

Assume that $I_{\bar \cS', p'} \subseteq I_{\bar \cS, p}$. Then there exists a surjective ring homomorphism $\phi$ from $B_n(G)/I_{\bar \cS', p'}\cong \ZZ/p'\ZZ$ to $B_n(G)/I_{\bar \cS, p}\cong \ZZ/p\ZZ$. Hence either $p'=p$ or $p'=0$ and $p>0$. If $p=p'$ then $\phi$ is a bijection, and so $I_{\bar \cS', p'}= I_{\bar \cS, p}$ which implies that $\bar \cS'$ and $\bar \cS$ are conjugate. If  $p'=0$ and $p>0$ then $\phi^G_{\bar \cS}$ is the reduction modulo $p$ of $\phi^G_{\bar \cS'}$, and so $I_{\bar \cS', p} \subseteq I_{\bar \cS, p}$. Hence  $\bar \cS'^{\to}_p$ and $\bar \cS$ are conjugate in $G$, by Corollary <ref>. Conversely, if $\bar \cS'$ and $\bar \cS$ are conjugate then $\phi^G_{\bar \cS}=\phi^G_{\bar \cS'}$, in particular $I_{\bar \cS', 0} = I_{\bar \cS, 0}$. If $p$ is a prime then Proposition <ref> implies that $\phi^G_{\bar \cS'}(x) \equiv \phi^G_{\bar \cS'^{\to}_p}(x)~{\mod ~p}$ for any $x \in B_n(G)$. So $I_{\bar \cS'^{\to}_p, 0} \subseteq I_{\bar \cS', p}$. And if $\bar \cS'^{\to}_p$ and $\bar \cS$ are conjugate in $G$, then  $I_{\bar \cS'^{\to}_p, p}=I_{\bar \cS', p} = I_{\bar \cS, p}$. So $I_{\bar \cS'^{\to}_p, 0} \subseteq I_{\bar \cS, p}$.

* Since for a prime $p$, the prime ideals of $\big( B_n(G) \big)_{(p)}$ are of the form $\ZZ_{(p)}I$ where $I$ is a prime ideal of $B_n(G)$ which does not meet $\ZZ-p\ZZ$, we have that $I=I_{\bar \cS, p}$ or $I=I_{\bar \cS, 0}$. Hence the connected  component  of $\big(B_n(G) \big)_{(p)} $ are indexed by $[\big(\Pi_n(G) \big)_{(p)} ]$. Moreover, the component  indexed  by the $n$-slice $\bar \cS$ consists  of a unique maximal element $\ZZ_{(p)}I_{\bar \cS, p}$ and of ideals $\ZZ_{(p)}I_{\bar \cS', 0}$, where $\bar \cS' \in \Pi_n(G)$ is such that $\bar \cS'^{\to}_p =_G\bar \cS$.

* For $\bar \cS$ in $\big(\Pi_n(G) \big)_{(p)}$  let $\tilde e_{\bar \cS}$ denote the sum $\sum e^G_{\bar \cS'}$ in $\QQ B_n(G)$ where $\bar \cS'$ ranges over all $n$-slices in $[\Pi_n(G)]$  such that $\bar \cS$ is conjugate to $\bar \cS'^{\to}_p$. Then the distinct
$\tilde e_{\bar \cS}$ are   the primitive idempotents of $\big( B_n(G) \big)_{(p)}$. This is because, for any commutative ring $R$ the connected component of $Spec R$ corresponding to a primitive idempotent $e$ in $R$ consists of all prime ideals of $R$ which contain $1-e$. If $e$ is a primitive idempotent of $\big( B_n(G) \big)_{(p)}$ corresponding to the connected component of $\big( B_n(G) \big)_{(p)}$ indexed by $\bar \cS, p$, then $1-e \in \ZZ_{(p)}I_{\bar \cS', 0}$ if and only if $\bar \cS'^{\to}_p=_G \bar \cS$.

Two ideals    $ I_{\bar \cS, p}$   and $I_{\bar \cS', p'} $  are in the same connected component of $Spec(B_n(G))$ if and only $D^{\infty}(\bar \cS)$ is conjugate to $ D^{\infty}(\bar \cS')$, where

$D^{\infty}(\bar \cK):= (D^{\infty}(\bar \cK_0), \dots, D^{\infty}(\bar \cK_n))$ and $D^{\infty}(\bar \cK_i)$ denotes the last term in the derived series of $\bar \cK_i$.
In particular, $Spec(B_n(G))$ is connected  if and only if $G$ is solvable.

Recall that if $R$ is a Noetherian ring. For any prime ideal $I \in Spec(R)$, let $\overline{I}= \{ P~\vert~ P \in Spec(R), ~P \supset I\}$ be the closure of $I$ in $Spec(R)$. Then, two ideals $P$ and $P'$ are in the same connected component of $Spec(R)$ if and only if there exists a series of minimal ideals $I_1, \dots, I_n$ with $P \in \overline{I_1}$,  $P' \in \overline{I_n}$ and $\overline{I_i }\cap \overline{I_{i+1}} \neq \emptyset$ for $i=1, \dots,n-1$. Now, if $R= B_n(G)$, then $\bar I_{\bar \cS, 0} \cap \bar I_{\bar \cS', 0} \neq \emptyset$ if and only if the $n$-slice $\bar \cS^{\to}_p$ is conjugate to $\bar \cS'^{\to}_p$, for some prime $p$. Hence, if  $I_{\bar \cS, p} $ and $I_{\bar \cS', p'} $ are in the same connected component of $Spec(B_n(G))$ then $ D^{\infty}(\bar \cS)$ is conjugate to $ D^{\infty}(\bar \cS')$.
The ideals $I_{\bar \cS, p}$ and $I_{D^{\infty}(\bar \cS), 0}$ are in the same connected component. Indeed,  one can find a series of normal subgroups of $S_0$  such that $D^{\infty}(S_0) = S_0^{(n)} \triangleleft S_0^{(n-1)}\triangleleft \dots \triangleleft S_0^{(1)} \triangleleft  S_0^{(0)}= S_0$ such that $S_0^{(i-1)}/ S_0^{(i)}$ is a $p_i$-group for some prime $p_i$ ( $i=1,\dots, n$). Hence letting,
$\bar \cK_i = (S_0^{(n)}, S_1, \dots, S_n)$
for $i=0,\dots,n$ one obtains

$$\bar \cS=\bar \cK_0 \overset{p_0}{\sim} \bar \cK_1\overset{p_1}{\sim} \dots \overset{p_{n-2}}{\sim}      \bar \cK_{n-1}    \overset{p_{n-1}}{\sim}   \bar \cK_n= (D^{\infty}(S_0), S_1, \dots, S_n),$$

$I_{\bar \cS, p} \in  \bar I_{\bar \cK_0, 0}$,  $I_{D^{\infty}(S_0), S_1, \dots, S_n, 0} \in  \bar I_{\bar \cK_n, 0}$ and  $\bar I_{\bar \cK_{i-1}, 0} \cap \bar I_{\bar \cK_{i}, 0} \neq \emptyset$. So, $I_{\bar \cS, p}$  and $I_{D^{\infty}(S_0), S_1, \dots, S_n, 0}$ are in the same connected component. By the same proceed, one prove that
$I_{(D^{\infty}(S_0), D^{\infty}(S_1), \dots, S_n), 0}$ and $I_{\bar \cS, p}$ are in the same component, and so one.

§ GREEN BISET FUNCTOR

Let $R$ be a commutative ring with identity.
The biset category over $R$ will be denoted by $R \cC$: its objects are all finite groups, and that for finite groups $G$ and H, the hom-set $\Hom_{R\cC}(G,H)$ is $RB(H,G)= R \otimes B(H,G)$, where $B(H,G)$ is the Grothendieck group of the category of finite $(H,G)$-bisets. The composition of morphisms in $R \cC$ is induced by $R$-bilinearity from the composition of bisets (see [2]  Definition 3.1.1).
A good choice of a family  $\cG$ of finite groups and for every $G,H \in \cG$, a set $\Gamma(G,H)$ of subgroups of $G \times H$ can  lead to an important category. For example  if we  fix a non-empty class $\cD$ of finite groups closed under subquotients and cartesian products and  we denote by $R \cD$ the full subcategory of $R \cC$  consisting of groups in $\cD$, then $R \cD$ is a replete subcategory of $R \cC$ in the sense of Bouc [2].The category of biset functors, i.e. the category of $R$-linear functors from $R\cC$  to the category $R-\Mod$ of all $R$-modules, will be denoted by $\cF_{R}$. The category $\cF_{\cD,R}$ of $\cD$-biset is the category of $R$-linear functors from $R\cD$ to $R-\Mod$.

Let $G$ be a finite group,  $H$ a subgroup of of $G$ and $N$ be a normal subgroup of $G$. One sets

* $\Res_H^G := [{}_HG_G]$ for the image in the biset Burnside group $B(H,G)$ of the isomorphism class of $G$ where $G$ is viewed as an $(H,G)$-biset via the left and right multiplication.

* $\Ind_H^G := [{}_GG_H]$ for the image in  $B(H,G)$ of the isomorphism class of $G$ where $G$ is viewed as a $(G,H)$-biset via the left and right multiplication.

* $\Inf^G_{G/N} := [{}_G(G/N)_{G/N}]$ for the image in  $B(G,G/N)$ of the isomorphism class of $G/N$ where $G/N$ is viewed as a $(G,G/N)$-biset via the the canonical epimorphism $G \ra G/N$ and left and right multiplication.

* $\Dof^G_{G/N} := [{}_{G/N}(G/N)_{G}]$ for the image in  $B(G/N,G)$ of the isomorphism class of $G/N$ where $G/N$ is viewed as a $(G,G/N)$-biset via the the canonical epimorphism $G \ra G/N$ and left and right multiplication.

* If $f:G\ra H$ is a group isomorphism, then we set $\Iso(f):= [{}_{H}H_{G}]$ for the image in  $B(H,G)$ of the isomorphism class of $H$ where $H$ is considered  as an $(H,G)$-biset via $hxg=hxf(g)$ for $h,x \in H$ and $g \in G$.

It is possible to verify that all elements in $B(H,G)$ are sums of $[H\times G/L]$ where $L$ runs through subgroups of $H\times G$ and  that these satisfy the following decomposition:

\begin{equation}
[H\times G/L]= \Ind^H_D \circ \Inf^D_{D/C} \circ \Iso (f) \circ  \Dof^B_{B/A}\circ \Res^G_B
\end{equation}

where $(D,C)$ and $(B,A)$ are $1$-slices of $H$ and $G$ respectively with the additional properties that $C \lhd D$ and $A \lhd B$, and $f$ is some the group isomorphism from $B/A$ to $D/C$ (see [2] Lemma 2.3.26 for more details).

A Green $\cD$-biset functor is defined as a monoid in $\cF_{\cD,R}$. This is equivalent to the following definitions:
[[2] Definition 8.5.1]
A $\cD$-biset functor $A$ is a Green $\cD$-biset  functor if it is equipped with a linear products $A(G) \times A(H) \ra A(G \times H)$ denoted by $(a,b) \mapsto a \times b$, for groups, $G$, $H$ in $\cD$, and an element $\epsilon_A \in A(1)$, satisfying the following conditions:

* (Associativity). Let $G,H$ and $K$ be groups in $\cD$. If we consider the canonical isomorphism from $G \times (H \times K)$ to $(G \times H) \times K$, then for any $a \in A(G)$, $b \in A(H)$ and $c \in A(K)$

$$(a\times b) \times c = A\bigg(\Iso^{(G \times H) \times K}_{G \times (H \times K)}\bigg) \big(a \times (b \times c) \big).$$

* (Identity element).  Let $G$ be a group in $\cD$ and consider the canonical isomorphisms $1 \times G \ra G$ and $G \times 1 \ra G$. Then for any $a \in A(G)$

$$a = A\big(\Iso^G_{1 \times G}\big)( \epsilon_A  \times a)=  A\big(\Iso^G_{ G\times 1}\big)( a \times \epsilon_A )$$

* (Functoriality). If $\phi: G \ra G'$ and $\psi: H \ra H'$ are morphisms in $R \cD$, then for any $a \in A(G)$ and $b \in A(H)$

$$A \big(\phi \times \psi\big)(a \times b)= A(\phi)(a) \times A(\psi)(b).$$

There is an equivalent way of defining a Green biset functor given by Romero in ([12] Lema 4.2.3):

[[12] Definiciòn 3.2.7]
A  $\cD$-Green biset functor is an object $A \in \cF_{\cD,R}$ together with the datum of an $R$-algebra sructure on each $A(H)$, $H \in \cD$, such that the following axioms are satisfied for all groups in $K$ and $G$ in $\cD$ and all group homomorphisms $K \ra G$:

* For the $(K,G)$-biset $G$, which we denote by $G_r$,  the morphism $A(G_r)$ is a ring homomorphism.

* For the $(G,K)$-biset $G$, denoted by $G_l$, the morphism $A(G_l)$ satisfies the Frobenius identities for all $b \in A(G)$ and $a \in A(K)$,

\begin{eqnarray*}
A(G_l)(a) \cdot b&=& A(G_l)\big(a \cdot A(G_r)(b) \big)\\
b\cdot A(G_l)(a)&=& A(G_l)\big(A(G_r)(b) \cdot a \big)
\end{eqnarray*}

where $\cdot$ denotes the ring product on $A(G)$, resp. $A(K)$.

If $A$ and $C$ are Green $\cD$-biset functors, a morphism of Green $\cD$-biset functors from $A$ to $C$ is a natural transformations $f: A \ra C$ such that $f_{H \times K}(a \times b)= f_H(a) \times f_K(b)$ for any groups $H$ and $K$ in $\cD$ and any $a \in A(H)$, $b \in A(K)$, and such that $f_1(\epsilon_A)=\epsilon_C$.

The correspondence
$$G \mapsto B_n(G)$$
defines a structure of Green biset functor.

There are several steps:

* Any $(n,G)$ simplex $\cX^f_n$ give rise a $(n, H)$-simplex $U \times_G \cX^f_n$ in the following rule (following the notion of Definition 2.3.11 in [2]):

$$U \times_G \cX^f_n: (\xymatrix{U\times_GX_0 \ar[r]^{Uf_1} &  U\times_GX_{1}\ar[r]^-{Uf_{2}} & \dots U\times_GX_{n-1} \ar[r]^-{Uf_{n}} &  U\times_GX_{n}}) $$

defined by  $Uf_i: U\times_GX_{i-1} \ra U\times_GX_i, (u,x) \mapsto (u, f_i(x)) $ for any $i= 1,\dots, n$.

Let $\mu= (\mu_i)$ be a morphism from $\cX^f_n$ to $\cY^g_n$. Then $\mu$ defines a morphism of $(n,H)$-simplices  from $U \times_G \cX^f_n$ to $U \times_G \cY^g_n$ given by
$$U\times_G\mu= ( U\mu_i: (u,x)\mapsto (u, \mu_i(x)))_i.$$
Indeed, for any $i$, $x_i \in X_i$ and $u \in U$, we have

\begin{eqnarray*}
(U\mu_i)\circ (Uf_i)(u,x_{i-1}) &=& (u, \mu_if_i(x_{i-1})) \\
&=&(u, g_{i}\mu_{i-1}(x_{i-1}))\\
&=& (Ug_i)\circ (U\mu_{i-1})(u,x_{i-1})
\end{eqnarray*}

Therefore, the correspondence $I_U: \cX^f_n \mapsto U \times_G \cX^f_n $ is a functor from the  $(n,G)$-simplices to $(n,H)$-simplices. On the other hand,  it is straightforward to show that the defining relations of $B_n(G)$ are mapped to the defining relations of  $B_n(H)$. Hence, the later functor induces a homomorphism of groups $B_n(U): B_n(G) \ra B_n(H)$.

* The  correspondence $G \mapsto B_n(G)$ defines a structure of  biset functor:

Clearly, if $U \cong U'$ (as (H,G)-bisets) then the functors $I_U$ and $I_{U'}$ are isomorphic. So, $B_n(U)=B_n(U')$. If $U$ has the form $U=U_1 \sqcup U_2$ (as (H,G)-bisets) then $I_U= I_{U_1} \sqcup I_{U_2}$, and so $B_n(U)= B_n(U_1) + B_n(U_2)$. We may state that for any $(K,H)$-biset $V$, we have a isomorphism of $(n,K)$-simplices  between $ V \times_H(U \times \cX^f_n)$ and $ (V \times_HU) \times \cX^f_n$ which induces an isomorphism $I_V \circ I_U \cong I_{V \times_HU}$ and so $B_n(V) \circ B_n(U)= B_n(V \times_HU)$. Finally, since $I_{\Id_G} \cong 1$(the functor identity, we have $B_n(Id_G)=1_{B_n(G)}$. This shows that we have a functor from the biset category to the category $\ZZ-\Mod$.

* Let  $\cY^g_n$ be a $(n,H)$-simplex. Then the product $\cX^f_n \times \cY^g_n$ is a $(n, G\times H)$-simplex in the obvious way. It induces a well-defined bilinear
$$\times: B_n(G) \times B_n(H) \ra B_n(G\times H), ~~((a,b) \mapsto a \times b),$$

and $(n,1)$-simplex $\textbf e$ is the identity of the product, up to identification $G\times 1=G$. One verifies easily that the axioms of Definition <ref> are satisfied.

The functors $d_j $, $j=1,\dots, n$   (resp.  $s_i $, $i=0,\dots, n-1$)
induce morphisms of Green biset functors
$$d_j : B_n \ra B_{n-1} ~~~~\big(\text{resp.}~   s_i: B_{n-1} \ra B_{n} \big)$$
such that the identities (<ref>), (<ref>), (<ref>) hold.

This is a simple verification.

§ ACKNOWLEDGEMENT
The author would like to thank to acknowledge support from CCM-UNAM-Morelia.

[1]
Serge Bouc.
Burnside rings.
Handbook of Algebra, vol. 2, (739-804), 2000.

[2]
Serge Bouc.
Bisets functors for finite groups, volume 1990.
Lecture Notes in Mathematics. Springer, 2010.

[3]
Serge Bouc.
The slice Burnside ring and the section Burnside ring of a finite
Compositio Mathematica, 148:868–906, 2012.

[4]
William Burnside.
The theory of groups of finite order.
Cambridge University Press, second edition, 1911.

[5]
Andreas Dress.
A characterization of solvable groups.
Math. Zath., (110: 213-217), 1969.

[6]
Andreas Dress and Dennis Kletzing.
On prime ideals in representation rings.
Math. Zath., (133: 285-300), 1973.

[7]
David Gluck.
Idempotents formula for the Burnside ring with applications to the
$p$- subgroup simplicial complex.
Illinois J. Math, (25: 63-67), 1981.

[8]
Saunders Mac Lane.
Categories for the working mathematician, volume 5.
Springer-Verlag, New York, Graduate Tests in Mathematics,

[9]
J Peter May.
Simplicial objects in algebraic topology.
University of Chicago Press, Chicago IL, 1992.

[10]
Fumihito Oda, Yugen Takegahara, and Tomoyuki Yoshida.
The lattice burnside rings.
Algebra Univers, (81,No.4, Paper N0.53,31), 2020.

[11]
Daniel Quillen.
Homotopy properties of the poset of nontrivial $p$-subgroups of a
Advances in Math, (28: 101-128), 1978.

[12]
Nadia Romero.
Funtores de Mackey.
Tesis de Doctorado, Universidad Nacional Autónoma de
México, 2011.

[13]
Richard P. Stanley.
Enumerative combinatorics.
Volume 49 of Cambrige Studies in Advanced Mathematics.
Cambridge University Press, Cambrige, second edition, 2012.

[14]
Tomoyuki Yoshida.
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Journal of Algebra, (80: 90-105), 1983.

Ibrahima Tounkara

Centro de Ciencias Matemáticas

C.P. 58089

Morelia Mich

e-mail<EMAIL_ADDRESS>


# Isolating systematic effects with beam polarisation at e+e- colliders

Jakob Beyer Talk presented at the International Workshop on Future Linear
Colliders (LCWS2021),
15-18 March 2021. C21-03-15.1. Deutsches Elektronen-Synchroton (DESY),
Hamburg, Germany  Universität Hamburg, Hamburg, Germany  Jenny List  Deutsches
Elektronen-Synchroton (DESY), Hamburg, Germany

###### Abstract

Future high-energy $e^{+}e^{-}$ colliders will provide some of the most
precise tests of the Standard Model. Statistical uncertainties are expected to
improve by orders of magnitude over current measurements. This provides a new
challenge in accurately assessing and minimizing systematic uncertainties.
Beam polarisation may hold a unique potential to isolate and determine the
size of systematic effects. This study aims to set this hypothesis by setting
up a combined fit of systematic effect and electroweak observables that can
test different configurations of available beam polarisations and
luminosities.

A framework for this fit is developed. In a first test it fits the luminosity
and polarisation with external constraints to 2-fermion and 4-fermion
differential distributions. An implementation of a first systematic - the muon
acceptance - shows consistent behaviour in this framework. This lays the
groundwork for testing the interplay of systematic and physical effects in the
presence or absence of beam polarisation.

##  Introduction

Measurements at the SLC demonstrated the physical insight that polarised beams
bring [1]. Similar advantages from beam polarisation are seen in studies for
future colliders [2]. The focus in the latter have so far been on the purely
physical advantages.

In addition to physical benefits, beam polarisation may have the potential to
also improve on systematic uncertainties. This thought bases on the
polarisation-independent nature of the majority of systematic effects. Such
systematic effects stay constant when the helicity of the beam is reversed,
while the chirality-dependent physical effects change.

A combined fit of the polarised datasets at future colliders can make use of
this qualitative difference between physical and systematic effects. The
systematic effects will stand out as global amongst the different polarised
datasets if they are explicitly included in the fit. Resulting from this
obvious difference between effects, the parameters of the systematic effects
would then not correlate with those of the physical effects. This uncorrelated
behaviour could minimize systematic biases and uncertainties in the physical
results.

Here, a framework is set up to study such a combined fit of polarised datasets
that explicitly includes systematic effects. First tests validate the
framework for different luminosity and polarisation configurations. A
systematic effect - the $\mu$ acceptance - is parametrised and for the first
time included in such a combined fit. This work lays the foundation for
testing the hypothesis that beam polarisation helps to reduce the impact of
systematic effects.

##  A combined fit setup

An assessment of the impact of beam polarisation on systematic uncertainties
at future colliders requires a combined fit of the polarised datasets. This
section describes the framework can perform such fits, the Monte-Carlo
distributions used in it, and a first validation test of the framework.

###  A fit framework for polarised datasets

A framework is set up that can use all polarised (or unpolarised)
distributions for multiple final states at a future collider in a simultaneous
fit. The fit performs a Poissonian log-likelihood maximisation using the
Minuit2 framework of ROOT. All polarisations and the luminosity can vary
freely, be given a Gaussian constraint, or set to a fixed value. The helicity
flip between polarised datasets is not assumed to be perfect, so that each
polarisation of each dataset is a separate parameter.

Proposed future colliders motivate the polarisation and luminosity
configurations used in this study [3]. The polarisation scenarios are

  * •

an unpolarised scenario ($\mathcal{P}_{e^{-}}=\mathcal{P}_{e^{+}}=0\%$) with
the fully luminosity in this unpolarised dataset,

  * •

a $e^{-}$-only polarised scenario with $\mathcal{P}_{e^{-}}=\pm 80\%$ and
$\mathcal{P}_{e^{+}}=0\%$, and a luminosity sharing of $50:50$ between the two
datasets,

  * •

a fully polarised scenario with $\mathcal{P}_{e^{-}}=\pm 80\%$ and
$\mathcal{P}_{e^{+}}=\pm 30\%$, and an optimized luminosity sharing of
$45:45:5:5$ where the first two are the opposite-sign and the second two the
same-sign datasets.

Two luminosity scenarios with a factor 5 difference are tested:
$\mathcal{L}=2\,\text{ab}^{-1}$ and $\mathcal{L}=10\,\text{ab}^{-1}$. All
scenarios use realistic Gaussian constraints for each polarisation and
luminosity parameter [4, 5].

$\begin{split}\frac{\Delta L}{L}&=3\times 10^{-3}\\\
\frac{\Delta\mathcal{P}_{non-0}}{\mathcal{P}_{non-0}}&=2.5\times 10^{-3}\text{
(for polarised beams)}\\\ \Delta\mathcal{P}_{0}&=2.5\times 10^{-3}\text{ (for
unpolarised beams)}\end{split}$

###  Processes and distributions

This work uses differential distributions of di-muon production and
semileptonic $W$-pair production at $250\,$GeV. It uses generator-level event
created for the $250\,$GeV ILD production [6], and are produced using
WHIZARD2.8 [7, 8]. Previous work within a similar framework for polarisation
measurements used a set of processes with two and four fermions in the final
state [9]. The first tests presented here select a subset of those processes
that all contain muons in order to test common muon-relevant systematic
effects later on. All distributions are purely on generator level and do not
make use of detector simulation or high-level analysis.

The di-muon distributions are one-dimensional distributions of the $\mu^{-}$
polar angle in the di-muon rest frame (example in fig. 1(a)). Initial state
radiation (ISR) in such events can cause the effective center-mass-energy
$\sqrt{s*}$ to change significantly. Events with strong ISR can land on the
$Z$-pole resonance. Such events are treated separately so that two distinct
distributions are used: one with $\sqrt{s*}\in[180,275]\,$GeV (“high-$Q^{2}$”)
and one with $\sqrt{s*}\in[81,101]\,$GeV (“return-to-Z”). The upper cut in the
high-$\sqrt{s*}$ region is higher than $250\,$GeV to account for the beam
spectrum.

The semileptonic $W$ pair production distributions describe the
$qq\mu\nu\,(q=u,d,s,c,b)$ final state with a three-dimensional angular
distribution (example in fig. 1(b)). These three angles are the $W^{-}$ angle
in the detector frame and the $\mu$ angles in the rest frame of the
leptonically decaying $W$. Events with negative $\mu^{-}$ are used separately
from events with positive $\mu^{+}$, leading to two distinct distributions.

(a)

(b)

Figure 1: Generator-level differential distributions for(1(a)) $\mu\mu$
($\sqrt{s*}\in[180,275]\,$GeV) and(1(b)) $qq\mu^{-}\bar{\nu}\,(q=u,d,s,c,b)$
($\sqrt{s*}\in[180,275]\,$GeV)events for $e^{-}_{L}e^{+}_{R}$ collisions at
250GeV from the ILD production [6].

###  First test of the framework

A first test of the framework uses the distributions and collider
configurations described above to extract just the polarisation and luminosity
parameters. The test does not implement any other physical or systematic
effects, and is a simple validation of the framework. It assumes all physical
constants and chiral cross sections, including chiral asymmetries, to be known
exactly.

The uncertainties extracted from this test show an increased precision gained
from higher luminosity (fig. 2). This gain does not always scale with the
square-root of the luminosity increase because the uncertainty is a
combination of the statistical uncertainty and the external constraint on the
parameter. In addition to the uncertainty on each parameter, the fit also
provides the correlation matrix for each configuration (fig. 3). All
correlations depend visibly on the collider scenario. The unpolarised scenario
does not allow an unambiguous determination of all 3 parameters from the
distributions and relies on the external constraints. Assuming the
differential cross sections as perfectly known, all polarisations can be
determined if at least one beam polarised.

This test lays the foundation for implementing further physical and systematic
effects.

Figure 2: Uncertainties for the luminosity and polarisation parameters for the
different tested collider configurations in the first validation test of the
framework.

Figure 3: Correlation matrices for the luminosity and polarisation parameters
in the different collider configurations of the first validation test. The
upper line uses $\mathcal{L}=2\,\text{ab}^{-1}$, the lower
$\mathcal{L}=10\,\text{ab}^{-1}$. The three rows are from-left-to-right:
unpolarised, electrons-only polarised, both beams polarised.

##  $\boldsymbol{\mu}$ acceptance parametrisation

A test of the behaviour of a systematic effect in a combined fit requires an
implementation of the effect into the fit. This section describes the choice
of the effect, the choice of the parametrisation in fit, and the validation of
the parametrisation.

The choice of the systematic effect falls on the muon acceptance. It is chosen
because it has been a challenging uncertainty in previous electroweak
measurements and it has a simple shape which is stable for high-momentum
muons. The tracking efficiency at future $e^{+}e^{-}$ detectors, which is the
core component of the muon acceptance, is almost $100\%$ in the central region
and largely independent of azimuthal angle, and sharply falls in very forward
region (fig. 4(a)).

This behaviour corresponds approximately to an acceptance range with an upper
edge $\cos\theta_{up}$ and a lower edge $\cos\theta_{low}$ of the muon polar
angle in the detector frame, between which the acceptance is perfect. A simple
box acceptance model is therefore chosen as a first implementation, with a
width $w=\cos\theta_{up}-\cos\theta_{low}$ and a center
$c=\frac{1}{2}(\cos\theta_{up}+\cos\theta_{low})$ parameter (fig. 4(b)). An
event passes this acceptance if all muons in the event are within the box
range. Here, the initial values for $c$ and $w$ correspond to edges at
$\theta_{low}=180^{\circ}-\theta_{up}=7^{\circ}$.

The combined fit will extract these two parameters from the variations they
cause on the differential distributions. These variations need a
parameterisation that can be implemented into the fit. A quadratic
parametrisation is chosen in terms of the variations $\left(\Delta
c=c^{\prime}-c,\Delta w=w^{\prime}-w\right)$ of the parameters

$\frac{d\sigma^{bin}}{\sigma^{bin}}=k_{0}^{bin}+k_{c}^{bin}\Delta
c+k_{w}^{bin}\Delta w+k_{c2}^{bin}\Delta c^{2}+k_{w2}^{bin}\Delta
w^{2}+k_{cw}^{bin}\Delta c\Delta w$ (1)

where the coefficients $k$ are bin-dependent.

The coefficients of the polynomial are determined by applying the box-shaped
acceptance on the generator-level events and slightly varying the cut
parameters. A common scale $\delta$ of these variations describes the order-
of-magnitude of the expected uncertainties. Here, $\delta=2\times 10^{-4}$ is
chosen. The coefficient determination tests 29 points
$(c^{\prime},w^{\prime})$ in the range $\sqrt{\Delta c^{2}+\Delta
w^{2}}\in[0,1.5]\delta$. It first chooses 6 points and solves the linear
equation system to get an initial estimate of the coefficients. Then, using
these starting values, it fits the polynomial to all 29 points and extracts
the final coefficients.

The parametrisation step requires a validation. Such a validation needs to
show that the quadratic polynomial approach accurately estimates the box-
shaped acceptance on all affected bins. A definition and comparison of the two
following quantities for each tested $\left(\Delta c,\Delta w\right)$ point
can provide such a validation. The first quantity evaluates the impact of
changing the cut parameters when applying the cut on the events.

$\chi^{2}_{shift}=\sum_{\text{affected bins}}\left(\frac{N_{cut}^{(\Delta
c,\Delta w)}-N_{cut}^{0}}{\sqrt{N_{cut}^{0}}}\right)^{2}$ (2)

$N_{cut}$ describes the number of events remaining after the cut, and
$N_{cut}^{0}$ uses the initial non-varied $c$ and $w$ values. The second
quantity tests the impact of the deviation that parametrisation shows with
respect to the box-shaped cut applied to the events.

$\chi^{2}_{par}=\sum_{\text{affected bins}}\left(\frac{N_{par}^{(\Delta
c,\Delta w)}-N_{cut}^{(\Delta c,\Delta w)}}{\sqrt{N_{cut}^{(\Delta c,\Delta
w)}}}\right)^{2}$ (3)

$N_{par}$ is the number of events that the parametrisation yields.

The validation tests many different combinations of $\Delta c$ and $\Delta w$
in range of $\sqrt{\Delta c^{2}+\Delta w^{2}}\in[0.5,2]\delta$. At each point,
the impact of cut can be considered more relevant than the deviation caused by
the parametrisation if the tested point has $\chi^{2}_{shift}>\chi^{2}_{par}$.
Most tested points are found to be far in $\chi^{2}_{shift}\gg\chi^{2}_{par}$
region, indicating that the parametrisation is safe to use (fig. 5). For the
$\mu\mu$ distributions, a few points are found at the
$\chi^{2}_{shift}\approx\chi^{2}_{par}$ line. These are unproblematic because
they have a low impact on the distribution and other distributions strongly
restrict the corresponding direction.

(a)

(b)

Figure 4: Comparison of (4(a)) a realistic tracking efficiency [10] and (4(b))
the simple box-shape model for the muon acceptance which is used in this
study.  Figure 5: Example for the validation test of the muon acceptance
parametrisation that is employed in the fit, using
$qq\mu^{-}\bar{\nu}\,(q=u,d,s,c,b)$ ($\sqrt{s*}\in[180,275]\,$GeV) events for
$e^{-}_{L}e^{+}_{R}$.

##  $\boldsymbol{\mu}$ acceptance in different collider setups

This section presents a first test of the implementation of the box-shaped
muon acceptance in the different collider configurations described above. Each
fit considers the corresponding luminosity and polarisation parameters as well
as the muon acceptance parameters. It does not consider any physical effects,
assumes all chiral cross sections to be known exactly, and does not include
backgrounds or high-level selection efficiencies.

Under these idealised conditions, the fits determine the $c$ and $w$
parameters with an accuracy of the order of $10^{-5}$ (fig. 6). Observed
differences come from changes in the statistics, either a factor $\sqrt{5}$
from the luminosity change, or the optimised luminosity sharing in the case of
where both beams are polarised. Fixing the luminosity and polarisation
parameters leaves the results unchanged.

The acceptance parameters do not correlate with each other or with the
polarisations (fig. 7). A slight anticorrelation of the width parameter $w$
with the luminosity originates from the common effect of increasing the
integrated number of events. Changes in the collider configuration do not
influence the correlations of the muon acceptance parameters.

This first test for the inclusion of systematic effects does not include any
physical effects. It cannot make any statement on the potential advantages of
beam polarisation with respect to systematic uncertainties. Such statements
will require the implementation of physical effects, which will be the next
steps in this study.

Figure 6: Uncertainties for the muon acceptance parameters for the different
tested collider configurations in the test that involved the muon acceptance
and the luminosity and polarisation parameters.

Figure 7: Correlation matrices for the parameters in the test that involved
the muon acceptance, the luminosity and polarisation. The upper line uses
$\mathcal{L}=2\,\text{ab}^{-1}$, the lower $\mathcal{L}=10\,\text{ab}^{-1}$.
The three rows are from-left-to-right: unpolarised, electrons-only polarised,
both beams polarised.

##  Conclusion & Outlook

Beam polarisation may help reduce the impact of systematic uncertainties when
all different polarised datasets are included in a combined fit. A fit
framework is set up that can perform combined fits on the different datasets
of a polarised collider. First tests in different collider configurations
validate the usage of the framework for the case of extracting the beam
polarisation and luminosity parameters from $\mu\mu$ and $qq\mu\nu$
differential distributions. The implementation and validation of a simplified
muon acceptance into the fit shows that it may be feasible to directly include
realistic systematic effects. Further studies that include physical effects
will allow tests of the impact of beam polarisation on systematic
uncertainties.

### Acknowledgments

This work was funded by the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. It has
benefited from the computing services provided by the German National Analysis
Facility (NAF)[11].

## References

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