Simplicial Burnside ring (original) (raw)

Simplicial Burnside ring

Ibrahima Tounkara

Abstract

This paper develops links between the Burnside ring of a finite group GG 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):19 A22,18G30,06 A11,20 J15\operatorname{MSC}(2010): 19 \mathrm{~A} 22,18 \mathrm{G} 30,06 \mathrm{~A} 11,20 \mathrm{~J} 15
Keywords: Burnside ring, simplicial sets, Poset, biset functor.

1 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 GG be a finite group. The Grothendieck ring constructed from the category of GG-sets is denoted by B(G)B(G) and is called the Burnside ring of GG. If XX is a finite GG-set, let [X][X] be its image in B(G)B(G). Additively, B(G)B(G) is the free abelian group on isomorphism classes of transitive GG-sets. Equivalently, an additive Z\mathbb{Z} basis is given by the [G/H][G / H] where HH runs through a set [C(G)][C(G)] of representatives of conjugacy classes of subgroups of GG. The multiplication comes from the decomposition of G/H×G/KG / H \times G / K into orbits. The ring B(G)B(G) is commutative with unit [G/G][G / G]. If HH is a subgroup of GG, then there is a unique linear form ϕH:B(G)→Z\phi_{H}: B(G) \rightarrow \mathbb{Z} such that ϕH([X])=∣X∣L∣\phi_{H}([X])=|X|^{L} \mid for any GG-set XX. It is clear moreover that ϕH\phi_{H} is a ring homomorphism, and Burnside’s theorem ( [4] Chap. XII Theorem I) is equivalent to the ghost map Φ=∏H∈[C(G)]ϕHG:B(G)→∏H∈[C(G)]Z\Phi=\prod_{H \in[C(G)]} \phi_{H}^{G}: B(G) \rightarrow \prod_{H \in[C(G)]} \mathbb{Z} being injective. The cokernel of the ghost map is finite, and has been explicitly described by Dress [5]. In particular, the ghost map QΦ:QB(G)→∏H∈[C(G)]Q\mathbb{Q} \Phi: \mathbb{Q} B(G) \rightarrow \prod_{H \in[C(G)]} \mathbb{Q} is an algebra isomorphism, where QB(G)=Q⊗ZB(G)\mathbb{Q} B(G)=\mathbb{Q} \otimes_{\mathbb{Z}} B(G). This shows that QB(G)\mathbb{Q} B(G) is a split semisimple commutative Q\mathbb{Q} 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 pp is 0 or a prime and IH,p={X∈B(G)∣ϕH(X)∈pZ}I_{H, p}=\{X \in B(G) \mid \phi_{H}(X) \in p \mathbb{Z}\}, then any prime ideal in B(G)B(G) is of the form IH,pI_{H, p} for some H,pH, 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)B(G). If XX is a finite GG-set, the Q\mathbb{Q}-vector space QX\mathbb{Q} X with basis XX has a natural QG\mathbb{Q} G-module structure, induced by the action of GG on XX. The construction X↦QXX \mapsto \mathbb{Q} X maps disjoint unions of GG-sets to direct sums of QG\mathbb{Q} G-modules, and so it induces a map Ch:B(G)→RQ(G)C h: B(G) \rightarrow R_{\mathbb{Q}}(G). There are important connections between the Burnside ring and the permutation representations. This latter map, leads to an associated map Spec⁡:Spec⁡(RQ(G))→Spec⁡B(G)\operatorname{Spec}: \operatorname{Spec}\left(R_{\mathbb{Q}}(G)\right) \rightarrow \operatorname{Spec} B(G)

which is always injective (See [6]).
The slice Burnside ring Ξ(G)\Xi(G) introduced by Serge Bouc, is built as the Grothendieck ring of the category of morphisms of finite GG-sets, instead of the category of finite GG-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 Z\mathbb{Z}-module. The investigation of the slices, that is the pairs of groups (T,S)(T, S) such that SS is a subgroup of TT, is a central subject in the study of the slice Burnside ring. One reason for considering slices is that if f:X→Yf: X \rightarrow Y is a morphism of finite GG-sets then in Ξ(G)\Xi(G)

[X→fY]=∑x∈[G\X][G/Gx→pG/Gf(x)][X \xrightarrow{f} Y]=\sum_{x \in[G \backslash X]}\left[G / G_{x} \xrightarrow{p} G / G_{f(x)}\right]

where square brackets to denote here the image of the isomorphism class of ff in Ξ(G)\Xi(G) and GxG_{x} denotes the stabilizer of xx. Thus, the group Ξ(G)\Xi(G) is generated by the elements [G/S0→pG/S1\left[G / S_{0} \xrightarrow{p} G / S_{1}\right. ] where (S1,S0)\left(S_{1}, S_{0}\right) runs through a set [Π(G)][\Pi(G)] of representatives of conjugacy classes of slices of GG. 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 Q\mathbb{Q}, the slice Burnside ring becomes a split semisimple Q\mathbb{Q}-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 Bn(G)B_{n}(G), for n∈Nn \in \mathbb{N} such that B0(G)=B(G)B_{0}(G)=B(G) and B1(G)=Ξ(G)B_{1}(G)=\Xi(G). We extend to Bn(G)B_{n}(G) most of the properties recalled above in the case n=0n=0 or n=1n=1.

Recall that any poset Π\Pi can be treated as a category in which the objects are the elements of Π\Pi and in which there is exactly one morphism x→yx \rightarrow y if x≤yx \leq y and there are no other morphisms. The nerve of Π\Pi is then the same as the ordered simplicial complex associated ( that is the vertices of Πn\Pi_{n} are the objects of Π\Pi and the nn-simplices are the chains of objects of Π\Pi of length nn, with face maps given by di(S0,…,Sn)=(S0,…,Sˇi,…,Sn)d_{i}\left(S_{0}, \ldots, S_{n}\right)=\left(S_{0}, \ldots, \check{S}_{i}, \ldots, S_{n}\right), where as usual, the term ∧{ }^{\wedge} denotes a term that is being omitted and the degeneracy maps given by sj(S0,…,Sn)=(S0,…,Sj,Sj,…,Sn)s_{j}\left(S_{0}, \ldots, S_{n}\right)=\left(S_{0}, \ldots, S_{j}, S_{j}, \ldots, S_{n}\right) ). In particular, if we consider the collection of all subgroups of GG ordered by inclusion, we get the nerve category Π∙(G)\Pi_{\bullet}(G) where elements in Πn(G)\Pi_{n}(G), whose are just chains

Sˇ:S0⊆S1⊆⋯⊆Sn\check{\mathcal{S}}: S_{0} \subseteq S_{1} \subseteq \cdots \subseteq S_{n}

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

(S0,…,Sn)⋅(T0,…,Tn)=∑g∈[S0\G/T0](S0∩ST0,…,Sn∩STn)\left(S_{0}, \ldots, S_{n}\right) \cdot\left(T_{0}, \ldots, T_{n}\right)=\sum_{g \in\left[S_{0} \backslash G / T_{0}\right]}\left(S_{0} \cap \mathcal{S} T_{0}, \ldots, S_{n} \cap \mathcal{S} T_{n}\right)

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 GG-sets (we abuse for using the term nerve of GSet{ }_{G} S e t ). It turns out that the obtained group Bn(G)B_{n}(G), called the nn-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 Bn(G)B_{n}(G), since then will known regard this ring as a “geometric realization” of some simplicial GG-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 nn-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 Bn(G)B_{n}(G) is a commutative semisimple algebra afer tensoring with Q\mathbb{Q}, isomorphic to a direct sum indexed by [Πn(G)]\left[\Pi_{n}(G)\right] of copies of Q\mathbb{Q}. In sect. 4 . we give an explicit formula for the primitive idempotents of QBn(G)\mathbb{Q} B_{n}(G). Sect. 5 is devoted to the study of the prime spectrum of Bn(G)B_{n}(G) by extending Dress’s characterization of solvable groups for B(G)B(G). The last section examines the Green biset functor structure of BnB_{n}.

2 Simplicial Burnside group

For GG a finite group, let C∗G\mathcal{C}_{*}^{G} be the nerve of the category G{ }_{G} Set of finite GG-sets (see [9] for more details, [8] P.177), that is, the simplicial set whose nn-simplices are diagrams

CnG={Xnf:=X0→f1X1→f2…Xn−1→fnXn}\mathcal{C}_{n}^{G}=\left\{\mathcal{X}_{n}^{f}:=X_{0} \xrightarrow{f_{1}} X_{1} \xrightarrow{f_{2}} \ldots X_{n-1} \xrightarrow{f_{n}} X_{n}\right\}

where the XiX_{i} are GG-sets and the fif_{i} are morphisms of GG-sets, and the degeneracy sis_{i} and face di1d_{i}{ }^{1} maps are defined by including an identity Xi→idXiX_{i} \xrightarrow{i d} X_{i}, and leaving out X0X_{0} if i=0i=0, contracting Xi−1→fiXi→fi+1Xi+1X_{i-1} \xrightarrow{f_{i}} X_{i} \xrightarrow{f_{i+1}} X_{i+1} to Xi−1→fi+1∘fiXi+1X_{i-1} \xrightarrow{f_{i+1} \circ f_{i}} X_{i+1} if 0<i<0<i< nn, leaving XnX_{n} if i=ni=n, respectively. Then the following identities may be verified directly

dj∘di=di∘dj+1 for i≤jsi∘sj=sj+1∘si for i≤jdi∘sj={sj−1∘di if i<jid[n] if i=j,j+1sj∘di−1 otherwise \begin{gathered} d_{j} \circ d_{i}=d_{i} \circ d_{j+1} \quad \text { for } i \leq j \\ s_{i} \circ s_{j}=s_{j+1} \circ s_{i} \quad \text { for } i \leq j \\ d_{i} \circ s_{j}= \begin{cases}s_{j-1} \circ d_{i} & \text { if } \quad i<j \\ i d_{[n]} & \text { if } \quad i=j, j+1 \\ s_{j} \circ d_{i-1} & \text { otherwise } \end{cases} \end{gathered}

Notation 2.1. Let CnG\mathcal{C}_{n}^{G} be the category defined as follows:

Proposition 2.2. For a non-negative integer nn, the category CnG\mathcal{C}_{n}^{G} has finite products ×\times and coproducts ⊔\sqcup induced by those of the category of finite GG-sets, respectively. It has also an initial object ∅={∅⟶∅⟶…∅⟶∅}\varnothing=\{\varnothing \longrightarrow \varnothing \longrightarrow \ldots \varnothing \longrightarrow \varnothing\}.

Proof. This is straightforward.
Definition 2.3. Let GG be a finite group. A nn-slice of GG is a (n+1)(n+1)-tuple (S0,S1,…,Sn)\left(S_{0}, S_{1}, \ldots, S_{n}\right) of subgroups of GG, with Si−1≤Si,∀i∈{1,…,n}S_{i-1} \leq S_{i}, \forall i \in\{1, \ldots, n\}. It is helpful to refer the tuple as S‾\overline{\mathcal{S}}, in which each composite Si\mathcal{S}_{i} as representing the group SiS_{i}. The set of all nn-slices of GG will be denoted by Πn(G)\Pi_{n}(G).

[1]

  1. 1{ }^{1} Normally, we might be careful to label the face maps from CnG\mathcal{C}_{n}^{G} to Cn−1G\mathcal{C}_{n-1}^{G} as d0n,…,dnnd_{0}^{n}, \ldots, d_{n}^{n}, similarly for the degeneracy maps, but this is rarely done in practice. ↩︎

Definition 2.4. For any nerve C⋆\mathcal{C}_{\star}, define a pre-ordering ≤\leq of Ob(Cn)O b\left(\mathcal{C}_{n}\right) by A≤BA \leq B if Hom⁡Cn(A,B)≠∅\operatorname{Hom}_{\mathcal{C}_{n}}(A, B) \neq \varnothing and an equivalence ≅\cong on Ob(Cn)O b\left(\mathcal{C}_{n}\right) by A≅BA \cong B if and only if A≤BA \leq B and B≤AB \leq A. So, on Πn(G)\Pi_{n}(G), we have the following relation

Tˇ:=(T0,…,Tn)≤Sˇ:=(S0,…,Sn)⟺Ti≤Si,∀i=0,…,n\check{T}:=\left(T_{0}, \ldots, T_{n}\right) \leq \check{\mathcal{S}}:=\left(\mathcal{S}_{0}, \ldots, \mathcal{S}_{n}\right) \Longleftrightarrow T_{i} \leq \mathcal{S}_{i}, \quad \forall i=0, \ldots, n

Recall that in G{ }_{G} Set, XX is indecomposable if and only if XX is simple and any simple GG-set is isomorphic to G/HG / H for some subgroup HH of GG.

Notation 2.5.

(G/Sˇ)n:=(G/S0→p1G/S1→p2…G/Sn−1→pnG/Sn)(G / \check{\mathcal{S}})_{n}:=\left(G / S_{0} \xrightarrow{p_{1}} G / S_{1} \xrightarrow{p_{2}} \ldots G / S_{n-1} \xrightarrow{p_{n}} G / S_{n}\right)

where pip_{i} are the projection morphisms.
The (n,G)(n, G)-simplices (G/Sˇ)n(G / \check{\mathcal{S}})_{n} are indecomposable in the sense that (G/Sˇ)n=(G / \check{\mathcal{S}})_{n}= YnS⊔ZnS⟹YnS=∅\mathcal{Y}_{n}^{\mathcal{S}} \sqcup \mathcal{Z}_{n}^{\mathcal{S}} \Longrightarrow \mathcal{Y}_{n}^{\mathcal{S}}=\varnothing or ZnS=∅\mathcal{Z}_{n}^{\mathcal{S}}=\varnothing.

ϕSˇ(Xnf):=∣Hom⁡CnS((G/Sˇ)n,Xnf)∣\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right):=\left|\operatorname{Hom}_{\mathcal{C}_{n}^{\mathrm{S}}}\left((G / \check{\mathcal{S}})_{n}, \mathcal{X}_{n}^{f}\right)\right|

the number of elements in the set Hom⁡CnS((G/Sˇ)n,Xnf)\operatorname{Hom}_{\mathcal{C}_{n}^{\mathrm{S}}}\left((G / \check{\mathcal{S}})_{n}, \mathcal{X}_{n}^{f}\right).
Proposition 2.6. Let Xnf,YnS\mathcal{X}_{n}^{f}, \mathcal{Y}_{n}^{\mathcal{S}} be (n,G)(n, G)-simplices

ϕSˇ(Xnf)=∣f1−1(f2−1(−fi−1(−fn−1−1(fn−1(XnSn)Sn−1)Sn−2…)Si…)S1)S0∣\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right)=\left|f_{1}^{-1}\left(f_{2}^{-1}\left(-f_{i}^{-1}\left(-f_{n-1}^{-1}\left(f_{n}^{-1}\left(X_{n}^{S_{n}}\right)^{S_{n-1}}\right)^{S_{n-2} \ldots}\right)^{S_{i}} \ldots\right)^{S_{1}}\right)^{S_{0}}\right|

in particular, for any nn-slice Tˇ\check{T}, one has

ϕSˇ((G/Tˇ)n)=∣{g∈G/T0∣Sˇg≤Tˇ}∣\phi_{\check{\mathcal{S}}}\left((G / \check{T})_{n}\right)=\left|\left\{g \in G / T_{0} \mid \check{\mathcal{S}}^{g} \leq \check{T}\right\}\right|

Proof. Observe that for any i=0,…,ni=0, \ldots, n, there is a bijection between Hom⁡(G/Si,Xi)\operatorname{Hom}\left(G / \mathcal{S}_{i}, X_{i}\right) and the set XiSi:={x∈Xi∣gx=xX_{i}^{\mathcal{S}_{i}}:=\left\{x \in X_{i} \mid g x=x\right. for all g∈Si}\left.g \in \mathcal{S}_{i}\right\}. Indeed, each ff in Hom⁡(G/Si,Xi)\operatorname{Hom}\left(G / \mathcal{S}_{i}, X_{i}\right) maps the element Si∈G/Si\mathcal{S}_{i} \in G / \mathcal{S}_{i} onto an element xi∈Xix_{i} \in X_{i} which is invariant by Si\mathcal{S}_{i}. Further, ff is completely determined by xix_{i}, since f(gSi)=gxif\left(g \mathcal{S}_{i}\right)=g x_{i} for all g∈Gg \in G. The correspondence f↦xif \mapsto x_{i} gives the desired bijection. Therefore, the set Hom⁡((G/Sˇ)n,Xnf)\operatorname{Hom}\left((G / \check{\mathcal{S}})_{n}, \mathcal{X}_{n}^{f}\right) is in bijection with de set

{(x0,x1,…,xn)∈X0S0×X1S1×…×XnSn∣f1(x0)=x1;…;fn(xn−1)=xn}\left\{\left(x_{0}, x_{1}, \ldots, x_{n}\right) \in X_{0}^{S_{0}} \times X_{1}^{S_{1}} \times \ldots \times X_{n}^{S_{n}} \mid f_{1}\left(x_{0}\right)=x_{1} ; \ldots ; f_{n}\left(x_{n-1}\right)=x_{n}\right\}

where the last equalities follow from the commutativity of the diagram
img-0.jpeg

If we take Xi=G/TiX_{i}=G / T_{i} for i=0,…,ni=0, \ldots, n, we note that the left coset gTig T_{i} is Si\mathcal{S}_{i}-invariant if and only if Si⋅gTi=gTi\mathcal{S}_{i} \cdot g T_{i}=g T_{i}, that is, g−1Sig≤Tig^{-1} \mathcal{S}_{i} g \leq T_{i}. This establishes that

(G/Ti)Si={gTi∣g∈G,g−1Sig≤Ti}\left(G / T_{i}\right)^{\mathcal{S}_{i}}=\left\{g T_{i} \mid g \in G, g^{-1} S_{i} g \leq T_{i}\right\}

and therefore (G/Ti)Si=∅\left(G / \mathcal{T}_{i}\right)^{S_{i}}=\varnothing unless Si≤GTi\mathcal{S}_{i} \leq_{G} \mathcal{T}_{i} ( by which we mean that SiS_{i} is conjugated to a subgroup of Ti\mathcal{T}_{i} ). In particular,

ϕSˇ((G/Tˇ)n)=∣{(gT0,…,gTn)∣g∈G,S0S≤T0,…,SnS≤Tn and gTi−1=gTi}∣=∣{g∈G/T0∣Sˇg≤Tˇ}∣\begin{aligned} \phi_{\check{\mathcal{S}}}\left((G / \check{\mathcal{T}})_{n}\right) & =\left|\left\{\left(g \mathcal{T}_{0}, \ldots, g \mathcal{T}_{n}\right) \mid g \in G, S_{0}^{\mathcal{S}} \leq \mathcal{T}_{0}, \ldots, S_{n}^{\mathcal{S}} \leq \mathcal{T}_{n} \text { and } g \mathcal{T}_{i-1}=g \mathcal{T}_{i}\right\}\right| \\ & =\left|\left\{g \in G / \mathcal{T}_{0} \mid \check{\mathcal{S}}^{g} \leq \check{\mathcal{T}}\right\}\right| \end{aligned}

Corollary 2.7. Let Xnf,Yng\mathcal{X}_{n}^{f}, \mathcal{Y}_{n}^{g} be (n,G)(n, G)-simplices

  1. If Xnf\mathcal{X}_{n}^{f} and Yng\mathcal{Y}_{n}^{g} are isomorphic then ϕSˇ(Xnf)=ϕSˇ(Yng)\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right)=\phi_{\check{\mathcal{S}}}\left(\mathcal{Y}_{n}^{g}\right).
  2. Let pp be a prime and let Sˇ\check{\mathcal{S}} be a nn-slice of GG. If PP is a p-subgroup of NG(Sˇ)=N_{G}(\check{\mathcal{S}})= ∩(=0nNG(Si)\cap_{(=0}^{n} N_{G}\left(S_{i}\right) and PSˇP \check{\mathcal{S}} denotes the nn-slice (PS0,…,PSn)\left(P S_{0}, \ldots, P S_{n}\right), then

ϕSˇ(Xnf)≡ϕPSˇ(Xnf)( mod .p)\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right) \equiv \phi_{P \check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right)(\bmod . p)

for any (n,G)(n, G)-simplex Xnf\mathcal{X}_{n}^{f}.
3. Two indecomposable (n,G)(n, G)-simplices (G/Sˇ)n(G / \check{\mathcal{S}})_{n} and (G/Tˇ)n(G / \check{\mathcal{T}})_{n} are isomorphic if and only if the nn-slices Sˇ\check{\mathcal{S}} and Tˇ\check{\mathcal{T}} are conjugate (we set Sˇ=GTˇ\check{\mathcal{S}}={ }_{G} \check{\mathcal{T}} ).
4. If Sˇ\check{\mathcal{S}} and Tˇ\check{\mathcal{T}} are two nn-slices, then ϕSˇ(Xnf)≤ϕTˇ(Xnf)\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right) \leq \phi_{\check{\mathcal{T}}}\left(\mathcal{X}_{n}^{f}\right) for any (n,G)(n, G)-simplex Xnf\mathcal{X}_{n}^{f} if and only if Tˇ≤GSˇ\check{\mathcal{T}} \leq_{G} \check{\mathcal{S}}.
In particular, ϕSˇ(Xnf)=ϕTˇ(Xnf)\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right)=\phi_{\check{\mathcal{T}}}\left(\mathcal{X}_{n}^{f}\right) for all (n,G)(n, G)-simplices Xnf\mathcal{X}_{n}^{f} if and only if Tˇ=GSˇ\check{\mathcal{T}}={ }_{G} \check{\mathcal{S}}.
5. For any two indecomposable (n,G)(n, G)-simplices (G/Sˇ)n(G / \check{\mathcal{S}})_{n} and (G/Tˇ)n(G / \check{\mathcal{T}})_{n}, one has ϕSˇ((G/Sˇ)n)\phi_{\check{\mathcal{S}}}\left((G / \check{\mathcal{S}})_{n}\right) divides ϕTˇ((G/Sˇ)n)\phi_{\check{\mathcal{T}}}\left((G / \check{\mathcal{S}})_{n}\right).

Proof.

  1. It is clear, since any isomorphism of GG-sets μi:Xi→Yi\mu_{i}: X_{i} \rightarrow Y_{i} induces a bijection XiS→YiSX_{i}^{S} \rightarrow Y_{i}^{S} on the sets of fixed points by any subgroup SS of GG.
  2. For any (n,G)(n, G)-simplex Xnf\mathcal{X}_{n}^{f}, the set

Inv⁡Sˇ(Xnf):=f1−1(f2−1(…fi−1(…fn−1−1(fn−1(XnSn)Sn−1)Sn−2…)Si…)S1)S0\operatorname{Inv}_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right):=f_{1}^{-1}\left(f_{2}^{-1}\left(\ldots f_{i}^{-1}\left(\ldots f_{n-1}^{-1}\left(f_{n}^{-1}\left(X_{n}^{S_{n}}\right)^{S_{n-1}}\right)^{S_{n-2}} \ldots\right)^{S_{i}} \ldots\right)^{S_{1}}\right)^{S_{0}}

is invariant by NG(Sˇ)N_{G}(\check{\mathcal{S}}), and so

∣Inv⁡Sˇ(Xnf)∣≡∣f1−1(…fi−1(…fn−1(XnSn)Sn−1…)Si…)PS0∣( mod p)\left|\operatorname{Inv}_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right)\right| \equiv\left|f_{1}^{-1}\left(\ldots f_{i}^{-1}\left(\ldots f_{n}^{-1}\left(X_{n}^{S_{n}}\right)^{S_{n-1}} \ldots\right)^{S_{i}} \ldots\right)^{P S_{0}}\right|(\bmod p)

and moreover

f1−1(…fi−1(…fn−1(XnSn)Sn−1…)Si…)PS0≡f1−1(…fi−1(…fn−1(XnPSn)PSn−1…)PSi…)PS0f_{1}^{-1}\left(\ldots f_{i}^{-1}\left(\ldots f_{n}^{-1}\left(X_{n}^{S_{n}}\right)^{S_{n-1}} \ldots\right)^{S_{i}} \ldots\right)^{P S_{0}} \equiv f_{1}^{-1}\left(\ldots f_{i}^{-1}\left(\ldots f_{n}^{-1}\left(X_{n}^{P S_{n}}\right)^{P S_{n-1}} \ldots\right)^{P S_{i}} \ldots\right)^{P S_{0}}

  1. If (G/Sˇ)n≅(G/Tˇ)n(G / \check{\mathcal{S}})_{n} \cong(G / \check{\mathcal{T}})_{n} then Hom⁡(G/Sˇ,G/Tˇ)≠∅\operatorname{Hom}(G / \check{\mathcal{S}}, G / \check{\mathcal{T}}) \neq \varnothing and so Sˇ≤GTˇ\check{\mathcal{S}} \leq_{G} \check{\mathcal{T}}. Therefore Sˇ=G\check{\mathcal{S}}={ }_{G} Tˇ\check{\mathcal{T}} by symmetry. Conversely if Sˇ=GTˇ\check{\mathcal{S}}={ }_{G} \check{\mathcal{T}}, for example Sˇg=Tˇ\check{\mathcal{S}}^{g}=\check{\mathcal{T}} with g∈Gg \in G, then there exists an isomorphism of (n,G)(n, G)-simplices (μi:G/Ti→G/Si)0≤i≤n\left(\mu_{i}: G / \mathcal{T}_{i} \rightarrow G / \mathcal{S}_{i}\right)_{0 \leq i \leq n}, given by

μi(g′Ti)=μi(g′g−1Sig)=g′g−1Si, for g′∈G\mu_{i}\left(g^{\prime} \mathcal{T}_{i}\right)=\mu_{i}\left(g^{\prime} g^{-1} S_{i} g\right)=g^{\prime} g^{-1} S_{i}, \text { for } g^{\prime} \in G

So (G/Sˇ)n≅(G/Tˇ)n(G / \check{\mathcal{S}})_{n} \cong(G / \check{\mathcal{T}})_{n} as (n,G)(n, G)-simplices if and only Sˇ=GTˇ\check{\mathcal{S}}={ }_{G} \check{\mathcal{T}}.

  1. If ϕSˇ(Xnf)≤ϕT(Xnf)\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right) \leq \phi_{\mathcal{T}}\left(\mathcal{X}_{n}^{f}\right) for any (n,G)(n, G)-simplex, then in particular ϕT((G/Sˇ)n)≠0\phi_{\mathcal{T}}\left((G / \check{\mathcal{S}})_{n}\right) \neq 0 since ϕSˇ(G/Sˇ)≠0\phi_{\check{\mathcal{S}}}(G / \check{\mathcal{S}}) \neq 0, and so Tˇ≤GSˇ\check{\mathcal{T}} \leq_{G} \check{\mathcal{S}}. On the other hand, if Tˇ≤gSˇ=Kˇ\check{\mathcal{T}} \leq^{\mathrm{g}} \check{\mathcal{S}}=\check{\mathcal{K}}, then ϕSˇ(Xnf)=∣Inv⁡Sˇ(Xnf)∣=∣Inv⁡Kˇ(Xnf)∣≤∣Inv⁡Tˇ(Xnf)∣=ϕT(Xnf)\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right)=\left|\operatorname{Inv}_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right)\right|=\left|\operatorname{Inv}_{\check{\mathcal{K}}}\left(\mathcal{X}_{n}^{f}\right)\right| \leq\left|\operatorname{Inv}_{\check{\mathcal{T}}}\left(\mathcal{X}_{n}^{f}\right)\right|=\phi_{\mathcal{T}}\left(\mathcal{X}_{n}^{f}\right).
  2. Consider the action of NG(Sˇ)N_{G}(\check{\mathcal{S}}) on G/S0G / S_{0} defined by

x.gS0=gx−1S0x . g S_{0}=g x^{-1} S_{0}

for x∈NG(Sˇ)x \in N_{G}(\check{\mathcal{S}}) and gS0∈G/S0g S_{0} \in G / S_{0}. Then S0S_{0} acts trivially on G/S0G / S_{0} and so it becomes a left NG(Sˇ)/S0N_{G}(\check{\mathcal{S}}) / S_{0}. Moreover, NG(Sˇ)/S0N_{G}(\check{\mathcal{S}}) / S_{0} acts freely on G/S0G / S_{0}. Note that for any (n,G)(n, G)-slice Tˇ\check{\mathcal{T}}, the set

Inv⁡Tˇ((G/Sˇ)n):={gS0∣Tˇ⋅gSˇ=gSˇ}\operatorname{Inv}_{\check{\mathcal{T}}}\left((G / \check{\mathcal{S}})_{n}\right):=\left\{g S_{0} \mid \check{\mathcal{T}} \cdot g \check{\mathcal{S}}=g \check{\mathcal{S}}\right\}

is an NG(Sˇ)/S0N_{G}(\check{\mathcal{S}}) / S_{0}-subset of G/S0G / S_{0}. So NG(Sˇ)/S0N_{G}(\check{\mathcal{S}}) / S_{0} acts freely on Inv⁡Tˇ((G/Sˇ)n)\operatorname{Inv}_{\check{\mathcal{T}}}\left((G / \check{\mathcal{S}})_{n}\right), and so ∣NG(Sˇ)/S0∣\left|N_{G}(\check{\mathcal{S}}) / S_{0}\right| divides ∣Inv⁡Tˇ((G/Sˇ)n)∣\left|\operatorname{Inv}_{\check{\mathcal{T}}}\left((G / \check{\mathcal{S}})_{n}\right)\right|.

Definition 2.8. A (n,G)(n, G)-simplex Xnf\mathcal{X}_{n}^{f} is called ii-sliceable with ii in {0,…,n}\{0, \ldots, n\} if Xi=X_{i}= Ai⊔BiA_{i} \sqcup B_{i} as disjoint union of two non-empty GG-sets.

Remark 2.9. Note that for any ii-sliceable (n,G)(n, G)-simplex Xnf\mathcal{X}_{n}^{f}, one has (n,G)(n, G)-simplices

defined inductively by

Aj−1:=fj−1(Aj) and Bj−1:=fj−1(Bj)A_{j-1}:=f_{j}^{-1}\left(A_{j}\right) \text { and } B_{j-1}:=f_{j}^{-1}\left(B_{j}\right)

for any 1≤j≤i1 \leq j \leq i.
We set [Xnf,Anf,Bnf]i\left[\mathcal{X}_{n}^{f}, \mathcal{A}_{n}^{f}, \mathcal{B}_{n}^{f}\right]_{i} to denote the corresponding triple. If Γ\Gamma denotes the class of such triple (Xnf,Anf,Bnf)\left(\mathcal{X}_{n}^{f}, \mathcal{A}_{n}^{f}, \mathcal{B}_{n}^{f}\right), then Γ\Gamma is closed by isomorphism.

Proposition 2.10.

For any (Xnf,Anf,Bnf)∈Γ\left(\mathcal{X}_{n}^{f}, \mathcal{A}_{n}^{f}, \mathcal{B}_{n}^{f}\right) \in \Gamma and for a nn-slice Sˇ\check{\mathcal{S}} fixed, we have

ϕSˇ(Xnf)=ϕSˇ(Anf)+ϕSˇ(Bnf)\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right)=\phi_{\check{\mathcal{S}}}\left(\mathcal{A}_{n}^{f}\right)+\phi_{\check{\mathcal{S}}}\left(\mathcal{B}_{n}^{f}\right)

Proof. Let xnx_{n} be an element of XnSnX_{n}^{S_{n}} and ii be an integer such that Xi=Ai⊔BiX_{i}=A_{i} \sqcup B_{i}.Then, either fi+1−1∘⋯∘fn−1(xn)∈Aif_{i+1}^{-1} \circ \cdots \circ f_{n}^{-1}\left(x_{n}\right) \in A_{i} and fj+1−1∘⋯∘fn−1(xn)∈Ajf_{j+1}^{-1} \circ \cdots \circ f_{n}^{-1}\left(x_{n}\right) \in A_{j} for any j<ij<i, or fi+1−1∘⋯∘fn−1(xn)∈Bif_{i+1}^{-1} \circ \cdots \circ f_{n}^{-1}\left(x_{n}\right) \in B_{i} and fj+1−1∘⋯∘fn−1(xn)∈Bjf_{j+1}^{-1} \circ \cdots \circ f_{n}^{-1}\left(x_{n}\right) \in B_{j} for any j<ij<i. Hence ϕSˇ(Xnf)=\phi_{\check{\mathcal{S}}}\left(\mathcal{X}_{n}^{f}\right)= ∣(A0∩f−1(Xnf)Sˇ)⊔(B0∩f−1(Xnf)Sˇ)∣=ϕSˇ(Anf)+ϕSˇ(Bnf)\left|\left(A_{0} \cap f^{-1}\left(\mathcal{X}_{n}^{f}\right)^{\check{\mathcal{S}}}\right) \sqcup\left(B_{0} \cap f^{-1}\left(\mathcal{X}_{n}^{f}\right)^{\check{\mathcal{S}}}\right)\right|=\phi_{\check{\mathcal{S}}}\left(\mathcal{A}_{n}^{f}\right)+\phi_{\check{\mathcal{S}}}\left(\mathcal{B}_{n}^{f}\right).

Definition 2.11. Let GG be a finite group.
We denote Bn(G)B_{n}(G) the Grothendieck group of CnG\mathcal{C}_{n}^{G} with respect to the relations Γ\Gamma that is, the quotient

Bn(G):=Ωn(G)/Ωˉn(G)B_{n}(G):=\Omega_{n}(G) / \bar{\Omega}_{n}(G)

of the free abelian group Ωn(G)\Omega_{n}(G) on the set of isomorphism classes of (n,G)(n, G)-simplices by the subgroup Ωˉn(G)\bar{\Omega}_{n}(G) generated by the formal differences [Xnf]−[Anf]−[Bnf]\left[\mathcal{X}_{n}^{f}\right]-\left[\mathcal{A}_{n}^{f}\right]-\left[\mathcal{B}_{n}^{f}\right].

Remark 2.12.

The function Xnf↦∣Hom⁡CG((G/Sˇ)n,Xnf)∣\mathcal{X}_{n}^{f} \mapsto\left|\operatorname{Hom}_{\mathcal{C}^{G}}\left((G / \check{\mathcal{S}})_{n}, \mathcal{X}_{n}^{f}\right)\right| defined on the class of (n,G)(n, G)-simplices (up to isomorphism), with values in Z\mathbb{Z} extends to a group homomorphism Bn(G)→ZB_{n}(G) \rightarrow \mathbb{Z}. So, to see that Bn(G)B_{n}(G) is non-trivial, it suffices to find Xnf\mathcal{X}_{n}^{f} with ∣Hom⁡((G/Sˇ)n,Xnf)∣≠0\left|\operatorname{Hom}\left((G / \check{\mathcal{S}})_{n}, \mathcal{X}_{n}^{f}\right)\right| \neq 0. Since

∣Hom⁡((G/Sˇ)n,(G/Sˇ)n)∣≠0\left|\operatorname{Hom}\left((G / \check{\mathcal{S}})_{n},(G / \check{\mathcal{S}})_{n}\right)\right| \neq 0

we have that Bn(G)≠0B_{n}(G) \neq 0.

Proposition 2.13. The functor dj,j=0,…,nd_{j}, j=0, \ldots, n (resp. si,i=0,…,n−1s_{i}, i=0, \ldots, n-1 ) induces a group homomorphism

dj:Bn(G)→Bn−1(G)( resp. si:Bn−1(G)→Bn(G))2d_{j}: B_{n}(G) \rightarrow B_{n-1}(G) \quad\left(\text { resp. } s_{i}: B_{n-1}(G) \rightarrow B_{n}(G)\right)^{2}

such that the identities (1), (2), (3) hold.
Proof. Indeed, for any (n,G)(n, G)-simplices Xnf\mathcal{X}_{n}^{f} with decomposition [Xnf,Anf,Bnf]k\left[\mathcal{X}_{n}^{f}, \mathcal{A}_{n}^{f}, \mathcal{B}_{n}^{f}\right]_{k}, i.e. A0⊔B0→f1A1⊔B1→f2…Ak⊔Bk→fk+1Xk+1…→fn−1Xn−1→fnXnA_{0} \sqcup B_{0} \xrightarrow{f_{1}} A_{1} \sqcup B_{1} \xrightarrow{f_{2}} \ldots A_{k} \sqcup B_{k} \xrightarrow{f_{k+1}} X_{k+1} \ldots \xrightarrow{f_{n-1}} X_{n-1} \xrightarrow{f_{n}} X_{n} then there is a decomposition of dj(Xnf)d_{j}\left(\mathcal{X}_{n}^{f}\right) as [dj(Xnf),dj(Anf),dj(Bnf)]k\left[d_{j}\left(\mathcal{X}_{n}^{f}\right), d_{j}\left(\mathcal{A}_{n}^{f}\right), d_{j}\left(\mathcal{B}_{n}^{f}\right)\right]_{k} if k<jk<j and [dj(Xnf),dj(Anf),dj(Bnf)]k−1\left[d_{j}\left(\mathcal{X}_{n}^{f}\right), d_{j}\left(\mathcal{A}_{n}^{f}\right), d_{j}\left(\mathcal{B}_{n}^{f}\right)\right]_{k-1} if k≥jk \geq j. So the map which assigns an (n,G)(n, G)-simplex Xnf\mathcal{X}_{n}^{f} to the image of dj(Xnf)d_{j}\left(\mathcal{X}_{n}^{f}\right) in Bn−1(G)B_{n-1}(G) induces, by the universal property, a well-defined map from Bn(G)B_{n}(G) to Bn−1(G)B_{n-1}(G).
Similarly, there is a decomposition of sj(Xnf)s_{j}\left(\mathcal{X}_{n}^{f}\right) as [sj(Xnf),sj(Anf),sj(Bnf)]k\left[s_{j}\left(\mathcal{X}_{n}^{f}\right), s_{j}\left(\mathcal{A}_{n}^{f}\right), s_{j}\left(\mathcal{B}_{n}^{f}\right)\right]_{k} if k<jk<j and [sj(Xnf),sj(Anf),sj(Bnf)]k+1\left[s_{j}\left(\mathcal{X}_{n}^{f}\right), s_{j}\left(\mathcal{A}_{n}^{f}\right), s_{j}\left(\mathcal{B}_{n}^{f}\right)\right]_{k+1} if k≥jk \geq j. So sjs_{j} maps the defining relations Bn−1(G)B_{n-1}(G) to those of Bn(G)B_{n}(G) ) and therefore gives a well-defined map (which is a group homomorphism) from Bn−1(G)B_{n-1}(G) to Bn(G)B_{n}(G). The equalities (1), (2), (3) may be verified easily.

Notation 2.14.

X0→f1f1(X0)→f2…→fnfn(fn−1(fn−2(…f1(X0)…))X_{0} \xrightarrow{f_{1}} f_{1}\left(X_{0}\right) \xrightarrow{f_{2}} \ldots \xrightarrow{f_{n}} f_{n}\left(f_{n-1}\left(f_{n-2}\left(\ldots f_{1}\left(X_{0}\right) \ldots\right)\right)\right.

Then,

[1]

  1. 2{ }^{2} Note that the maps djd_{j} and sis_{i} should more appropriately be labeled, but we stick common practice and use djd_{j} and sis_{i} for face and degeneracy wherever we find them. ↩︎

Lemma 2.15. Let Xnf\mathcal{X}_{n}^{f} be a (n,G)(n, G)-simplex. Then in the group Bn(G)B_{n}(G), we have

π(Xnf)=∑x∈[G\X0]⟨G^fx⟩G\pi\left(\mathcal{X}_{n}^{f}\right)=\sum_{x \in\left[G \backslash X_{0}\right]}\left\langle\hat{\mathcal{G}}_{f}^{x}\right\rangle_{G}

where G^fx\hat{\mathcal{G}}_{f}^{x} denotes the nn-slice (Gx,Gf1(x),…,Gfn−1f2f1(x))\left(G_{x}, G_{f_{1}(x)}, \ldots, G_{f_{n-1} f_{2} f_{1}(x)}\right) and G∙G_{\bullet} denotes the stabilizer of the element ∙\bullet.
Proof. Note first that in the group Bn(G)B_{n}(G) we have that π(Xnf)=π(f(Xnf))\pi\left(\mathcal{X}_{n}^{f}\right)=\pi\left(f\left(\mathcal{X}_{n}^{f}\right)\right). Indeed writing Xn=Xn⊔QX_{n}=X_{n} \sqcup \mathfrak{Q}, we get by the defining relations of Bn(G)B_{n}(G) that

π(Xnf)=π(Q)+π(f1−1(X1)→f1f2−1(X2)→f2…fn−1(Xn)→fnXn)=π(f1−1(X1)→f1f2−1(X2)→f2…fn−1(Xn)→fnXn)=π(f(Xnf))\begin{aligned} \pi\left(\mathcal{X}_{n}^{f}\right) & =\pi(\mathfrak{Q})+\pi\left(f_{1}^{-1}\left(X_{1}\right) \xrightarrow{f_{1}} f_{2}^{-1}\left(X_{2}\right) \xrightarrow{f_{2}} \ldots f_{n}^{-1}\left(X_{n}\right) \xrightarrow{f_{n}} X_{n}\right) \\ & =\pi\left(f_{1}^{-1}\left(X_{1}\right) \xrightarrow{f_{1}} f_{2}^{-1}\left(X_{2}\right) \xrightarrow{f_{2}} \ldots f_{n}^{-1}\left(X_{n}\right) \xrightarrow{f_{n}} X_{n}\right) \\ & =\pi\left(f\left(\mathcal{X}_{n}^{f}\right)\right) \end{aligned}

Further,

π(f(Xnf))=π(∐x∈[G\X0]Ox→f1∐x∈[G\X0]Of1(x)→f2…∐x∈[G\X0]Ofn(fn−1(…f1(x)…)))=∑x∈[G\X0]π(Ox→f1Of1(x)→f2…Ofn(fn−1(…f1(x)…))⏟xnf′′)\begin{aligned} \pi\left(f\left(\mathcal{X}_{n}^{f}\right)\right) & =\pi\left(\coprod_{x \in\left[G \backslash X_{0}\right]} \mathcal{O}_{x} \xrightarrow{f_{1}} \coprod_{x \in\left[G \backslash X_{0}\right]} \mathcal{O}_{f_{1}(x)} \xrightarrow{f_{2}} \ldots \coprod_{x \in\left[G \backslash X_{0}\right]} \mathcal{O}_{f_{n}\left(f_{n-1}\left(\ldots f_{1}(x) \ldots\right)\right)}\right) \\ & =\sum_{x \in\left[G \backslash X_{0}\right]} \pi\left(\underbrace{\mathcal{O}_{x} \xrightarrow{f_{1}} \mathcal{O}_{f_{1}(x)} \xrightarrow{f_{2}} \ldots \mathcal{O}_{f_{n}\left(f_{n-1}\left(\ldots f_{1}(x) \ldots\right)\right)}}_{x_{n}^{f^{\prime \prime}}}\right) \end{aligned}

where the last equality follows from the defining relations of Bn(G)B_{n}(G). Now, the (n,G)(n, G)-simplex Xnf′′\mathcal{X}_{n}^{f^{\prime \prime}} is obviously isomorphic to the indecomposable (n,G)(n, G)-simplex

G/Gx→p1G/Gf1(x)→p2…G/Gfn(fn−1(…f1(x)…))G / G_{x} \xrightarrow{p_{1}} G / G_{f_{1}(x)} \xrightarrow{p_{2}} \ldots G / G_{f_{n}\left(f_{n-1}\left(\ldots f_{1}(x) \ldots\right)\right)}

Since two indecomposable (n,G)(n, G)-simplices (G/S^)n(G / \hat{\mathcal{S}})_{n} and (G/T^)n(G / \hat{\mathcal{T}})_{n} are isomorphic if and only if the nn-slices S^\hat{\mathcal{S}} and T^\hat{\mathcal{T}} are conjugate we have the following corollary:

Corollary 2.16. The group Bn(G)B_{n}(G) is generated by the elements ⟨S^⟩G\langle\hat{\mathcal{S}}\rangle_{G} where S^\hat{\mathcal{S}} runs through a set [Πn(G)]\left[\Pi_{n}(G)\right] of representatives of conjugacy classes of nn-slices of GG.

Proposition 2.17. The product of (n,G)(n, G)-simplices induces a commutative ring structure on Bn(G)B_{n}(G) with identity en:=[∙⟶⋯∙⟶∙]\boldsymbol{e}_{n}:=[\bullet \longrightarrow \cdots \bullet \longrightarrow \bullet], where ∙\bullet is a GG-set of cardinality 1 . This ring is called the nn-simplicial ring of the finite group GG.
Moreover, the morphisms dj(j=1,…,n)d_{j}(j=1, \ldots, n) and si(i=0,…,n−1)s_{i}(i=0, \ldots, n-1) in Proposition 2.13 define morphisms of rings.

Proof. We have to show that the product of (n,G)(n, G)-simplices induces a well-defined bilinear product Bn(G)×Bn(G)→Bn(G)B_{n}(G) \times B_{n}(G) \rightarrow B_{n}(G). Let Xnf\mathcal{X}_{n}^{f} be a (n,G)(n, G)-simplex such that there is a decomposition [Xnf,Anf,Bnf]i\left[\mathcal{X}_{n}^{f}, \mathcal{A}_{n}^{f}, \mathcal{B}_{n}^{f}\right]_{i}, and let YnS\mathcal{Y}_{n}^{\mathcal{S}} be any (n,G)(n, G)-simplex. We set Znb:=\mathcal{Z}_{n}^{b}:= Xnf×YnS\mathcal{X}_{n}^{f} \times \mathcal{Y}_{n}^{\mathcal{S}}. Then the (n,G)(n, G)-simplex Znb\mathcal{Z}_{n}^{b} can be visualized as follows

(A0×Y0)⊔(B0×Y0)→f0×g0…(Ai×Yi)⊔(Bi×Yi)→fi×giXi+1×Yi+1…→Rn−1Xn×Yn\left(A_{0} \times Y_{0}\right) \sqcup\left(B_{0} \times Y_{0}\right) \xrightarrow{f_{0} \times g_{0}} \ldots\left(A_{i} \times Y_{i}\right) \sqcup\left(B_{i} \times Y_{i}\right) \xrightarrow{f_{i \times g_{i}}} X_{i+1} \times Y_{i+1} \ldots \xrightarrow{R_{n-1}} X_{n} \times Y_{n}

since (Ai⊔Bi)×Yi=(Ai×Yi)⏟Ci⊔(Bi×Yi)⏟Di\left(A_{i} \sqcup B_{i}\right) \times Y_{i}=\underbrace{\left(A_{i} \times Y_{i}\right)}_{C_{i}} \sqcup \underbrace{\left(B_{i} \times Y_{i}\right)}_{D_{i}}. Hence

π(Znh)=π(C0⊔D0→h0…Ci⊔Di→hiXi+1×Yi+1…→hn−1Xn×Yn))=π(C0→h0…Ci→hiXi+1×Yi+1…→hn−1Xn×Yn))+π(D0→h0…Di→hiXi+1×Yi+1…→hn−1Xn×Yn))=π(f1−1(A1)×g1−1(Y1)→h0f2−1(A2)×g2−1(Y2)…→hiAi×Yi…→hn−1Xn×Yn))+π(f1−1(B1)×g1−1(Y1)→h0f2−1(B2)×g2−1(Y2)…→hiAi×Yi…→hn−1Xn×Yn))\begin{aligned} & \pi\left(\mathcal{Z}_{n}^{h}\right)=\pi\left(C_{0} \sqcup D_{0} \xrightarrow{h_{0}} \ldots C_{i} \sqcup D_{i} \xrightarrow{h_{i}} X_{i+1} \times Y_{i+1} \ldots \xrightarrow{h_{n-1}} X_{n} \times Y_{n}\right)) \\ & =\pi\left(C_{0} \xrightarrow{h_{0}} \ldots C_{i} \xrightarrow{h_{i}} X_{i+1} \times Y_{i+1} \ldots \xrightarrow{h_{n-1}} X_{n} \times Y_{n}\right)) \\ & +\pi\left(D_{0} \xrightarrow{h_{0}} \ldots D_{i} \xrightarrow{h_{i}} X_{i+1} \times Y_{i+1} \ldots \xrightarrow{h_{n-1}} X_{n} \times Y_{n}\right)) \\ & =\pi\left(f_{1}^{-1}\left(A_{1}\right) \times g_{1}^{-1}\left(Y_{1}\right) \xrightarrow{h_{0}} f_{2}^{-1}\left(A_{2}\right) \times g_{2}^{-1}\left(Y_{2}\right) \ldots \xrightarrow{h_{i}} A_{i} \times Y_{i} \ldots \xrightarrow{h_{n-1}} X_{n} \times Y_{n}\right)) \\ & +\pi\left(f_{1}^{-1}\left(B_{1}\right) \times g_{1}^{-1}\left(Y_{1}\right) \xrightarrow{h_{0}} f_{2}^{-1}\left(B_{2}\right) \times g_{2}^{-1}\left(Y_{2}\right) \ldots \xrightarrow{h_{i}} A_{i} \times Y_{i} \ldots \xrightarrow{h_{n-1}} X_{n} \times Y_{n}\right)) \end{aligned}

Clearly, we have (it suffices to set Yi=Yi⊔∅Y_{i}=Y_{i} \sqcup \varnothing )

π(g1−1(Y1)→g0g2−1(Y2)…→giYi…→gn−1Yn)=π(YnS)\pi\left(g_{1}^{-1}\left(Y_{1}\right) \xrightarrow{g_{0}} g_{2}^{-1}\left(Y_{2}\right) \ldots \xrightarrow{g_{i}} Y_{i} \ldots \xrightarrow{g_{n-1}} Y_{n}\right)=\pi\left(\mathcal{Y}_{n}^{\mathcal{S}}\right)

So π(Znh)=π(Anf×YnS)+π(Bnf×YnS)\pi\left(\mathcal{Z}_{n}^{h}\right)=\pi\left(\mathcal{A}_{n}^{f} \times \mathcal{Y}_{n}^{\mathcal{S}}\right)+\pi\left(\mathcal{B}_{n}^{f} \times \mathcal{Y}_{n}^{\mathcal{S}}\right), this shows that the product preserves the defining relations of Bn(G)B_{n}(G). Now it is obvious to see that the product is associative, commutative and admits en\mathbf{e}_{n} as an identity element.
Let YnS\mathcal{Y}_{n}^{\mathcal{S}} be (n,G)(n, G)-simplex. Since (fj+1∘fj)×(gj+1∘gj)=(fj+1×gj+1)∘(fj×gj)\left(f_{j+1} \circ f_{j}\right) \times\left(g_{j+1} \circ g_{j}\right)=\left(f_{j+1} \times g_{j+1}\right) \circ\left(f_{j} \times g_{j}\right), we have that

dj(Xnf×YnS)=dj(Xnf)×dj(YnS),sj(Xnf×YnS)=sj(Xnf)×sj(YnS)d_{j}\left(\mathcal{X}_{n}^{f} \times \mathcal{Y}_{n}^{\mathcal{S}}\right)=d_{j}\left(\mathcal{X}_{n}^{f}\right) \times d_{j}\left(\mathcal{Y}_{n}^{\mathcal{S}}\right), \quad s_{j}\left(\mathcal{X}_{n}^{f} \times \mathcal{Y}_{n}^{\mathcal{S}}\right)=s_{j}\left(\mathcal{X}_{n}^{f}\right) \times s_{j}\left(\mathcal{Y}_{n}^{\mathcal{S}}\right)

and so the induced maps djd_{j} and sjs_{j} are ring homomorphisms. Furthermore dj(en)=d_{j}\left(\mathbf{e}_{n}\right)= en−1\mathbf{e}_{n-1} and sj(en)=en+1s_{j}\left(\mathbf{e}_{n}\right)=\mathbf{e}_{n+1}.
Remark 2.18. Letting Δ\Delta the simplicial category that is Δ\Delta has as objects all finite ordinal numbers [n]=[0,1,…,n][n]=[0,1, \ldots, n] and as morphisms f:[n]→[m]f:[n] \rightarrow[m] all monotone maps; that is, the maps ff such that f(i)≤f(j)f(i) \leq f(j) if i<ji<j.
Then an induced functor CnG→CmG\mathcal{C}_{n}^{G} \rightarrow \mathcal{C}_{m}^{G} for any finite GG and an induced GG-equivariant map Πn(G)→Πm(G)\Pi_{n}(G) \rightarrow \Pi_{m}(G) and a group homomorphism Bn(G)→Bm(G)B_{n}(G) \rightarrow B_{m}(G) given by,

Xnf↦sikn−1…sj1n−kdi1n−k…dikmXnf\mathcal{X}_{n}^{f} \mapsto s_{i_{k}}^{n-1} \ldots s_{j_{1}}^{n-k} d_{i_{1}}^{n-k} \ldots d_{i_{k}}^{m} \mathcal{X}_{n}^{f}

where kk and hh satisfy the following conditions:

Proposition 2.19. Using the generators of Bn(G)B_{n}(G), the multiplication is given by

⟨S^⟩G⋅⟨T^⟩G=∑g∈[S0∩G/T0]⟨S^∩S⟩G\langle\hat{\mathcal{S}}\rangle_{G} \cdot\langle\hat{\mathcal{T}}\rangle_{G}=\sum_{g \in\left[S_{0} \cap G / T_{0}\right]}\left\langle\hat{\mathcal{S}} \cap{ }^{\mathcal{S}}\right\rangle_{G}

Proof. Note that each GG-orbit of the GG-set (G/S0)×(G/T0)\left(G / \mathcal{S}_{0}\right) \times\left(G / \mathcal{T}_{0}\right) determines a double coset S0gT0\mathcal{S}_{0} g \mathcal{T}_{0}, in the following way:

O(xS0,gT0)→S0x−1γT0\mathcal{O}_{\left(x S_{0}, g \mathcal{T}_{0}\right)} \rightarrow S_{0} x^{-1} \gamma T_{0}

Conversely, the GG-orbit of (G/S0)×(G/T0)\left(G / \mathcal{S}_{0}\right) \times\left(G / \mathcal{T}_{0}\right) corresponding to S0aT0\mathcal{S}_{0} a \mathcal{T}_{0} consists of all distinct pairs in the collection {(xS0,γT0)∣x−1γ∈S0aT0}\left\{\left(x \mathcal{S}_{0}, \gamma \mathcal{T}_{0}\right) \mid x^{-1} \gamma \in \mathcal{S}_{0} a \mathcal{T}_{0}\right\}. The stabilizer of the pair (S0,gT0)\left(\mathcal{S}_{0}, g \mathcal{T}_{0}\right) is precisely S0∩PT0\mathcal{S}_{0} \cap \mathbb{P} \mathcal{T}_{0}. Therefore, the orbit of (G/S0)×(G/T0)\left(G / \mathcal{S}_{0}\right) \times\left(G / \mathcal{T}_{0}\right) corresponding to S0gT0\mathcal{S}_{0} g \mathcal{T}_{0} is isomorphic (as GG-set) to G/(S0∩PT0)G /\left(\mathcal{S}_{0} \cap \mathbb{P} \mathcal{T}_{0}\right), and therefore

(G/S0)×(G/T0)≅∐g∈[S0\G/T0]G/(S0∩PT0)\left(G / \mathcal{S}_{0}\right) \times\left(G / \mathcal{T}_{0}\right) \cong \coprod_{g \in\left[\mathcal{S}_{0} \backslash G / \mathcal{T}_{0}\right]} G /\left(\mathcal{S}_{0} \cap \mathbb{P} \mathcal{T}_{0}\right)

On the other hand, the image of (Si,gTi)\left(\mathcal{S}_{i}, g \mathcal{T}_{i}\right) by the map

(G/Si)×(G/Ti)→(G/Si+1)×(G/Ti+1)\left(G / \mathcal{S}_{i}\right) \times\left(G / \mathcal{T}_{i}\right) \rightarrow\left(G / \mathcal{S}_{i+1}\right) \times\left(G / \mathcal{T}_{i+1}\right)

is the pair (Si+1,gTi+1)\left(\mathcal{S}_{i+1}, g \mathcal{T}_{i+1}\right), whose stabilizer is Si+1∩PTi+1\mathcal{S}_{i+1} \cap \mathbb{P} \mathcal{T}_{i+1}. Hence inductively the result follows from Lemma 2.15.

3 Ghost maps

In this section, we examine a map ΦnG\Phi_{n}^{G} that this intrinsically related with the nn simplicial ring Bn(GB_{n}(G in the sense that ΦnG\Phi_{n}^{G} may be discovered from the ring structure of Bn(G)B_{n}(G).

Proposition 3.1.

  1. For a nn-slice A‾\overline{\mathcal{A}} fixed, the correspondence Xnf↦ϕA‾(Xnf)\mathcal{X}_{n}^{f} \mapsto \phi_{\overline{\mathcal{A}}}\left(\mathcal{X}_{n}^{f}\right) extends to a ring homomorphism still denoted by

ϕA‾:Bn(G)⟶Z\phi_{\overline{\mathcal{A}}}: B_{n}(G) \longrightarrow \mathbb{Z}

  1. (An analogue of Burnside’s Theorem) We let

ΦnG=(ϕA‾):Bn(G)⟶∏A‾∈[Πn(G)]Z=Cn(G)\Phi_{n}^{G}=\left(\phi_{\overline{\mathcal{A}}}\right): B_{n}(G) \longrightarrow \prod_{\overline{\mathcal{A}} \in\left[\Pi_{n}(G)\right]} \mathbb{Z}=C_{n}(G)

be the product of the ϕA‾\phi_{\overline{\mathcal{A}}}. Then ΦnG\Phi_{n}^{G} is an injective ring homomorphism, with finite cokernel as morphism of abelian groups. The set

{⟨A‾⟩G∣A‾∈[Πn(G)]}\left\{\langle\overline{\mathcal{A}}\rangle_{G} \mid \overline{\mathcal{A}} \in\left[\Pi_{n}(G)\right]\right\}

form a basis of Bn(G)B_{n}(G).
3. Let RR be an integral domain, ϕ:Bn(G)→R\phi: B_{n}(G) \rightarrow R any ring homomorphism, and

T(ϕ):={A‾∈Πn(G)∣ϕ(⟨A‾⟩G)≠0}T(\phi):=\left\{\overline{\mathcal{A}} \in \Pi_{n}(G) \mid \phi\left(\langle\overline{\mathcal{A}}\rangle_{G}\right) \neq 0\right\}

Then, there exists exactly one element K‾∈[Πn(G)]\overline{\mathcal{K}} \in\left[\Pi_{n}(G)\right] that is minimal with respect to ⪯\preceq in T(ϕ)T(\phi). Moreover, one has ϕ(x)=ϕK‾(x)⋅1R\phi(x)=\phi_{\overline{\mathcal{K}}}(x) \cdot 1_{R} for all x∈Bn(G)x \in B_{n}(G) and this minimal K‾\overline{\mathcal{K}} in T(ϕ)T(\phi).
Proof. 1. Corollary 2.10 shows that ϕA‾\phi_{\overline{\mathcal{A}}} the defining relations of Bn(G)B_{n}(G) are mapped to 0 by ϕA‾\phi_{\overline{\mathcal{A}}}. So it induces a well defined map from Bn(G)B_{n}(G) to Z\mathbb{Z}. Now, since the product of (n,G)(n, G)-simplices is the product of the category of (n,G)(n, G)-simplices, it follows that

ϕA‾(Xnf⋅YnS)=ϕA‾(Xnf)ϕA‾(YnS)\phi_{\overline{\mathcal{A}}}\left(\mathcal{X}_{n}^{f} \cdot \mathcal{Y}_{n}^{\mathcal{S}}\right)=\phi_{\overline{\mathcal{A}}}\left(\mathcal{X}_{n}^{f}\right) \phi_{\overline{\mathcal{A}}}\left(\mathcal{Y}_{n}^{\mathcal{S}}\right)

The image of the identity en\mathbf{e}_{n} by ϕA‾\phi_{\overline{\mathcal{A}}} is obviously 1 .

  1. By definition ΦnG\Phi_{n}^{G} is a ring homomorphism. Suppose that u≠0u \neq 0 is in the kernel of ΦnG\Phi_{n}^{G}. We write uu in terms of the generators

u=∑S~∈[Πn(G)]aS~(S~)Gu=\sum_{\tilde{\mathcal{S}} \in\left[\Pi_{n}(G)\right]} a_{\tilde{\mathcal{S}}}(\tilde{\mathcal{S}})_{G}

We have a partial ordering on the ⟨S0…Sn⟩G\left\langle S_{0} \ldots S_{n}\right\rangle_{G} induced by ≤\leq. Let ⟨S~⟩G\langle\tilde{\mathcal{S}}\rangle_{G} be maximal among the generators with aS~≠0a_{\tilde{\mathcal{S}}} \neq 0. Then ϕS~(⟨T~⟩G)≠0\phi_{\tilde{\mathcal{S}}}(\langle\tilde{\mathcal{T}}\rangle_{G}) \neq 0 implies that ⟨S~⟩G≤⟨T~⟩G\langle\tilde{\mathcal{S}}\rangle_{G} \leq\langle\tilde{\mathcal{T}}\rangle_{G}. Hence

0=ϕS~(u)=aS~ϕS~(⟨S~⟩G)≠00=\phi_{\tilde{\mathcal{S}}}(u)=a_{\tilde{\mathcal{S}}} \phi_{\tilde{\mathcal{S}}}(\langle\tilde{\mathcal{S}}\rangle_{G}) \neq 0

a contradiction.
Now by the first part of the proof, Bn(G)B_{n}(G) has Z\mathbb{Z}-rank {[Πn(G)]}=rank⁡ZCn(G)\left\{\left[\Pi_{n}(G)\right]\right\}=\operatorname{rank}_{\mathbb{Z}} C_{n}(G), and since ΦnG\Phi_{n}^{G} is injective, then Im⁡(ΦnG)\operatorname{Im}\left(\Phi_{n}^{G}\right) and Cn(G)C_{n}(G) have the same rank, and so, the cokernel of ΦnG\Phi_{n}^{G} is finite.
3. The set T(ϕ)T(\phi) is not empty because en∉Ker⁡ϕ\mathbf{e}_{n} \notin \operatorname{Ker} \phi.

Let S~\tilde{\mathcal{S}} be minimal in T(ϕ)T(\phi) with respect to the relation ≤\leq. Then by Proposition 2.19 , for any nn-slice T~\tilde{\mathcal{T}}

⟨S~⟩G⋅⟨T~⟩G=∑g∈[S0⟨G/T0]⟨S0∩gT0,…,Sn∩gTn⟩G\langle\tilde{\mathcal{S}}\rangle_{G} \cdot\langle\tilde{\mathcal{T}}\rangle_{G}=\sum_{g \in\left[S_{0}\left\langle G / T_{0}\right]\right.}\left\langle S_{0} \cap^{g} T_{0}, \ldots, S_{n} \cap^{g} T_{n}\right\rangle_{G}

Since ϕ(⟨S~⟩G⋅⟨T~⟩G)=ϕ(⟨S~⟩G)⋅ϕ(⟨T~⟩G)≠0\phi\left(\langle\tilde{\mathcal{S}}\rangle_{G} \cdot\langle\tilde{\mathcal{T}}\rangle_{G}\right)=\phi\left(\langle\tilde{\mathcal{S}}\rangle_{G}\right) \cdot \phi\left(\langle\tilde{\mathcal{T}}\rangle_{G}\right) \neq 0 in RR, there exists g∈S0\G/T0g \in S_{0} \backslash G / T_{0} such that ϕ(S~∩T~)≠0\phi(\tilde{\mathcal{S}} \cap \tilde{\mathcal{T}}) \neq 0. Hence, by minimality of S~\tilde{\mathcal{S}}, we have that those gg ranges over the set {g∈G/T0∣Sn≤gTn……S0≤gT0}\left\{g \in G / T_{0} \mid S_{n} \leq{ }^{g} T_{n} \ldots \ldots S_{0} \leq{ }^{g} T_{0}\right\}. So ϕ(⟨S~⟩G)⋅ϕ(⟨T~⟩G)=\phi\left(\langle\tilde{\mathcal{S}}\rangle_{G}\right) \cdot \phi\left(\langle\tilde{\mathcal{T}}\rangle_{G}\right)= ϕS~(⟨T~⟩G)ϕ(⟨S~⟩G\phi_{\tilde{\mathcal{S}}}(\langle\tilde{\mathcal{T}}\rangle_{G}) \phi\left(\langle\tilde{\mathcal{S}}\rangle_{G}\right.. Since RR is an integral domain, we can divide both sides by ϕ(⟨S~⟩G)\phi\left(\langle\tilde{\mathcal{S}}\rangle_{G}\right) and we then have ϕ(⟨T~⟩G)=ϕS~(⟨T~⟩G)1R\phi\left(\langle\tilde{\mathcal{T}}\rangle_{G}\right)=\phi_{\tilde{\mathcal{S}}}\left(\langle\tilde{\mathcal{T}}\rangle_{G}\right) 1_{R}. So by linearity, ϕ(x)=\phi(x)= ϕS~(x)⋅1R\phi_{\tilde{\mathcal{S}}}(x) \cdot 1_{R} for all x∈Bn(G)x \in B_{n}(G). Now, if K~\tilde{\mathcal{K}} is another minimal element, then ϕ(⟨K~⟩G)=ϕS~(⟨K~⟩G)\phi\left(\langle\tilde{\mathcal{K}}\rangle_{G}\right)=\phi_{\tilde{\mathcal{S}}}\left(\langle\tilde{\mathcal{K}}\rangle_{G}\right) and ϕ(⟨K~⟩G)=ϕK~(⟨K~⟩G)\phi\left(\langle\tilde{\mathcal{K}}\rangle_{G}\right)=\phi_{\tilde{\mathcal{K}}}\left(\langle\tilde{\mathcal{K}}\rangle_{G}\right). So S~≤GK~\tilde{\mathcal{S}} \leq_{G} \tilde{\mathcal{K}} and by symmetry S~=GK~\tilde{\mathcal{S}}={ }_{G} \tilde{\mathcal{K}}.

Remark 3.2.

{⟨S~⟩G∣S~∈[Πn(G)]}\left\{\langle\tilde{\mathcal{S}}\rangle_{G} \mid \tilde{\mathcal{S}} \in\left[\Pi_{n}(G)\right]\right\}

form a basis of Bn(G)B_{n}(G) and a (1,G)(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,
Corollary 3.3. The map

QΦnG:QBn(G)→∏Sˇ∈[Πn(G)]Q\mathbb{Q} \Phi_{n}^{G}: \mathbb{Q} B_{n}(G) \rightarrow \prod_{\check{\mathcal{S}} \in\left[\Pi_{n}(G)\right]} \mathbb{Q}

where QΦnG\mathbb{Q} \Phi_{n}^{G} is the rational extension of ΦnG\Phi_{n}^{G}, is an isomorphism of Q\mathbb{Q}-algebras.
Note that every Q\mathbb{Q}-algebra homomorphism QBn(G)→Q\mathbb{Q} B_{n}(G) \rightarrow \mathbb{Q} is of the form QϕSˇ\mathbb{Q} \phi_{\check{\mathcal{S}}} for some Sˇ∈Πn(G)\check{\mathcal{S}} \in \Pi_{n}(G). For Sˇ,Tˇ∈Πn(G)\check{\mathcal{S}}, \check{\mathcal{T}} \in \Pi_{n}(G), we have QϕSˇ=QϕTˇ\mathbb{Q} \phi_{\check{\mathcal{S}}}=\mathbb{Q} \phi_{\check{\mathcal{T}}} if and only Sˇ=GTˇ\check{\mathcal{S}}={ }_{G} \check{\mathcal{T}}. More generally,

Notation 3.4. Put WG(Sˇ)=NG(Sˇ)/S0W_{G}(\check{\mathcal{S}})=N_{G}(\check{\mathcal{S}}) / S_{0}. Let pp be a prime, and ∞\infty be just a symbol. For each Z\mathbb{Z}-module MM, we shall set M(p)=Z(p)⊗ZMM_{(p)}=\mathbb{Z}_{(p)} \otimes_{\mathbb{Z}} M, where Z(p)\mathbb{Z}_{(p)} is the localisation of Z\mathbb{Z} at pp, and M(∞)=MM_{(\infty)}=M. For a nn-slice Sˇ\check{\mathcal{S}}, we denote WG(Sˇ)pW_{G}(\check{\mathcal{S}})_{p} a Sylow pp-subgroup of WG(Sˇ)W_{G}(\check{\mathcal{S}}), and set WG(Sˇ)∗∗=WG(Sˇ)W_{G}(\check{\mathcal{S}})_{* *}=W_{G}(\check{\mathcal{S}}). Let (ΦnG)p\left(\Phi_{n}^{G}\right)_{p} or simply Φ(p)G\Phi_{(p)}^{G} (if there is no risk of confusion) denote the homomorphism of Z(p)\mathbb{Z}_{(p)}-modules from Bn(G)(p)B_{n}(G)_{(p)} to Cn(G)(p)C_{n}(G)_{(p)} induced by ΦnG\Phi_{n}^{G}.

Proposition 3.5. Then

  1. We consider Bn(G)(p)B_{n}(G)_{(p)} and Cn(G)(p)C_{n}(G)_{(p)} as subrings of ∏Sˇ∈[Πn(G)]Q\prod_{\check{\mathcal{S}} \in\left[\Pi_{n}(G)\right]} \mathbb{Q}.

The set J′:={1ϕSˇ(Sˇ)Sˇ:=(ϕTˇ(Sˇ)ϕSˇ(Sˇ))Tˇ∈[Πn(G)]∣Sˇ∈[Πn(G)]}J^{\prime}:=\left\{\frac{1}{\phi_{\check{\mathcal{S}}}(\check{\mathcal{S}})} \check{\mathcal{S}}:=\left(\frac{\phi_{\check{\mathcal{T}}}(\check{\mathcal{S}})}{\phi_{\check{\mathcal{S}}}(\check{\mathcal{S}})}\right)_{\check{\mathcal{T}} \in\left[\Pi_{n}(G)\right]} \mid \check{\mathcal{S}} \in\left[\Pi_{n}(G)\right]\right\} is a basis of Cn(G)C_{n}(G).
2. If we define Obs⁡(G)\operatorname{Obs}(G) as ⨁Sˇ∈[Πn(G)]Z/∣WG(Sˇ)∣Z\bigoplus_{\check{\mathcal{S}} \in\left[\Pi_{n}(G)\right]} \mathbb{Z} / \mid W_{G}(\check{\mathcal{S}}) \mid \mathbb{Z}, then

Cn(G)(p)/Im⁡Φ(p)G≅Obs(G)(p)C_{n}(G)_{(p)} / \operatorname{Im} \Phi_{(p)}^{G} \cong O b s(G)_{(p)}

  1. Define a homomorphism of Z(p)\mathbb{Z}_{(p)}-modules Ψ(p)G:Cn(G)(p)→Obs⁡(G)(p)\Psi_{(p)}^{G}: C_{n}(G)_{(p)} \rightarrow \operatorname{Obs}(G)_{(p)} by

(xSˇ)Sˇ∈[Πn(G)]↦(∑ZˇT0∈WG(Tˇ)(p)xZˇ≥Tˇ mod ∣WG(Tˇ)(p))Tˇ∈[Πn(G)]\left(x_{\check{\mathcal{S}}}\right)_{\check{\mathcal{S}} \in\left[\Pi_{n}(G)\right]} \mapsto\left(\sum_{\check{\mathcal{Z}} T_{0} \in W_{G}(\check{\mathcal{T}})_{(p)}} x_{\check{\mathcal{Z}} \geq \check{\mathcal{T}}} \bmod \mid W_{G}(\check{\mathcal{T}})_{(p)}\right)_{\check{\mathcal{T}} \in\left[\Pi_{n}(G)\right]}

Then Ψ(p)G\Psi_{(p)}^{G} is surjective.
4. The sequence

0⟶Bn(G)(p)→Φ(p)GCn(G)(p)→Ψ(p)GObs⁡(G)(p)⟶00 \longrightarrow B_{n}(G)_{(p)} \xrightarrow{\Phi_{(p)}^{G}} C_{n}(G)_{(p)} \xrightarrow{\Psi_{(p)}^{G}} \operatorname{Obs}(G)_{(p)} \longrightarrow 0

of Z(p)\mathbb{Z}_{(p)}-modules is exact.
Proof. 1. By Lemma 2.7 4), we have that ϕSˇ(Sˇ)\phi_{\check{\mathcal{S}}}(\check{\mathcal{S}}) divides ϕTˇ(Sˇ)\phi_{\check{\mathcal{T}}}(\check{\mathcal{S}}), and so, (ϕT(Sˇ)ϕS(Sˇ))Tˇ∈[Πn(G)]\left(\frac{\phi_{\mathcal{T}}(\check{\mathcal{S}})}{\phi_{\mathcal{S}}(\check{\mathcal{S}})}\right)_{\check{\mathcal{T}} \in\left[\Pi_{n}(G)\right]} is an element of Cn(G)(p)C_{n}(G)_{(p)}, for any Sˇ∈[Πn(G)]\check{\mathcal{S}} \in\left[\Pi_{n}(G)\right]. Now, compare the set J′J^{\prime} with the canonical basis

J:={iSˇ=(δ(Sˇ,Tˇ))Tˇ∈[Πn]∣Sˇ∈[Πn]}J:=\left\{i_{\check{\mathcal{S}}}=\left(\delta(\check{\mathcal{S}}, \check{\mathcal{T}})\right)_{\check{\mathcal{T}} \in\left[\Pi_{n}\right]} \mid \check{\mathcal{S}} \in\left[\Pi_{n}\right]\right\}

of Cn(G)(p)C_{n}(G)_{(p)} where δ\delta is the Kronecker’s symbol, i.e. δ(Sˇ,Tˇ)={1 for Sˇ=Tˇ0 for Sˇ≠Tˇ\delta(\check{\mathcal{S}}, \check{\mathcal{T}})= \begin{cases}1 & \text { for } \check{\mathcal{S}}=\check{\mathcal{T}} \\ 0 & \text { for } \check{\mathcal{S}} \neq \check{\mathcal{T}}\end{cases} Since ∣J∣=∣J′∣=∣Πn(G)∣|J|=|J^{\prime}|=\left|\Pi_{n}(G)\right|, it suffices to prove that each iSˇi_{\check{\mathcal{S}}} can be written as an

integral combination of the 1ϕS(S‾)S‾\frac{1}{\phi_{\mathcal{S}}(\overline{\mathcal{S}})} \overline{\mathcal{S}}. This is done by induction with respect to ≤\leq. If S‾=(1,…,1)\overline{\mathcal{S}}=(1, \ldots, 1) then ϕT(S‾)ϕS(S‾)=δ(S‾,T‾)\frac{\phi_{T}(\overline{\mathcal{S}})}{\phi_{\mathcal{S}}(\overline{\mathcal{S}})}=\delta(\overline{\mathcal{S}}, \overline{\mathcal{T}}), for any T‾\overline{\mathcal{T}}, and so, iS‾∈J′i_{\overline{\mathcal{S}}} \in J^{\prime}. For arbitrary S‾\overline{\mathcal{S}}, we have ϕS(S‾)ϕS(S‾)=1\frac{\phi_{\mathcal{S}}(\overline{\mathcal{S}})}{\phi_{\mathcal{S}}(\overline{\mathcal{S}})}=1 and ϕT(S‾)ϕS(S‾)=0\frac{\phi_{T}(\overline{\mathcal{S}})}{\phi_{\mathcal{S}}(\overline{\mathcal{S}})}=0 for T‾∉S‾\overline{\mathcal{T}} \notin \overline{\mathcal{S}}, and so, 1ϕS(S‾)S‾=\frac{1}{\phi_{\mathcal{S}}(\overline{\mathcal{S}})} \overline{\mathcal{S}}= iS‾+∑T∈[Πn(G)]T<S‾nT‾,S‾iT‾i_{\overline{\mathcal{S}}}+\sum_{\substack{T \in\left[\Pi_{n}(\mathrm{G})\right] \\ T<\overline{\mathcal{S}}}} n_{\overline{\mathcal{T}}, \overline{\mathcal{S}}} i_{\overline{\mathcal{T}}} with nT‾,S‾∈Zn_{\overline{\mathcal{T}}, \overline{\mathcal{S}}} \in \mathbb{Z}.
Now by induction hypothesis, any iT‾i_{\overline{\mathcal{T}}} with T‾<S‾\overline{\mathcal{T}}<\overline{\mathcal{S}} is an integral linear combination of the elements of J′J^{\prime}. So, the same is true for iS‾i_{\overline{\mathcal{S}}}.
2. By Proposition 3.1 2), we have Im⁡Φ(p)G=⨁S‾∈[Πn(G)]Φ(p)G((S‾)G)\operatorname{Im} \Phi_{(p)}^{G}=\bigoplus_{\overline{\mathcal{S}} \in\left[\Pi_{n}(\mathrm{G})\right]} \Phi_{(p)}^{G}\left((\overline{\mathcal{S}})_{\mathrm{G}}\right) and by Assertion 1)

Cn(G)(p)=⨁S‾∈[Πn(G)]1[WG(S‾)p]Φ(p)G((S‾)G)ZC_{n}(G)_{(p)}=\bigoplus_{\overline{\mathcal{S}} \in\left[\Pi_{n}(\mathrm{G})\right]} \frac{1}{\left[W_{G}(\overline{\mathcal{S}})_{p}\right]} \Phi_{(p)}^{G}\left((\overline{\mathcal{S}})_{\mathrm{G}}\right) \mathbb{Z}

Hence Cn(G)(p)/Im⁡Φ(p)G≅Obs(G)(p)C_{n}(G)_{(p)} / \operatorname{Im} \Phi_{(p)}^{G} \cong O b s(G)_{(p)}.
3. Let i~S‾=(δ(S‾,T‾) mod ∣WG(S‾)p∣)T‾∈[Πn]\tilde{i}_{\overline{\mathcal{S}}}=\left(\delta(\overline{\mathcal{S}}, \overline{\mathcal{T}}) \bmod \left|W_{G}(\overline{\mathcal{S}})_{p}\right|\right)_{\overline{\mathcal{T}} \in\left[\Pi_{n}\right]}. Obviously, the elements i~S‾\tilde{i}_{\overline{\mathcal{S}}} for S‾∈[Πn]\overline{\mathcal{S}} \in\left[\Pi_{n}\right] form a Z(p)\mathbb{Z}_{(p)}-basis of Obs(G)(p)O b s(G)_{(p)}. Now set

R0={S‾∈[Πn]∣i~S‾∉Im⁡Ψ(p)G}R_{0}=\left\{\overline{\mathcal{S}} \in\left[\Pi_{n}\right] \mid \tilde{i}_{\overline{\mathcal{S}}} \notin \operatorname{Im} \Psi_{(p)}^{G}\right\}

Suppose that R0≠∅R_{0} \neq \varnothing, and let S‾\overline{\mathcal{S}} be a minimal element of R0R_{0} with respect to ≤G\leq_{G}. Then no element T‾\overline{\mathcal{T}} of R0∼{S‾}R_{0} \sim\{\overline{\mathcal{S}}\} satisfies T‾≤GS‾\overline{\mathcal{T}} \leq_{G} \overline{\mathcal{S}}, and so Ψ(p)G((iT‾)T‾∈[Πn])=\Psi_{(p)}^{G}\left(\left(i_{\overline{\mathcal{T}}}\right)_{\overline{\mathcal{T}} \in\left[\Pi_{n}\right]}\right)= (ψT‾)T‾∈[Πn]\left(\psi_{\overline{\mathcal{T}}}\right)_{\overline{\mathcal{T}} \in\left[\Pi_{n}\right]}, where

ψT‾={1 mod ∣WG(S‾)p∣ for T‾=S‾0 mod ∣WG(S‾)p∣ for T‾∈R0∼{S‾}\psi_{\overline{\mathcal{T}}}= \begin{cases}1 \bmod \left|W_{G}(\overline{\mathcal{S}})_{p}\right| & \text { for } \quad \overline{\mathcal{T}}=\overline{\mathcal{S}} \\ 0 \bmod \left|W_{G}(\overline{\mathcal{S}})_{p}\right| & \text { for } \quad \overline{\mathcal{T}} \in R_{0} \sim\{\overline{\mathcal{S}}\}\end{cases}

But, i~T‾∈Im⁡Ψ(p)G\tilde{i}_{\overline{\mathcal{T}}} \in \operatorname{Im} \Psi_{(p)}^{G} for any T‾∉R0\overline{\mathcal{T}} \notin R_{0}, which yields i~S‾∈Im⁡Ψ(p)G\tilde{i}_{\overline{\mathcal{S}}} \in \operatorname{Im} \Psi_{(p)}^{G}. This is a contradiction. Consequently, we have R0=∅R_{0}=\varnothing, and so Ψ(p)G\Psi_{(p)}^{G} is surjective.
4. Let S‾∈[Πn(G)]\overline{\mathcal{S}} \in\left[\Pi_{n}(\mathrm{G})\right]. Then

Ψ(p)G(Φ(p)G((S‾)G))=(∑rT0∈WG(T‾)(p)∣Inv⁡<r>T‾(S‾)∣ mod ∣WG(T‾)(p)∣)T‾∈[Πn(G)]\Psi_{(p)}^{G}\left(\Phi_{(p)}^{G}\left((\overline{\mathcal{S}})_{\mathrm{G}}\right)\right)=\left(\sum_{r T_{0} \in W_{G}(\overline{\mathcal{T}})_{(p)}}\left|\operatorname{Inv}_{\overline{\mathcal{T}}}(\overline{\mathcal{S}})\right| \bmod \left|W_{G}(\overline{\mathcal{T}})_{(p)}\right|\right)_{\overline{\mathcal{T}} \in\left[\Pi_{n}(\mathrm{G})\right]}

where Inv⁡<r>T‾(S‾)={gS0∈G/S0∣<r>T‾≤S‾}:=I<r>T‾\operatorname{Inv}_{\overline{\mathcal{T}}}(\overline{\mathcal{S}})=\left\{g S_{0} \in G / S_{0} \mid\overline{\mathcal{T}} \leq \overline{\mathcal{S}}\right\}:=I_{\overline{\mathcal{T}}}. Set W=WG(S‾)pW=W_{G}(\overline{\mathcal{S}})_{p}. Then Inv⁡T(S‾)\operatorname{Inv}_{\mathcal{T}}(\overline{\mathcal{S}}) can be view as a left WW-set by the action given by rT0⋅gS0=r T_{0} \cdot g S_{0}= rgS0r g S_{0}. With this action, one has Inv⁡<r>T‾(S‾)={gS0∈Inv⁡T(S‾)∣rT0⋅gS0=gS0}\operatorname{Inv}_{\overline{\mathcal{T}}}(\overline{\mathcal{S}})=\left\{g S_{0} \in \operatorname{Inv}_{\mathcal{T}}(\overline{\mathcal{S}}) \mid r T_{0} \cdot g S_{0}=g S_{0}\right\}. So

∑rT0∈W∣Inv⁡<r>T‾(S‾)∣=∑gS0∈IT‾{rT0∈W∣rT0⋅gS0=gS0}=∑gS0∈IT‾∣WgˉS0∣=∑gS0∈[W\IT‾]∣OgK0∣∣WgS0∣≡0 mod ∣W∣\begin{aligned} \sum_{r T_{0} \in W}\left|\operatorname{Inv}_{\overline{\mathcal{T}}}(\overline{\mathcal{S}})\right| & =\sum_{g S_{0} \in I_{\overline{\mathcal{T}}}}\left\{r T_{0} \in W \mid r T_{0} \cdot g S_{0}=g S_{0}\right\} \\ & =\sum_{g S_{0} \in I_{\overline{\mathcal{T}}}}\left|W_{\bar{g} S_{0}}\right| \\ & =\sum_{g S_{0} \in\left[W \backslash I_{\overline{\mathcal{T}}}\right]}\left|O_{g K_{0}}\right|\left|W_{g S_{0}}\right| \\ & \equiv 0 \bmod |W| \end{aligned}

and so Im⁡Φ(p)G⊆Ker⁡Ψ(p)G\operatorname{Im} \Phi_{(p)}^{G} \subseteq \operatorname{Ker} \Psi_{(p)}^{G}. It remains to prove that Ker⁡Ψ(p)G⊆Im⁡Φ(p)G\operatorname{Ker} \Psi_{(p)}^{G} \subseteq \operatorname{Im} \Phi_{(p)}^{G}. Now, since Ψ(p)G\Psi_{(p)}^{G} is surjective and Ψ(p)G∘Φ(p)G=0\Psi_{(p)}^{G} \circ \Phi_{(p)}^{G}=0, we have that Ψ(p)G\Psi_{(p)}^{G} factorizes through

the cokernel of Φ(p)G\Phi_{(p)}^{G}, which is isomorphic by Obs(p)O b s_{(p)} by 2 ). So, we obtain a surjective map Coker⁡Φ(p)G→Obs(p)\operatorname{Coker} \Phi_{(p)}^{G} \rightarrow O b s_{(p)} between two isomorphic groups. This map is then an isomorphism, and so Ker⁡Ψ(p)G\operatorname{Ker} \Psi_{(p)}^{G} is equal to Im⁡Φ(p)G\operatorname{Im} \Phi_{(p)}^{G}.

4 Idempotents elements

By the isomorphism QΦ\mathbb{Q} \Phi, there is an element eTG∈QBn(G)e_{\mathcal{T}}^{G} \in \mathbb{Q} B_{n}(G) for each nn-slice Tˇ\check{\mathcal{T}} of GG such that

QϕSˇG(eTˇG)={1 if Sˇ=GTˇ0 otherwise \mathbb{Q} \phi_{\check{\mathcal{S}}}^{G}\left(e_{\check{\mathcal{T}}}^{G}\right)=\left\{\begin{array}{lll} 1 & \text { if } & \check{\mathcal{S}}=_{G} \check{\mathcal{T}} \\ 0 & \text { otherwise } & \end{array}\right.

Clearly, the set {eTˇG∣Tˇ∈[Πn(G)]}\left\{e_{\check{T}}^{G} \mid \check{T} \in\left[\Pi_{n}(G)\right]\right\} is the set of primitive idempotents of QBn(G)\mathbb{Q} B_{n}(G).
In order to give the explicit formula of the primitive idempotent eTˇGe_{\check{T}}^{G}, we need the Möbius function of a finite poset (X,≤)(\mathcal{X}, \leq). The Möbius function μX:X×X→Z\mu_{\mathcal{X}}: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{Z} of a finite poset is defined inductively as follows (see [13]):

μX(x,x)=1,μX(x,y)=0 if x≤y,∑t≤yμX(x,t)=0 if x<y\mu_{\mathcal{X}}(x, x)=1, \quad \mu_{\mathcal{X}}(x, y)=0 \text { if } x \leq y, \quad \sum_{t \leq y} \mu_{\mathcal{X}}(x, t)=0 \text { if } x<y

Proposition 4.1. Let Sˇ\check{\mathcal{S}} be a n-slice of GG. Then the explicit formula of the primitive idempotent eSˇGe_{\check{\mathcal{S}}}^{G} is given by

eSˇG=1∣NG(Sˇ)∣∑Tˇ≤Sˇ∣T0∣μΠ(Tˇ,Sˇ)(Tˇ)Ge_{\check{\mathcal{S}}}^{G}=\frac{1}{\left|N_{G}(\check{\mathcal{S}})\right|} \sum_{\check{T} \leq \check{\mathcal{S}}}\left|\mathcal{T}_{0}\right| \mu_{\Pi}(\check{\mathcal{T}}, \check{\mathcal{S}})\left(\check{\mathcal{T}}\right)_{G}

where μΠ\mu_{\Pi} is the Möbius function of the poset (Πn(G),≤)\left(\Pi_{n}(G), \leq\right).
Proof. Let Kˇ\check{\mathcal{K}} be any nn-slice of GG. Then

QϕKˇG(∣G∣eSˇG)=∣G∣∣NG(Sˇ)∣∑Tˇ≤Sˇ∣Tˇ0∣μΠ(Tˇ,Sˇ)QϕKˇG(⟨Tˇ⟩G)=∣G∣∣NG(Sˇ)∣∑Tˇ≤Sˇ∣Tˇ0∣μΠ(Tˇ,Sˇ)∣(g∈G/Tˇ0∣Kˇ≤gTˇ)∣=∣G∣∣NG(Sˇ)∣∑gˇ∈G∑Kˇ≤Tˇ≤SˇμΠ(Tˇ,Sˇ)=∣G∣δ(Sˇ,Tˇ)\begin{aligned} \mathbb{Q} \phi_{\check{\mathcal{K}}}^{G}\left(|G| e_{\check{\mathcal{S}}}^{G}\right) & =\frac{|G|}{\left|N_{G}(\check{\mathcal{S}})\right|} \sum_{\check{T} \leq \check{\mathcal{S}}}\left|\check{\mathcal{T}}_{0}\right| \mu_{\Pi}(\check{\mathcal{T}}, \check{\mathcal{S}}) \mathbb{Q} \phi_{\check{\mathcal{K}}}^{G}\left(\left\langle\check{T}\right\rangle_{G}\right) \\ & =\frac{|G|}{\left|N_{G}(\check{\mathcal{S}})\right|} \sum_{\check{T} \leq \check{\mathcal{S}}}\left|\check{\mathcal{T}}_{0}\right| \mu_{\Pi}(\check{\mathcal{T}}, \check{\mathcal{S}})\left|\left(g \in G / \check{\mathcal{T}}_{0} \mid \check{\mathcal{K}} \leq^{g} \check{\mathcal{T}}\right)\right| \\ & =\frac{|G|}{\left|N_{G}(\check{\mathcal{S}})\right|} \sum_{\check{g} \in G} \sum_{\check{\mathcal{K}} \leq \check{\mathcal{T}} \leq \check{\mathcal{S}}} \mu_{\Pi}(\check{\mathcal{T}}, \check{\mathcal{S}}) \\ & =|G| \delta(\check{\mathcal{S}}, \check{\mathcal{T}}) \end{aligned}

where the second equality follows from Lemma 2.7. By the property of the Möbius function, we have that the sum ∑Kˇ≤Tˇ≤SˇμΠ(Tˇ,Sˇ)\sum_{\check{\mathcal{K}} \leq \check{\mathcal{T}} \leq \check{\mathcal{S}}} \mu_{\Pi}(\check{\mathcal{T}}, \check{\mathcal{S}}) is zero unless Kˇ=gSˇ\check{\mathcal{K}}=g \check{\mathcal{S}}, for any g∈Gg \in G. Therefore,

ϕKˇG(eSˇG)={1 if in 0 otherwise Kˇ=GSˇ\phi_{\check{\mathcal{K}}}^{G}\left(e_{\check{\mathcal{S}}}^{G}\right)=\left\{\begin{array}{lll} 1 & \text { if } & \text { in } \\ 0 & \text { otherwise } \end{array} \quad \check{\mathcal{K}}=_{G} \check{\mathcal{S}}\right.

In the case n=0n=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=1n=1 was stated by Bouc in [3]. The idempotent eSˇGe_{\check{\mathcal{S}}}^{G} is the only idempotent of QBn(G)\mathbb{Q} B_{n}(G) with the following property:

Proposition 4.2 (Characterization of eSGe_{\mathcal{S}}^{G} ).
Let Sˇ\check{\mathcal{S}} be a n-slice of GG. Then X.eSˇG=QϕSˇG(X)eSˇG\mathcal{X} . e_{\check{\mathcal{S}}}^{G}=\mathbb{Q} \phi_{\check{\mathcal{S}}}^{G}(\mathcal{X}) e_{\check{\mathcal{S}}}^{G}, for any X∈QBn(G)\mathcal{X} \in \mathbb{Q} B_{n}(G).
Conversely, if Y∈QBn(G)\mathcal{Y} \in \mathbb{Q} B_{n}(G) is such that X⋅Y=QϕSˇG(X)Y\mathcal{X} \cdot \mathcal{Y}=\mathbb{Q} \phi_{\check{\mathcal{S}}}^{G}(\mathcal{X}) \mathcal{Y}, then Y∈QeSˇG\mathcal{Y} \in \mathbb{Q} e_{\check{\mathcal{S}}}^{G} (that is Y\mathcal{Y} is a rational multiple of eSˇGe_{\check{\mathcal{S}}}^{G} ).

Proof.

Note that a Q\mathbb{Q}-basis of QBn(G)\mathbb{Q} B_{n}(G) is given by the eTˇGe_{\check{T}}^{G} where Tˇ\check{T} runs through the set Πn(G)\Pi_{n}(G) of conjugacy classes of nn-slices of GG. So for any X∈QBn(G)\mathcal{X} \in \mathbb{Q} B_{n}(G), we have that

X=∑Tˇ∈{Πn(G)}λTˇeTˇG\mathcal{X}=\sum_{\check{T} \in\left\{\Pi_{n}(G)\right\}} \lambda_{\check{T}} e_{\check{T}}^{G}

where λTˇ\lambda_{\check{T}} are rational numbers. Since the elements eTˇGe_{\check{T}}^{G} are orthogonal, it follows that for any nn-slice Sˇ∈[Πn(G)]\check{\mathcal{S}} \in\left[\Pi_{n}(G)\right] we have,

ϕSˇG(X)=λSˇ and X.eSˇG=ϕSˇG(X)eSˇG\phi_{\check{\mathcal{S}}}^{G}(\mathcal{X})=\lambda_{\check{\mathcal{S}}} \text { and } \mathcal{X} . e_{\check{\mathcal{S}}}^{G}=\phi_{\check{\mathcal{S}}}^{G}(\mathcal{X}) e_{\check{\mathcal{S}}}^{G}

Conversely, let Y\mathcal{Y} be an element of QBn(G)\mathbb{Q} B_{n}(G) verifying XY=ϕSˇG(X)Y\mathcal{X} \mathcal{Y}=\phi_{\check{\mathcal{S}}}^{G}(\mathcal{X}) \mathcal{Y} for any X∈\mathcal{X} \in QBn(G)\mathbb{Q} B_{n}(G). Then in particular eTˇGY=0e_{\check{T}}^{G} \mathcal{Y}=0 if Sˇ≠GTˇ\check{\mathcal{S}} \neq_{G} \check{T}, thus Y=ϕSˇG(Y)eSˇG\mathcal{Y}=\phi_{\check{\mathcal{S}}}^{G}(\mathcal{Y}) e_{\check{\mathcal{S}}}^{G} is a rational multiple of eSˇGe_{\check{\mathcal{S}}}^{G}.

Proposition 4.3. Let (X,≤)(\mathcal{X}, \leq) be a finite poset. Let Πn(X)\Pi_{n}(\mathcal{X}) denote the set of nn-tuples (x0,…,xn)\left(x_{0}, \ldots, x_{n}\right) of elements of X\mathcal{X} such that x0≤⋯≤xnx_{0} \leq \cdots \leq x_{n}. Define a partial order ≤\leq on Πn(X)\Pi_{n}(\mathcal{X}) by

(x0,…,xn)≤(y0,…,yn)⇔xi≤yi,∀i=0,…,n\left(x_{0}, \ldots, x_{n}\right) \leq\left(y_{0}, \ldots, y_{n}\right) \Leftrightarrow x_{i} \leq y_{i}, \forall i=0, \ldots, n

Then the Möbius function μΠ\mu_{\Pi} of the poset (Πn(X),≤)\left(\Pi_{n}(\mathcal{X}), \leq\right) can be computed as follows, for any xˉ:=(x0,…,xn),yˉ:=(y0,…,yn)∈Πn(X)\bar{x}:=\left(x_{0}, \ldots, x_{n}\right), \bar{y}:=\left(y_{0}, \ldots, y_{n}\right) \in \Pi_{n}(\mathcal{X}) :

μΠ(xˉ,yˉ)={∫i=0nμX(xi,yi) if x0≤y0≤x1≤y1⋯≤xn≤yn0 otherwise \mu_{\Pi}(\bar{x}, \bar{y})=\left\{\begin{array}{cc} \int_{i=0}^{n} \mu_{\mathcal{X}}\left(x_{i}, y_{i}\right) & \text { if } \quad x_{0} \leq y_{0} \leq x_{1} \leq y_{1} \cdots \leq x_{n} \leq y_{n} \\ 0 & \text { otherwise } \end{array}\right.

where μX\mu_{\mathcal{X}} is the Möbius function of the poset (X,≤)(\mathcal{X}, \leq).
Proof. Let m(xˉ,yˉ)m(\bar{x}, \bar{y}) denote the expression defined by the right hand side of (5). Then if xˉ≤yˉ\bar{x} \leq \bar{y}

∑i∈Πn(X)xˉ≤i≤yˉm(xˉ,iˉ)=∑(t0,…,tn)∈PnμX(x0,t0)…μX(xn,tn)\sum_{\substack{i \in \Pi_{n}(\mathcal{X}) \\ \bar{x} \leq i \leq \bar{y}}} m(\bar{x}, \bar{i})=\sum_{\left(t_{0}, \ldots, t_{n}\right) \in \mathcal{P}_{n}} \mu_{\mathcal{X}}\left(x_{0}, t_{0}\right) \ldots \mu_{\mathcal{X}}\left(x_{n}, t_{n}\right)

where Pn:={f∈Πn(X)∣{xi≤ti≤yix0≤⋯≤xny0≤⋯≤ynti≤xi+1∣ for all i=0,…,n−1\mathcal{P}_{n}:=\left\{f \in \Pi_{n}(\mathcal{X})\left|\left\{\begin{array}{c}x_{i} \leq t_{i} \leq y_{i} \\ x_{0} \leq \cdots \leq x_{n} \\ y_{0} \leq \cdots \leq y_{n} \\ t_{i} \leq x_{i+1}\end{array}\right| \text { for all } i=0, \ldots, n-1\right.\right..
So,

∑(t0,…,tn)∈PnμX(x0,t0)…μX(xn,tn)=\sum_{\left(t_{0}, \ldots, t_{n}\right) \in \mathcal{P}_{n}} \mu_{\mathcal{X}}\left(x_{0}, t_{0}\right) \ldots \mu_{\mathcal{X}}\left(x_{n}, t_{n}\right)=

(∑xn≤tn≤ynμX(xn,tn))(∑(t0,…,tn−1)∈Pn−1μX(x0,t0)…μX(xn−1,tn−1))\left(\sum_{x_{n} \leq t_{n} \leq y_{n}} \mu_{\mathcal{X}}\left(x_{n}, t_{n}\right)\right)\left(\sum_{\left(t_{0}, \ldots, t_{n-1}\right) \in \mathcal{P}_{n-1}} \mu_{\mathcal{X}}\left(x_{0}, t_{0}\right) \ldots \mu_{\mathcal{X}}\left(x_{n-1}, t_{n-1}\right)\right)

where Pn−1′:={(t0,…,tn−1)∈Πn−1(X)∣{xi≤ti≤yiti≤xi+1 for all i=0,…,n−1 for all i=0,…,n−1}\mathcal{P}_{n-1}^{\prime}:=\left\{\left(t_{0}, \ldots, t_{n-1}\right) \in \Pi_{n-1}(\mathcal{X})\left|\left\{\begin{array}{c}x_{i} \leq t_{i} \leq y_{i} \\ t_{i} \leq x_{i+1}\end{array} \begin{array}{c}\text { for all } i=0, \ldots, n-1 \\ \text { for all } i=0, \ldots, n-1\end{array}\right.\right\}\right..
The first factor of (⋆)(\star) is equal to 0 if xn≠ynx_{n} \neq y_{n}, and 1 if xn=ynx_{n}=y_{n}. Hence, if xn=ynx_{n}=y_{n}, then the second factor is equal to

∑(t0,…,tn−1)∈Pn−1μX(x0,t0)…μX(xn−1,tn−1)\sum_{\left(t_{0}, \ldots, t_{n-1}\right) \in P_{n-1}} \mu_{\mathcal{X}}\left(x_{0}, t_{0}\right) \ldots \mu_{\mathcal{X}}\left(x_{n-1}, t_{n-1}\right)

Hence inductively, we have

∑i∈Πn(X)x≤t≤ym(xˉ,tˉ)=∫i=0nδ(xi,yi)\sum_{\substack{i \in \Pi_{n}(\mathcal{X}) \\ x \leq t \leq y}} m(\bar{x}, \bar{t})=\int_{i=0}^{n} \delta\left(x_{i}, y_{i}\right)

and the proposition follows.
Corollary 4.4. Let S‾=(S0,…,Sn)\overline{\mathcal{S}}=\left(S_{0}, \ldots, S_{n}\right) and T‾=(T0,…,Tn)\overline{\mathcal{T}}=\left(T_{0}, \ldots, T_{n}\right) be nn-slices of GG. Then

μΠ(T‾,S‾)={∏i=0nμ(Ti,Si) if T0≤S0≤T1≤S1⋯≤Tn≤Sn0 otherwise \mu_{\Pi}(\overline{\mathcal{T}}, \overline{\mathcal{S}})=\left\{\begin{array}{cc} \prod_{i=0}^{n} \mu\left(T_{i}, S_{i}\right) & \text { if } \quad T_{0} \leq S_{0} \leq T_{1} \leq S_{1} \cdots \leq T_{n} \leq S_{n} \\ 0 & \text { otherwise } \end{array}\right.

where μ\mu is the Möbius function of the poset of subgroups of GG.
In particular

eS‾G=1∣NG(S‾)∣∑T0≤S0≤⋯≤Tn≤Sn∣T0∣μ(T0,S0)…μ(Tn,Sn)(T‾)Ge_{\overline{\mathcal{S}}}^{G}=\frac{1}{\left|N_{G}(\overline{\mathcal{S}})\right|} \sum_{T_{0} \leq S_{0} \leq \cdots \leq T_{n} \leq S_{n}}\left|T_{0}\right| \mu\left(T_{0}, S_{0}\right) \ldots \mu\left(T_{n}, S_{n}\right)(\overline{\mathcal{T}})_{G}

5 Prime ideals

The aim of this part is the proof that a finite group GG is solvable if and only if the prime ideal spectrum of Bn(G)B_{n}(G) is connected, i.e., if and only if 0 and 1 are the only idempotents in Bn(G)B_{n}(G), like established by A.Dress in [5] for B0(G)B_{0}(G) and S.Bouc in [3]. Note that Cn(G)C_{n}(G) is integral over Bn(G)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 Bn(G)B_{n}(G) comes from Cn(G)C_{n}(G).

Proposition 5.1. Let pp denote either 0 or a prime number. If S‾∈Πn(G)\overline{\mathcal{S}} \in \Pi_{n}(G), let IS‾,pI_{\overline{\mathcal{S}}, p} be the prime ideal of Bn(G)B_{n}(G) defined as the kernel of the ring homomorphism

Bn(G)→ϕS‾GZ→pZ/pZB_{n}(G) \xrightarrow{\phi_{\overline{\mathcal{S}}}^{G}} \mathbb{Z} \xrightarrow{p} \mathbb{Z} / p \mathbb{Z}

Then every prime ideal II of Bn(G)B_{n}(G) has the form IS‾,pI_{\overline{\mathcal{S}}, p} for a suitable S‾∈Πn(G)\overline{\mathcal{S}} \in \Pi_{n}(G). Moreover, given a prime ideal II there exists a unique K‾∈[Πn(G)]\overline{\mathcal{K}} \in\left[\Pi_{n}(G)\right] with I=IK,pI=I_{\mathcal{K}, p} and ϕK‾G(⟨K‾⟩G)≠\phi_{\overline{\mathcal{K}}}^{G}\left(\left\langle\overline{\mathcal{K}}\right\rangle_{G}\right) \neq 0( mod p)0(\bmod p) where pp is the characteristic of the ring R:=Bn(G)/IR:=B_{n}(G) / I.

Proof. Consider the natural map ϕ:Bn(G)→Bn(G)/I\phi: B_{n}(G) \rightarrow B_{n}(G) / I. Then ϕ(x)=ϕS‾(x)⋅1R\phi(x)=\phi_{\overline{\mathcal{S}}}(x) \cdot 1_{R} for some nn-slice S‾\overline{\mathcal{S}} by Proposition 3.1. So

I={x∈Bn(G)∣ϕ(x)=0}={x∈Bn(G)∣ϕS‾(x)⋅1R=0}={x∈Bn(G)∣ϕS‾(x)≡0( mod p)}=IS‾,p\begin{aligned} I & =\left\{x \in B_{n}(G) \mid \phi(x)=0\right\} \\ & =\left\{x \in B_{n}(G) \mid \phi_{\overline{\mathcal{S}}}(x) \cdot 1_{R}=0\right\} \\ & =\left\{x \in B_{n}(G) \mid \phi_{\overline{\mathcal{S}}}(x) \equiv 0(\bmod p)\right\}=I_{\overline{\mathcal{S}}, p} \end{aligned}

where pp is the characteristic of the ring Bn(G)/IB_{n}(G) / I.
Let In:={Sˇ∈Πn(G)∣⟨Sˇ⟩G∉I}\mathcal{I}_{n}:=\left\{\check{\mathcal{S}} \in \Pi_{n}(G) \mid\langle\check{\mathcal{S}}\rangle_{G} \notin I\right\}. If Tˇ\check{\mathcal{T}} and Kˇ\check{\mathcal{K}} are both minimal in I\mathcal{I}, then

⟨Tˇ⟩G⋅⟨Kˇ⟩G=∑g∈[I0\G/K0]⟨Tˇ∩SˇKˇ⟩G=ϕTˇ(⟨Kˇ⟩G)⟨Tˇ⟩G( mod I)\begin{aligned} \langle\check{\mathcal{T}}\rangle_{G} \cdot\langle\check{\mathcal{K}}\rangle_{G} & =\sum_{g \in\left[I_{0} \backslash G / K_{0}\right]}\langle\check{\mathcal{T}} \cap \check{\mathcal{S}}\check{\mathcal{K}}\rangle_{G} \\ & =\phi_{\check{\mathcal{T}}}\left(\langle\check{\mathcal{K}}\rangle_{G}\right)\langle\check{\mathcal{T}}\rangle_{G}(\bmod I) \end{aligned}

(This relation must hold for any ⟨Kˇ⟩G∉I\langle\check{\mathcal{K}}\rangle_{G} \notin I )
Since ⟨Tˇ⟩G⋅⟨Kˇ⟩G∉I\langle\check{\mathcal{T}}\rangle_{G} \cdot\langle\check{\mathcal{K}}\rangle_{G} \notin I, we have ϕTˇ(⟨Kˇ⟩G)≠0\phi_{\check{\mathcal{T}}}\left(\langle\check{\mathcal{K}}\rangle_{G}\right) \neq 0 and ϕKˇ(⟨Tˇ⟩G)≠0\phi_{\check{\mathcal{K}}}\left(\langle\check{\mathcal{T}}\rangle_{G}\right) \neq 0 by symmetry, and so Kˇ=GTˇ\check{\mathcal{K}}=_{G} \check{\mathcal{T}}. So the set In\mathcal{I}_{n} has a unique minimal element, up to isomorphism. On the other hand, the quotient ring Bn(G)/IB_{n}(G) / I is an integral domain, we have that either p=0p=0 or pp is a prime number. If p=0p=0, then the projection ϕ:Bn(G)→Bn(G)/I\phi: B_{n}(G) \rightarrow B_{n}(G) / I is equal to ϕKˇ\phi_{\check{\mathcal{K}}} and I=IK,pI=I_{\mathcal{K}, p}. If p≠0p \neq 0, then ϕ\phi is equal to the reduction of ϕKˇ\phi_{\check{\mathcal{K}}} modulo pp. If Gˇ\check{\mathcal{G}} is any nn-slice of GG with I=IG,pI=I_{\mathcal{G}, p} and ϕG(⟨Gˇ⟩G)≠0( mod p)\phi_{\mathcal{G}}\left(\langle\check{\mathcal{G}}\rangle_{G}\right) \neq 0(\bmod p) then for a Kˇ\check{\mathcal{K}} minimal in In\mathcal{I}_{n}

ϕKˇ(⟨Kˇ⟩G)≡ϕGˇ(⟨Kˇ⟩G)≠0( mod p)\phi_{\check{\mathcal{K}}}\left(\langle\check{\mathcal{K}}\rangle_{G}\right) \equiv \phi_{\check{\mathcal{G}}}\left(\langle\check{\mathcal{K}}\rangle_{G}\right) \neq 0(\bmod p)

In particular ϕKˇ(⟨Gˇ⟩G)\phi_{\check{\mathcal{K}}}\left(\langle\check{\mathcal{G}}\rangle_{G}\right) is non zero; and similarly ϕGˇ(⟨Kˇ⟩G)\phi_{\check{\mathcal{G}}}\left(\langle\check{\mathcal{K}}\rangle_{G}\right) is non zero. This can only occur if Kˇ=GGˇ\check{\mathcal{K}}=_{G} \check{\mathcal{G}}.

Notation 5.2. Let pp be a prime number.

Define inductively an increasing sequence (Sˇi)i\left(\check{\mathcal{S}}^{i}\right)_{i} in (Πn(G),≤)\left(\Pi_{n}(G), \leq\right) by Sˇ0=Sˇ\check{\mathcal{S}}^{0}=\check{\mathcal{S}} and Sˇi+1=(Sˇi)p∗\check{\mathcal{S}}^{i+1}=\left(\check{\mathcal{S}}^{i}\right)_{p}^{*}, for i∈Ni \in \mathbb{N}. We set Sˇ∞\check{\mathcal{S}}^{\infty} for the largest term of the sequence (Sˇi)i\left(\check{\mathcal{S}}^{i}\right)_{i}.

Note that if we define a relation " ∼p\stackrel{p}{\sim} " on Πn(G)\Pi_{n}(G) by Sˇ∼pSˇ\check{\mathcal{S}} \stackrel{p}{\sim} \check{\mathcal{S}} ’ if Sˇp→=GSˇp↪\check{\mathcal{S}}_{p}^{\rightarrow}=_{G} \check{\mathcal{S}}_{p}^{\hookrightarrow} then " ∼p\stackrel{p}{\sim} " is an equivalence relation.

Proposition 5.3. The nn-slice Sˇp→\check{\mathcal{S}}_{p}^{\rightarrow} is conjugate to Sˇ∞\check{\mathcal{S}}^{\infty}.
Proof. By definition, the nn-slice Sˇp→\check{\mathcal{S}}_{p}^{\rightarrow} is a minimal element Kˇ\check{\mathcal{K}} of the (Πn(G),≤)\left(\Pi_{n}(G), \leq\right) such that

ϕSˇ(⟨Kˇ⟩G):={[g∈G/K0∣Sˇg≤Kˇ]}≢0( mod p)\phi_{\check{\mathcal{S}}}\left(\langle\check{\mathcal{K}}\rangle_{G}\right):=\left\{\left[g \in G / K_{0} \mid \check{\mathcal{S}}^{g} \leq \check{\mathcal{K}}\right]\right\} \not \equiv 0(\bmod p)

Thus one can assume that Sˇ≤Kˇ\check{\mathcal{S}} \leq \check{\mathcal{K}}. Since ϕSˇ≡ϕPSˇ( mod p)\phi_{\check{\mathcal{S}}} \equiv \phi_{P \check{\mathcal{S}}}(\bmod p) for any pp-subgroup PP of NG(Sˇ)N_{G}(\check{\mathcal{S}}) by Corollary 2 , one can also assume that Sˇp→≤Kˇ\check{\mathcal{S}}_{p}^{\rightarrow} \leq \check{\mathcal{K}}, and inductively, that Sˇ∞≤Kˇ\check{\mathcal{S}}^{\infty} \leq \check{\mathcal{K}}. Moreover ϕSˇ∞≡ϕKˇ( mod p)\phi_{\check{\mathcal{S}}^{\infty}} \equiv \phi_{\check{\mathcal{K}}}(\bmod p). As NG(Sˇ∞)/S0∞N_{G}\left(\check{\mathcal{S}}^{\infty}\right) / S_{0}^{\infty} is a p′p^{\prime}-group, it follows that Sˇ∞=GKˇ\check{\mathcal{S}}^{\infty}=_{G} \check{\mathcal{K}}.

Proposition 5.4. Let (Sˇ,p),(Sˇ′,p′)(\check{\mathcal{S}}, p),\left(\check{\mathcal{S}}^{\prime}, p^{\prime}\right) be elements of Θn(G)\Theta_{n}(G).
Then ISˇ′,p′⊆ISˇ,pI_{\check{\mathcal{S}}^{\prime}, p^{\prime}} \subseteq I_{\check{\mathcal{S}}, p} if and only if

Proof. Assume that ISˇ′,p′⊆ISˇ,pI_{\check{\mathcal{S}}^{\prime}, p^{\prime}} \subseteq I_{\check{\mathcal{S}}, p}. Then there exists a surjective ring homomorphism ϕ\phi from Bn(G)/ISˇ′,p′≅Z/p′ZB_{n}(G) / I_{\check{\mathcal{S}}^{\prime}, p^{\prime}} \cong \mathbb{Z} / p^{\prime} \mathbb{Z} to Bn(G)/ISˇ,p≅Z/pZB_{n}(G) / I_{\check{\mathcal{S}}, p} \cong \mathbb{Z} / p \mathbb{Z}. Hence either p′=pp^{\prime}=p or p′=0p^{\prime}=0 and p>0p>0. If p=p′p=p^{\prime} then ϕ\phi is a bijection, and so ISˇ′,p′=ISˇ,pI_{\check{\mathcal{S}}^{\prime}, p^{\prime}}=I_{\check{\mathcal{S}}, p} which implies that Sˇ′\check{\mathcal{S}}^{\prime} and Sˇ\check{\mathcal{S}} are conjugate. If p′=0p^{\prime}=0 and p>0p>0 then ϕSˇG\phi_{\check{\mathcal{S}}}^{G} is the reduction modulo pp of ϕSˇG\phi_{\check{\mathcal{S}}}^{G}, and so ISˇ′,p⊆ISˇ,pI_{\check{\mathcal{S}}^{\prime}, p} \subseteq I_{\check{\mathcal{S}}, p}. Hence Sˇps→\check{\mathcal{S}}_{p}^{s \rightarrow} and Sˇ\check{\mathcal{S}} are conjugate in GG, by Corollary 5.1. Conversely, if Sˇ′\check{\mathcal{S}}^{\prime} and Sˇ\check{\mathcal{S}} are conjugate then ϕSˇG=ϕSˇG\phi_{\check{\mathcal{S}}}^{G}=\phi_{\check{\mathcal{S}}}^{G}, in particular ISˇ′,0=ISˇ,0I_{\check{\mathcal{S}}^{\prime}, 0}=I_{\check{\mathcal{S}}, 0}. If pp is a prime then Proposition 5.4 implies that ϕSˇG(x)≅ϕSˇps→G(x) mod p\phi_{\check{\mathcal{S}}}^{G}(x) \cong \phi_{\check{\mathcal{S}}_{p}^{s \rightarrow}}^{G}(x) \bmod p for any x∈Bn(G)x \in B_{n}(G). So ISˇps→,0⊆ISˇ,pI_{\check{\mathcal{S}}_{p}^{s \rightarrow}, 0} \subseteq I_{\check{\mathcal{S}}, p}. And if Sˇps→\check{\mathcal{S}}_{p}^{s \rightarrow} and Sˇ\check{\mathcal{S}} are conjugate in GG, then ISˇps→,p=ISˇ′,p=ISˇ,pI_{\check{\mathcal{S}}_{p}^{s \rightarrow}, p}=I_{\check{\mathcal{S}}^{\prime}, p}=I_{\check{\mathcal{S}}, p}. So ISˇps→,0⊆ISˇ,pI_{\check{\mathcal{S}}_{p}^{s \rightarrow}, 0} \subseteq I_{\check{\mathcal{S}}, p}.

Remark 5.5.

Proposition 5.6. Two ideals ISˇ,pI_{\check{\mathcal{S}}, p} and ISˇ′,p′I_{\check{\mathcal{S}}^{\prime}, p^{\prime}} are in the same connected component of Spec⁡(Bn(G))\operatorname{Spec}\left(B_{n}(G)\right) if and only D∞(Sˇ)D^{\infty}(\check{\mathcal{S}}) is conjugate to D∞(Sˇ′)D^{\infty}\left(\check{\mathcal{S}}^{\prime}\right), where
D∞(Kˇ):=(D∞(Kˇ0),…,D∞(Kˇn))D^{\infty}(\check{\mathcal{K}}):=\left(D^{\infty}\left(\check{\mathcal{K}}_{0}\right), \ldots, D^{\infty}\left(\check{\mathcal{K}}_{n}\right)\right) and D∞(Kˇi)D^{\infty}\left(\check{\mathcal{K}}_{i}\right) denotes the last term in the derived series of Kˇi\check{\mathcal{K}}_{i}. In particular, Spec⁡(Bn(G))\operatorname{Spec}\left(B_{n}(G)\right) is connected if and only if GG is solvable.
Proof. Recall that if RR is a Noetherian ring. For any prime ideal I∈Spec⁡(R)I \in \operatorname{Spec}(R), let Iˉ=\bar{I}= {P∣P∈Spec⁡(R),P⊃I}\{P \mid P \in \operatorname{Spec}(R), P \supset I\} be the closure of II in Spec⁡(R)\operatorname{Spec}(R). Then, two ideals PP and P′P^{\prime} are in the same connected component of Spec⁡(R)\operatorname{Spec}(R) if and only if there exists a series of minimal ideals I1,…,InI_{1}, \ldots, I_{n} with P∈Iˉ1,P′∈IˉnP \in \bar{I}_{1}, P^{\prime} \in \bar{I}_{n} and Iˉi∩Ii+1‾≠∅\bar{I}_{i} \cap \overline{I_{i+1}} \neq \varnothing for i=1,…,n−1i=1, \ldots, n-1. Now, if R=Bn(G)R=B_{n}(G), then IˇSˇ,0∩IˇSˇ′,0≠∅\check{I}_{\check{\mathcal{S}}, 0} \cap \check{I}_{\check{\mathcal{S}}^{\prime}, 0} \neq \varnothing if and only if the nn-slice Sˇps→\check{\mathcal{S}}_{p}^{s \rightarrow} is conjugate to Sˇps→\check{\mathcal{S}}_{p}^{s \rightarrow}, for some prime pp. Hence, if ISˇ,pI_{\check{\mathcal{S}}, p} and ISˇ′,p′I_{\check{\mathcal{S}}^{\prime}, p^{\prime}} are in the same connected component of Spec⁡(Bn(G))\operatorname{Spec}\left(B_{n}(G)\right) then D∞(Sˇ)D^{\infty}(\check{\mathcal{S}}) is conjugate to D∞(Sˇ′)D^{\infty}\left(\check{\mathcal{S}}^{\prime}\right). The ideals ISˇ,pI_{\check{\mathcal{S}}, p} and ID∞(Sˇ),0I_{D^{\infty}\left(\check{\mathcal{S}}\right), 0} are in the same connected component. Indeed, one can find a series of normal subgroups of S0S_{0} such that D∞(S0)=S0(n)∗S0(n−1)∗⋯∗S0(1)∗S0(0)=S0D^{\infty}\left(S_{0}\right)=S_{0}^{(n)} * S_{0}^{(n-1)} * \cdots * S_{0}^{(1)} * S_{0}^{(0)}=S_{0} such that S0(i−1)/S0(i)S_{0}^{(i-1)} / S_{0}^{(i)} is a pip_{i}-group for some prime pi(i=1,…,n)p_{i}(i=1, \ldots, n). Hence letting, Kˇi=(S0(n),S1,…,Sn)\check{\mathcal{K}}_{i}=\left(S_{0}^{(n)}, S_{1}, \ldots, S_{n}\right) for i=0,…,ni=0, \ldots, n one obtains

Sˇ=Kˇ0∼p0Kˇ1∼p1…∼pn−2Kˇn−1∼pn−1Kˇn=(D∞(S0),S1,…,Sn)\check{\mathcal{S}}=\check{\mathcal{K}}_{0} \stackrel{p_{0}}{\sim} \check{\mathcal{K}}_{1} \stackrel{p_{1}}{\sim} \ldots \stackrel{p_{n-2}}{\sim} \check{\mathcal{K}}_{n-1} \stackrel{p_{n-1}}{\sim} \check{\mathcal{K}}_{n}=\left(D^{\infty}\left(S_{0}\right), S_{1}, \ldots, S_{n}\right)

and
Iδ,p∈I^κˉ0,0,ID=(S0),S1,…,Sn,0∈I^κˉn,0I_{\delta, p} \in \hat{I}_{\bar{\kappa}_{0}, 0}, I_{D=\left(S_{0}\right), S_{1}, \ldots, S_{n}, 0} \in \hat{I}_{\bar{\kappa}_{n}, 0} and I^κˉi−1,0∩I^κˉi,0≠∅\hat{I}_{\bar{\kappa}_{i-1}, 0} \cap \hat{I}_{\bar{\kappa}_{i}, 0} \neq \varnothing. So, Iδ,pI_{\delta, p} and ID=(S0),S1,…,Sn,0I_{D=\left(S_{0}\right), S_{1}, \ldots, S_{n}, 0} are in the same connected component. By the same proceed, one prove that I(D=(S0),D=(S1),…,Sn),0I_{\left(D=\left(S_{0}\right), D=\left(S_{1}\right), \ldots, S_{n}\right), 0} and Iδ,pI_{\delta, p} are in the same component, and so one.

6 Green biset functor

Let RR be a commutative ring with identity. The biset category over RR will be denoted by RCR \mathcal{C} : its objects are all finite groups, and that for finite groups GG and H , the hom-set Hom⁡RC(G,H)\operatorname{Hom}_{R C}(G, H) is RB(H,G)=R⊗B(H,G)R B(H, G)=R \otimes B(H, G), where B(H,G)B(H, G) is the Grothendieck group of the category of finite (H,G)(H, G)-bisets. The composition of morphisms in RCR \mathcal{C} is induced by RR-bilinearity from the composition of bisets (see [2] Definition 3.1.1).

A good choice of a family G\mathcal{G} of finite groups and for every G,H∈GG, H \in \mathcal{G}, a set Γ(G,H)\Gamma(G, H) of subgroups of G×HG \times H can lead to an important category. For example if we fix a non-empty class D\mathcal{D} of finite groups closed under subquotients and cartesian products and we denote by RDR \mathcal{D} the full subcategory of RCR \mathcal{C} consisting of groups in D\mathcal{D}, then RDR \mathcal{D} is a replete subcategory of RCR \mathcal{C} in the sense of Bouc [2].

The category of biset functors, i.e. the category of RR-linear functors from RCR \mathcal{C} to the category R−Mod⁡R-\operatorname{Mod} of all RR-modules, will be denoted by FR\mathcal{F}_{R}. The category FD,R\mathcal{F}_{\mathcal{D}, R} of D\mathcal{D}-biset is the category of RR-linear functors from RDR \mathcal{D} to R−Mod⁡R-\operatorname{Mod}.

Let GG be a finite group, HH a subgroup of of GG and NN be a normal subgroup of G. One sets

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

[H×G/L]=Ind⁡DH∘Inf⁡D/CD∘Iso⁡(f)∘Def⁡B/AB∘Res⁡BG[H \times G / L]=\operatorname{Ind}_{D}^{H} \circ \operatorname{Inf}_{D / C}^{D} \circ \operatorname{Iso}(f) \circ \operatorname{Def}_{B / A}^{B} \circ \operatorname{Res}_{B}^{G}

where (D,C)(D, C) and (B,A)(B, A) are 1-slices of HH and GG respectively with the additional properties that C◃DC \triangleleft D and A◃BA \triangleleft B, and ff is some the group isomorphism from B/AB / A to D/CD / C (see [2] Lemma 2.3.26 for more details).

A Green D\mathcal{D}-biset functor is defined as a monoid in FD,R\mathcal{F}_{\mathcal{D}, R}. This is equivalent to the following definitions:
Definition 6.1 ([2] Definition 8.5.1). A D\mathcal{D}-biset functor AA is a Green D\mathcal{D}-biset functor if it is equipped with a linear products A(G)×A(H)→A(G×H)A(G) \times A(H) \rightarrow A(G \times H) denoted by (a,b)↦a×b(a, b) \mapsto a \times b, for groups, G,HG, H in D\mathcal{D}, and an element eA∈A(1)e_{A} \in A(1), satisfying the following conditions:

  1. (Associativity). Let G,HG, H and KK be groups in D\mathcal{D}. If we consider the canonical isomorphism from G×(H×K)G \times(H \times K) to (G×H)×K(G \times H) \times K, then for any a∈A(G),b∈A(H)a \in A(G), b \in A(H) and c∈A(K)c \in A(K)

(a×b)×c=A(Iso⁡G×(H×K)(G×H)×K)(a×(b×c))(a \times b) \times c=A\left(\operatorname{Iso}_{G \times(H \times K)}^{(G \times H) \times K}\right)(a \times(b \times c))

  1. (Identity element). Let GG be a group in D\mathcal{D} and consider the canonical isomorphisms 1×G→G1 \times G \rightarrow G and G×1→GG \times 1 \rightarrow G. Then for any a∈A(G)a \in A(G)

a=A(Iso⁡1×GG)(ϵA×a)=A(Iso⁡G×1G)(a×ϵA)a=A\left(\operatorname{Iso}_{1 \times G}^{G}\right)\left(\epsilon_{A} \times a\right)=A\left(\operatorname{Iso}_{G \times 1}^{G}\right)\left(a \times \epsilon_{A}\right)

  1. (Functoriality). If ϕ:G→G′\phi: G \rightarrow G^{\prime} and ψ:H→H′\psi: H \rightarrow H^{\prime} are morphisms in RDR \mathcal{D}, then for any a∈A(G)a \in A(G) and b∈A(H)b \in A(H)

A(ϕ×ψ)(a×b)=A(ϕ)(a)×A(ψ)(b)A(\phi \times \psi)(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):

Definition 6.2 ([12] Definiciòn 3.2.7). A D\mathcal{D}-Green biset functor is an object A∈A \in FD,B\mathcal{F}_{\mathcal{D}, B} together with the datum of an RR-algebra sructure on each A(H),H∈DA(H), H \in \mathcal{D}, such that the following axioms are satisfied for all groups in KK and GG in D\mathcal{D} and all group homomorphisms K→GK \rightarrow G :

  1. For the (K,G)(K, G)-biset GG, which we denote by GrG_{r}, the morphism A(Gr)A\left(G_{r}\right) is a ring homomorphism.
  2. For the (G,K)(G, K)-biset GG, denoted by GlG_{l}, the morphism A(Gl)A\left(G_{l}\right) satisfies the Frobenius identities for all b∈A(G)b \in A(G) and a∈A(K)a \in A(K),

A(Gl)(a)⋅b=A(Gl)(a⋅A(Gr)(b))b⋅A(Gl)(a)=A(Gl)(A(Gr)(b)⋅a)\begin{aligned} & A\left(G_{l}\right)(a) \cdot b=A\left(G_{l}\right)\left(a \cdot A\left(G_{r}\right)(b)\right) \\ & b \cdot A\left(G_{l}\right)(a)=A\left(G_{l}\right)\left(A\left(G_{r}\right)(b) \cdot a\right) \end{aligned}

where ⋅\cdot denotes the ring product on A(G)A(G), resp. A(K)A(K).
Definition 6.3. If AA and CC are Green D\mathcal{D}-biset functors, a morphism of Green D\mathcal{D} biset functors from AA to CC is a natural transformations f:A→Cf: A \rightarrow C such that fH×K(a×f_{H \times K}(a \times b)=fH(a)×fK(b)b)=f_{H}(a) \times f_{K}(b) for any groups HH and KK in D\mathcal{D} and any a∈A(H),b∈A(K)a \in A(H), b \in A(K), and such that f1(ϵA)=ϵCf_{1}\left(\epsilon_{A}\right)=\epsilon_{C}.

Proposition 6.4. The correspondence

G↦Bn(G)G \mapsto B_{n}(G)

defines a structure of Green biset functor.
Proof. There are several steps:

U×GXnf:(U×GX0→Uf1U×GX1→Uf2…U×GXn−1→UfnU×GXn)U \times_{G} \mathcal{X}_{n}^{f}:\left(U \times_{G} X_{0} \xrightarrow{U f_{1}} U \times_{G} X_{1} \xrightarrow{U f_{2}} \ldots U \times_{G} X_{n-1} \xrightarrow{U f_{n}} U \times_{G} X_{n}\right)

defined by Ufi:U×GXi−1→U×GXii(u,x)↦(u,fi(x))U f_{i}: U \times_{G} X_{i-1} \rightarrow U \times_{G} X_{i i}(u, x) \mapsto\left(u, f_{i}(x)\right) for any i=1,…,ni=1, \ldots, n. Let μ=(μi)\mu=\left(\mu_{i}\right) be a morphism from Xnf\mathcal{X}_{n}^{f} to YnZ\mathcal{Y}_{n}^{\mathcal{Z}}. Then μ\mu defines a morphism of (n,H)(n, H)-simplices from U×GXnfU \times_{G} \mathcal{X}_{n}^{f} to U×GYnZU \times_{G} \mathcal{Y}_{n}^{\mathcal{Z}} given by

U×Gμ=(Uμi:(u,x)↦(u,μi(x)))iU \times_{G} \mu=\left(U \mu_{i}:(u, x) \mapsto\left(u, \mu_{i}(x)\right)\right)_{i}

Indeed, for any i,xi∈Xii, x_{i} \in X_{i} and u∈Uu \in U, we have

(Uμi)∘(Ufi)(u,xi−1)=(u,μifi(xi−1))=(u,giμi−1(xi−1))=(Ugi)∘(Uμi−1)(u,xi−1)\begin{aligned} \left(U \mu_{i}\right) \circ\left(U f_{i}\right)\left(u, x_{i-1}\right) & =\left(u, \mu_{i} f_{i}\left(x_{i-1}\right)\right) \\ & =\left(u, g_{i} \mu_{i-1}\left(x_{i-1}\right)\right) \\ & =\left(U g_{i}\right) \circ\left(U \mu_{i-1}\right)\left(u, x_{i-1}\right) \end{aligned}

Therefore, the correspondence IU:Xnf↦U×GXnfI_{U}: \mathcal{X}_{n}^{f} \mapsto U \times_{G} \mathcal{X}_{n}^{f} is a functor from the (n,G)(n, G)-simplices to (n,H)(n, H)-simplices. On the other hand, it is straightforward to show that the defining relations of Bn(G)B_{n}(G) are mapped to the defining relations of Bn(H)B_{n}(H). Hence, the later functor induces a homomorphism of groups Bn(U):Bn(G)→Bn(H)B_{n}(U): B_{n}(G) \rightarrow B_{n}(H).

Clearly, if U≅U′U \cong U^{\prime} (as (H,G)-bisets) then the functors IUI_{U} and IU′I_{U^{\prime}} are isomorphic. So, Bn(U)=Bn(U′)B_{n}(U)=B_{n}\left(U^{\prime}\right). If UU has the form U=U1⊔U2U=U_{1} \sqcup U_{2} (as (H,G)bisets) then IU=IU1⊔IU2I_{U}=I_{U_{1}} \sqcup I_{U_{2}}, and so Bn(U)=Bn(U1)+Bn(U2)B_{n}(U)=B_{n}\left(U_{1}\right)+B_{n}\left(U_{2}\right). We may state that for any (K,H)(K, H)-biset VV, we have a isomorphism of (n,K)(n, K)-simplices between V×H(U×Xnf)V \times_{H}\left(U \times \mathcal{X}_{n}^{f}\right) and (V×HU)×Xnf\left(V \times_{H} U\right) \times \mathcal{X}_{n}^{f} which induces an isomorphism IV∘IU≅IV×HUI_{V} \circ I_{U} \cong I_{V \times_{H} U} and so Bn(V)∘Bn(U)=Bn(V×HU)B_{n}(V) \circ B_{n}(U)=B_{n}\left(V \times_{H} U\right). Finally, since IIdG≅1I_{\mathrm{Id}_{G}} \cong 1 (the functor identity, we have Bn(IdG)=1Bn(G)B_{n}\left(I d_{G}\right)=1_{B_{n}(G)}. This shows that we have a functor from the biset category to the category Z−Mod\mathbb{Z}-\mathrm{Mod}.

×:Bn(G)×Bn(H)→Bn(G×H),((a,b)↦a×b)\times: B_{n}(G) \times B_{n}(H) \rightarrow B_{n}(G \times H), \quad((a, b) \mapsto a \times b)

and (n,1)(n, 1)-simplex e\mathbf{e} is the identity of the product, up to identification G×1=G \times 1= GG. One verifies easily that the axioms of Definition 6.1 are satisfied.

Proposition 6.5. The functors dj,j=1,…,nd_{j}, j=1, \ldots, n (resp. si,i=0,…,n−1s_{i}, i=0, \ldots, n-1 ) induce morphisms of Green biset functors

dj:Bn→Bn−1( resp. si:Bn−1→Bn)d_{j}: B_{n} \rightarrow B_{n-1} \quad\left(\text { resp. } s_{i}: B_{n-1} \rightarrow B_{n}\right)

such that the identities (1), (2), (3) hold.
Proof. This is a simple verification.

Acknowledgement

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

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Ibrahima Tounkara
Centro de Ciencias Matemáticas
UNAM,
C.P. 58089

Morelia Mich
Mexico
e-mail: tounkara@matmor.unam.mx