A ghost ring for the left-free double Burnside ring and an application to fusion systems^{∗}

Robert Boltje Department of Mathematics University of California Santa Cruz, CA 95064 U.S.A. boltje@ucsc.edu

Susanne Danz Mathematical Institute, University of Oxford 24-29 St Giles’ Oxford, OX1 3LB United Kingdom danz@maths.ox.ac.uk

April 11, 2011

Abstract

For a finite group G, we define a ghost ring and a mark homomorphism for the double Burnside ring B^{◁}(G, G) of left-free (G, G)-bisets. In analogy to the case of the Burnside ring B(G), the ghost ring has a much simpler ring structure, and after tensoring with ℚ one obtains an isomorphism of ℚ-algebras. As an application of a key lemma, we obtain a very general formula for the Brauer construction applied to a tensor product of two p-permutation bimodules M and N in terms of Brauer constructions of the bimodules M and N. Over a field of characteristic 0 we determine the simple modules of the left-free double Burnside algebra and prove semisimplicity results for the bifree double Burnside algebra. These results carry over to results about biset-functor categories. Finally, we apply the ghost ring and mark homomorphism to fusion systems on a finite p-group. We extend a remarkable bijection, due to Ragnarsson and Stancu, between saturated fusion systems and certain idempotents of the bifree double Burnside algebra over ℤ_{(p)}, to a bijection between all fusion systems and a larger set of idempotents in the bifree double Burnside algebra over ℚ.

Introduction

The Burnside ring B(G) of a finite group G, the Grothendieck ring of the category of finite G-sets, has proved to be an object of central importance for the representation theory of finite groups. It is an initial object in functorial approaches to the representation theory of finite groups, in the sense that there is a unique functorial homomorphism from the Burnside ring to any other representation ring of G, for example to the character ring. Induction theorems that are now cornerstones of representation theory are proved through the study of these homomorphisms, see [D1]. The Burnside ring is also an important invariant of the group G itself. For instance, a fundamental theorem of Dress, cf. [D2], states that G is soluble if and only if B(G) is connected, i.e., 0 and 1 are the only idempotents of B(G). The main tool in achieving these results and studying the ring structure of B(G) is the so-called mark homomorphism Φ: B(G) → ∏_{H⩽G} ℤ. It is an injective ring homomorphism whose image is contained in the ring B̃(G) of G-fixed points of the latter product ring, where G acts by permutation of the components with respect to the conjugation action on the set of subgroups H of G. The ring B̃(G) is often called the ghost ring of B(G). It can be interpreted as the integral closure of B(G) in ℚB(G) := ℚ ⊗_{ℤ} B(G). The image of Φ has finite index in B̃(G), so that Φ induces an isomorphism of ℚ-algebras ℚB(G) → (∏_{H⩽G} ℚ)^{G} = ℚB̃(G) after scalar extension to ℚ.

∗MR Subject Classification: 19A22, 20C20. Keywords: Burnside ring, double Burnside ring, biset, biset functor, fusion system, Brauer construction, mark homomorphism, ghost ring

The main purpose of this paper is to construct a ghost ring and a mark homomorphism for certain subrings of the double Burnside ring B(G, G) such that these ghost rings have a simpler multiplicative structure. The ring B(G, G) is the Grothendieck group of the category of finite (G, G)-bisets. More generally, for finite groups G and H, one constructs B(G, H) as the Grothendieck group of the category of finite (G, H)-bisets. If K is another finite group, a construction on bisets that is similar to the tensor product of bimodules induces a bilinear map B(G, H) × B(H, K) → B(G, K). This defines the ring structure on B(G, G). Considering the subcategories of left-free and bifree (G, H)-bisets, one obtains subgroups B^{Δ}(G, H) ⊆ B^{◁}(G, H) ⊆ B(G, H) that are stable under the bilinear map. In particular, one has unitary subrings B^{Δ}(G, G) ⊆ B^{◁}(G, G) ⊆ B(G, G). One should point out that these rings are not commutative, in contrast to B(G). We refer the reader to work of Bouc, cf. [Bc96a] and [Bc10], for motivations and fundamentals on the double Burnside group. More recently, this group has become a focus of attention through applications to modular representation theory (cf. [Bc06]), connections with algebraic topology (cf. [MP], [BLO03]) and connections with fusion systems on p-groups (cf. [BLO04],

[R] and [RS]).

In this paper we construct bifree and left-free ghost groups B̃^{Δ}(G, H) ⊆ B̃^{◁}(G, H) and a mark homomorphism ρ_{G,H}: B^{◁}(G, H) → B̃^{◁}(G, H), which restricts to a homomorphism B^{Δ}(G, H) → B̃^{Δ}(G, H). We also construct a bilinear map B̃^{◁}(G, H) × B̃^{◁}(H, K) → B̃^{◁}(G, K) such that the maps ρ commute with the bilinear maps. We show that ρ_{G,H} is injective and has finite cokernel. The construction of the ghost ring and the mark homomorphism is carried out in Section 4, and the main properties are proved in Theorem 4.7.

The key observation to the construction of the ghost rings and mark homomorphisms is Theorem 2.3, which shows how to express fixed points of the tensor product of two left-free bisets X and Y in terms of fixed points of X and Y. As an immediate application we derive a formula (see Theorem 3.2) for the Brauer construction of the tensor product of two p-permutation bimodules M and N in terms of the Brauer constructions of M and N. Special cases and variations of this formula were known before, see for instance [Ri, Section 4], [BX, Corollary 3.6] and [L2, Theorem 9.2].

One obvious application of the ghost rings and mark homomorphisms is that they give a different perspective on the ring structures of B^{Δ}(G, G) and B^{◁}(G, G), especially if tensored with ℚ. The ghost ring of B^{Δ}(G, G) decomposes into a direct product of rings, indexed by the isomorphism classes of subgroups of G. We are able to give three alternative descriptions of this ghost ring. One of them is a direct product of endomorphism rings of permutation modules over outer automorphism groups of subgroups of G. These constructions are given in Section 5. Their main properties are stated in Theorem 5.2, Theorem 5.5 and Theorem 5.7, respectively. The last one of these variations is applied later to fusion systems on finite p-groups. The ghost ring B̃^{◁}(G, G) carries a natural grading whose component in degree 0 is the subring B̃^{Δ}(G, G). Tensoring with ℚ leads to a direct sum decomposition ℚB̃^{◁}(G, G) = ℚB̃^{Δ}(G, G) ⊕ J(ℚB̃^{◁}(G, G)), where the second summand denotes the Jacobson radical. Via the mark isomorphism this leads to a decomposition ℚB^{◁}(G, G) = ℚB^{Δ}(G, G) ⊕ J, where J denotes the Jacobson radical of ℚB^{◁}(G, G). Therefore, there is a natural bijection between the simple modules of ℚB^{◁}(G, G) and those of ℚB^{Δ}(G, G).

In [Bc96a] Bouc defines a category whose objects are the finite groups, whose morphism sets are the groups B(G, H) and whose composition law is the bilinear map mentioned above. Representation groups of finite groups can be considered as additive functors (biset functors) on this category. The structure of a biset functor was a key tool in the classification of endo-permutation modules of p-groups, achieved by Bouc, Thévenaz, Carlson, and others. The category of left-free (respectively, bifree) biset functors is equivalent to a module category of the algebra A^{◁} := ⨁_{G,H} B^{◁}(G, H) (respectively A^{Δ} := ⨁_{G,H} B^{Δ}(G, H)), endowed with a natural multiplication. Through the above mark homomorphisms these algebras embed into the ghost algebras Ã^{◁} (respectively Ã^{Δ}) whose multiplicative structure is much cleaner. Over the rational numbers we even obtain isomorphisms of algebras shedding new light on the biset functors. For instance, known results about the semisimplicity of the bifree-biset-functor category, cf. [W, Corollary 9.2], become very clear from this point of view, cf. Remark 5.10. Turning to left-free biset functors, one can see an obvious structure of graded algebra on ℚÃ^{◁}, which is not apparent on ℚA^{◁}. This leads to a filtration on any left-free biset functor over ℚ, cf. Remark 6.9. We do not go into any detailed study of this line of application, and hope to explore this in a future paper. But we want to mention recent work of Webb in this direction, cf. [W]

All the above results we actually state and prove in vastly greater generality. For exposition reasons we have only described them here in the case of the bifree and left-free double Burnside groups. More generally, one can fix a class D of finite groups and, for every pair (G, H) of groups in D, a set S_{G,H} of subgroups of G × H and then consider the Grothendieck group B^{S}(G, H) of finite bisets whose point stabilizers belong to S_{G,H}. We need to require some axioms on the selection of the sets S_{G,H}, which are stated in Hypothesis 1.13. The reason for formulating results in this generality is to allow, for future applications, more flexible classes of biset functors, and to be able to apply the theory developed so far to fusion systems in the last section of the current paper.

The notion of a fusion system F on a finite p-group originated from work of Puig about 40 years ago, and has recently got a lot of attention in modular representation theory of finite groups, since every block defines a fusion system on a defect group, in algebraic topology (cf. [BLO03] and [BLO04]), and in abstract group theory (cf. [A] and [AC]) as a potential new avenue to the classification of finite simple groups. The so-called saturated fusion systems play a particularly fundamental role. In this paper we observe (cf. Theorem 7.3) that the set of fusion systems on a finite p-group S is isomorphic (as a partially ordered set) to the set of systems of subgroups of S × S satisfying the axioms in Hypothesis 1.13. We propose to study the associated subring B^{F}(S, S) of the bifree double Burnside ring B^{Δ}(S, S) as an algebraic invariant of a fusion system. In [RS], Ragnarsson and Stancu described a remarkable bijection between the set of saturated fusion systems on S and a certain set of idempotents in ℤ_{(p)}B^{Δ}(S, S). Using the mark homomorphism for B^{Δ}(S, S), we are able to explicitly determine these idempotents as elements in the ghost algebra ℚB̃^{Δ}(S, S), and extend the bijection of Ragnarsson–Stancu to a bijection between the set of all fusion systems on S and a set of idempotents in ℚB̃^{Δ}(S, S), cf. Theorem 7.15. In the end, we consider the class of fusion systems whose associated idempotents are mapped, under the mark homomorphism, to idempotents in ℤ_{(p)}B̃^{Δ}(S, S). By definition, this class contains the class of saturated fusion systems. We show that it is strictly bigger (cf. Example 7.18) and that it shares some properties with the class of saturated fusion systems (cf. Proposition 7.19).

The present paper is arranged as follows. In Section 1 we introduce the necessary notions and recall the necessary results on the double Burnside group and biset functors. In Section 2 we show how one can express the fixed points of the tensor product of two bisets X and Y in terms of fixed points of X and fixed points of Y. This has an immediate application, expressing the Brauer construction applied to a tensor product of p-permutation bimodules M and N in terms of Brauer constructions of M and N, which is carried out in Section 3. In Section 4 we introduce the construction of the ghost group B̃^{◁}(G, H) and the mark homomorphism ρ_{G,H}: B^{◁}(G, H) → B̃^{◁}(G, H), and we prove their main properties. Section 5 continues with a closer study of the group B^{Δ}(G, H), for which we introduce an alternative ghost group consisting of homomorphism groups and an alternative mark homomorphism σ_{G,H}. This leads to the semisimplicity results regarding ℚB^{Δ}(G, G) and related biset functor categories, and to the determination of simple modules of ℚB^{Δ}(G, G) and of simple biset functors. In Section 6 we show that ℚB^{◁}(G, H) has a natural grading, turning ℚB^{◁}(G, G) and the biset algebra ℚA^{◁} into graded ℚ-algebras. This enables us to see that the simple modules for ℚB^{◁}(G, G) (respectively ℚA^{◁}) are the same as for ℚB^{Δ}(G, G) (respectively ℚA^{Δ}). Finally, in Section 7 we give an application to fusion systems.

Acknowledgements. The authors’ research on this project has been supported through a Marie Curie Intra-European Fellowship (grant PIEF-GA-2008-219543). Both authors wish to thank the Mathematical Institute of the University of Oxford and the Mathematics Department of UC Santa Cruz for their hospitality during mutual visits.

1 Bisets and the double Burnside group

Throughout this section, G, H, and K denote finite groups. Moreover, R denotes an associative unitary commutative ring. We recall the definition and basic properties of (G, H)-bisets and their Grothendieck group, the double Burnside group B(G, H). As a convention throughout this paper, G-sets without further specification will always be assumed to be finite left G-sets. We refer the reader to [Bc96a], [Bc00], [Bc10], and [CR, §80] for more details concerning the results of this section.

1.1 Notation. We will write U ⩽ G to indicate that U is a subgroup of G. For g ∈ G we denote the inner automorphism x ↦ gxg^{−1} of G by c_{g}, and for U ⩽ G we set ^{g}U := gUg^{−1} and U^{g} := g^{−1}Ug. If U and V are subgroups of G we write U ⩽_{G} V if U is G-conjugate to a subgroup of V, and we write U =_{G} V if U and V are G-conjugate. Moreover, we denote by Hom_{G}(U, V) the set of group homomorphisms φ: U → V such that there exists some g ∈ G with φ(u) = c_{g}(u) for all u ∈ U. We also write Aut_{G}(U) instead of Hom_{G}(U, U). Note that the homomorphism N_{G}(U) → Aut(U), g ↦ c_{g}, induces an isomorphism N_{G}(U)/C_{G}(U) → Aut_{G}(U). Here, N_{G}(U) and C_{G}(U) denote the normalizer and centralizer of U in G, respectively. We write Σ_{G} for the set of subgroups of G. If T is any other group then Σ_{G}(T) denotes the set of subgroups of G that are isomorphic to T.

For any finite subset C of an abelian group or a module we write C^{+} for the sum of the elements of C.

The set of positive (respectively, non-negative) integers will be denoted by ℕ (respectively, ℕ_{0}).

1.2 Bisets. Recall that a (G, H)-biset is a finite set X, endowed with a left G-action and a right Haction that commute with each other. Together with the (G, H)-equivariant maps, the (G, H)-bisets form a category. Every G × H-set X can be regarded as a (G, H)-biset and vice versa, by defining

gxh := (g, h^{−1})x and (g, h)x := gxh^{−1}

for x ∈ X, g ∈ G, h ∈ H. We freely use these identifications without further notice. Thus, if L ⩽ G × H and if X is a (G, H)-biset we can speak of the L-fixed points X^{L} of X and of the G × H-orbits (or just orbits) of X.

Note that the group G is an (H, K)-biset for any two subgroups H and K of G, by using left and right multiplication.

A (G, H)-biset is called left-free if the left G-action is free (i.e., if every element has trivial G-stabilizer), it is called right-free if the right H-action is free, and it is called bifree if it is left-free and right-free.

1.3 Tensor product of bisets. Let X be a (G, H)-biset and let Y be an (H, K)-biset. The cartesian product X × Y becomes a (G, K)-biset, by setting g(x, y)k := (gx, yk) for x ∈ X, y ∈ Y, g ∈ G, and k ∈ K. Moreover, X × Y is a left H-set via h(x, y) := (xh^{−1}, hy). The H-action commutes with the G × K-action, and the set X ×_{H} Y of H-orbits of X × Y inherits a (G, K)-biset structure. The H-orbit of the element (x, y) ∈ X × Y will be denoted by x ×_{H} y ∈ X ×_{H} Y. If L is another group and Z is a (K, L)-biset then (X ×_{H} Y)×_{K} Z ≅ X ×_{H} (Y ×_{K} Z) as (G, L)-bisets, under (x×_{H} y)×_{K} z ↦ x×_{H} (y×_{K} z). Note also that for a (G, H)-biset X one has isomorphisms

G ×_{G} X → X, g ×_{G} x ↦ gx, and X ×_{H} H → X, x ×_{H} h ↦ xh ,

of (G, H)-bisets.

1.4 Opposite biset. For a (G, H)-biset X we denote by X^{∘} its opposite biset. Thus, X^{∘} is an (H, G)- biset whose underlying set is equal to X and whose biset structure is given by hx^{∘}g := (g^{−1}xh^{−1})^{∘}, for x ∈ X, g ∈ G, and h ∈ H. Here, we write x^{∘} if we view the element x of X as an element in X^{∘}. Note that X and (X^{∘})^{∘} are isomorphic as (G, H)-bisets. If G = H and X is isomorphic to X^{∘} as (G, G)-biset then we say that X is symmetric.

The opposite of a subgroup L of G × H is defined by

L^{∘} := {(h, g) ∈ H × G | (g, h) ∈ G × H} .

Note that L^{∘} is a subgroup of H × G and that

(H × G)/L^{∘} → (G × H/L)^{∘} , (h, g)L^{∘} ↦ ((g, h)L)^{∘} ,

is an isomorphism of (H, G)-bisets.

1.5 Subgroups of G × H. We denote the canonical projections G × H → G and G × H → H by p_{1} and p_{2}, respectively. For a subgroup L of G × H we also set

k_{1}(L) := {g ∈ G | (g, 1) ∈ L} and k_{2}(L) := {h ∈ H | (1, h) ∈ L} .

Then k_{i}(L) ⊴ p_{i}(L) for i = 1, 2, and the projection p_{i} induces an isomorphism p̄_{i}: L/(k_{1}(L) × k_{2}(L)) → p_{i}(L)/k_{i}(L). Thus,

η_{L} = p̄_{1} ∘ p̄_{2}^{−1} : p_{2}(L)/k_{2}(L) →^{∼} p_{1}(L)/k_{1}(L)

is an isomorphism satisfying η_{L}(hk_{2}(L)) = gk_{1}(L) for all (g, h) ∈ L. This construction defines a bijection between the set of subgroups of G × H and the set of quintuples (A_{1}, B_{1}, η, A_{2}, B_{2}), where A_{1} ⊴ B_{1} ⩽ G, A_{2} ⊴ B_{2} ⩽ H, and η: B_{2}/A_{2} ⟶^{∼} B_{1}/A_{1} is an isomorphism.

In this paper we are mostly interested in the set ◁_{G,H} of subgroups L of G×H with k_{1}(L) = 1. These are related to left-free bisets by the next proposition. As a special case of the above (with A_{1} = 1) these groups can be described as follows. Let E_{G,H} denote the set of triples (U, α, V), where U ⩽ G, V ⩽ H and α: V → U is an epimorphism. For (U, α, V) ∈ E_{G,H} we set

◁(U, α, V) := {(α(h), h) ∈ G × H | h ∈ V } .

Note that if L = ◁(U, α, V) then p_{1}(L) = U, p_{2}(L) = V, k_{1}(L) = 1 and k_{2}(L) = ker(α). This construction defines a bijection

E_{G,H} ⟶^{∼} ◁_{G,H} , (U, α, V) ↦ ◁(U, α, V) .  (1)

The group G × H acts on both sets via conjugation. More precisely, on E_{G,H} it acts via

(g,h)(U, α, V) := (^{g}U, c_{g}αc_{h}^{−1}, ^{h}V) .

With this definition, the bijection in (1) is G × H-equivariant.

If α: V → U from above is an isomorphism then we call the subgroup ◁(U, φ, V) a twisted diagonal subgroup. In this case we sometimes write Δ(U, α, V) to indicate that α is an isomorphism. The set of twisted diagonal subgroups of G × H will be denoted by Δ_{G,H}. Note that a subgroup L of G × H is twisted diagonal if and only if k_{1}(L) = 1 and k_{2}(L) = 1. The G × H-equivariant bijection in (1) restricts to a G × H-equivariant bijection

I_{G,H} ⟶^{∼} Δ_{G,H} , (U, α, V) ↦ Δ(U, α, V) ,  (2)

where I_{G,H} denotes the set of elements (U, α, V) ∈ E_{G,H} such that α is an isomorphism. As a particular case, if G = H and U ⩽ G then we set Δ(U) := Δ(U, id, U).

The following two propositions are easy consequences of the definitions. We leave the proofs to the reader. We denote the stabilizer of an element x of a G-set X by stab_{G}(x). If U ⊴ V ⩽ G then we write C_{G}(U, V) for the set of all elements g ∈ N_{G}(U) ∩ N_{G}(V) for which the conjugation map c_{g} induces the identity on V/U.

1.6 Proposition Let X be a (G, H)-biset. Then X is left-free if and only if k_{1}(stab_{G×H}(x)) = 1 for all x ∈ X, and X is right-free if and only if k_{2}(stab_{G×H}(x)) = 1 for all x ∈ X. Thus, X is bifree if and only if stab_{G×H}(x) is a twisted diagonal subgroup of G × H for all x ∈ X.

1.7 Proposition (a) Let L ⩽ G × H be arbitrary. Assume that either (i) A ⊆ G is a transversal for G/k_{1}(L) and B ⊆ H is a transversal for H/p_{2}(L), or that (ii) A ⊆ G is a transversal for G/p_{1}(L) and B ⊆ H is a transversal for H/k_{2}(L). Then, A × B ⊆ G × H is a transversal for G × H/L.

(b) Let L ⩽ G × H be arbitrary. Then, for i = 1, 2,

k_{i}(N_{G×H}(L)) = C_{G}(k_{i}(L), p_{i}(L)) .

In particular, if U ⩽ G, V ⩽ H and if α: V → U is an epimorphism then

k_{1}(N_{G×H}(◁(U, α, V))) = C_{G}(U) and k_{2}(N_{G×H}(◁(U, α, V))) = C_{H}(ker(α), V) .

In particular, one has C_{H}(V) ⊴ k_{2}(N_{G×H}(◁(U, α, V))).

(c) Let U ⩽ G, V ⩽ H, and let α: V → U be an isomorphism. Then

k_{1}(N_{G×H}(Δ(U, α, V))) = C_{G}(U) , k_{2}(N_{G×H}(Δ(U, α, V))) = C_{H}(V) ,
p_{1}(N_{G×H}(Δ(U, α, V))) = {g ∈ N_{G}(U) | ∃h ∈ N_{H}(V): c_{g} = αc_{h}α^{−1}} and
p_{2}(N_{G×H}(Δ(U, α, V))) = {h ∈ N_{H}(V) | ∃g ∈ N_{G}(U): c_{h} = α^{−1}c_{g}α} .

Usually, the second projection group in Proposition 1.7(c) is denoted by N_{α} and the first one is denoted by N_{α^{−1}}. We will also use this notation in Section 7 in connection with fusion systems.

1.8 The double Burnside group B(G, H) of G and H is defined as the Grothendieck group of the category of (G, H)-bisets. Using the identification in 1.2, we may identify B(G, H) with the Burnside group B(G × H). The group B(G, H) is defined as the factor group F/U, where F is the free abelian group on the set of isomorphism classes {X} of (G, H)-bisets X, and U is generated by the elements {X ∐ X′} − {X} − {X′}, with arbitrary (G, H)-bisets X and X′. Here, X ∐ X′ denotes the coproduct (or disjoint union) of X and X′. The coset of the element {X} ∈ F will be denoted by [X] ∈ B(G, H). Thus [X ∐ X′] = [X] + [X′]. If L is a transversal of the conjugacy classes of subgroups of G × H then {[G × H/L] | L ∈ L} is a ℤ-basis, the standard basis, of B(G, H). One has [X] = [X′] ∈ B(G, H) if and only if X and X′ are isomorphic (G, H)-bisets.

The tensor product construction for bisets in 1.3 induces a bilinear map

− ·_{H} −: B(G, H) × B(H, K) → B(G, K) , ([X], [Y]) ↦ [X ×_{H} Y] ,

where X denotes a (G, H)-biset and Y denotes an (H, K)-biset. In the case that G = H = K, this map defines a multiplication on B(G, G) establishing a ring structure with identity element [G] = [G × G/Δ(G)]. This ring is called the double Burnside ring of G.

The construction of the opposite biset induces a group isomorphism

−^{∘}: B(G, H) → B(H, G) , [X] ↦ [X^{∘}] ,

satisfying

([X] ·_{H} [Y])^{∘} = [Y]^{∘} ·_{H} [X]^{∘} ∈ B(K, G) and ([X]^{∘})^{∘} = [X] ∈ B(G, H) ,

for every (G, H)-biset X and every (H, K)-biset Y. In particular, if G = H = K, this implies that −^{∘}: B(G, G) → B(G, G) is an anti-involution of B(G, G).

1.9 The ∗-product of subgroups. The following proposition, due to Bouc, gives an explicit description of the bilinear map − ·_{H} − on the standard basis elements. It requires the following notation: for subgroups L ⩽ G × H and M ⩽ H × K one defines the subgroup

L ∗ M ⩽ G × K

as the set of all pairs (g, k) ∈ G × K for which there exists some h ∈ H such that (g, h) ∈ L and (h, k) ∈ M. Viewing L as a relation between G and H, and M as a relation between H and K, the subgroup L ∗ M is the composition of these two relations. Note that

(L ∗ M)^{∘} = M^{∘} ∗ L^{∘}

as subgroups of K × G. We emphasize that, in general, the double Burnside ring B(G, G) is not commutative, as can be easily seen from the following proposition.

1.10 Proposition (cf. [Bc10, 2.3.24]) For L ⩽ G × H and M ⩽ H × K one has

[(G × H)/L] ·_{H} [(H × K)/M] = ∑_{\begin{matrix}p_{2}(L)hp_{1}(M)∈ \\ p_{2}(L)\H/p_{1}(M) \end{matrix}}[(G × K)/(L ∗ ^{(h,1)}M)] ∈ B(G, K) .

1.11 Classical ghost group and mark homomorphism. Recall from [CR, Proposition 80.12] that for every subgroup L ⩽ G × H, one has a group homomorphism Φ_{L}: B(G, H) → ℤ, [X] ↦ |X^{L}|, with the following properties: if L and L′ are conjugate subgroups of G × H then Φ_{L} = Φ_{L′}, and if L denotes a transversal of the conjugacy classes of subgroups of G × H then the map

Φ = (Φ_{L})_{L∈L}: B(G, H) → ∏_{L∈L} ℤ , [X] ↦ (|X^{L}|)_{L∈L} ,

is an injective group homomorphism with finite cokernel. It is called the classical mark homomorphism, and its codomain is called the classical ghost group of B(G, H). One of the goals of this paper is to construct ghost groups and mark homomorphisms for B^{◁}(G, H) that naturally come with bilinear maps that correspond, under the mark homomorphism, to the tensor product of bisets.

For any commutative ring R, the map Φ induces an R-module homomorphism

Φ: RB(G, H) → ∏_{L∈L} R,  (3)

where RB(G, H) := R ⊗_{ℤ} B(G, H) will often be identified with the free R-module with basis [G × H/L], L ∈ L. If |G × H| is invertible in R then (3) is an R-module isomorphism. Moreover, if R is a field of characteristic 0 then we can view B(G, H) as a subgroup of RB(G, H).

1.12 The group B^{S}(G, H). For a set S of subgroups of G × H, we define B^{S}(G, H) as the subgroup of B(G, H) spanned by the standard basis elements [G × H/L] with L ∈ S. In the case that S = ◁_{G,H} (respectively, S = Δ_{G,H}) we also use the notation B^{◁}(G, H) (respectively, B^{Δ}(G, H)). We call B^{◁}(G, H) (respectively, B^{Δ}(G, H)) the left-free double Burnside group (respectively, bifree double Burnside group) of G and H. Proposition 1.6 justifies the terminology. Clearly one has B^{△}(G, H) ⊆ B^{◁}(G, H).

1.13 Hypothesis Assume that we are given a class D of finite groups and that for any two groups G and H belonging to D we are given a subset S_{G,H} of subgroups of G × H. We often write S for the collection of sets S_{G,H}, G, H ∈ D. For any groups G and H in D, we define the R-module RB^{S_{G,H}} (G, H), or for short RB^{S}(G, H), as in 1.12 above. In subsequent results we will put further restrictions on the sets S_{G,H}. For this, we say that (D, S) satisfies Condition (I) if the following hold:

(i) For all G, H ∈ D, the set S_{G,H} is closed under G × H-conjugation.

(ii) For all G, H ∈ D, the set S_{G,H} is closed under taking subgroups.

(iii) For all G, H, K ∈ D and all L ∈ S_{G,H} and M ∈ S_{H,K} one has L ∗ M ∈ S_{G,K}.

(iv) For all G ∈ D one has Δ(G) ∈ S_{G,G}. Moreover, we say that (D, S) satisfies Condition (II) if

(v) for all G, H ∈ D one has (S_{G,H})^{∘} = S_{H,G}.

1.14 Proposition Let D and S_{G,H} (for G, H ∈ D) be as in Hypothesis 1.13, and assume further that (D, S) satisfies Condition (I) in Hypothesis 1.13. Let G, H, K ∈ D.

(a) The bilinear map − ·_{H} −: B(G, H) × B(H, K) → B(G, K) restricts to a bilinear map

− ·_{H} −: B^{S}(G, H) × B^{S}(H, K) → B^{S}(G, K) .

In particular, B^{S}(G, G) is a unitary subring of B(G, G).

(b) Assume that (D, S) additionally satisfies Condition (II) in Hypothesis 1.13. Then the group isomorphism −^{∘}: B(G, H) → B(H, G) restricts to an isomorphism

B^{S}(G, H) → B^{S}(H, G) .

(c) If L is a transversal for the conjugacy classes of subgroups of G × H and if L^{S} := L ∩ S_{G,H} then the classical mark homomorphism Φ: B(G, H) → ∏_{L∈L} ℤ restricts to a group monomorphism B^{S}(G, H) → ∏_{L∈L^{S}} ℤ. Its cokernel is a finite group of order ∏_{L∈L^{S}} [N_{G×H}(L) : L]. In particular, if |G × H| ∈ R^{×} then the latter homomorphism induces an R-module isomorphism

RB^{S}(G, H) → ∏_{L∈L^{S}} R .

Proof Part (a) follows immediately from the explicit formula in Proposition 1.10 and Hypothesis 1.13(i), (iii), and (iv). Part (b) is immediate from Hypothesis 1.13(v). For Part (c), note that the representing matrix (also called the table of marks) of Φ: B(G, H) → ∏_{L∈L} ℤ with respect to the standard basis elements in appropriate order is an upper triangular square matrix, with rows and columns both indexed by L and with diagonal entries [N_{G×H}(L) : L], L ∈ L. This follows immediately from the fact that if L ∈ L is not conjugate to a subgroup of L′ ∈ L then (G × H/L′)^{L} is empty. Omitting rows and columns from this matrix indexed by elements L ∈ L that do not belong to S_{G,H}, results again in an upper triangular square matrix whose diagonal entries are units in R. This matrix is the representing matrix of the restricted morphism considered in (c). This completes the proof.

1.15 Note that (i)–(iv) in Hypothesis 1.13 are satisfied for S_{G,H} = ◁_{G,H}, and that (i)–(v) are satisfied for S_{G,H} = Δ_{G,H}. Therefore, B^{◁}(G, G) and B^{Δ}(G, G) are unitary subrings of B(G, G), which we call the left-free and the bifree double Burnside ring of G, respectively.

1.16 Biset functors. Let D be a set of finite groups, and for each pair (G, H) of groups in D let S_{G,H} be a set of subgroups of G×H satisfying Condition (I) in Hypothesis 1.13. Moreover, let R be a commutative ring.

(a) Following Bouc, cf. [Bc96a], we can define the category C^{D,S} whose objects are the groups in D, whose morphisms are given by Hom_{C^{D,S}} (H, G) = B^{S}(G, H) for G, H ∈ D, and whose composition is induced by the tensor product construction of bisets. This is an additive category. In this setting, a biset functor over R is an additive functor from C^{D,S} to the category of left R-modules. Together with natural transformations as morphisms, the biset functors form an abelian category Func_{R}^{D,S}.

(b) The category Func_{R}^{D,S} is isomorphic to a module category. This construction works for any additive category in place of C^{D,S} and goes back to Gabriel in [Ga, Chapter II], see also [W, §2]. Define the R-module

A_{R}^{D,S} := ⨁_{G,H∈D} RB^{S}(G, H)

and define a multiplication on A_{R}^{D,S} as follows: if a ∈ RB^{S}(G, H) and b ∈ RB^{S}(H′, K) then set a · b := a ·_{H} b if H = H′, and a · b := 0 if H ≠ H′. This way, A_{R}^{D,S} is an associative ring. If R = ℤ then we also use the notation A^{D,S}. For G ∈ D, we set e_{G} := [G × G/Δ(G)], the identity element in RB^{S}(G, G). Then the elements e_{G}, G ∈ D, form a set of mutually orthogonal idempotents in A_{R}^{D,S}. The ring A_{R}^{D,S} has an identity if and only if D is finite. In this case the identity is equal to ∑_{G∈D} e_{G}. Now consider the category _{A_{R}^{D,S}} Mod^{∗} of all left A_{R}^{D,S}-modules M with the property that M = ∑_{G∈D}(e_{G}M). This is an

abelian category and the obvious functor, which sends F ∈ Func_{R}^{D,S} to the module M := ⨁_{G∈D} F (G), defines a category equivalence. An inverse is given by mapping an A_{R}^{D,S}-module M to the obvious functor F, defined on objects by F (G) := e_{G}M for G ∈ D.

2 Fixed points of products of left-free bisets

Throughout this section, G, H, and K denote finite groups. For a (G, H)-biset X, an (H, K)-biset Y, and a subgroup N ∈ ◁_{G,K}, we will give an explicit description of the fixed point set (X ×_{H} Y)^{N} in terms of the fixed point sets X^{L} and Y^{M} with L ∈ ◁_{G,H} and M ∈ ◁_{H,K}.

For any two groups U and V we will abbreviate by E(U, V) the set of epimorphisms α: V → U, and by I(U, V) the set of isomorphisms α: V →^{∼} U.

2.1 Let U ⩽ G, V ⩽ H, W ⩽ K and assume that α ∈ E(U, V) and β ∈ E(V, W). We will consider the subgroups ◁(U, α, V) ⩽ G × H and ◁(V, β, W) ⩽ H × K. Note that E(U, V) (respectively, I(U, V)) is a left-free (respectively, bifree) (Aut_{G}(U), Aut_{H}(V))-biset and that E(V, W) (respectively, I(V, W)) is a left-free (respectively, bifree) (Aut_{H}(V), Aut_{K}(W))-biset by composition. Now let X be a left-free (G, H)-biset and let Y be a left-free (H, K)-biset. Note that X^{◁(U,α,V)} is a left-free (C_{G}(U), C_{H}(V))-biset and that Y^{◁(V,β,W)} is a left-free (C_{H}(V), C_{K}(W))-biset, by restriction of the structures of X and Y. In the case that X and Y are bifree and α and β are isomorphisms, these bisets are bifree. Also note that

x ∈ X^{◁(U,α,V)} ⟺ xv = α(v)x for all v ∈ V.

If x ∈ X^{◁(U,α,V)} and y ∈ Y^{◁(V,β,W)} then x ×_{H} y ∈ (X ×_{H} Y)^{◁(U,αβ,W)}. In fact, for all w ∈ W one has

x ×_{H} yw = x ×_{H} β(w)y = xβ(w) ×_{H} y = α(β(w))x ×_{H} y .

Since also xh ×_{H} y = x ×_{H} hy for all h ∈ C_{H}(V), we obtain a well-defined map

\overline{μ}: X^{◁(U,α,V)} ×_{C_{H}(V)} Y^{◁(V,β,W)} → (X ×_{H} Y)^{◁(U,αβ,W)}
x ×_{C_{H}(V)} y ↦ x ×_{H} y

between left-free (C_{G}(U), C_{K}(W))-bisets. Moreover, the map \overline{μ} is injective. In fact, if x, x′ ∈ X^{◁(U,α,V)} and y, y′ ∈ Y^{◁(V,β,W)} satisfy x ×_{H} y = x′ ×_{H} y′ then there exists some h ∈ H such that x′ = xh^{−1} and y′ = hy. The fixed point properties of y and y′ imply

hβ(w)y = hyw = y′w = β(w)y′ = β(w)hy ,

and since Y is left-free, we obtain c_{h}β = β and then h ∈ C_{H}(V). Thus, x′ ×_{C_{H}(V)} y′ = x ×_{C_{H}(V)} y, and the injectivity of \overline{μ} is proved.

2.2 (a) For U ⩽ G, W ⩽ K, and γ ∈ E(U, W) we denote by Γ_{H}(U, γ, W) the set of all triples (α, V, β) with V ⩽ H, α ∈ E(U, V) and β ∈ E(V, W) such that αβ = γ. In other words, Γ_{H}(U, γ, W) consists of all factorizations of γ as two epimorphisms via subgroups of H.

(b) Note that H acts on Γ_{H}(U, γ, W) by

h(α, V, β) := (αc_{h}^{−1}, ^{h}V, c_{h}β)

and that the stabilizer of (α, V, β) is equal to C_{H}(V).

Note also that if (α, V, β) and (α̃, Ṽ , β̃) lie in the same H-orbit of Γ_{H}(U, γ, W) and if h ∈ H satisfies (α̃, Ṽ , β̃) = h(α, V, β) = (αc_{h}^{−1}, ^{h}V, c_{h}β) then one has an isomorphism of (C_{G}(U), C_{K}(W))-bisets,

φ_{h}: X^{◁(U,α,V)} × Y^{◁(V,β,W)} →^{∼} X^{◁(U,α̃,Ṽ)} × Y^{◁(Ṽ ,β̃,W)} , (x, y) ↦ (xh^{−1}, hy) ,

which induces an isomorphism

\overline{φ}_{h}: X^{◁(U,α,V)} ×_{C_{H}(V)} Y^{◁(V,β,W)} →^{∼} X^{◁(U,α̃,Ṽ)} ×_{C_{H}(Ṽ)} Y^{◁(Ṽ ,β̃,W)} ,
x ×_{C_{H}(V)} y ↦ xh^{−1} ×_{C_{H}(Ṽ)} hy ,

of (C_{G}(U), C_{K}(W))-bisets such that \overline{μ} ∘ \overline{φ}_{h} = \overline{μ}. The isomorphism \overline{φ}_{h} does not depend on the choice of the element h ∈ H. In fact, if also h′(α, V, β) = (α̃, Ṽ , β̃) then h′ = ch for some c ∈ C_{H}(Ṽ).

The following result will be crucial for studying the ring structures of B^{◁}(G, G) and B^{Δ}(G, G).

2.3 Theorem Let U ⩽ G and W ⩽ K, and let γ ∈ E(U, W). Let Γ_{H}(U, γ, W) be defined as in 2.2 and let Γ̃_{H}(U, γ, W) ⊆ Γ_{H}(U, γ, W) be a set of representatives of the H-orbits of Γ_{H}(U, γ, W). Then the maps \overline{μ} from 2.1 induce an isomorphism of (C_{G}(U), C_{K}(W))-bisets

∐_{(α,V,β)∈Γ̃_{H}(U,γ,W)} X^{◁(U,α,V)} ×_{C_{H}(V)} Y^{◁(V,β,W)} ⟶^{∼} (X ×_{H} Y)^{◁(U,γ,W)} .

Moreover, if Σ̃_{H} ⊆ Σ_{H} is a transversal of the H-conjugacy classes of subgroups of H then

|(X ×_{H} Y)^{◁(U,γ,W)}| = ∑_{V ⩽H} |H|^{−1} ∑_{\begin{matrix}(α,β)∈E(U,V)×E(V,W) \\ αβ=γ \end{matrix}} |X^{◁(U,α,V)}| · |Y^{◁(V,β,W)}|

= ∑_{V ∈Σ̃_{H}} |N_{H}(V)|^{−1} ∑_{\begin{matrix}(α,β)∈E(U,V)×E(V,W) \\ αβ=γ \end{matrix}} |X^{◁(U,α,V)}| · |Y^{◁(V,β,W)}| .

Proof We first show that the map in the theorem is surjective. So let x ∈ X, y ∈ Y be such that x ×_{H} y ∈ (X ×_{H} Y)^{◁(U,γ,W)}. Then x ×_{H} yw = γ(w)x ×_{H} y for all w ∈ W. This implies that, for every w ∈ W, there exists an element h_{w} ∈ H such that

xh_{w} = γ(w)x and h_{w}y = yw .

Since Y is left-free, h_{w} is uniquely determined by w. Thus, we obtain a function β̃: W → H such that

xβ̃(w) = γ(w)x and β̃(w)y = yw  (4)

for all w ∈ W. Moreover, for w, w′ ∈ W we have

β̃(ww′)y = yww′ = β̃(w)yw′ = β̃(w)β̃(w′)y ,

and since Y is left-free, we see that β̃ is a group homomorphism. We set Ṽ := β̃(W) and have β̃ ∈ E(Ṽ , W). By the second equation in (4), we have y ∈ Y^{◁(Ṽ ,β̃,W)}. Next we define a function α̃: Ṽ → U as follows. For ṽ ∈ Ṽ, choose w ∈ W with β̃(w) = ṽ, and set α̃(ṽ) := γ(w). This is well defined. For if also w′ ∈ W satisfies β̃(w′) = ṽ then, by the first equation in (4),

γ(w)x = xβ̃(w) = xβ̃(w′) = γ(w′)x .

Since X is left-free, we obtain γ(w) = γ(w′). By construction, α̃(Ṽ) = γ(W) = U and γ = α̃β̃. Since γ and β̃ are surjective homomorphisms, also α̃ is a surjective homomorphism. Now, the first equation in (4) implies that x ∈ X^{◁(U,α̃,Ṽ)}. Since (α̃, Ṽ , β̃) is an element in Γ_{H}(U, γ, W), there exist h ∈ H and (α, V, β) ∈ Γ̃_{H}(U, γ, W) such that (α, V, β) = h(α̃, Ṽ , β̃). This implies that xh^{−1} ∈ X^{◁(U,α,V)} and hy ∈ Y^{◁(V,β,W)}. Thus, x ×_{H} y = xh^{−1} ×_{H} hy lies in the image of the map in the theorem.

Next we show that the map in the theorem is injective. Let (α, V, β), (α̃, Ṽ , β̃) ∈ Γ̃_{H}(U, γ, W) and let x ∈ X^{◁(U,α,V)}, y ∈ Y^{◁(V,β,W)}, x̃ ∈ X^{◁(U,α̃,Ṽ)}, and ỹ ∈ Y^{◁(Ṽ ,β̃,W)} be such that x ×_{H} y = x̃ ×_{H} ỹ. By the injectivity of the map \overline{μ} in 2.1, it suffices to show that (α, V, β) = (α̃, Ṽ , β̃), or, equivalently, that h(α, V, β) = (α̃, Ṽ , β̃) for some h ∈ H. Now, since x ×_{H} y = x̃ ×_{H} ỹ, there exists h ∈ H such that x̃ = xh^{−1} and ỹ = hy. Moreover, for all w ∈ W, we have

β(w)y = yw and β̃(w)hy = β̃(w)ỹ = ỹw = hyw .

These two equations imply h^{−1}β̃(w)hy = yw = β(w)y. Since Y is left-free, we obtain β̃ = c_{h}β and Ṽ = β̃(W) = (c_{h}β)(W) = ^{h}V. In order to see that α̃ = αc_{h}^{−1}, let ṽ ∈ Ṽ and choose w ∈ W such that β̃(w) = ṽ. Then

α̃(ṽ) = (α̃β̃)(w) = γ(w) = (αβ)(w) = (αc_{h}^{−1}β̃)(w) = (αc_{h}^{−1})(ṽ) .

This implies that (α̃, Ṽ , β̃) = h(α, V, β) and completes the proof of the injectivity of the map in the theorem.

Finally, we will show the equations in the theorem. First note that

|X^{◁(U,α,V)} ×_{C_{H}(V)} Y^{◁(V,β,W)}| = |C_{H}(V)|^{−1} · |X^{◁(U,α,V)} × Y^{◁(V,β,W)}| ,

since Y is left-free. Next, recall from 2.2(b), that if (α, V, β) and (α̃, Ṽ , β̃) lie in the same H-orbit of Γ_{H}(U, γ, W) then X^{◁(U,α,V)} × Y^{◁(V,β,W)} and X^{◁(U,α̃,Ṽ)} × Y^{◁(Ṽ ,β̃,W)} are in bijective correspondence. Since the H-orbit of (α, V, β) has size [H : C_{H}(V)], these two facts and the isomorphism in the theorem imply the first equation. The second equation is immediate.

2.4 Remark (a) Note that the left-hand side of the isomorphism in Theorem 2.3 does not depend on the choice of the set Γ̃_{H}(U, γ, W), in the sense that any other choice would lead to canonically isomorphic components, the isomorphism being given by the maps \overline{φ}_{h}, cf. 2.2(b).

(b) If the epimorphism γ in Theorem 2.3 is an isomorphism and if (α, V, β) is an element in Γ_{H}(U, γ, W), then also α and β are isomorphisms, since γ = αβ. In particular, V is isomorphic to U and W. Therefore, Theorem 2.3 implies immediately the following result.

2.5 Theorem Let U ⩽ G and W ⩽ K be isomorphic subgroups of G and K, respectively, and let γ ∈ I(U, W) be an isomorphism between them. Let Γ_{H}(U, γ, W) be defined as in 2.2 and let Γ̃_{H}(U, γ, W) ⊆ Γ_{H}(U, γ, W) be a set of representatives of the H-orbits of Γ_{H}(U, γ, W). Then the maps \overline{μ} from 2.1 induce an isomorphism of (C_{G}(U), C_{K}(W))-bisets

∐_{(α,V,β)∈Γ̃_{H}(U,γ,W)} X^{Δ(U,α,V)} ×_{C_{H}(V)} Y^{Δ(V,β,W)} ⟶^{∼} (X ×_{H} Y)^{Δ(U,γ,W)} .

Moreover, if Σ̃_{H}(U) ⊆ Σ_{H}(U) is a transversal of the H-conjugacy classes of Σ_{H}(U) then

|(X ×_{H} Y)^{Δ(U,γ,W)}| = ∑_{V ⩽H} |H|^{−1} ∑_{\begin{matrix}(α,β)∈I(U,V)×I(V,W) \\ αβ=γ \end{matrix}} |X^{Δ(U,α,V)}| · |Y^{Δ(V,β,W)}|

= ∑_{V ∈Σ̃_{H}(U)} |N_{H}(V)|^{−1} ∑_{\begin{matrix}(α,β)∈I(U,V)×I(V,W) \\ αβ=γ \end{matrix}} |X^{Δ(U,α,V)}| · |Y^{Δ(V,β,W)}| .

3 An application to the Brauer constructions of tensor products of p-permutation bimodules

Throughout this section let F be a field of prime characteristic p. Also, G, H, and K will denote finite groups. We will denote by F G the group algebra of G over F, and for any G-set X we will denote by F X the F -vector space with basis X. The left G-action on X induces a left F G-module structure on F X. Similarly, if X is a (G, H)-biset then we obtain an (F G, F H)-bimodule F X. All modules over group algebras will be assumed to have finite F -dimension.

3.1 In this subsection we recall some concepts and results from modular representation theory. We refer the reader to [Br] for the statements concerning p-permutation modules and the Brauer construction, and to [N-T] for the theory of vertices of indecomposable modules.

(a) Similarly as for bisets, we will identify left F [G × H]-modules M with (F G, F H)-bimodules by defining gmh := (g, h^{−1})m and (g, h)m := gmh^{−1}, for m ∈ M, g ∈ G and h ∈ H.

(b) An F G-module that is isomorphic to a direct summand of a module of the form F X, for some G-set X, is called a p-permutation module. Thus, we call an (F G, F H)-bimodule M a p-permutation bimodule if it is isomorphic to a direct summand of a bimodule of the form F X for a (G, H)-biset X.

(c) It is well known and easy to check that if X is a (G, H)-biset and Y is an (H, K)-biset then the map

F [X ×_{H} Y] → F X ⊗_{F H} F Y , x ×_{H} y ↦ x ⊗_{F H} y ,

is an isomorphism of (F G, F K)-bimodules.

(d) A vertex of an indecomposable F G-module M is a subgroup P of G that is minimal with respect to inclusion and the property that M is isomorphic to a direct summand of the F G-module F G ⊗_{F P} M. The set of vertices of M is a single conjugacy class of p-subgroups of G. The vertices of an indecomposable (F G, F H)-bimodule M are considered as subgroups of G × H, by viewing M as an F [G × H]-module.

(e) If M is an indecomposable p-permutation F G-module and P is a vertex of M then M is isomorphic to a direct summand of F [G/P], where G/P is viewed as a G-set.

(f) For an F G-module M and a p-subgroup P of G, the Brauer construction of M with respect to P is defined as

M(P) := M^{P} / ∑_{Q<P} tr_{Q}^{P}(M^{Q})

where M^{P} denotes the set of P -fixed points of M and tr_{Q}^{P}: M^{Q} → M^{P} denotes the relative trace map, which maps a Q-fixed point m of M to ∑_{xQ∈P/Q} xm. The vector space M(P) inherits an F N_{G}(P)-module structure from the F G-module structure of M. Moreover P acts trivially on M(P). It is a well-known fact that if M(P) ≠ 0 then P is contained in a vertex of some indecomposable direct summand of M.

(g) It is well known that if M = F X, for a G-set X, then the composition

F (X^{P}) → (F X)^{P} → M(P)

of the canonical maps defines an isomorphism of F N_{G}(P)-modules.

(h) Let M be an (F G, F H)-bimodule and let N be an (F H, F K)-bimodule. Moreover let U ⩽ G and W ⩽ K be p-subgroups and let γ ∈ E(U, W). If V ⩽ H and if α ∈ E(U, V) and β ∈ E(V, W) satisfy αβ = γ then one has a well-defined bilinear map

M^{◁(U,α,V)} × N^{◁(V,β,W)} → (M ⊗_{F H} N)^{◁(U,γ,W)} , (m, n) ↦ m ⊗ n .

Moreover, assume that m = tr_{◁(U′,α′,V′)}^{◁(Uα,V)}(m′), with ◁(U′, α′, V′) < ◁(U, α, V) and m′ ∈ M^{◁(U′,α′,V′)}. Then V′ < V, U′ ⩽ U and α′ = α|_{V′}. We set W′ := β^{−1}(V′) and obtain a bijection W/W′ → V/V′, induced by β. We also set γ′ := γ|_{W′} and obtain (U′, γ′, W′) ∈ E(U′, W′) with γ′ = α′β|_{W′} = γ|_{W′}, and ◁(U′, γ′, W′) < ◁(U, γ, W). For n ∈ N^{◁(V,β,W)} we have

m ⊗ n = tr_{◁(U′,α′,V′)}^{◁(U,α,V)}(m′) ⊗ n = ∑_{v∈V/V′} α(v)m′v^{−1} ⊗ n = ∑_{v∈V/V′} α(v)m′ ⊗ v^{−1}n

= ∑_{w∈W/W′} γ(w)m′ ⊗ β(w)^{−1}n = ∑_{w∈W/W′} γ(w)m′ ⊗ nw^{−1} = tr_{◁(U′,γ′,W′)}^{◁(U,γ,W)}(m′ ⊗ n) .

Similarly, if n is a trace from a proper subgroup of ◁(V, β, W) and m is arbitrary then m ⊗ n is again a trace from a proper subgroup of ◁(U, γ, W). Thus, the above map induces a bilinear map

M(◁(U, α, V)) × N(◁(V, β, W)) → (M ⊗_{F H} N)(◁(U, γ, W)) , (\overline{m}, \overline{n}) ↦ \overline{m ⊗ n} .

Since M(◁(U, α, V)) is an F N_{G×H}(◁(U, α, V))-module and C_{G}(U) × C_{H}(V) ⩽ N_{G×H}(◁(U, α, V)), we can regard M(◁(U, α, V)) as (F C_{G}(U), F C_{H}(V))-bimodule. Similarly, we can view M(◁(V, β, W)) as (F C_{H}(V), F C_{K}(W))-bimodule and M(◁(U, γ, W)) as (F C_{G}(U), F C_{K}(W))-bimodule. The above map now induces a homomorphism

M(◁(U, α, V)) ⊗_{F C_{H}(V)} N(◁(V, β, W)) → (M ⊗_{F H} N)(◁(U, γ, W)) , \overline{m} ⊗ \overline{n} ↦ \overline{m ⊗ n} ,

of (F C_{G}(U), F C_{K}(W))-bimodules. As a special case, if α and β are isomorphisms, we obtain a map

M(Δ(U, α, V)) ⊗_{F C_{H}(V)} N(Δ(V, β, W)) → (M ⊗_{F H} N)(Δ(U, γ, W)) , \overline{m} ⊗ \overline{n} ↦ \overline{m ⊗ n} ,

of (F C_{G}(U), F C_{K}(W))-bimodules.

3.2 Theorem Let M be a p-permutation (F G, F H)-bimodule and let N be a p-permutation (F H, F K)- bimodule. Assume that the vertices of every indecomposable direct summand of M and N lie in ◁_{G,H} and ◁_{H,K}, respectively. Let U ⩽ G and W ⩽ K be p-subgroups, and let γ ∈ E(U, W). Furthermore, let Γ̃_{H}(U, γ, W) be a set of representatives of the H-orbits of Γ_{H}(U, γ, W). Then the canonical maps in 3.1(h) induce an isomorphism

⨁_{(α,V,β)∈Γ̃_{H}(U,γ,W)} M(◁(U, α, V)) ⊗_{F C_{H}(V)} N(◁(V, β, W)) →^{∼} (M ⊗_{F H} N)(◁(U, γ, W)) ,
\overline{m} ⊗ \overline{n} ↦ \overline{m ⊗ n} ,

of (F C_{G}(U), F C_{K}(W))-bimodules.

Proof We first note that the left-hand side and the right-hand side can be considered as the evaluation of a functor from the category product of the category of (F G, F H)-bimodules and the category of (F H, F K)-bimodules to the category of (F C_{G}(U), F C_{K}(W))-bimodules. Moreover, it is easy to see that the map in the theorem gives a natural transformation between these two functors. Both functors respect direct sums in both arguments in a bilinear way. It follows immediately that the map in the theorem is an isomorphism for M and N if and only if it is an isomorphism for every pair of indecomposable direct summands of M and N, respectively. Therefore and by 3.1(e), it suffices to show that the map is an isomorphism in the case where M and N are of the form F X and F Y, respectively, for a left-free (G, H)-biset X and a left-free (H, K)-biset Y. But in this case, the map in the theorem is induced by the map in Theorem 2.3, using the canonical identifications from 3.1(c) and (g).

Remark 2.4(b) and the last sentence in 3.1 imply immediately the following theorem.

3.3 Theorem Let M be a p-permutation (F G, F H)-bimodule and let N be a p-permutation (F H, F K)- bimodule. Assume that the vertices of every indecomposable direct summand of M and N lie in Δ_{G,H} and Δ_{H,K}, respectively. Let U ⩽ G and W ⩽ K be isomorphic p-subgroups of G and K, respectively, and let γ ∈ I(U, W). Furthermore, let Γ̃_{H}(U, γ, W) be a set of representatives of the H-orbits of Γ_{H}(U, γ, W). Then the canonical maps in 3.1(h) induce an isomorphism

⨁_{(α,V,β)∈Γ̃_{H}(U,γ,W)} M(Δ(U, α, V)) ⊗_{F C_{H}(V)} N(Δ(V, β, W)) →^{∼} (M ⊗_{F H} N)(Δ(U, γ, W)) ,
\overline{m} ⊗ \overline{n} ↦ \overline{m ⊗ n} ,

of (F C_{G}(U), F C_{K}(W))-bimodules.

4 A ghost group B̃^{◁}(G, H) for B^{◁}(G, H) and a mark homomorphism

In this section we will construct, for any two finite groups G and H, a ghost group B̃^{◁}(G, H) of the left-free double Burnside group B^{◁}(G, H), together with a mark homomorphism ρ_{G,H}^{◁}: B^{◁}(G, H) → B̃^{◁}(G, H) that is injective and has finite cokernel. For any finite groups G, H, and K, we define a bilinear map − ·_{H} −: B̃^{◁}(G, H) × B̃^{◁}(H, K) → B̃^{◁}(G, K) that corresponds under the mark homomorphism to the tensor product on the double Burnside groups.

We will adopt a more functorial and general approach by assuming throughout this section that D is a class of finite groups and that, for any G, H ∈ D, we are given a set S_{G,H} ⊆ ◁_{G,H} of subgroups such that D together with the system S of all these sets S_{G,H} satisfies Condition (I) in Hypothesis 1.13. Note that D could be an arbitrary set of finite groups, ranging from just one group (as considered in Section 7) to the case of all finite groups. Using a system S will enable us to carry out the construction of the ghost groups and mark homomorphism at the same time for left-free and bifree double Burnside groups, as well as for the situation of a fusion system, which will be considered in Section 7. The constructions are functorial in S in the sense that if S′ is a subsystem of S then B̃^{S′}(G, H) is a subset of B̃^{S}(G, H) and the mark homomorphism ρ_{G,H}^{S′} is the restriction of ρ_{G,H}^{S}.

As before, R denotes an associative unitary commutative ring.

4.1 The group A^{S}(G, H). For G, H ∈ D, we set

E_{G,H}^{S} := {(U, α, V) ∈ E_{G,H} | ◁(U, α, V) ∈ S_{G,H}} .

The G × H-equivariant bijection in (1) restricts, by construction, to a G × H-equivariant bijection

E_{G,H}^{S} → S_{G,H} , (U, α, V) ↦ ◁(U, α, V) .  (5)

If S_{G,H} ⊆ Δ_{G,H} then we also write I_{G,H}^{S} for E_{G,H}^{S}.

We define A^{S}(G, H) as the free ℤ-module with basis E_{G,H}^{S}. In the case that S_{G,H} = ◁_{G,H} we write A^{◁}(G, H) for A^{S}(G, H). If S_{G,H} = Δ_{G,H} then we also write A^{Δ}(G, H) for A^{S}(G, H). We identify the Rmodule R⊗_{ℤ}A^{S}(G, H) = RA^{S}(G, H) with the free R-module with R-basis E_{G,H}^{S}. The G×H-conjugation action on E_{G,H}^{S} induces a left R[G × H]-module structure on RA^{S}(G, H).

4.2 Definition For G, H ∈ D we define the ghost group B̃^{S}(G, H) of B^{S}(G, H) by

B̃^{S}(G, H) := A^{S}(G, H)^{G×H} ,

the set of G × H-fixed points of the ℤ[G × H]-module A^{S}(G, H). If [U, α, V]_{G×H} denotes the G × H-orbit of the element (U, α, V) ∈ E_{G,H}^{S} then the orbit sums [U, α, V]_{G×H}^{+} form a ℤ-basis of B̃^{S}(G, H). We set RB̃^{S}(G, H) := R ⊗_{ℤ} B̃^{S}(G, H), and identify this R-module in the canonical way with the free R-module with basis [U, α, V]_{G×H}^{+}. We will call this basis the standard basis of RB̃^{S}(G, H).

We define the mark homomorphism ρ_{G,H}^{S} as the ℤ-linear map, given by

ρ_{G,H}^{S}: B^{S}(G, H) → B̃^{S}(G, H)
[X] ↦ ∑_{(U,α,V)∈E_{G,H}^{S}} \frac{|X^{◁(U,α,V)}|}{|C_{G}(U)|}(U, α, V) ,

where X is any (G, H)-biset with point stabilizers in S_{G,H}. Note that |C_{G}(U)| divides |X^{◁(U,α,V)}|, since X^{◁(U,α,V)} is a left-free C_{G}(U)-set, cf. 2.1. Tensoring with R over ℤ induces an R-module homomorphism

ρ_{G,H}^{S}: RB^{S}(G, H) → RB̃^{S}(G, H) .

4.3 Remark If S′ is another collection of systems of subgroups of ◁_{G,H}, for G, H ∈ D, and if S′ ⊆ S (i.e., if S′_{G,H} ⊆ S_{G,H} for all G, H ∈ D) then, for every G, H ∈ D, the R-module RB̃^{S′}(G, H) is a submodule of RB̃^{S}(G, H) and the map ρ_{G,H}^{S′}: RB^{S′}(G, H) → RB̃^{S′}(G, H) is the restriction of the map ρ_{G,H}^{S}: RB^{S}(G, H) → RB̃^{S}(G, H). In fact, since S′_{G,H} is closed under taking subgroups, one has Φ_{L}(a) = 0 for every a ∈ RB^{S′}(G, H) and L ∈ S_{G,H} ∖ S′_{G,H}.

4.4 Tensor product on B̃^{S}(G, H) and B̃^{S}(H, K). We first define a ℚ-bilinear map

− ·_{H} −: ℚA^{S}(G, H) × ℚA^{S}(H, K) → ℚA^{S}(G, K) ,  (6)

by setting

(U, α, V) ·_{H} (V′, β, W) := \begin{cases}0, & if V ≠ V′, \\ \frac{|C_{H}(V)|}{|H|}(U, αβ, W) & if V = V′. \end{cases}  (7)

It will turn out that the unmotivated factor [H : C_{H}(V)]^{−1} needs to be there in order for the mark homomorphism to translate the tensor product of bisets into this product. For a ∈ ℚA^{S}(G, H), b ∈ ℚA^{S}(H, K), and c ∈ ℚA^{S}(K, L) one has

(a ·_{H} b) ·_{K} c = a ·_{H} (b ·_{K} c) .  (8)

In particular, the vector space ℚA^{S}(G, G), together with the multiplication − ·_{G} −, is a ℚ-algebra with identity element ∑_{U⩽G}[G : C_{G}(U)](U, id_{U}, U). Moreover, for a and b as above, g ∈ G, h ∈ H, and k ∈ K, one has

g(a ·_{H} b)k = (ga) ·_{H} (bk) and (ah) ·_{H} b = a ·_{H} (hb) .  (9)

The first of these two equations implies that the above bilinear map restricts to a ℚ-bilinear map

− ·_{H} −: ℚA^{S}(G, H)^{G×H} × ℚA^{S}(H, K)^{H×K} → ℚA^{S}(G, K)^{G×K} .  (10)

Recall that B̃^{S}(G, H) = A^{S}(G, H)^{G×H}. By the next lemma, this bilinear map restricts to a bilinear map

− ·_{H} −: B̃^{S}(G, H) × B̃^{S}(H, K) → B̃^{S}(G, K) ,

which we call the tensor product. Again, by tensoring with R over ℤ we obtain an R-bilinear tensor product − ·_{H} −: RB̃^{S}(G, H) × RB̃^{S}(H, K) → RB̃^{S}(G, K). If S′ ⊆ S, then the tensor product map on RB̃^{S}(G, H) × RB̃^{S}(H, K) restricts to the one on RB̃^{S′}(G, H) × RB̃^{S′}(H, K).

The following lemma describes the result of the bilinear map − ·_{H} − in (6) on two standard basis elements of B̃^{S}(G, H) and B̃^{S}(H, K), respectively.

4.5 Lemma Let G, H, K ∈ D, and let (U, α, V) ∈ E_{G,H}^{S} and (V′, β, W) ∈ E_{H,K}^{S}.

(a) If V and V′ are not H-conjugate then [U, α, V]_{G×H}^{+} ·_{H} [V′, β, W]_{H×K}^{+} = 0.

(b) If V = V′ then the following equation holds in ℚA^{S}(G, H):

[U, α, V]_{G×H}^{+} ·_{H} [V, β, W]_{H×K}^{+} = ∑_{(g,h,k)∈A×B×C} ^{(g,k)}(U, αc_{h}^{−1}β, W) .

Here, A ⊆ G is a transversal for G/p_{1}(N_{G×H}(◁(U, α, V))), B ⊆ N_{H}(V) is a transversal for N_{H}(V)/C_{H}(ker(α), V), and C ⊆ K is a transversal for K/p_{2}(N_{H×K}(◁(V, β, W))). The right-hand side of the above equation does not depend on the choice of A, B, and C.

(c) One has [U, α, V]_{G×H}^{+} ·_{H} [V, β, W]_{H×K}^{+} ∈ B̃^{S}(G, K).

(d) The free abelian group B̃^{S}(G, G) is a ℤ-order in the ℚ-algebra ℚA^{S}(G, G)^{G×G}, with identity element

∑_{U∈Σ̃_{G}}[U, id_{U}, U]_{G×G}^{+} ,

where Σ̃_{G} ⊆ Σ_{G} is a transversal of the conjugacy classes of subgroups of G.

Proof (a) This follows immediately from the definition of − ·_{H} − in (7).

(b) It is straightforward to verify that the right-hand side of the equation does not depend on the choice of A, B and C. In order to prove the equation, let B_{1} ⊆ H be a transversal for H/C_{H}(ker(α), V) and let B_{2} ⊆ H be a transversal for H/C_{H}(V). Then, by Proposition 1.7, one has

[U, α, V]_{G×H}^{+} = ∑_{(g,h_{1})∈A×B_{1}} g(U, α, V)h_{1}^{−1} and [V, β, W]_{H×K}^{+} = ∑_{(h_{2},k)∈B_{2}×C} h_{2}(V, β, W)k^{−1} .

Thus, by (7) and (9), we obtain

[U, α, V]_{G×H}^{+} ·_{H} [V, β, W]_{H×K}^{+} = ∑_{(g,k)∈A×C} g( ∑_{\begin{matrix}(h_{1},h_{2})∈B_{1}×B_{2} \\ h_{1}^{−1}h_{2}∈N_{H}(V) \end{matrix}} \frac{|C_{H}(V)|}{|H|}(U, αc_{h_{1}^{−1}h_{2}}β, W))k^{−1} .

Now let B_{3} ⊆ H be a transversal of H/N_{H}(V), let B′_{1} ⊆ N_{H}(V) be a transversal of N_{H}(V)/C_{H}(ker(α), V) and let B′_{2} ⊆ N_{H}(V) be a transversal of N_{H}(V)/C_{H}(V). Then, using the relations h_{1} = hx and h_{2} = hy, we can rewrite the inner sum as

∑_{h∈B_{3}} ∑_{(x,y)∈B′_{1}×B′_{2}} \frac{|C_{H}(V)|}{|H|}(U, αc_{x^{−1}y}β, W) = \frac{|C_{H}(V)|}{|N_{H}(V)|} ∑_{(x,y)∈B′_{1}×B′_{2}}(U, αc_{x^{−1}y}β, W) .

Finally, since for every y ∈ B′_{2} the elements y^{−1}x, x ∈ B′_{1}, form again a transversal of N_{H}(V)/C_{H}(ker(α), V) and since C_{H}(ker(α), V) is equal to the set of all h ∈ N_{H}(V) such that αc_{h} = α, we obtain

∑_{(x,y)∈B′_{1}×B′_{2}}(U, αc_{x^{−1}y}β, W) = [N_{H}(V) : C_{H}(V)] ∑_{x∈B′_{1}}(U, αc_{x}^{−1}β, W) .

Altogether, we obtain the desired equation with B′_{1} = B.

(c) To see that [U, α, V]_{G×H}^{+} ·_{H} [V, β, W]_{H×K}^{+} ∈ B̃^{S}(G, K), recall that B̃^{S}(G, K) = A^{S}(G, K)^{G×K}. Since the right-hand side of the equation in (b) is contained in A^{S}(G, K), it suffices to show that it is a G × K-fixed point. But, for every (x, y) ∈ G × K, the set xA (respectively yC) is again a transversal for G/p_{1}(N_{G×H}(◁(U, α, V))) (respectively K/p_{2}(N_{H×K}(◁(V, β, W)))).

(d) An easy computation, using Proposition 1.7 again, shows that the given element is the identity element. This completes the proof.

4.6 Let G, H, K ∈ D and assume that S_{G,H} ⊆ Δ_{G,H}, that S_{H,K} ⊆ Δ_{H,K}, and that S_{G,H} and S_{H,K} also satisfy the symmetry condition (v) in Hypothesis 1.13. Assume further that R is a commutative ring such that |G|, |H|, and |K| are invertible in R. Then the R-module homomorphism given by

−^{∘}: RA^{S}(G, H) → RA^{S}(H, G), (U, α, V) ↦ \frac{|C_{H}(V)|}{|C_{G}(U)|}(V, α^{−1}, U) ,

is an isomorphism satisfying

(a^{∘})^{∘} = a and (a ·_{H} b)^{∘} = b^{∘} ·_{H} a^{∘},

for all a ∈ RA^{S}(G, H) and all b ∈ RA^{S}(H, K). It restricts to an R-module isomorphism

−^{∘}: RB̃^{S}(G, H) → RB̃^{S}(H, G), [U, α, V]_{G×H}^{+} ↦ \frac{|C_{H}(V)|}{|C_{G}(U)|}[V, α^{−1}, U]_{H×G}^{+} .

We are now ready to prove the main theorem of this section.

4.7 Theorem Let G, H, K ∈ D and let R be a commutative ring.

(a) For a ∈ B̃^{S}(G, H) and b ∈ B̃^{S}(H, K), one has

ρ_{G,K}^{S}(a ·_{H} b) = ρ_{G,H}^{S}(a) ·_{H} ρ_{H,K}^{S}(b) .

(b) The map ρ_{G,H}^{S}: B^{S}(G, H) → B̃^{S}(G, H) is an injective group homomorphism with finite cokernel whose order divides a power of |G×H|. If |G×H| is a unit in R then the induced R-module homomorphism is an isomorphism.

(c) The map ρ_{G,G}^{S}: B^{S}(G, G) → B̃^{S}(G, G) is an injective ring homomorphism with image of finite index. If |G| is a unit in R then the induced R-algebra homomorphism ρ_{G,G}^{S}: RB^{S}(G, G) → RB̃^{S}(G, G) is an isomorphism.

(d) Assume that S_{G,H} ⩽ Δ_{G,H}, that S_{G,H} satisfies the symmetry condition (v) in Hypothesis 1.13, and that |G × H| is invertible in R. Then

ρ_{H,G}^{S}(a^{∘}) = ρ_{G,H}^{S}(a)^{∘}

for all a ∈ RB^{S}(G, H).

Proof (a) We may assume that a = [X] and b = [Y], for a (G, H)-biset X and an (H, K)-biset Y such that stab_{G×H}(x) ∈ S_{G,H} for every x ∈ X, and stab_{H×K}(y) ∈ S_{H,K} for every y ∈ Y. Then

ρ_{G,H}^{S}(a) ·_{H} ρ_{H,K}^{S}(b)
= ( ∑_{(U,α,V)∈E_{G,H}^{S}} \frac{|X^{◁(U,α,V)}|}{|C_{G}(U)|}(U, α, V)) ·_{H} (  ∑_{(V′,β,W)∈E_{H,K}^{S}} \frac{|Y^{◁(V′,β,W)}|}{|C_{H}(V′)|}(V′, β, W))

= ∑_{\begin{matrix}(U,α,V)∈E_{G,H}^{S} \\ (V,β,W)∈E_{H,K}^{S} \end{matrix}} \frac{|X^{◁(U,α,V)}| · |Y^{◁(V,β,W)}|}{|C_{G}(U)| · |H|}(U, αβ, W)

= ∑_{\begin{matrix}(U,α,V)∈E_{G,H} \\ (V,β,W)∈E_{H,K} \end{matrix}} \frac{|X^{◁(U,α,V)}| · |Y^{◁(V,β,W)}|}{|C_{G}(U)| · |H|}(U, αβ, W) ,

since |X^{L}| = 0 = |Y^{M}| for all L ∈ ◁_{G,H} ∖ S_{G,H} and all M ∈ ◁_{H,K} ∖ S_{H,K}.

We fix a standard basis element (U, γ, W) ∈ E_{G,K}^{S} of A^{S}(G, K). By the above, the coefficient of ρ_{G,H}^{S}(a) ·_{H} ρ_{H,K}^{S}(b) at the basis element (U, γ, W) is equal to the number

\frac{1}{|C_{G}(U)| · |H|} ∑_{(α,V,β)∈Γ_{H}(U,γ,W)} |X^{◁(U,α,V)}| · |Y^{◁(V,β,W)}| ,

where Γ_{H}(U, γ, W) is defined as in Subsection 2.2(b). On the other hand, the coefficient of ρ_{G,K}^{S}(a·_{H} b) = ρ_{G,K}^{S}([X ×_{H} Y]) at the basis element (U, γ, W) is equal to the number

\frac{|(X ×_{H} Y)^{◁(U,γ,W)}|}{|C_{G}(U)|} .

But, by Theorem 2.3, these two numbers are equal.

(b) Using the bijection (5), we see that the standard bases of B^{S}(G, H) and B̃^{S}(G, H) are both parametrized by the set of G × H-conjugacy classes of S_{G,H}. Arranging basis elements with respect to ascending group order, the matrix describing ρ_{G,H}^{S} with respect to these bases is upper triangular with diagonal entries of the form [N_{G×H}(L) : L]/|C_{G}(U)|, where L = ◁(U, α, V), cf. the proof of Proposition 1.14(c). The statements in Part (b) follow easily.

(c) This is an immediate consequence of Parts (a) and (b). Note that B̃^{S}(G, G) is a ring, by Lemma 4.5(d).

(d) We may assume that a = [X], for a bifree (G, H)-biset X such that stab_{G×H}(x) ∈ S_{G,H} for every x ∈ X. Then we have

ρ_{G,H}^{S}([X])^{∘} = [ ∑_{(U,α,V)∈I_{G,H}^{S}} \frac{|X^{Δ(U,α,V)}|}{|C_{G}(U)|}(U, α, V)]^{∘}

= ∑_{(U,α,V)∈I_{G,H}^{S}} \frac{|X^{Δ(U,α,V)}|}{|C_{H}(V)|}(V, α^{−1}, U)

= ∑_{(U,α,V)∈I_{G,H}^{S}} \frac{|(X^{∘})^{Δ(V,α^{−1},U)}|}{|C_{H}(V)|}(V, α^{−1}, U)

= ρ_{H,G}^{S}([X]^{∘}) .

Here we used that the map I_{G,H}^{S} → I_{H,G}^{S}, (U, α, V) ↦ (V, α^{−1}, U), is a bijection by Hypothesis 1.13(v), and that x ∈ X^{L} if and only if x^{∘} ∈ (X^{∘})^{L^{∘}}, for all x ∈ X and all L ⩽ G × H.

4.8 Remark The inverse of the isomorphism ρ_{G,H}^{◁}: ℚB^{◁}(G, H) → ℚB̃^{◁}(G, H) can be given explicitly. In fact, one can use the inversion formula in [Gl] to see that

(ρ_{G,H}^{◁})^{−1}([U, α, V]_{G×H}^{+}) = \frac{|C_{G}(U)|}{|N_{G×H}(◁(U, α, V))|} ∑_{W ⩽V} |W |μ(W, V) · [G × H/ ◁ (α(W), α, W)] .

Here, we used that the map W ↦ (α(W), α|_{W} , W) defines an isomorphism between the partially ordered sets of subgroups of V and subgroups of ◁(U, α, V). Also, μ denotes the Möbius function of the partially ordered set of subgroups of H.

4.9 Remark Given a set D, and S_{G,H} ⊆ ◁_{G,H} for G, H ∈ D, such that Condition (I) in Hypothesis 1.13 is satisfied, we can define a ring Ã^{D,S} by

Ã^{D,S} := ⨁_{G,H∈D} B̃^{S}(G, H) ,

with multiplication defined as follows: for a ∈ B̃^{S}(G, H) and b ∈ B̃^{S}(H′, K) set a · b := a ·_{H} b if H = H′, and a · b := 0 if H ≠ H′. By Theorem 4.7, the collection of the maps ρ_{G,H}^{S}: B^{S}(G, H) → B̃^{S}(G, H), G, H ∈ D, defines an injective ring homomorphism

ρ: A^{D,S} → Ã^{D,S} .

Here, A^{D,S} is as in 1.16. For a commutative ring R, we set Ã_{R}^{D,S} := R ⊗_{ℤ} Ã^{D,S} and identify Ã_{R}^{D,S} canonically with ⨁_{G,H∈D} RB̃^{S}(G, H). If, for every G ∈ D, the number |G| is a unit in R then the above ring homomorphism induces an R-algebra isomorphism

ρ: A_{R}^{D,S} ⟶^{∼} Ã_{R}^{D,S} .

In this case, the abelian categories _{A_{R}^{D,S}} Mod^{∗} and Ã_{R}^{D,S} Mod^{∗} are isomorphic via ρ. Here, the category Ã_{R}^{D,S} Mod^{∗} ist defined in analogy to the category _{A_{R}^{D,S}} Mod^{∗} in 1.16. Combining this with the category

equivalence in 1.16(b) we obtain a category equivalence between the biset functor category Func_{R}^{D,S} and the category Ã_{R}^{D,S} Mod^{∗}. This is useful, since the ring structure of Ã_{R}^{D,S} is much more transparent than

the ring structure of A_{R}^{D,S}, as we will see in the following sections.

5 The multiplicative structure of B̃^{Δ}(G, G)

In this section we introduce a natural direct-product decomposition of the ghost ring B̃^{Δ}(G, G) into subrings indexed by the isomorphism classes of subgroups of G. We also give two descriptions of these components in terms of Hecke algebras, i.e., endomorphism rings of permutation modules.

Throughout this section let T denote a set of representatives of the isomorphism classes of finite groups. Recall that, for a finite group G and T ∈ T , we denote by Σ_{G}(T) the set of subgroups of G that are isomorphic to T. We assume again that D is a class of finite groups and that, for every G, H ∈ D, S_{G,H} ⊆ Δ_{G,H} is a set of subgroups of G × H satisfying Condition (I) in Hypothesis 1.13. We emphasize that, in this section, we assume that S_{G,H} is contained in Δ_{G,H}. The constructions we present will not work for the larger ghost ring B̃^{◁}(G, G) of the left-free double Burnside ring.

Again, R will denote a commutative ring.

5.1 A decomposition of B̃^{Δ}(G, H). Let G, H, K ∈ D. Recall from 4.1 that I_{G,H}^{S} consists of all triples (U, α, V) ∈ I_{G,H} such that Δ(U, α, V) ∈ S_{G,H}. The set I_{G,H}^{S} decomposes into a disjoint union

I_{G,H}^{S} = ⋃_{T ∈T} I_{G,H,T}^{S} ,

where

I_{G,H,T}^{S} := {(U, α, V) ∈ I_{G,H} | U ≅ T ≅ V } .

This decomposition gives rise to direct-sum decompositions

A^{S}(G, H) = ⨁_{T ∈T} A_{T}^{S} (G, H) and RA^{S}(G, H) = ⨁_{T ∈T} RA_{T}^{S} (G, H) ,

where A_{T}^{S} (G, H) (respectively RA_{T}^{S} (G, H)) is the ℤ-span (respectively R-span) of the subset I_{G,H,T}^{S} of I_{G,H}^{S}. Note that in the above direct sum all but finitely many summands are equal to {0}, since I_{G,H,T}^{S} is non-empty only if G and H have subgroups that are isomorphic to T. Note further that I_{G,H,T}^{S} is a G×H-stable subset of I_{G,H}^{S}, for all T ∈ T . Therefore, taking G×H-fixed points, we obtain decompositions

B̃^{S}(G, H) = ⨁_{T ∈T} B̃_{T}^{S}(G, H) and RB̃^{S}(G, H) = ⨁_{T ∈T} RB̃_{T}^{S}(G, H) ,  (11)

where B̃_{T}^{S}(G, H) := A_{T}^{S} (G, H)^{G×H} for T ∈ T . By the definition of the multiplication in A^{Δ}(G, H), cf. (7), the decomposition (11) satisfies

RB̃_{T_{1}}^{S}(G, H) ·_{H} RB̃_{T_{2}}^{S}(H, K) = 0 if T_{1} ≠ T_{2}.  (12)

Thus, the bilinear map − ·_{H} − is the collection of componentwise bilinear maps with respect to the decomposition in (11).

We can now write the mark homomorphism as a collection

ρ_{G,H}^{S} = (ρ_{G,H,T}^{S})_{T ∈T} : B^{S}(G, H) ⟶^{∼} ⨁_{T ∈T} B̃_{T}^{S}(G, H)  (13)

of homomorphisms ρ_{G,H,T}^{S} : B^{S}(G, H) → B̃_{T}^{S}(G, H).

If S_{G,H} = Δ_{G,H} then we will use the notation A_{T}^{Δ}(G, H) and B̃_{T}^{Δ}(G, H) for A_{T}^{S} (G, H) and B̃_{T}^{S}(G, H), respectively.

Combining the above statements with those from Lemma 4.5 and Theorem 4.7, we obtain the following theorem.

5.2 Theorem Let G, H, K ∈ D.

(a) Let a = (a_{T}) ∈ RB̃^{S}(G, H), and b = (b_{T}) ∈ RB̃^{S}(H, K). Then one has

(a_{T}) ·_{H} (b_{T}) = (a_{T} ·_{H} b_{T}).

(b) For T ∈ T , the R-module RB̃_{T}^{S}(G, G) is an R-subalgebra of RA_{T}^{S} (G, G) with identity element

e_{G,T} := ∑_{U∈Σ̃_{G}(T)}[U, id_{U}, U]_{G×G}^{+} ,

where Σ̃_{G}(T) ⊆ Σ_{G}(T) denotes a set of representatives of the G-conjugacy classes of Σ_{G}(T). If |G| is invertible in R then

ρ_{G,G}^{S}: RB^{S}(G, G) ⟶^{∼} ⨁_{T ∈T} RB̃_{T}^{S}(G, G) .

is an isomorphism of R-algebras.

Next, we will give an alternative construction of ghost groups and mark homomorphisms for B^{S}(G, H) ⊆ B^{Δ}(G, H) in terms of homomorphism groups between permutation modules.

5.3 The mark homomorphism σ_{G,H}^{S}. (a) For T ∈ T , let Inj(T, G) denote the set of injective homomorphisms λ: T → G. Note that Inj(T, G) is a (G, Aut(T))-biset via gλω := c_{g} ∘λ∘ω for g ∈ G, λ ∈ Inj(T, G) and ω ∈ Aut(T). We denote by \overline{Inj}(T, G) the set of G-orbits of Inj(T, G), and denote by [λ] the G-orbit of an element λ ∈ Inj(T, G). The set \overline{Inj}(T, G) is still a right Aut(T)-set and the group Inn(T) of inner automorphisms of T acts trivially on \overline{Inj}(T, G), since λ ∘ c_{t} = c_{λ(t)} ∘ λ for t ∈ T and λ ∈ Inj(T, G). Thus, we may consider \overline{Inj}(T, G) as a right Out(T)-set, where Out(T) := Aut(T)/Inn(T) denotes the group of outer automorphisms of T. Note that, of course, Inj(T, G) is empty if T is not isomorphic to a subgroup of G.

(b) For T ∈ T consider the map

f: Inj(T, G) × Inj(T, H) → I_{G,H,T} , (λ, μ) ↦ Δ(λ(T), λμ^{−1}, μ(T)) .

This map is clearly surjective, and satisfies f(c_{g}λω, c_{h}μω) = ^{(g,h)}f(λ, μ) for all g ∈ G, h ∈ H and ω ∈ Aut(T). Thus, we obtain an induced G × H-equivariant surjective map

Inj(T, G) ×_{Aut(T)} Inj(T, H) → I_{G,H,T} .

Strictly speaking we should write Inj(T, H)^{∘} in order to view Inj(T, H) as a left Aut(T)-set, but we prefer to keep the notation simple. It is straightforward to see that this map is also injective. Consequently it induces a bijection

\overline{Inj}(T, G) ×_{Aut(T)} \overline{Inj}(T, H) ⟶^{∼} I_{G,H,T} /(G × H) .  (14)

Recall from (2) that I_{G,H,T} is also in G×H-equivariant bijection with the set of twisted diagonal subgroups of G × H that are isomorphic to T.

(c) For T ∈ T and finite groups G and H, we define the ℤ-linear map

σ_{G,H,T} : B^{Δ}(G, H) → Hom_{ℤOut(T)}(ℤ\overline{Inj}(T, H), ℤ\overline{Inj}(T, G)) ,
[X] ↦ ([μ] ↦ ∑_{[λ]∈\overline{Inj}(T,G)} \frac{|X^{Δ(λ(T),λμ^{−1},μ(T))|}}{|C_{G}(λ(T))|} [λ]) ,

where X is any bifree (G, H)-biset. Note that the integer |X^{Δ(λ(T),λμ^{−1},μ(T))}| is divisible by |C_{G}(U)|, since X is left-free. Since the map σ_{G,H,T} ([X]) is defined on a ℤ-basis, and since the definition does not depend on the choices of λ and μ in their classes, it is a well-defined group homomorphism. Moreover it is easy to verify that it respects the ℤAut(T)-module structures. Collecting all these maps we obtain a map

σ_{G,H}: B^{Δ}(G, H) → ⨁_{T ∈T} Hom_{ℤOut(T)}(ℤ\overline{Inj}(T, H), ℤ\overline{Inj}(T, G)) .  (15)

(d) Assume that G, H ∈ D. Then the map σ_{G,H,T} in Part (c) restricts to a map

σ_{G,H,T}^{S} : B^{S}(G, H) → Hom_{ℤOut(T)}^{S}(ℤ\overline{Inj}(T, H), ℤ\overline{Inj}(T, G)) ,  (16)

where the latter set consists of those ℤOut(T)-module homomorphisms f: ℤ\overline{Inj}(T, H) → ℤ\overline{Inj}(T, G) with the property that, for every μ ∈ Inj(T, H), the element f([μ]) lies in the ℤ-span of those standard basis elements [λ] satisfying (λ(T), λμ^{−1}, μ(T)) ∈ I_{G,H}^{S}. In fact, since S_{G,H} is closed under G × Hconjugation and under taking subgroups, we have X^{Δ(λ(T),λμ^{−1},μ(T))} = ∅ for [X] ∈ B^{S}(G, H), unless Δ(λ(T), λμ^{−1}, μ(T)) ∈ S_{G,H}. Moreover, the map σ_{G,H} in (15) restricts to a map

σ_{G,H}^{S} = (σ_{G,H,T}^{S})_{T ∈T} : B^{S}(G, H) → ⨁_{T ∈T} Hom_{ℤOut(T)}^{S}(ℤ\overline{Inj}(T, H), ℤ\overline{Inj}(T, G)) .  (17)

Finally, tensoring the above maps with R over ℤ yields an R-module homomorphism

σ_{G,H}^{S} = (σ_{G,H,T}^{S})_{T ∈T} : RB^{S}(G, H) → ⨁_{T ∈T} Hom_{ROut(T)}^{S}(R\overline{Inj}(T, H), R\overline{Inj}(T, G)) .  (18)

In fact, since ℤ\overline{Inj}(T, H) and ℤ\overline{Inj}(T, G) are permutation modules, the canonical map between the tensor product of R with the homomorphism group in (16) to the latter homomorphism module is an isomorphism.

Note also that all but finitely many summands are trivial in all the above direct sums running over T ∈ T .

5.4 The connecting map τ_{G,H,T}^{S}. Let G, H ∈ D. We define the group homomorphism

τ_{G,H,T}^{S} : B̃_{T}^{S}(G, H) → Hom_{ℤOut(T)}^{S}(ℤ\overline{Inj}(T, H), ℤ\overline{Inj}(T, G)) ,  (19)

by setting

(τ_{G,H,T}^{S} (a))([μ]) := ∑_{[λ]∈\overline{Inj}(T,G)} a_{(λ(T),λμ^{−1},μ(T))} [λ] ,

for a = ∑_{(U,α,V)∈I_{G,H,T}^{S}} a_{(U,α,V)}(U, α, V) ∈ B̃_{T}^{S}(G, H) and [μ] ∈ \overline{Inj}(T, H). It is straightforward to check that this map is well defined.

The following theorem shows that the direct sum of homomorphism groups in (17) can serve as an alternative ghost group, that σ_{G,H} can serve as a mark homomorphism translating the tensor product construction on bisets into componentwise composition of homomorphisms, and that the map τ_{G,H}^{S} is an isomorphism that translates between these two constructions.

A special case of Part (c) of the following theorem can be derived from [Bc96b, Théorème 2] or [Bc96b, Proposition 7], using the statement from the second paragraph on page 753 in [Bc96b] and translating our setting via duality (cf. 1.4) to the realm of right-free bisets.

5.5 Theorem Let G, H, K ∈ D and let R be a commutative ring.

(a) For T ∈ T , a ∈ B^{S}(G, H) and b ∈ B^{S}(H, K), one has

σ_{G,K,T}^{S} (a ·_{H} b) = σ_{G,H,T}^{S} (a) ∘ σ_{H,K,T}^{S} (b) .

(b) The group homomorphism σ_{G,H}^{S} in (17) is injective with finite cokernel. If |G × H| is invertible in R then the induced R-module homomorphism in (18) is an isomorphism.

(c) The map

σ_{G,G}^{S}: B^{S}(G, G) → ∏_{T ∈T} End_{ℤOut(T)}^{S}(ℤ\overline{Inj}(T, G))

is an injective ring homomorphism with image of finite index, where the multiplication in the codomain is given by componentwise composition. If |G| is invertible in R then the induced map

σ_{G,G}^{S}: RB^{S}(G, G) → ∏_{T ∈T} End_{ROut(T)}^{S}(R\overline{Inj}(T, G))

is an R-algebra isomorphism.

(d) For T ∈ T , the map τ_{G,H,T}^{S} is a group isomorphism and it satisfies

τ_{G,H,T}^{S} ∘ ρ_{G,H,T}^{S} = σ_{G,H,T}^{S} .

In particular, the diagram

[figure: B^{S}(G, H); ρ_{G,H}^{S}; σ_{G,H}^{S}; ⨁_{T ∈T} B̃^{S}(G, H); (τ_{G,H,T}^{S}); ⨁_{T ∈T} Hom_{ℤOut(T)}^{S}(ℤ\overline{Inj}(T, H), ℤ\overline{Inj}(T, G))]

is commutative. Moreover, for T ∈ T , a ∈ B̃_{T}^{S}(G, H), and b ∈ B̃_{T}^{S}(H, K), one has

τ_{G,K,T}^{S} (a ·_{H} b) = τ_{G,H,T}^{S} (a) ∘ τ_{H,K,T}^{S} (b) .

Proof (a) We may assume that a = [X] and b = [Y] for bifree bisets X and Y. Let ν ∈ Inj(T, K). Then, by Theorem 2.5, we obtain

σ_{G,K,T}^{S} ([X] ·_{H} [Y])([ν]) = ∑_{[λ]∈\overline{Inj}(T,G)} \frac{|(X ×_{H} Y)^{Δ(λ(T),λν^{−1},ν(T))}|}{|C_{G}(λ(T))|}[λ]

= ∑_{[λ]∈\overline{Inj}(T,G)} \frac{1}{|H|} ∑_{(α,V,β)∈Γ_{H}(λ(T),λν^{−1},ν(T))} \frac{|X^{Δ(λ(T),α,V)}| · |Y^{Δ(V,β,ν(T))}|}{|C_{G}(λ(T))}[λ]

= ∑_{[λ]∈\overline{Inj}(T,G)} ∑_{[μ]∈\overline{Inj}(T,H)} \frac{|X^{Δ(λ(T),λμ^{−1},μ(T))}| · |Y^{Δ(μ(T),μν^{−1},ν(T))}|}{|C_{G}(λ(T))| · |C_{H}(μ(T))|}[λ]

= σ_{G,H,T}^{S} ([X])( ∑_{[μ]∈\overline{Inj}(T,H)} \frac{|Y^{Δ(μ(T),μν^{−1},ν(T))}|}{|C_{H}(μ(T))|}[μ]) = (σ_{G,H,T}^{S} ([X]) ∘ σ_{H,K,T}^{S} ([Y]))([ν]) .

Here we used that, for a fixed λ ∈ Inj(T, G), the map μ ↦ (λμ^{−1}, μ(T), μν^{−1}) defines a bijection between Inj(T, H) and Γ_{H}(λ(T), λν^{−1}, ν(T)) with inverse (α, V, β) ↦ α^{−1}λ, and that stab_{H}(μ) = C_{H}(μ(T)).

(d) We prove Part (d) before we prove Parts (b) and (c). Define the map

τ′_{G,H,T}^{S} : Hom_{ℤOut(T)}^{S}(ℤ\overline{Inj}(T, H), ℤ\overline{Inj}(T, G)) → B̃_{T}^{S}(G, H)

as follows. If f ∈ Hom_{ℤOut(T)}^{S}(ℤ\overline{Inj}(T, H), ℤ\overline{Inj}(T, G)) is a ℤOut(T)-homomorphism and if f is represented by the integral matrix (a_{[λ],[μ]}) with respect to the standard bases \overline{Inj}(T, G) and \overline{Inj}(T, H) then we set

τ′_{G,H,T}^{S} (f) := ∑_{λ×_{Aut(T)}μ∈} Inj(T,G)×_{Aut(T)}Inj(T,H) a_{[λ],[μ]} (λ(T), λμ^{−1}, μ(T)) .

Note that a_{[λ],[μ]} = 0, unless (λ(T), λμ^{−1}, μ(T)) ∈ I_{G,H,T}^{S}. Also note that, since f is a ℤOut(T)-module homomorphism, one has a_{[λω],[μω]} = a_{[λ],[μ]}, for ω ∈ Aut(T). A straightforward verification shows that τ_{G,H,T}^{S} and τ′_{G,H,T}^{S} are inverses of each other. Moreover, by the very definitions of ρ_{G,H,T}^{S}, σ_{G,H,T}^{S} and τ_{G,H,T}^{S}, we see that τ_{G,H,T}^{S} ∘ ρ_{G,H,T}^{S} = σ_{G,H,T}^{S}.

For the last statement of Part (d), we apply Theorem 4.7(b) to choose a′ ∈ ℚB^{S}(G, H) and b′ ∈ ℚB^{S}(H, K) satisfying ρ_{G,H,T}^{S} (a′) = a and ρ_{H,K,T}^{S} (b′) = b. Then, using the second statement of Part (d), Part (a), and Theorem 4.7(a), we have

τ_{G,H,T}^{S} (a) ∘ τ_{H,K,T}^{S} (b) = σ_{G,H,T}^{S} (a′) ∘ σ_{H,K,T}^{S} (b′) = σ_{G,K,T}^{S} (a′ ·_{H} b′)
=(τ_{G,K,T}^{S} ∘ ρ_{G,K,T}^{S})(a′ ·_{H} b′) = τ_{G,K,T}^{S} (ρ_{G,H,T}^{S} (a′) ·_{H} ρ_{H,K,T}^{S} (b′)) = τ_{G,K,T}^{S} (a ·_{H} b) .

(b) This follows immediately from the corresponding statement for ρ_{G,H}^{S} in Theorem 4.7(b), the commutativity of the triangle diagram in Part (d), and the fact that τ_{G,H,T}^{S} is an isomorphism for all T ∈ T .

(c) The map σ_{G,G}^{S} is a ring homomorphism by Part (a). The remaining statements follow immediately from Part (b).

5.6 The mark homomorphism σ̃_{G}^{S}. Next we consider the case where G = H more closely. For G ∈ D we define a category S_{G} whose objects are the subgroups of G and where any morphism set Hom_{S_{G}}(V, U) is defined as the set of all group homomorphisms ϕ: V → U such that {(ϕ(v), v)) | v ∈ V } ∈ S_{G,G}. Note that automatically each such ϕ is injective. The conditions in Hypothesis 1.13 imply that this is in fact a category with the usual composition of homomorphisms, and that every conjugation map c_{g}: V → U, for g ∈ G, between subgroups V and U of G is a morphism in this category. We denote by S̃_{G} a set of representatives of the isomorphism classes of objects of S_{G}. For U ⩽ G we set Aut_{S_{G}}(U) := Hom_{S_{G}}(U, U) and Out_{S_{G}}(U) := Aut_{S_{G}}(U)/Inn(U). Moreover we set \overline{Hom}_{S_{G}}(U, V) := Inn(V)\Hom_{S_{G}}(U, V). This set can be considered as a right Out_{S_{G}}(U)-set by composition. Thus, specializing to V = G, we obtain a permutation ℤOut_{S_{G}}(U)-module ℤ\overline{Hom}_{S_{G}}(U, G) and its endomorphism ring

End_{ℤOut_{S_{G}} (U)}(ℤ\overline{Hom}_{S_{G}}(U, G)) .  (20)

Similarly as in 5.3(c), we obtain, for every U ⩽ G, a well-defined group homomorphism

σ̃_{G,U}^{S}: B^{S}(G, G) → End_{ℤOut_{S_{G}} (U)}(ℤ\overline{Hom}_{S_{G}}(U, G)) ,
[X] ↦ ([ψ] ↦ ∑_{[ϕ]∈\overline{Hom}_{S_{G}} (U,G)} \frac{|X^{Δ(ϕ(U),ϕψ^{−1},ψ(U))}|}{|C_{G}(ϕ(U))|}[ϕ]) .

The collection of these maps, for U ∈ S̃_{G}, defines a group homomorphism

σ̃_{G}^{S}: B^{S}(G, G) → ⨁_{U∈S̃_{G}} End_{ℤOut_{S_{G}} (U)}(ℤ\overline{Hom}_{S_{G}}(U, G))  (21)

which induces an R-module homomorphism

σ̃_{G}^{S}: RB^{S}(G, G) → ⨁_{U∈S̃_{G}} End_{ROut_{S_{G}} (U)}(R\overline{Hom}_{S_{G}}(U, G)) .  (22)

5.7 Theorem Let G ∈ D and assume the notation from 5.6.

(a) The map in (21) is an injective ring homomorphism with image of finite index.

(b) If |G| is a unit in R then the map in (22) is an isomorphism of R-algebras.

Proof (a) First we show that σ̃_{G}^{S} is injective. Assume that a ∈ B^{S}(G, G) is such that σ̃_{G}^{S}(a) = 0. Then Φ_{L}(a) = 0 for all L = Δ(ϕ(U), ϕψ^{−1}, ψ(U)), where U ∈ S̃_{G}, and ϕ, ψ ∈ Hom_{S_{G}}(U, G). However, it is straightforward to show that the G × G-equivariant maps

Hom_{S_{G}}(U, G) ×_{Aut_{S_{G}} (U)} Hom_{S_{G}}(U, G) → S_{G,G} , ϕ ×_{Aut_{S_{G}} (U)} ψ ↦ Δ(ϕ(U), ϕψ^{−1}, ψ(U)) ,  (23)

induce a bijection

∐_{U∈S̃_{G}} \overline{Hom}_{S_{G}}(U, G) ×_{Aut_{S_{G}} (U)} \overline{Hom}_{S_{G}}(U, G) → S_{G,G}/(G × G)  (24)

on the disjoint union, cf. (14). By the surjectivity of the map in (24) and by Proposition 1.14(c), we obtain a = 0. Thus, σ̃_{G}^{S} is injective.

We still need to show that σ̃_{G}^{S} is multiplicative. But this is a straightforward variation of the proof of Part (a) in Theorem 5.5.

(b) Note that the endomorphism ring in (20) is isomorphic to the set of integral matrices (a_{[ϕ],[ψ]}) with rows and colums indexed by \overline{Hom}_{S_{G}}(U, G) with the property that a_{[ϕα],[ψα]} = a_{[ϕ],[ψ]} for all ϕ, ψ ∈ Hom_{S_{G}}(U, G) and all α ∈ Aut_{S_{G}}(U). Therefore, it has a standard basis indexed by the elements of the U-component in the coproduct in (24). Using the bijection in (24) and Proposition 1.14(c), it follows that, with respect to suitable bases, the map σ̃_{G}^{S} is represented by an upper triangular square matrix with diagonal entries equal to [N_{G×G}(L) : L]/|C_{G}(p_{1}(L))|, where L runs through a transversal of S_{G,G}/(G×G). This proves part (b).

Since endomorphism rings of semisimple artinian modules are semisimple, the following corollary is an immediate consequence of Theorem 5.7(b) and Maschke’s Theorem. Independent proofs for the semisimplicity of RB^{Δ}(G, G), for a field R of characteristic 0, can for instance be found in [Bc96b, Corollaire 7] and [W, Theorem 9.6(1)].

5.8 Corollary Let G ∈ D and let R be a field such that the numbers |G| and |Out_{S_{G}}(U)| are invertible in R for all U ∈ S̃_{G}. Then the R-algebra RB^{S}(G, G) is semisimple. The isomorphism classes of simple RB^{S}(G, G)-modules are in bijective correspondence with the pairs (U, [V]), where U ∈ S̃_{G} and [V] is the isomorphism class of a simple ROut_{S_{G}}(U)-module V that occurs as a direct summand in the permutation module R\overline{Hom}_{S_{G}}(U, G).

5.9 Remark Assume again the notation from 5.6. We mention, without proof, that, for every T ∈ T , there exists a ring isomorphism

τ̃_{G,T}^{S} : End_{ℤOut(T)}^{S}(ℤ\overline{Inj}(T, G)) → ⨁_{U∈S̃_{G}(T)} End_{ℤOut_{S_{G}} (U)}(ℤ\overline{Hom}_{S_{G}}(U, G))

with the property that

(τ̃_{G,T}^{S})_{T ∈T} ∘ σ_{G,G}^{S} = σ̃_{G}^{S} .

In the above direct sum, S̃_{G}(T) denotes a transversal of the S_{G}-isomorphism classes of Σ_{G}(T), the set of subgroups of G that are abstractly isomorphic to T. The homomorphism τ̃_{G,T}^{S} is defined as follows. If f ∈ End_{ℤOut(T)}^{S}(ℤ\overline{Inj}(T, G)) is represented by the matrix (a_{[λ],[μ]}), then the U-component of τ̃_{G,T}^{S} (f) is represented by the matrix (b_{[ϕ],[ψ]}), where b_{[ϕ],[ψ]} := a_{[ϕ∘θ],[ψ∘θ]} for some fixed isomorphism θ: T ⟶^{∼} U. This definition is independent of the choice of θ. We leave the verifications of the properties of τ̃_{G,T}^{S} to the reader.

5.10 Remark Assume now that D is a set. Moreover, for simplicity, assume that for every G ∈ D and every subgroup U ⩽ G there exists some H ∈ D with H ≅ U, and assume that S_{G,H} = Δ_{G,H} for all G, H ∈ D. Assume further that R is a field such that |G| and |Out(G)| are invertible in R, for every G ∈ D. By 1.16, the module category _{A_{R}^{D,S}} Mod^{∗} is then equivalent to the category of Mackey functors on D over R. Note that, by Theorem 5.5, the collection of the maps σ_{G,H}^{S}, G, H ∈ D, defines an isomorphism of R-algebras

σ: A_{R}^{D,S} ⟶^{∼} ⨁_{T ∈T} ⨁_{G,H∈D} Hom_{ROut(T)}(R\overline{Inj}(T, H), R\overline{Inj}(T, G)) ,

where the latter direct sum has componentwise multiplication with respect to T. Thus, the latter algebra is the direct sum of the R-algebras

˜Ã_{T,R}^{D,S} := ⨁_{G,H∈D} Hom_{ROut(T)}(R\overline{Inj}(T, H), R\overline{Inj}(T, G))

with multiplication of two components given by composition of homomorphisms if they are composable, and by the 0-product otherwise. From this point of view one could quickly show that the category Func_{R}^{D,S} of Mackey functors on D over R is semisimple and that the isomorphism classes of simple objects in Func_{R}^{D,S} are parametrized by pairs (T, [V]), where T ∈ T such that D contains a group isomorphic to T, and were [V] is the isomorphism class of a simple ROut(T)-module V, cf. [W, Section 9]. For general D and S one could still show semisimplicity, and work out a parametrizing set of simple functors, but we do not want to expand on it in this paper.

6 The multiplicative structure of B̃^{◁}(G, G)

In this section we will study the structure of the ghost ring B̃^{◁}(G, G). We show that it has a natural grading B̃^{◁}(G, G) = ⨁_{n⩾0} B̃_{n}^{◁}(G, G) and that the ideal ⨁_{n⩾1} B̃_{n}^{◁}(G, G) is nilpotent. This allows us to fully understand the simple modules of RB̃^{◁}(G, G), provided that R is a field such that |G| and |Out(U)|, for all U ⩽ G, are invertible in R. These simple modules are then the same as those for RB̃^{Δ}(G, G).

Again, we will prove more general results, by considering the following situation: assume that D is a class of finite groups and that, for each choice of G, H ∈ D, we are given a subset S_{G,H} ⊆ ◁_{G,H} of subgroups of G × H satisfying Condition (I) in Hypothesis 1.13. With these assumptions, B^{S}(G, H) is a subgroup of B^{◁}(G, H), for G, H ∈ D. Throughout this section, R denotes a commutative ring.

6.1 A grading on B̃^{S}(G, H). For a finite group G let l(G) denote the composition length of G. For finite groups G, H ∈ D and n ∈ ℕ_{0}, we define

E_{G,H,n}^{S} := {(U, α, V) ∈ E_{G,H}^{S} | l(ker(α)) = n}

and

A_{n}^{S}(G, H) := ⟨(U, α, V) | (U, α, V) ∈ E_{G,H,n}^{S}⟩_{ℤ} ⊆ A^{S}(G, H) .

Since E_{G,H}^{S} is the disjoint union of the subsets E_{G,H,n}^{S}, we obtain a direct-sum decomposition

A^{S}(G, H) = ⨁_{n⩾0} A_{n}^{S}(G, H) .

Clearly, A_{n}^{S}(G, H) is a G × H-invariant subgroup of A^{S}(G, H), and we obtain direct-sum decompositions

B̃^{S}(G, H) = ⨁_{n⩾0} B̃_{n}^{S}(G, H) and RB̃^{S}(G, H) = ⨁_{n⩾0} RB̃_{n}^{S}(G, H) ,  (25)

where B̃_{n}^{S}(G, H) := A_{n}^{S}(G, H)^{G×H} = B̃^{S}(G, H) ∩ A_{n}^{S}(G, H). Note that

RB̃_{0}^{S}(G, H) = RB̃^{Δ(S)}(G, H) ,

where Δ(S)_{G,H} := S_{G,H} ∩ Δ_{G,H} for G, H ∈ D.

Recall from 5.6 that we can define a category Δ(S)_{G} whose objects are the subgroups of G and whose morphism sets are determined by the groups in Δ(S)_{G,G}. In the following, the Jacobson radical of any ring A will be denoted by J(A).

6.2 Lemma Let G, H, K ∈ D.

(a) For any m, n ∈ ℕ_{0} one has

RB̃_{m}^{S}(G, H) ·_{H} RB̃_{n}^{S}(H, K) ⊆ RB̃_{m+n}^{S}(G, K) .

(b) The decomposition in (25) provides the R-algebra RB̃^{S}(G, G) with the structure of a graded R-algebra such that RB̃_{0}^{S}(G, G) = RB̃^{Δ(S)}(G, G).

(c) Assume that R is a field such that |G| and |Out_{Δ(S)_{G}}(U)| are units in R, for every subgroup U ⩽ G. Then

J(RB̃^{S}(G, G)) = ⨁_{n⩾1} RB̃_{n}^{S}(G, G) .

In particular, one has a decomposition

RB̃^{S}(G, G) = RB̃^{Δ(S)}(G, G) ⊕ J(RB̃^{S}(G, G)) .

Proof (a) It suffices to show that

A_{m}^{◁}(G, H) ·_{H} A_{n}^{◁}(H, K) ⊆ A_{m+n}^{◁}(G, H) .

So let (U, α, V) ∈ E_{G,H,m} and let (V, β, W) ∈ E_{H,K,n}. We need to show that l(ker(αβ)) = l(ker(α)) + l(ker(β)). But this follows from the short exact sequence

1  [figure]ker(β)  [figure]ker(αβ) [figure: β]ker(α)  [figure]1

where the second arrow is the inclusion map.

(b) This follows immediately from Part (a).

(c) By Part (a), the subspace I := ⨁_{n⩾1} RB̃_{n}^{S}(G, G) is an ideal of RB̃^{S}(G, G). Since RB̃_{n}^{S}(G, G) = 0 for n > max{l(H) | H ⩽ G}, Part (a) also implies that this is a nilpotent ideal, therefore contained in J(RB̃^{S}(G, G)). On the other hand, the factor algebra modulo the ideal I is isomorphic to RB̃_{0}^{S}(G, G) = RB̃^{Δ(S)}(G, G), which is semisimple by Corollary 5.8. This implies J(RB̃^{S}(G, G)) ⊆ I and we have equality.

6.3 A grading on RB^{S}(G, G). Let G, H ∈ D and let n ∈ ℕ_{0}. We define the subgroup

B_{n}^{S}(G, H) ⊆ B^{S}(G, H)

as the subset of those elements a in B^{S}(G, H) satisfying Φ_{◁(U,α,V)}(a) = 0 for all (U, α, V) ∈ ⋃_{i∈ℕ_{0}∖{n}} E_{G,H,i}.

Now assume that |G × H| is a unit in R. Then the isomorphism in Theorem 4.7(b) induces an isomorphism

ρ_{G,H}^{S}: RB_{n}^{S}(G, H) ⟶^{∼} RB̃_{n}^{S}(G, H)

and we obtain a direct sum decomposition

RB^{S}(G, H) = ⨁_{n⩾0} RB_{n}^{S}(G, H)  (26)

with

RB_{0}^{S}(G, H) = RB^{Δ(S)}(G, H) .

It seems worth mentioning that, in general, the sum ∑_{n⩾0} B_{n}^{S}(G, H) is a proper subgroup of B^{S}(G, H).

The following theorem is now immediate from Lemma 6.2 and the fact that the isomorphism ρ_{G,H}^{S} respects the tensor product construction of bisets, cf. Theorem 4.7(a).

6.4 Theorem Let G, H, K ∈ D.

(a) For any m, n ∈ ℕ_{0} one has

RB_{m}^{S}(G, H) ·_{H} RB_{n}^{S}(H, K) ⊆ RB_{m+n}^{S}(G, K) .

(b) Assume that |G| is invertible in R. Then the R-algebra RB^{S}(G, G) is a graded R-algebra, with the grading given in (26).

(c) Assume that R is a field and that |G| and |Out_{Δ(S)_{G}}(U)| are invertible in R, for every subgroup U ⩽ G. Moreover set J := J(RB^{S}(G, G)). One has J = ⨁_{n⩾1} RB_{n}^{S}(G, G), and J consists of precisely

those elements a ∈ RB^{S}(G, G) satisfying Φ_{Δ(U,α,V)}(a) = 0 for all (U, α, V) ∈ I_{G,H}^{Δ(S)}. In particular, one has

RB^{S}(G, G) = RB^{Δ(S)}(G, G) ⊕ J .

The following corollary is an immediate consequence of Theorem 6.4(c).

6.5 Corollary Let G ∈ D and let R be a field such that |G| and |Out_{Δ(S)_{G}}(U)| are units in R, for every subgroup U ⩽ G. Then the isomorphism classes of simple RB^{S}(G, G)-modules and the isomorphism classes of simple RB^{Δ(S)}(G, G)-modules are in natural bijective correspondence. More precisely, the correspondence is given by restriction from RB^{S}(G, G) to the subalgebra RB^{Δ(S)}(G, G). Its inverse is given by inflation from RB^{Δ(S)}(G, G) to RB^{S}(G, G) with respect to the ideal ⨁_{n⩾1} RB_{n}^{S}(G, G).

For the proof of the next Theorem we first need a well-known result about Hecke algebras.

6.6 Lemma Let R be a field of positive characteristic p, let G be a finite group and let H be a subgroup of G such that |H| is not divisible by p but |G| is divisible by p. Then J(End_{RG}(Ind_{H}^{G}(R))) ≠ {0}.

Proof Write e for the idempotent |H|^{−1} ∑_{h∈H} h ∈ RG. Note that Ind_{H}^{G}(R) = R⊗_{RH}RG is isomorphic to eRG as right RG-modules and that End_{RG}(Ind_{H}^{G}(R)) is isomorphic to eRGe as R-algebras. Furthermore, set a := ∑_{g∈G} g ∈ RG. Then 0 ≠ a = ea = eae ∈ eRGe, and Ra is a non-zero ideal of eRGe. Since a^{2} = |G|a = 0, we have Ra ⊆ J(eRGe).

The following theorem gives a criterion for the semisimplicity of RB^{S}(G, G) for an arbitrary field R. It can also be interpreted as a converse of the semisimplicity result in Corollary 5.8. For what follows, recall from 5.6 the definition of the category S_{G}.

6.7 Theorem Let R be a field, let G ∈ D be a non-trivial group and let S̃_{G} be a transversal of the S_{G}-isomorphism classes of subgroups of G. Then the following statements are equivalent:

(i) The R-algebra RB^{S}(G, G) is semisimple.

(ii) One has S_{G,G} ⊆ Δ_{G,G} and the numbers |G| and |Out_{S_{G}}(U)|, for all U ∈ S̃_{G}, are invertible in R.

Proof The statement in (ii) implies the statement in (i), by Corollary 5.8.

Next assume that RB^{S}(G, G) is semisimple. We first show that |G| is invertible in R. Assume that this is not the case and consider the element a := [G × G] = [(G × G)/{1}] ∈ RB^{S}(G, G), the class of the regular G × G-set. Let b ∈ RB^{S}(G, G) be arbitrary. Since the trivial subgroup {1} ⩽ G × G satisfies L ∗ {1} = {1} for every L ∈ ◁_{G,G}, the multiplication formula in Proposition 1.10 implies that b ·_{G} a = ra for some r ∈ R. Moreover the multiplication formula implies that a ·_{G} a = |G|a = 0. Thus, in the ring RB^{S}(G, G) we have (1 + ra)(1 − ba) = (1 + ra)(1 − ra) = 1 − r^{2}a^{2} = 1. This shows that 1 − ba has a left inverse. Since b was arbitrary, this shows that a ∈ J(RB^{S}(G, G)). This is a contradiction to the semisimplicity of RB^{S}(G, G). Therefore, we have proved that |G| is invertible in R.

Now Theorem 6.4(c) applies, and we derive that S_{G,G} ⊆ Δ_{G,G}.

With this established, Theorem 5.7(b) applies, and we obtain that RB^{S}(G, G) is a direct product of the R-algebras End_{ROut_{S_{G}} (U)}(R\overline{Hom}_{S_{G}}(U, G)), where U varies over S̃_{G}. Therefore, also these endomorphism algebras are semisimple. Now fix U ∈ S̃_{G} and let ϕ: U → G be the inclusion map, which is contained in Hom_{S_{G}}(U, G). Lemma 6.6 implies that the index of the stabilizer of [ϕ] in Out_{S_{G}}(U) is invertible in R. But this stabilizer is isomorphic to N_{G}(U)/(UC_{G}(U)). Its order is a divisor of |G|, and hence also invertible in R. Thus, |Out_{S_{G}}(U)| is invertible in R, and the proof of the theorem is complete.

The following corollary extends a computational result of Webb (cf. [W, Thm. 9.6(2)]) for the group of order 2 and the case S_{G,G} = ◁_{G,G}.

6.8 Corollary Let R be a non-zero commutative ring, let G ∈ D, and assume that S_{G,G} ⊈ Δ_{G,G}. Then RB^{S}(G, G) is not semisimple.

Proof Assume, for a contradiction, that RB^{S}(G, G) is semisimple. Let R̄ be the factor ring of R modulo some maximal ideal. Then R̄B^{S}(G, G) is a factor ring of RB^{S}(G, G). Since RB^{S}(G, G) is assumed to be semisimple, so is R̄B^{S}(G, G). Theorem 6.7 now implies S_{G,G} ⊆ Δ_{G,G}, a contradiction.

6.9 Remark Let again D be a set. Assume for simplicity that R is a field of characteristic 0, that S_{G,H} = ◁_{G,H} for all G, H ∈ D, and that D is closed under taking subgroups in the sense that every subgroup of a group G ∈ D is isomorphic to a group in D. Then Δ(S)_{G,H} = Δ_{G,H} for all G, H ∈ D. Theorem 6.4 implies that the R-algebra Ã_{R}^{D,S} = Ã_{R}^{D,◁} is graded by

Ã_{R}^{D,◁} = ⨁_{n⩾0} Ã_{n,R}^{D,◁}

with

Ã_{n,R}^{D,◁} := ⨁_{G,H∈D} RB̃_{n}^{◁}(G, H) ,

for n ⩾ 0. From Lemma 6.2(c) it is also straightforward to see that every simple object in Ã_{R}^{D,◁}Mod^{∗}

is annihilated by the ideal J̃_{R}^{D,◁} := ⨁_{n⩾1} Ã_{n,R}^{D,◁}. Note that the subalgebra Ã_{0,R}^{D,◁} = Ã_{R}^{D,Δ} is the algebra considered in Remark 5.10. Now the decomposition

Ã_{R}^{D,◁} = Ã_{R}^{D,Δ} ⊕ J̃_{R}^{D,◁},

together with the category equivalences in 1.16 and Remark 4.9, implies that the isomorphism classes of simple objects in Func_{R}^{D,◁} and those in Func_{R}^{D,Δ} are in natural bijective correspondence via restriction and inflation with respect to the above direct sum decomposition.

Moreover, one has a filtration of Ã_{R}^{S,◁} by the ideals J̃_{n}^{S,◁} := ⨁_{i⩾n} Ã_{i,R}^{S,◁}, for n ⩾ 0. This leads to a natural filtration of every functor in Func_{R}^{D,◁} with successive semisimple factors.

7 Fusion systems

Throughout this section, p denotes a prime number, ℤ_{(p)} ⊂ ℚ denotes the localization of ℤ with respect to the prime ideal (p) = pℤ, and S a finite p-group. As before, R is a commutative ring.

In this section we will show that fusion systems F on S are in bijective correspondence with subsets S_{S,S} of Δ_{S,S} that satisfy the Axioms (i)–(v) in Hypothesis 1.13. Thus, we can consider the ring B^{F}(S, S) = B^{S}(S, S) as an invariant of the fusion system F. We identify the characteristic idempotent ω_{F} of a saturated fusion system F in the ghost ring ℤ_{(p)}B̃^{F}(S, S) and are able to compute its marks, i.e., its numbers of fixed points with respect to the subgroups in Δ_{S,S}. Moreover, we extend the bijection between the set of saturated fusion systems on S and a certain set of idempotents in ℤ_{(p)}B^{Δ}(S, S), which was observed by Ragnarsson and Stancu in [RS], to a bijection between the set of all fusion systems on S and a certain set of idempotents in ℚB^{Δ}(S, S).

Recall that whenever P and Q are subgroups of S, we denote by Hom_{S}(P, Q) the set of homomorphisms P → Q that are induced by conjugations with elements in S. Moreover, we again denote by Inj(P, Q) the set of all injective group homomorphisms P → Q.

First we recall the definition of a fusion system and of a saturated fusion system on S, cf. [L] for instance.

7.1 Definition (a) A fusion system on S is a category F whose objects are the subgroups of S, and whose morphism sets Hom_{F}(P, Q) satisfy the following conditions:

(i) If P, Q ⩽ S then Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q).

(ii) If P, Q ⩽ S and if ϕ ∈ Hom_{F}(P, Q) then the resulting isomorphism P →^{∼} ϕ(P) as well as its inverse are morphisms in F. The composition of morphisms in F is the usual composition of maps. Whenever subgroups P and Q of S are isomorphic in F, we write P =_{F} Q.

(b) For a fusion system F on S one introduces the following three notions:

(i) A subgroup P of S is called fully F-centralized if, for every Q ⩽ S with Q =_{F} P, one has |C_{S}(Q)| ⩽ |C_{S}(P)|.

(ii) A subgroup P of S is called fully F-normalized if, for every Q ⩽ S with Q =_{F} P, one has |N_{S}(Q)| ⩽ |N_{S}(P)|.

(iii) For every P ⩽ S and every ϕ ∈ Hom_{F}(P, S), one sets

N_{ϕ}: = {y ∈ N_{S}(P) | ∃z ∈ N_{S}(ϕ(P)) : ϕ(^{y}u) = ^{z}ϕ(u) ∀u ∈ P }.

(c) A fusion system F on S is called saturated if the following axioms are satisfied:

(i) (Sylow Axiom) Aut_{S}(S) ∈ Syl_{p}(Aut_{F}(S)).

(ii) (Extension Axiom) Every morphism ϕ ∈ Hom_{F}(P, S) such that ϕ(P) is fully F-normalized extends to a morphism ψ ∈ Hom_{F}(N_{ϕ}, S).

7.2 Fusion systems on S and subsystems of Δ_{S,S}. (a) Suppose that F is a fusion system on S, and consider

S := S(F) := {Δ(ϕ(P), ϕ, P) | P ⩽ S, ϕ ∈ Hom_{F}(P, S)} .

Then the class D := {S}, together with the set S, satisfies Conditions (I) and (II) in Hypothesis 1.13. This follows immediately from the definition of a fusion system. Thus, we can introduce the notation B^{F}(S, S) for B^{S}(S, S).

(b) Suppose, conversely, we are given a set S ⊆ Δ_{S,S} such that D := {S}, together with this set S, satisfies Conditions (I) and (II) in Hypothesis 1.13. Then we define a category F = F(S) as follows: the objects in F are the subgroups of S. For P, Q ⩽ S, we define

Hom_{F}(P, Q) := {ι ∘ ϕ | Δ(ϕ(P), ϕ, P) ∈ S and ϕ(P) ⩽ Q} ,

where ι : ϕ(P) → Q denotes the inclusion map. The composition of morphisms in F shall be the usual composition of maps. Then F is a fusion system on S.

We denote the set of fusion systems on S by Fus(S). This is a finite poset, with the partial order given by the subcategory relation. The set of subsets S of Δ_{S,S} such that the class D = {S} together with S_{S,S} = S satisfies Conditions (I) and (II) in Hypothesis 1.13 will be denoted by Sys(S). This is a finite poset with respect to inclusion. The following theorem is an easy exercise, and is left to the reader.

7.3 Theorem The constructions in 7.2 are mutually inverse isomorphisms between the partially ordered sets Fus(S) and Sys(S).

7.4 Examples (a) The following is a standard example of a saturated fusion system. Let G be a group such that S ∈ Syl_{p}(G). Then the category F_{S}(G) with objects given by the set of subgroups of S and with morphisms

Hom_{F_{S}(G)}(P, Q) := Hom_{G}(P, Q), (P, Q ⩽ S)

is a saturated fusion system on S, called the fusion system of G on S. For a proof, see [L]. The corresponding set S is thus given as

S = {Δ(ϕ(P), ϕ, P) | P ⩽ S, ϕ ∈ Hom_{G}(P, S)},

or as the set of subgroups L ∈ Δ_{S,S} such that L ⩽_{G×G} Δ(S).

(b) Consider the alternating group 𝔄_{4} of degree 4, and let S be the unique Sylow 2-subgroup of 𝔄_{4}. That is, S = ⟨(1, 2)(3, 4), (1, 4)(2, 3)⟩ is a Klein four-group. The automorphism group Aut(S) of S is isomorphic to the symmetric group 𝔖_{3} of degree 3. Let A ⊴ Aut(S) with A ≅ 𝔄_{3}. Then the saturated fusion system F_{S}(𝔄_{4}) corresponds to the set

{Δ(ϕ(P), ϕ|_{P} , P) | P ⩽ S, ϕ ∈ A} ⊆ Δ_{S,S}.

Namely,

Aut_{F_{S}(𝔄_{4})}(S) = Aut_{𝔄_{4}}(S) ≅ N_{𝔄_{4}}(S)/C_{𝔄_{4}}(S) = 𝔄_{4}/S ≅ 𝔄_{3},

so that Aut_{F_{S}(𝔄_{4})}(S) = A. Now suppose that P ⩽ S and ψ ∈ Hom_{F}(P, S). Then, since S is abelian, we have N_{ψ} = S. Hence the Extension Axiom forces Hom_{F}(P, S) = {ψ|_{P} | ψ ∈ A} so that {Δ(ψ(P), ψ|_{P} , P) | P ⩽ S, ψ ∈ A} = S(F_{S}(𝔄_{4})).

(c) Let S and A be as in Part (b) above, and this time consider the set

S := {Δ(S)} ∪ {Δ(ϕ(P), ϕ|_{P} , P) | P < S, ϕ ∈ A} ⊆ Δ_{S,S}.

The class D := {S}, together with the set S, obviously satisfies Conditions (I) and (II) in Hypothesis 1.13. Thus S gives rise to a fusion system F := F(S) on S. However, F is not saturated. To see this, let Q := ⟨(1, 2)(3, 4)⟩ and P := ⟨(1, 3)(2, 4)⟩. Let further ϕ ∈ A be the automorphism of S induced by conjugation with (1, 2, 3) ∈ 𝔄_{3}. By definition, the restriction ψ := ϕ|_{P} belongs to Hom_{F}(P, Q). Since S is abelian, we have N_{ψ} = S. Since Aut_{F}(S) = {id_{S}}, the morphism ψ does not extend to N_{ψ}, and F is, therefore, not saturated.

In [RS], K. Ragnarsson and R. Stancu established a bijective correspondence between saturated fusion systems on S and certain idempotents in the ℤ_{(p)}-algebra ℤ_{(p)}B^{Δ}(S, S). We next aim to use our results from Section 5 to extend the bijection of Ragnarsson–Stancu to a bijection between the set of all fusion systems on S and certain idempotents in the ℚ-algebra ℚB^{Δ}(S, S). In order to do so, we recall the basic notions from [RS] needed in subsequent statements.

7.5 Consider the natural ℤ-bilinear map − × −, given by

− × −: B^{Δ}(S, S) × B^{Δ}(S, S) → B^{Δ}(S × S, S × S), ([X], [Y]) ↦ [X × Y],

where X and Y are bifree (S, S)-bisets. We view X × Y as (S × S, S × S)-biset by (s_{1}, s_{2})(x, y)(s′_{1}, s′_{2}) = (s_{1}xs′_{1}, s_{2}ys′_{2}), for s_{1}, s_{2}, s′_{1}, s′_{2} ∈ S and (x, y) ∈ X × Y. This map induces, for every commutative ring R, an R-bilinear map

RB^{Δ}(S, S) × RB^{Δ}(S, S) → RB^{Δ}(S × S, S × S), ([X], [Y]) ↦ [X × Y].

Moreover, if X and Y are bifree (S, S)-bisets then X × Y carries, via restriction along the diagonal map s ↦ (s, s), a bifree (S × S, S)-biset structure (respectively, a bifree (S, S × S)-biset structure) with

(g, h)(x, y)k := (gxk, hyk) (respectively, g(x, y)(h, k) := (gxh, gyk)),

for all x ∈ X, y ∈ Y, g, h, k ∈ S.

7.6 Definition Let a ∈ RB^{Δ}(S, S). We say that (a) a is a right Frobenius element if the following equality holds in RB^{Δ}(S × S, S):

a × a = (a × [S]) ·_{S} a.  (27)

(b) a is a left Frobenius element if the following equality holds in RB^{Δ}(S, S × S):

a × a = a ·_{S} (a × [S]).  (28)

(c) a is a Frobenius element if it is a left Frobenius element and a right Frobenius element.

7.7 Remark Let a ∈ RB^{Δ}(S, S). Then the equality a × a = (a × [S]) ·_{S} a is equivalent to the equality a × a = ([S] × a) ·_{S} a in RB^{Δ}(S × S, S). Similarly, the equality a × a = a ·_{S} (a × [S]) is equivalent to the equality a × a = a ·_{S} ([S] × a) in RB^{Δ}(S, S × S). This follows quickly by applying the natural function X ×Y ⟶^{∼} Y ×X, (x, y) ↦ (y, x) for (S, S)-bisets X and Y, and the group isomorphism S ×S ⟶^{∼} S ×S, (s, t) ↦ (t, s).

As an immediate consequence of Definition 7.6, we get

7.8 Proposition Let R be a commutative ring, and let a ∈ RB^{Δ}(S, S). Then a is a right Frobenius element if and only if a^{∘} is a left Frobenius element.

Our next aim is to establish a bijection between the set Fus(S) of fusion systems on S and a set Idem(S) of certain idempotents ω ∈ ℚB^{Δ}(S, S).

7.9 Definition We denote by Idem(S) the set of idempotents ω ∈ ℚB^{Δ}(S, S) satisfying the following properties:

(i) ω is a Frobenius element,

(ii) Fix(ω) := {L ∈ Δ_{S,S} | Φ_{L}(ω) ≠ 0} is closed under taking subgroups,

(iii) Δ(S) ∈ Fix(ω).

We will construct the bijection between Idem(S) and Fus(S) in several steps, and we begin by recalling from [RS] a criterion for a ∈ ℚB^{Δ}(S, S) being a Frobenius element. The result [RS, L. 7.4] is formulated for ℤ_{(p)}, and one cannot directly lift it to ℚ, since the two sides of the equation defining the Frobenius property are not linear in the element a. However, the proof below makes sure that the result remains true when replacing ℤ_{(p)} by ℚ, or any commutative ring R.

7.10 Proposition (cf. [RS], §7) Let a ∈ ℚB^{Δ}(S, S). Then the following hold:

(a) The element a is a right Frobenius element if and only if, for every P ⩽ S and all ϕ, ψ ∈ Inj(P, S), one has

Φ_{Δ(ϕ(P),ϕ,P)}(a)Φ_{Δ(ψ(P),ψ,P)}(a) = Φ_{Δ(ϕ(P),ϕψ^{−1},ψ(P))}(a)Φ_{Δ(ψ(P),ψ,P)}(a) .

(b) The element a is a left Frobenius element if and only if, for every P ⩽ S and all ϕ, ψ ∈ Inj(P, S), one has

Φ_{Δ(P,ϕ^{−1},ϕ(P))}(a)Φ_{Δ(P,ψ^{−1},ψ(P))}(a) = Φ_{Δ(ψ(P),ψϕ^{−1},ϕ(P))}(a)Φ_{Δ(P,ψ^{−1},ψ(P))}(a) .

Proof Suppose that L ∈ Δ_{S×S,S}. Then there exist P ⩽ S and ϕ, ψ ∈ Inj(P, S) such that L = Δ((ϕ × ψ)(P), ϕ × ψ, P), where (ϕ × ψ)(P) := {(ϕ(g), ψ(g)) | g ∈ P } ⩽ S × S.

The proof of [RS, L. 7.4] shows that, for all a, b ∈ ℚB^{Δ}(S, S) and all L = Δ((ϕ × ψ)(P), ϕ × ψ, P) ∈ Δ_{S×S,S}, we have

Φ_{L}(a × b) = Φ_{Δ(ϕ(P),ϕ,P)}(a)Φ_{Δ(ψ(P),ψ,P)}(b),  (29)

and

Φ_{L}((a × [S]) ·_{S} b) = Φ_{Δ(ϕ(P),ϕψ^{−1},ψ(P))}(a)Φ_{Δ(ψ(P),ψ,P)}(b).  (30)

Specializing a = b and using Proposition 1.14, Assertion (a) follows.

As for (b), let again a ∈ ℚB^{Δ}(S, S). By Proposition 7.8, a is a left Frobenius element if and only if a^{∘} is a right Frobenius element. By Part (a), this in turn is equivalent to requiring

Φ_{Δ(P,ϕ^{−1},ϕ(P))}(a)Φ_{Δ(P,ψ^{−1},ψ(P))}(a) = Φ_{Δ(ϕ(P),ϕ,P)}(a^{∘})Φ_{Δ(ψ(P),ψ,P)}(a^{∘})
= Φ_{Δ(ϕ(P),ϕψ^{−1},ψ(P))}(a^{∘})Φ_{Δ(ψ(P),ψ,P)}(a^{∘})
= Φ_{Δ(ψ(P),ψϕ^{−1},ϕ(P))}(a)Φ_{Δ(P,ψ^{−1},ψ(P))}(a),

for all P ⩽ S and all ϕ, ψ ∈ Inj(P, S). This settles (b).

7.11 The idempotent ω_{F}. Let F be a fusion system on S, and let F̃ be a transversal for the Fisomorphism classes of subgroups of S.

(a) Recall from Theorem 5.7 that we have a ℚ-algebra isomorphism

σ̃^{F}: ℚB^{F}(S, S) → ∏_{P ∈F̃} End_{ℚOut_{F} (P)}(ℚ\overline{Hom}_{F}(P, S)),

a ↦ ([ψ] ↦ ∑_{[ϕ]∈\overline{Hom}_{F} (P,S)} \frac{Φ_{Δ(ϕ(P),ϕψ^{−1},ψ(P))}(a)}{|C_{S}(ϕ(P))|}[ϕ])_{P} .  (31)

For every P ∈ F̃ we define ε_{P} ∈ \overline{Hom}_{F}(P, S) such that

ε_{P} ([ψ]) := ∑_{[ϕ]∈\overline{Hom}_{F} (P,S)} \frac{|S|}{|C_{S}(ϕ(P))||Hom_{F}(P, S)|}[ϕ] ,  (32)

for every [ψ] ∈ \overline{Hom}_{F}(P, S). We then define ω_{F} ∈ ℚB^{F}(S, S) via

σ̃^{F}(ω_{F}) := (ε_{P})_{P ∈F̃}.  (33)

(b) Let S := S(F) and note (cf. (24)) that we have a bijection

∐_{P ∈F̃} Hom_{F}(P, S) ×_{Aut_{F} (P)} Hom_{F}(P, S) → S ,
ϕ ×_{Aut_{F} (P)} ψ ↦ Δ(ϕ(P), ϕψ^{−1}, ψ(P)) ,  (34)

of S × S-sets. Thus, by (31) and (32), the element ω_{F} ∈ ℚB^{Δ}(S, S) is characterized by

Φ_{L}(ω_{F}) = \begin{cases}\frac{|S|}{|Hom_{F} (p_{1}(L),S)|} & if L ∈ S, \\ 0 & if L ∉ S. \end{cases}  (35)

In particular, ω_{F}^{∘} = ω_{F} is symmetric, since |Hom_{F}(P, S)| = |Hom_{F}(Q, S)| if P =_{F} Q.

The following lemma gives a list of properties of the element ω_{F}, including a uniqueness statement.

7.12 Lemma Let F be a fusion system on S, let S := S(F), and let ω_{F} ∈ ℚB^{F}(S, S) be the element defined in (33). Then

(a) ω_{F} is an idempotent in ℚB^{Δ}(S, S);

(b) ω_{F} is a Frobenius element;

(c) Fix(ω_{F}) = S. In particular, ω_{F} ∈ Idem(S).

Moreover, ω_{F} is the unique element in ℚB^{Δ}(S, S) satisfying Properties (a)–(c).

Proof (a) A quick computation shows that, for every P ∈ F̃, the endomorphism ε_{P} is an idempotent. In fact this comes down to the equation

∑_{[ϕ]∈\overline{Hom}_{F} (P,S)} \frac{|S|}{|C_{S}(ϕ(P)||Hom_{F}(P, S)|} = 1 ,

which holds, since C_{S}(ϕ(P)) is the stabilizer of ϕ in the S-set Hom_{F}(P, S).

(b) This follows immediately from Equation (35) and Proposition 7.10.

(c) This follows immediately from Equation (35).

To show the uniqueness statement, let ω ∈ ℚB^{Δ}(S, S) satisfy (a)–(c). Since ω satisfies (c), we obtain Φ_{L}(ω) = 0 for L ∉ S. We fix P ∈ F̃. By (35), it suffices to show that

a_{ϕ,ψ} := Φ_{Δ(ϕ(P),ϕψ^{−1},ψ(P))}(ω) = \frac{|S|}{|Hom_{F}(P, S)|} ,  (36)

for all P ∈ F̃ and ϕ, ψ ∈ Hom_{F}(P, S). Here, we made use of the bijection in (34) and the equality |Hom_{F}(ϕ(P), S)| = |Hom_{F}(P, S)|. We first show that a_{ϕ,ψ} = a_{id_{P} ,id_{P}} for all ϕ, ψ ∈ Hom_{F}(P, S). Since Φ_{Δ(ψ(P),ψ,P)}(ω) ≠ 0 by Property (c), Proposition 7.10(a) implies that a_{ϕ,ψ} = a_{ϕ,id_{P}}, since ω is right Frobenius. Similarly, replacing ϕ and ψ in the statement of Proposition 7.10(b) with id_{P} and ϕ, respectively, implies a_{id_{P} ,id_{P}} = a_{ϕ,id_{P}}. Now we can abbreviate a_{ϕ,ψ} by a constant rational number c. Writing ζ_{P} for the P -component of σ̃^{F}(ω), the endomorphism ζ_{P} satisfies

ζ_{P} ([ψ]) = ∑_{[ϕ]∈\overline{Hom}_{F} (P,S)} \frac{c}{|C_{S}(ϕ(P))|} [ϕ] ,

for all ψ ∈ Hom_{F}(P, S). Since ω is an idempotent, so is ζ_{P}. Evaluating the equation ζ_{P} ∘ ζ_{P} = ζ_{P}, yields

∑_{[ϕ]∈\overline{Hom}_{F} (P,S)} \frac{c^{2}}{|C_{S}(ϕ(P))|} = c .

Since c ≠ 0 by Property (c), and since C_{S}(ϕ(P)) is the stabilizer of the element ϕ in the S-set Hom_{F}(P, S), we obtain

c = ( ∑_{[ϕ]∈\overline{Hom}_{F} (P,S)} \frac{1}{|C_{S}(ϕ(P))|})^{−1} = ( ∑_{ϕ∈Hom_{F} (P,S)} \frac{1}{|S|})^{−1} = \frac{|S|}{|Hom_{F}(P, S)|}

as desired in Equation (36). This completes the proof of the lemma.

7.13 Lemma Let ω ∈ ℚB^{Δ}(S, S) be a right Frobenius element such that Δ(S) ∈ Fix(ω) and such that Fix(ω) is closed under taking subgroups. Then S := Fix(ω) ∈ Sys(S).

Proof We verify that S satisfies (i)–(v) in Hypothesis 1.13. By definition, S_{S,S} is closed under S × Sconjugation, and by our hypotheses it is closed under taking subgroups and contains Δ(S). In particular, Φ_{Δ(P)}(ω) ≠ 0, for all P ⩽ S.

Let Δ(ψ(P), ψ, P) ∈ S, so that Φ_{Δ(ψ(P),ψ,P)}(ω) ≠ 0. Since ω is a right Frobenius element, Proposition 7.10(a) with φ = id_{P} gives

0 ≠ Φ_{Δ(P)}(ω) = Φ_{Δ(P,ψ^{−1},ψ(P))},

thus Δ(ψ(P), ψ, P)^{∘} = Δ(P, ψ^{−1}, ψ(P)) ∈ S.

It remains to show that if Δ(ϕ(P), ϕ, P) ∈ S and Δ(ψ(Q), ψ, Q) ∈ S then also Δ(ϕ(P), ϕ, P) ∗ Δ(ψ(Q), ψ, Q) ∈ S_{S,S}.

Suppose first that ψ(Q) = P. Since ω is a right Frobenius element, Proposition 7.10(a) implies Φ_{Δ(ϕ(P),ϕ,P)}(ω)Φ_{Δ(Q,ψ^{−1},ψ(Q))}(ω) = Φ_{Δ(ϕ(P),ϕψ,Q)}(ω)Φ_{Δ(Q,ψ^{−1},ψ(Q))}(ω). As we have just shown, Φ_{Δ(Q,ψ^{−1},ψ(Q))}(ω) ≠ 0, so that 0 ≠ Φ_{Δ(ϕ(P),ϕ,P)}(ω) = Φ_{Δ(ϕ(P),ϕψ,Q)}(ω), thus Δ(ϕ(P), ϕ, P) ∗ Δ(ψ(Q), ψ, Q) = Δ(ϕ(P), ϕψ, Q) ∈ S.

In the general case we have

Δ(ϕ(P), ϕ, P) ∗ Δ(ψ(Q), ψ, Q) = Δ(ϕ(P ∩ ψ(Q)), ϕψ, ψ^{−1}(P ∩ ψ(Q)))
= Δ(ϕ(P ∩ ψ(Q)), ϕ, P ∩ ψ(Q)) ∗ Δ(P ∩ ψ(Q), ψ, ψ^{−1}(P ∩ ψ(Q))).

Since S is closed under taking subgroups, our considerations in the special case above show that also Δ(ϕ(P), ϕ, P) ∗ Δ(ψ(Q), ψ, Q) ∈ S, and the proof is complete.

7.14 Remark By Lemma 7.12 and Lemma 7.13 we obtain maps

f: Fus(S) → Idem(S) , F ↦ ω_{F} , and g: Idem(S) → Sys(S) , ω ↦ Fix(ω) .

Moreover, we write

h: Fus(S) ⟶^{∼} Sys(S) , F ↦ S(F) ,

for the bijection in Theorem 7.3. Then we obtain a triangle diagram

[figure: Fus(S); f; Idem(S); h; g; Sys(S)]

Summarizing our previous considerations, we obtain the following theorem.

7.15 Theorem The maps f, g and h in Remark 7.14 are bijections, and the triangle diagram in Remark 7.14 is commutative.

Proof We already know from Theorem 7.3 that h is a bijection.

Let F be a fusion system on S. By Lemma 7.12(c), we have g(f(F)) = g(ω_{F}) = Fix(ω_{F}) = S(F) = h(F). Therefore, g ∘ f = h. Since h is bijective, f is injective and g is surjective.

It suffices now to show that f is surjective. So let ω ∈ Idem(S) and set F := h^{−1}(g(ω)). Then Fix(ω) = g(ω) = h(F) = S(F) and ω satisfies the Properties (a)–(c) in Lemma 7.12. By the uniqueness statement in Lemma 7.12, we obtain ω = ω_{F} = f(F). This shows that the map f is surjective and the proof is complete.

7.16 Remark In [RS] Ragnarsson and Stancu proved that there is a bijection between the set Fus^{∗}(S) of saturated fusion systems on S and the set Idem^{∗}(S) := ℤ_{(p)}B^{Δ}(S, S) ∩ Idem(S). The correspondence associates to ω ∈ Idem^{∗}(S) the fusion system F satisfying S(F) = Fix(ω). Thus, the bijection f: Fus(S) ⟶^{∼} Idem(S) is an extension of their bijection Fus^{∗}(S) → Idem^{∗}(S). Note that Equation (35) yields a very explicit description of ω_{F} in terms of the fixed points of ω_{F} that was not apparent in [RS].

If F is a saturated fusion system then

σ̃^{F}(ω_{F}) ∈ ∏_{P ∈F̃} End_{ℤ_{(p)}Out_{F} (P)}(ℤ_{(p)}\overline{Hom}_{F}(P, S)) .

In fact, the ℚ-algebra isomorphism from Theorem 5.7 (with R = ℚ) restricts to the injective ℤ_{(p)}-algebra homomorphism in (22) with R = ℤ_{(p)}.

The previous remark and our description of σ̃^{F}(ω_{F}) in 7.11 lead to the following corollary.

7.17 Corollary Let F be a saturated fusion system on S, and let P ⩽ S. Then

\frac{|S|}{|Hom_{F}(P, S)||C_{S}(P)|} ∈ ℤ_{(p)}.  (37)

One might ask whether the converse of Corollary 7.17 is also true, that is, whether every fusion system F on S satisfying the Condition (37) for every P ⩽ S has to be saturated. However, this is not the case, as the following example shows.

7.18 Example Suppose that p = 2 and that S = ⟨x, y | x^{2} = y^{2} = 1, xy = yx⟩ is a Klein four-group. Let F be the fusion system on S, defined in Examples 7.4(c). Hence, Aut_{F}(S) = 1 and, for every Q < S, we have Hom_{F}(Q, S) = {α|_{Q} | α ∈ A} where 𝔄_{3} ≅ A ⊴ Aut(S) ≅ 𝔖_{3}. We set Q_{1} := ⟨x⟩, Q_{2} := ⟨y⟩, and Q_{3} := ⟨xy⟩. Then, for i, j ∈ {1, 2, 3}, we have |Hom_{F}(Q_{j}, Q_{i})| = 1 and therefore |Hom_{F}(Q_{i}, S)| = 3. Moreover, Hom_{F}(S, S) = {id_{S}}, and Hom_{F}(1, 1) = {id_{1}}. Thus, for Q ⩽ S, we have

\frac{|S|}{|Hom_{F}(Q, S)||C_{S}(Q)|} = \begin{cases}1, & if Q ∈ \{1, S\}, \\ \frac{1}{3}, & if Q ∈ \{Q_{1}, Q_{2}, Q_{3}\}. \end{cases}

Hence F satisfies (37) in Corollary 7.17, but F is not saturated, as we have already seen in Examples 7.4(c).

Although fusion systems on S satisfying Condition (37) in Corollary 7.17 need not be saturated, they still do share some properties with saturated fusion systems on S, as for instance Part (b) of the following lemma, cf. [L, Prop. 2.5]. We do not know whether Part (a) of the next lemma was observed before for saturated fusion systems. In the following, for every natural number n, we denote by n_{p} the highest p-power dividing n.

7.19 Proposition Let F be a fusion system on S satisfying (37) in Corollary 7.17 and let P ⩽ S.

(a) The number f_{F}(P) of S-conjugacy classes of fully F-normalized subgroups of S that are Fisomorphic to P is not divisible by p.

(b) The subgroup P is fully F-normalized if and only if P is fully F-centralized and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

Proof Let [P]_{F} denote the set of subgroups of S that are F-isomorphic to P. Moreover, let {P_{1}, …, P_{n}} be a transversal for the S-conjugacy classes of subgroups in [P]_{F} and assume that P_{1} is fully F-normalized. We consider the function

c: [P]_{F} → ℕ , Q ↦ |C_{S}(Q)| · |Hom_{F}(Q, S)|_{p} .

Note that the positive integer c(Q) is a p-power and that c(Q) ⩽ |S| for all Q ∈ [P]_{F}, by the condition in (37). Since the number |Hom_{F}(Q, S)| is independent of Q ∈ [P]_{F}, the function c takes its maximum value precisely at the fully F-centralized subgroups in [P]_{F}. Let Q ∈ [P]_{F} be arbitrary. Then

|Hom_{F}(Q, S)| = |Aut_{F}(Q)| · |[P]_{F}| = |Out_{F}(Q)| · |N_{S}(Q) : C_{S}(Q)| · |[P]_{F}|

and

|[P]_{F}| = ∑_{i=1}^{n} |S : N_{S}(P_{i})| = |S : N_{S}(P_{1})|(f_{F}(P) + pm_{P})

for some integer m_{P}. Therefore,

c(Q) = |Out_{F}(Q)|_{p} · \frac{|N_{S}(Q)|}{|N_{S}(P_{1})|} · |S| · (f_{F}(P) + pm_{P})_{p} .  (38)

Now assume that Q is fully F-normalized. Then the last equation becomes

c(Q) = |Out_{F}(Q)|_{p} · (f_{F}(P) + pm_{P})_{p} · |S| .

Since c(Q) ⩽ |S|, we obtain |Out_{F}(Q)|_{p} = 1, i.e., Aut_{S}(Q) ∈ Syl_{p}(Aut_{F}(Q)), and p ∤ f_{F}(P). Moreover, we obtain c(Q) = |S| and, therefore, that the maximal value of c is equal to |S| and that it is achieved at Q. Thus, Q is fully F-centralized. This shows Part (a) and one direction of Part (b).

Conversely, assume that Q ∈ [P]_{F} is fully F-centralized and that |Out_{F}(Q)|_{p} = 1. Then c(Q) = |S|, the maximal value of c. On the other hand, by Equation (38) and by Part (a), we have

c(Q) = \frac{|N_{S}(Q)|}{|N_{S}(P_{1})|} · |S| .

This implies |N_{S}(Q)| = |N_{S}(P_{1})|, and Q is fully F-normalized. This completes the proof of Part (b).

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