The center of a Green biset functor

Serge Bouc and Nadia Romero

Abstract: For a Green biset functor A, we define the commutant and the center of A and we study some of their properties and their relationship. This leads in particular to the main application of these constructions: the possibility of splitting the category of A-modules as a direct product of smaller abelian categories. We give explicit examples of such decompositions for some classical shifted representation functors. These constructions are inspired by similar ones for Mackey functors for a fixed finite group. Keywords: biset functor, Green functor, functor category, center. AMS MSC (2010): 16Y99, 18D10, 18D15, 20J15.

Introduction

This paper is devoted to the construction of two analogues of the center of a ring in the realm of Green biset functors, that is “biset functors with a compatible ring structure”. For a Green biset functor A, we present the commutant CA of A, defined from a commutation property, and the center ZA of A, defined from the structure of the category of A-modules. Both CA and ZA are again Green biset functors. These constructions are inspired by similar ones for Mackey functors for a fixed finite group made in Chapter 12 of [1].

The commutant CA is always a Green biset subfunctor of A, and we say that A is commutative if CA = A. Most of the classical representation functors are commutative in that sense. One of them plays a fundamental - we should say initial - role, namely the Burnside biset functor B, as biset functors are nothing but modules over the Burnside functor. An important feature of the category B-Mod is its monoidal structure: given two biset functors M and N, one can build their tensor product M ⊗N, which is again a biset functor. For this tensor product, the category B-Mod becomes a symmetric monoidal category, and a Green biset functor A is a monoid object in B-Mod.

More generally, for any Green biset functor A, we consider the category A-Mod of A-modules. We will make a heavy use of the equivalence of categories between A-Mod and the category of linear representations of the category P_{A} introduced in Chapter 8 of [2] (see also Definition 9 below), which has finite groups as objects, and in which the set of morphisms from G to H is equal to A(H × G). The category P_{B} associated to the Burnside functor is precisely the biset category of finite groups. It is a symmetric monoidal category (for the product given by the direct product of groups), and this monoidal structure induces via Day convolution ([6]) the monoidal structure of B-Mod mentioned before.

A natural question is then to know when the cartesian product of groups endows the category P_{A} with a symmetric monoidal structure, and we show that this is the case precisely when A is commutative. In this case the category A-Mod also becomes a symmetric monoidal category.

Even though the definition of the center ZA of a Green biset functor A is fairly natural, showing that it is endowed with a Green biset functor structure (even showing that ZA(G) is indeed a set!) is not an easy task, it requires several and sometimes rather nasty computations. On the other hand, one of the rewarding consequences of this laborious process is that we obtain a description of ZA(G) in terms also of a commutation condition, this time on the morphisms of P_{A}. Once we have that ZA is indeed a Green biset functor, we show some nice properties of it, for instance that there is an injective morphism of Green biset functors from CA to ZA. This implies in particular that ZA is a CA-module. We show also that in case A is commutative, it is a direct summand of ZA as A-modules.

In the last section we work within ZA(1), which, as we will see, coincides with the center of the category A-Mod. Any decomposition of the identity element of ZA(1) as a sum of orthogonal idempotents, fulfilling certain finiteness conditions, allows us to decompose A-Mod as a direct product of smaller abelian categories. Moreover, since CA(1) is generally easier to compute than ZA(1), we can also use similar decompositions of the identity element of CA(1) instead, thanks to the inclusion CA ↪ ZA. We give then a series of explicit examples. The first one is the Burnside p-biset functor A = RB_{p} over a ring R where the prime p is invertible. In this case, we obtain an infinite series of orthogonal idempotents in ZA(1), and this shows in particular that ZA can be much bigger that CA. Next we consider some classical representation functors, shifted by some fixed finite group L via the Yoneda-Dress functor. In this series of examples, we will see that the smaller abelian categories obtained in the decomposition are also module categories for Green biset functors arising from the functor A, the shifting group L, and the above-mentioned idempotents.

1 Preliminaries

Throughout the paper, we fix a commutative unital ring R. All referred groups will be finite. The center of a ring S will be denoted by Z(S).

1.1 Green biset functors

The biset category over R will be denoted by RC. Recall that its objects are all finite groups, and that for finite groups G and H, the hom-set Hom_{RC}(G, H) is RB(H, G) = R ⊗_{ℤ} B(H, G), where B(H, G) is the Grothendieck group of the category of finite (H, G)-bisets. The composition of morphisms in RC is induced by R-bilinearity from the composition of bisets, which will be denoted by ∘.

We fix a non-empty class D of finite groups closed under subquotients and cartesian products, and a set D of representatives of isomorphism classes of groups in D. We denote by RD the full subcategory of RC consisting of groups in D, so in particular RD is a replete subcategory of RC, in the sense of [2], Definition 4.1.7. The category of biset functors, i.e. the category of R-linear functors from RC to the category R-Mod of all R-modules, will be denoted by Fun_{R}. The category Fun_{D,R} of D-biset functors is the category of R-linear functors from RD to R-Mod.

A Green D-biset functor is defined as a monoid in Fun_{D,R} (see Definition 8.5.1 in [2]). This is equivalent to the following definition:

Definition 1. A D-biset functor A is a Green D-biset functor if it is equipped with bilinear products A(G) × A(H) → A(G × H) denoted by (a, b) ↦ a × b, for groups G, H in D, and an element ε_{A} ∈ A(1), satisfying the following conditions:

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

(a × b) × c = A (Iso_{G×(H×K)}^{(G×H)×K}) (a × (b × c)).

2. Identity element. Let G be a group in D and consider the canonical isomorphisms 1 × G → G and G × 1 → G. Then for any a ∈ A(G)

a = A (Iso_{1×G}^{G}) (ε_{A} × a) = A (Iso_{G×1}^{G}) (a × ε_{A}).

3. Functoriality. If φ : G → G′ and ψ : H → H′ are morphisms in RD, then for any a ∈ A(G) and b ∈ A(H)

A(φ × ψ)(a × b) = A(φ)(a) × A(ψ)(b).

The identity element of A will be denoted simply by ε if there is no risk of confusion. If A and C are Green D-biset functors, a morphism of Green D-biset functors from

A to C is a natural transformation f : A → C such that f_{H×K}(a × b) = f_{H}(a) × f_{K}(b) for any groups H and K in D and any a ∈ A(H), b ∈ A(K), and such that f_{1}(ε_{A}) = ε_{C}. We will denote by Green_{D,R} the category of Green D-biset functors with morphisms given in this way.

There is an equivalent way of defining a Green biset functor, as we see in the next lemma.

Definition 2. A D-biset functor A is a Green D-biset functor provided that for each group H in D, the R-module A(H) is an R-algebra with unity that satisfies the following. If K and G are groups in D and K → G is a group homomorphism, then:

1. For the (K, G)-biset G, which we denote by G_{r}, the morphism A(G_{r}) is a ring homomorphism.

2. For the (G, K)-biset G, denoted by G_{l}, the morphism A(G_{l}) satisfies the Frobenius identities for all b ∈ A(G) and a ∈ A(K),

A(G_{l})(a) · b = A(G_{l})(a · A(G_{r})(b))
b · A(G_{l})(a) = A(G_{l})(A(G_{r})(b) · a),

where · denotes the ring product on A(G), resp. A(K).

Lemma 3 (Lema 4.2.3 in [12]). The two previous definitions are equivalent. Starting by Definition 1, the ring structure of A(H) is given by

a · b = A (Iso_{Δ(H)}^{H} ∘ Res_{Δ(H)}^{H×H}) (a × b),

for a and b in A(H), with the unity given by A(Inf_{1}^{H})(ε). Conversely, starting by Definition 2, the product of A(G) × A(H) → A(G × H) is given by

a × b = A(Inf_{G}^{G×H})(a) · A(Inf_{H}^{G×H})(b)

for a ∈ A(G) and b ∈ A(H), with the identity element given by the unity of A(1).

In what follows, the ring structure on A(G) will be understood as (A(G), ·).

Observe that in the case of A(1), the product × : A(1) × A(1) → A(1) coincides with the ring product · : A(1) × A(1) → A(1), up to identification of 1 × 1 with 1, and the unity coincides with the identity element.

Remark 4. A morphism of Green D-biset functors f : A → C induces, in each component G, a unital ring homomorphism f_{G} : A(G) → C(G). Conversely, a morphism of biset functors f : A → C such that f_{G} is a unital ring homomorphism for every G in D, is a morphism of Green D-biset functors.

Example 5. Classical examples of Green biset functors are the following:

• The Burnside functor B. The Burnside group of a finite group G is known to define a biset functor. The cross product of sets defines the bilinear products B(G) × B(H) → B(G × H) that make B a Green biset functor. The functor B can also be considered with coefficients in R, and denoted by RB = R⊗_{ℤ}B( ). It is shown in Proposition 8.6.1 of [2] that RB is an initial object in Green_{D,R}. More precisely, for a D-Green biset functor A, the unique morphism of Green functors υ : RB → A is defined at G ∈ D as the linear map υ_{G} sending a G-set X to A(_{G}X_{1})(ε_{A}), where _{G}X_{1} is the set X viewed as a (G, 1)-biset.

• The functor of 𝕂-linear representations, R_{𝕂}, where 𝕂 is a field of characteristic 0. That is, the functor which sends a finite group G to the Grothendieck group R_{𝕂}(G) of the category of finitely generated 𝕂G-modules. Also known to be a biset functor, it has a Green biset functor structure given by the tensor product over 𝕂. We will consider the scalar extension 𝔽R_{𝕂} = 𝔽 ⊗_{ℤ} R_{𝕂}( ), where 𝔽 is a field of characteristic 0.

• The functor of p-permutation representations pp_{k}, for k an algebraically closed field of positive characteristic p. This is the functor sending a finite group G to the Grothendieck group pp_{k}(G) of the category of finitely generated p-permutation kG-modules (also known as trivial source modules), for relations given by direct sum decompositions. The biset functor pp_{k} is a Green biset functor with products given by the tensor product over the field k. When considering coefficients for this functor, we will assume that 𝔽 is a field of characteristic 0 containing all the p′-roots of unity, and we write similarly 𝔽pp_{k} = 𝔽 ⊗_{ℤ} pp_{k}( ).

When p is a prime number, and D is the full subcategory of C consisting of finite p-groups, the D-biset functors are simply called p-biset functors, and their category is denoted by Fun_{p,R}. Similarly, the Green D-biset functors will be called Green p-biset functors, and their category will be denoted by Green_{p,R}.

An important element in what follows will be the Yoneda-Dress construction. We recall some of the basic results about it, more details can be found in Section 8.2 of [2]. If G is a fixed group in D and F is a D-biset functor, then the Yoneda-Dress construction of F at G is D-the biset functor F_{G} that sends each group K in D to F (K × G). The morphism F_{G}(φ) : F (H × G) → F (K × G) associated to an element φ in RB(K, H) is defined as F (φ × G). In turn F (φ × G) is defined by R-bilinearity from the case where φ is represented by a (K, H)-biset U: in this case φ × G denotes the cartesian product U ×G, endowed with its obvious (K ×G, H ×G)-biset structure. We also call F_{G} the functor shifted by G.

If f : F → T is a morphism of D-biset functors, then f_{G} : F_{G} → T_{G} is defined in its component K as (f_{G})_{K} = f_{K×G}. It is shown in Proposition 8.2.7 of [2] that this construction is a self-adjoint exact R-linear endofunctor of Fun_{D,R}.

When A is a Green D-biset functor, the particular shifted functor A_{G} is also a Green biset functor (Lemma 4.4 in [13]) with product given in the following way:

A_{G}(H) × A_{G}(K) → A_{G}(H × K) (a, b) ↦ A(α)(a × b)

where α is the biset Iso_{D}^{H×K×G}Res_{D}^{H×G×K×G} and D ≅ H × K × G is the subgroup of H × G × K × G consisting of elements of the form (h, g, k, g). Usually, by an abuse of notation, we will denote this biset simply by Res_{H×K×Δ(G)}^{H×G×K×G}. To avoid confusion with the product × of A we denote the product of A_{G} by ×^{d}, where the exponent d stands for diagonal.

Remark 6. It is not hard to show that the ring structure of Lemma 3 in A_{G}(H) induced by the product ×^{d} of A_{G} coincides with the ring structure of A(H × G) induced by the product × of A. So there is no risk of confusion when talking about the ring A_{G}(H), since the ring structure we are considering is unique. In particular, the isomorphism A_{G}(1) ≅ A(G) is an isomorphism of rings.

1.2 A-modules

Definition 7 (Definition 8.5.5 in [2]). Given a Green biset functor A, a left A-module M is defined as a biset functor, together with bilinear products

× : A(G) × M(H) ⟶ M(G × H)

for every pair of groups G and H in D, that satisfy analogous conditions to those of Definition 1. The notion of right A-module is defined similarly, from bilinear products M(G) × A(H) ⟶ M(G × H).

We use the same notation × for the product of A and the action of A on A-modules, as long as there is no risk of confusion.

If M and N are A-modules, a morphism of A-modules is defined as a morphism of biset functors f : M → N such that f_{G×H}(a × m) = a × f_{H}(m) for all groups G and H in D, a ∈ A(G) and m ∈ M(H). With these morphisms, the A-modules form a category, denoted by A-Mod. The category A-Mod is an abelian subcategory of Fun_{D,R}. Actually, the direct sum of biset functors is as well the direct sum of Amodules. Also, the kernel, the image and the cokernel of a morphism of A-modules are A-modules. Basic results on modules over a ring can be stated for A-modules.

In particular, a left (resp. right) ideal of a Green D-biset functor A is an Asubmodule of the left (resp. right) A-module A. A two sided ideal of A is a left ideal which is also a right ideal.

Example 8. If A is the Burnside functor RB, then an A-module is nothing but a biset functor with values in R-Mod.

From Proposition 8.6.1 of [2], or Proposition 2.11 of [13], an equivalent way of defining an A-module is as an R-linear functor from the category P_{A} to R-Mod, the category P_{A} being defined next.

Definition 9. Let A be a D-Green functor over R. The category P_{A} is defined in the following way:

• The objects of P_{A} are all finite groups in D.

• If G and H are groups in D, then Hom_{P_{A}}(H, G) = A(G × H).

• Let H, G and K be groups in D. The composition of β ∈ A(H × G) and α ∈ A(G × K) in P_{A} is the following:

β ∘ α = A (Def_{H×K}^{H×Δ(G)×K} ∘ Res_{H×Δ(G)×K}^{H×G×G×K}) (β × α).

• For a group G in D, the identity morphism ε_{G} of G in P_{A} is A(Ind_{Δ(G)}^{G×G}∘Inf_{1}^{Δ(G)})(ε).

Observe that the biset Def_{H×K}^{H×Δ(G)×K} ∘ Res_{H×Δ(G)×K}^{H×G×G×K} can also be written as

H × (Def_{1}^{Δ(G)} ∘ Res_{Δ(G)}^{G×G}) × K.

Another way of denoting the (1, G × G)-biset Def_{1}^{Δ(G)} ∘ Res_{Δ(G)}^{G×G} is as \overleftarrow{G}. In some cases it will be more convenient to use this notation.

The category P_{A} is essentially small, as it has a skeleton consisting of our chosen set D of representatives of isomorphism classes of groups in D. Hence, the category Fun_{R}(P_{A}, R-Mod) of R-linear functors is an abelian category. The above-mentioned equivalence of categories between A-Mod and Fun_{R}(P_{A}, R-Mod) is built as follows:

• If M is an A-module, let M̃ ∈ Fun_{R}(P_{A}, R-Mod) be the functor defined by:

1. For G ∈ D, we have M̃(G) = M(G).

2. For G, H ∈ D and a morphism α ∈ A(H × G) from G to H in P_{A}, the map α̃ : M̃(G) → M̃(H) is the map sending

m ∈ M(G) ↦ M(H × \overleftarrow{G})(α × m).

• Conversely if F ∈ Fun_{R}(P_{A}, R-Mod), let F̂ be the A-module defined by:

1. If G ∈ D, then F̂(G) = F (G).

2. For G, H ∈ D, a ∈ A(G) and m ∈ F (H), set

a × m = F (A(Ind_{G×Δ(H)}^{G×H×H}Inf_{G}^{G×H})(a))(m) ∈ F (G × H),

where A(Ind_{G×Δ(H)}^{G×H×H}Inf_{G}^{G×H})(a) ∈ A(G × H × H) is viewed as a morphism from H to G × H in the category P_{A}.

Then M ↦ M̃ and F ↦ F̂ are well defined equivalences of categories between A-Mod and Fun_{R}(P_{A}, R-Mod), inverse to each other.

Finally, we extend to A-modules our previous definition of the Yoneda-Dress construction.

Definition 10. Let A be a Green D-biset functor. For L ∈ D, consider the assignment ρ_{L} = − × L defined for objects G, H of P_{A} and morphisms α ∈ Hom_{P_{A}}(G, H) = A(H × G) by:

\{ \begin{matrix}ρ_{L}(G) & = & G × L \\ ρ_{L}(α) & = & α × L := Iso_{H×G×L×L}^{H×L×G×L}(α × υ_{A,L×L}(L)) \end{matrix}

where υ_{A,L×L}(L) is the image in A(L × L) of the identity (L, L)-biset L under the canonical morphism υ_{A,L×L}, and the isomorphism H × G × L × L → H × L × G × L maps (h, g, l_{1}, l_{2}) to (h, l_{1}, g, l_{2}).

A straightforward computation shows that

ρ_{L}(α) = A(Ind_{H×G×L}^{H×L×G×L}Inf_{H×G}^{H×G×L})(α),

and this form may be more convenient for calculations. Here H × G × L embeds in H × L × G × L via the map (h, g, l) ↦ (h, l, g, l), and maps surjectively onto H × G via (h, g, l) ↦ (h, g).

It is easy to check that ρ_{L} is in fact an endofunctor of P_{A}, called the (right) L-shift. It induces by precomposition an endofunctor of the category Fun_{R}(P_{A}, R-Mod), that is, up to the above equivalence of categories, an endofunctor of the category A-Mod, which can be described as follows. It maps an A-module M to the shifted D-biset functor M_{L}, endowed with the following product: for G, H ∈ D, α ∈ A(H) and m ∈ M_{L}(G) = M(G×L), the element α×m of M_{L}(H ×G) = M(H ×G×L) is simply the element α × m obtained from the A-module structure of M.

This endofunctor M ↦ M_{L} of the category A-Mod will be denoted by Id_{L}. It is the Yoneda-Dress construction for A-modules. Remark 11. For L ∈ D, there is another obvious endofunctor λ_{L} = L×− of P_{A} defined for objects G, H of P_{A} and morphisms α ∈ Hom_{P_{A}}(G, H) = A(H × G) by

\{ \begin{matrix}λ_{L}(G) & = & L × G \\ λ_{L}(α) & = & L × α := Iso_{L×L×H×G}^{L×H×L×G}(υ_{A,L×L}(L) × α) \end{matrix}

where the isomorphism L×L×H ×G → L×H ×L×G maps (l_{1}, l_{2}, h, g) to (l_{1}, h, l_{2}, g). As before, it is easy to see that L × α = A(Ind_{L×H×G}^{L×H×L×G}Inf_{H×G}^{L×H×G})(α).

It is then natural to ask if the assignment × : P_{A} × P_{A} → P_{A} sending (G, K) to G × K and (α, β) ∈ A(H × G) × A(L × K) to (α × L) ∘ (G × β) ∈ A(H × L × G × K) is a functor. We will answer this question at the end of Section 3 (Corollary 26).

2 Adjoint functors

Let A and C be Green D-biset functors. A morphism f : A → C of Green D-biset functors induces an obvious functor P_{f} : P_{A} → P_{C}, which is the identity on objects, and maps α ∈ Hom_{P_{A}}(G, H) = A(H × G) to f_{H×G}(α) ∈ C(H × G) = Hom_{P_{C}}(G, H).

Let L be a fixed group in D. The inflation morphism Inf_{L} : A → A_{L}, introduced in [8], is the morphism of Green biset functors defined for each G ∈ D and each α ∈ A(G) by Inf_{L}(α) = A(Inf_{G}^{G×L})(α) ∈ A(G × L) = A_{L}(G), where G is identified with (G × L)/({1} × L). The corresponding functor P_{A} → P_{A_{L}} will be denoted by ψ_{L}. Explicitely, for each G ∈ D, we have ψ_{L}(G) = G, and for a morphism α ∈ A(H × G), we have

ψ_{L}(α) = A(Inf_{H×G}^{H×G×L})(α) ∈ A(H × G × L) = A_{L}(H × G) = Hom_{P_{A_{L}}}(ψ_{L}(G), ψ_{L}(H)).

We introduce another functor θ_{L} : P_{A_{L}} → P_{A}, defined as follows: for an object G of P_{A_{L}}, wet set θ_{L}(G) = G × L, wiewed as an object of P_{A}. For a morphism α ∈ Hom_{P_{A_{L}}}(G, H) = A_{L}(H × G) = A(H × G × L), we define

θ_{L}(α) = A(Ind_{H×G×L}^{H×L×G×L})(α) ∈ A(H × L × G × L) = Hom_{P_{A}}(θ_{L}(G), θ_{L}(H)),

where H × G × L is viewed as a subgroup of H × L × G × L via the injective group homomorphism (h, g, l) ∈ H × G × L ↦ (h, l, g, l) ∈ H × L × G × L.

Notation 12. In what follows, we will use a convenient abuse of notation, and generally drop the symbols × of cartesian products of groups, writing e.g. HLGL instead of H × L × G × L.

Theorem 13. 1. ψ_{L} is an R-linear functor from P_{A} to P_{A_{L}}.

2. θ_{L} is an R-linear functor from P_{A_{L}} to P_{A}.

3. The functors ψ_{L} and θ_{L} are left and right adjoint to one another. In other words, for any G and H in D, there are R-module isomorphisms

Hom_{P_{A_{L}}}(G, ψ_{L}(H)) ≅ Hom_{P_{A}}(θ_{L}(G), H)
Hom_{P_{A_{L}}}(ψ_{L}(G), H) ≅ Hom_{P_{A}}(G, θ_{L}(H))

which are natural in G and H.

Proof. Point (1) is clear, since the functor ψ_{L} is built from a morphism of Green biset functors Inf_{L} : A → A_{L}.

To prove (2), let G, H, K ∈ D. If α ∈ A_{L}(HG) and β ∈ A_{L}(KH), then

θ_{L}(β) ∘ θ_{L}(α) = A(Def_{KLGL}^{KLHLGL}Res_{KLHLGL}^{KLHLHLGL})(A(Ind_{KHL}^{KLHL})(β) × A(Ind_{HGL}^{HLGL})(α))
= A(Def_{KLGL}^{KLHLGL}Res_{KLHLGL}^{KLHLHLGL}Ind_{KHLHGL}^{KLHLHLGL})(β × α).

In the restriction Res_{KLHLGL}^{KLHLHLGL}, the group KLHLGL maps into KLHLHLGL via

f : (k, l_{1}, h, l_{2}, g, l_{3}) ∈ KLHLGL ↦ (k, l_{1}, h, l_{2}, h, l_{2}, g, l_{3}) ∈ KLHLHLGL,

and in the induction Ind_{KHLHGL}^{KLHLHLGL}, the group KHLHGL maps into KLHLHLGL via

f′ : (k′, h′_{1}, l′_{1}, h′_{2}, g, l′_{2}) ∈ KHLHGL ↦ (k′, l′_{1}, h′_{1}, l′_{1}, h′_{2}, l′_{2}, g′, l′_{2}) ∈ KLHLHLGL.

Then one checks readily that Im(f)Im(f′) = KLHLHLGL, and that Im(f) ∩ Im(f′) is isomorphic to KHGL. Hence by the Mackey formula, there is an isomorphism of bisets

Res_{KLHLGL}^{KLHLHLGL}Ind_{KHLHGL}^{KLHLHLGL} ≅ Ind_{KHGL}^{KLHLGL}Res_{KHGL}^{KHLHGL},

where in Ind_{KHGL}^{KLHLGL}, the inclusion KHGL ↪ KLHLGL is (k, h, g, l) ↦ (k, l, h, l, g, l), and in Res_{KHGL}^{KHLHGL}, the inclusion KHGL ↪ KHLHGL is (k, h, g, l) ↦ (k, h, l, h, g, l).

Now in the deflation Def_{KLGL}^{KLHLGL}, the group KLHLGL maps onto KLGL via (k, l_{1}, h, l_{2}, g, l_{3}) ↦ (k, l_{1}, g, l_{3}). It follows that there is an isomorphism of bisets

Def_{KLGL}^{KLHLGL}Ind_{KHGL}^{KLHLGL} ≅ Ind_{KGL}^{KLGL}Def_{KGL}^{KHGL},

which gives

θ_{L}(β) ∘ θ_{L}(α) = A(Ind_{KGL}^{KLGL})A(Def_{KGL}^{KHGL}Res_{KHGL}^{KHLHGL})(β × α)
= A(Ind_{KGL}^{KLGL})A_{L}(Def_{KG}^{KHG}Res_{KHG}^{KHHG})A(Res_{KHHGL}^{KHLHGL})(β × α)
= A(Ind_{KGL}^{KLGL})A_{L}(Def_{KG}^{KHG}Res_{KHG}^{KHHG})(β ×^{d} α)
= A(Ind_{KGL}^{KLGL})(β ∘^{d} α) = θ_{L}(β ∘^{d} α).

This shows that θ_{L} is compatible with composition of morphisms. A straightforward computation shows that it maps identity morphisms to identity morphisms. This completes the proof of Assertion 2, since θ_{L} is obviously R-linear.

(3) Since the complete proof of Assertion 3 demands the verification of many technical details, we only include the full proof that θ_{L} is left adjoint to ψ_{L}. We next simply give the description of the bijection involved in the other direction, and leave the corresponding verifications to the reader.

For G and H in D, we have

Hom_{P_{A_{L}}}(G, ψ_{L}(H)) = A_{L}(ψ_{L}(H)G) = A(HGL) and Hom_{P_{A}}(θ_{L}(G), H) = A(HGL),

so an obvious candidate for an isomorphism Hom_{P_{A_{L}}}(G, ψ_{L}(H)) → Hom_{P_{A}}(θ_{L}(G), H) is the identity map of A(HGL). To avoid confusion, for α ∈ Hom_{P_{A_{L}}}(G, ψ_{L}(H)), we denote by α̃ the element α viewed as an element of Hom_{P_{A}}(θ_{L}(G), H).

We now check that the map α ↦ α̃ is natural in G and H. For naturality in G, if G′ ∈ D and u ∈ Hom_{A_{L}}(G′, G), we have the diagrams

\xymatrix{G \ar[r]^{α} & ψ_{L}(H) & θ_{L}(G) \ar[r]^{α̃} & H \\ G′ \ar[u]^{u} \ar[ur]_{αu} &  & θ_{L}(G′) \ar[u]^{θ_{L}(u)} \ar[ur]_{{ \atop α∘^{d}u}}}

and we have to show that the right-hand side diagram is commutative, i.e. that { \atop α ∘^{d} u} = α̃ ∘ θ_{L}(u). But

α̃ ∘ θ_{L}(u) = α ∘ A(Ind_{GG′L}^{GLG′L})(u)
= A(Def_{HG′L}^{HGLG′L}Res_{HGLG′L}^{HGLGLG′L})(α × A(Ind_{GG′L}^{GLG′L})(u))
= A(Def_{HG′L}^{HGLG′L}Res_{HGLG′L}^{HGLGLG′L}Ind_{HGLGG′L}^{HGLGLG′L})(α × u).

In the restriction Res_{HGLG′L}^{HGLGLG′L}, the inclusion HGLG′L ↪ HGLGLG′L is the map

f : (h, g, l_{1}, g′, l_{2}) ∈ HGLG′L ↦ (h, g, l_{1}, g, l_{1}, g′, l_{2}) ∈ HGLGLG′L,

and in the induction Ind_{HGLGG′L}^{HGLGLG′L}, the inclusion HGLGG′L ↪ HGLGLG′L is the map

f′ : (η, γ_{1}, λ_{1}, γ_{2}, γ′, λ_{2}) ∈ HGLGG′L ↦ (η, γ_{1}, λ_{1}, γ_{2}, λ_{2}, γ′, λ_{2}) ∈ HGLGLG′L.

Then clearly Im(f)Im(f′) = HGLGLG′L, and Im(f) ∩ Im(f′) ≅ HGG′L. By the Mackey formula, this gives an isomorphism of bisets

Res_{HGLG′L}^{HGLGLG′L}Ind_{HGLGG′L}^{HGLGLG′L} ≅ Ind_{HGG′L}^{HGLG′L}Res_{HGG′L}^{HGLGG′L},

where, in Ind_{HGG′L}^{HGLG′L}, the inclusion HGG′L ↪ HGLG′L is (h, g, g′, l) ↦ (h, g, l, g′, l), and in Res_{HGG′L}^{HGLGG′L}, the inclusion HGG′L ↪ HGLGG′L is (h, g, g′, l) ↦ (h, g, l, g, g′, l).

Now in Def_{HG′L}^{HGLG′L}, the quotient map HGLG′L → HG′L sends (h, g, l_{1}, g′, l_{2}) to (h, g′, l_{2}), so the image of the subgroup HGG′L is the whole of HG′L. It follows that there is an isomorphism of bisets

Def_{HG′L}^{HGLG′L}Ind_{HGG′L}^{HGLG′L} ≅ Def_{HG′L}^{HGG′L},

which gives finally

α̃ ∘ θ_{L}(u) = A(Def_{HG′L}^{HGG′L}Res_{HGG′L}^{HGLGG′L})(α × u)
= A_{L}(Def_{HG′}^{HGG′})A_{L}(Res_{HGG′}^{HGGG′})A(Res_{HGGG′L}^{HGLGG′L})(α × u)
= A_{L}(Def_{HG′}^{HGG′})A_{L}(Res_{HGG′}^{HGGG′})(α ×^{d} u)
= α ∘^{d} u,

as was to be shown.

We now check that the map α ↦ α̃ is natural in H. If H′ ∈ D and v ∈ Hom_{P_{A}}(H, H′) = A(H′H), we have the diagrams

\xymatrix{A \ar[r]^{α} \ar[dr]_{ψ_{L}(v)∘^{d}α} & ψ_{L}(H) \ar[d]^{ψ_{L}(v)} & θ_{L}(G) \ar[r]^{α̃} \ar[dr]_{{ \atop ψ_{L}(v)∘^{d}α}} & H \ar[d]^{v} \\  & ψ_{L}(H′) &  & H′}

and we have to show that the right-hand side diagram is commutative, i.e. that

{ \atop ψ_{L}(v) ∘^{d} α} = v ∘ α̃. But

ψ_{L}(v) ∘^{d} α = A(Inf_{H′H}^{H′HL})(v) ∘^{d} α = A_{L}(Def_{H′G}^{H′HG}Res_{H′HG}^{H′HHG})(A(Inf_{H′H}^{H′HL})(v) ×^{d} α)
= A(Def_{H′GL}^{H′HGL}Res_{H′HGL}^{H′HHGL})A(Res_{H′HHGL}^{H′HLHGL}Inf_{H′HHGL}^{H′HLHGL})(v × α)
= A(Def_{H′GL}^{H′HGL}Res_{H′HGL}^{H′HLHGL}Inf_{H′HHGL}^{H′HLHGL})(v × α).

In Res_{H′HGL}^{H′HLHGL}, the inclusion H′HGL ↪ H′HLHGL is (h′, h, g, l) ↦ (h′, h, l, h, g, l), and in Inf_{H′HHGL}^{H′HLHGL}, the quotient map H′HLHGL → H′HHGL is (h′, h_{1}, l_{1}, h_{2}, g, l_{2}) ↦ (h′, h_{1}, h_{2}, g, l_{2}). The composition of these two maps sends (h′, h, g, l) to (h′, h, h, g, l), hence it is injective. This gives an isomorphim of bisets

Res_{H′HGL}^{H′HLHGL}Inf_{H′HHGL}^{H′HLHGL} ≅ Res_{H′HGL}^{H′HHGL},

from which we get

ψ_{L}(v) ∘^{d} α = A(Def_{H′GL}^{H′HGL}Res_{H′HGL}^{H′HHGL})(v × α) = v ∘ α̃,

as was to be shown.

Hence the isomorphism α ∈ Hom_{P_{A_{L}}}(G, ψ_{L}(H)) ↦ α̃ ∈ Hom_{P_{A}}(θ_{L}(G), H) is natural in G and H, so θ_{L} is left adjoint to ψ_{L}.

We now describe the bijection implying that θ_{L} is also right adjoint to ψ_{L}. So, for G, H ∈ D, we have to build an isomorphism

α ∈ Hom_{P_{A_{L}}}(ψ_{L}(G), H) ↦ α̂ ∈ Hom_{P_{A}}(G, θ_{L}(H))

of R-modules, natural in G and H. But

Hom_{P_{A_{L}}}(ψ_{L}(G), H) = A_{L}(HG) = A(HGL) and Hom_{P_{A}}(G, θ_{L}(H)) = A(HLG),

so an obvious candidate for the above isomorphism is to set α̂ = A(Iso_{HGL}^{HLG})(α). The verification that this isomorphism is functorial in G and H is similar to the proof of the first adjunction, and we omit it.

Definition 14. Let A be a Green D-biset functor and L ∈ D. We denote by

Ψ_{L} : Fun_{R}(P_{A_{L}}, R-Mod) → Fun_{R}(P_{A}, R-Mod)

the functor induced by precomposition with ψ_{L}, and by

Θ_{L} : Fun_{R}(P_{A}, R-Mod) → Fun_{R}(P_{A_{L}}, R-Mod)

the functor induced by precomposition with θ_{L}.

Proposition 15. The functors Ψ_{L} and Θ_{L} are mutual left and right adjoint functors between Fun_{R}(P_{A_{L}}, R-Mod) and Fun_{R}(P_{A}, R-Mod).

Proof. This follows from Theorem 13, by standard category theory.

Remark 16. Using the above equivalences of categories between Fun_{R}(P_{A}, R-Mod) and A-Mod, and Fun_{R}(P_{A_{L}}, R-Mod) and A_{L}-Mod, we will consider Ψ_{L} as a functor from A_{L}-Mod to A-Mod and Θ_{L} as a functor from A-Mod to A_{L}-Mod. One can check that, from this point of view, if N is an A_{L}-module, then Ψ_{L}(N) is the A-module defined as follows:

• If G ∈ D, then Ψ_{L}(N)(G) = N(G).

• If G, H ∈ D, a ∈ A(G) and v ∈ N(H), then

a × v = A(Inf_{G}^{G×L})(a) ×^{d} v

where ×^{d} denotes the action of A_{L} on N, and A(Inf_{G}^{G×L})(a) ∈ A(G×L) is viewed as an element of A_{L}(G).

Conversely, if M is an A-module, then Θ_{L}(M) is the A_{L}-module defined as follows:

• If G ∈ D, then Θ_{L}(M)(G) = M(G × L).

• If G, H ∈ D, a ∈ A_{L}(G) and m ∈ M(H × L), then

a ×^{d} m = M(Res_{G×H×L}^{G×L×H×L})(a × m),

where a × m is the product of a ∈ A(G × L) and m ∈ M(H × L), and H × G × L is viewed as a subgroup of G × L × H × L via the map (g, h, l) ↦ (g, l, h, l).

Theorem 17. Let A be a Green D-biset functor, and L ∈ D. The endofunctor ρ_{L} of P_{A} is isomorphic to θ_{L} ∘ψ_{L} and so the endofunctor Ψ_{L} ∘Θ_{L} of A-Mod is isomorphic to the Yoneda-Dress functor Id_{L}. In particular Id_{L} is self adjoint.

Proof. One checks readily that ρ_{L} is isomorphic to the composition θ_{L} ∘ ψ_{L}. The other assertions follow by Theorem 13, as the Yoneda-Dress functor Id_{L} is obtained by precomposition with ρ_{L} = − × L.

We observe that the L-shift of the A-module A is the representable functor A(−, L) of the category P_{A}, so it is projective. More generally, the L-shift of the representable functor A(−, X) is the representable functor A(−, L × X). Hence the Yoneda-Dress construction maps a representable functor to a representable functor.

3 The commutant

Definition 18. Let A be a Green D-biset functor.

1. For G, H ∈ D, we say that an element a ∈ A(G) and an element b ∈ A(H)

commute if

a × b = A(Iso_{H×G}^{G×H})(b × a).

2. For a group G in D, we denote by CA(G) the set of elements of A(G) which commute with any element of A(H), for any H ∈ D, i.e.

{a ∈ A(G) | ∀H ∈ D, ∀b ∈ A(H), a × b = A(Iso_{H×G}^{G×H})(b × a)},

and call it the commutant of A at G.

Observe that CA(G) is an R-submodule of A(G), since the product × is bilinear.

Lemma 19. The commutant of A is a Green D-biset subfunctor of A.

Proof. To see it is a biset functor, let Y be a (K, G)-biset for groups K and G in D, and a be in CA(G). If b is in A(H) for a given group H in D, we have

A(Y)(a) × b = A ((Y × H) ∘ Iso_{H×G}^{G×H}) (b × a)

where Y × H is seen as a (K × H, G × H)-biset. If we show that (Y × H) ∘ Iso_{H×G}^{G×H} is isomorphic to Iso_{H×K}^{K×H} ∘ (H × Y), where H × Y is seen as a (H × K, H × G)-biset, the right-hand side of the equality above will be equal to

A(Iso_{H×K}^{K×H})(b × A(Y)(a)),

which is what we want. Now, Iso_{H×G}^{G×H} is the group H × G, seen as a (G × H, H × G)- biset, and Iso_{H×K}^{K×H} is the group H × K, seen as a (K × H, H × K)-biset. So, it is not hard to see that (Y × H) ∘ Iso_{H×G}^{G×H} is isomorphic to Y × H as (K × H, H × G)-biset, where the right action of H × G is given by (y, h)(h_{1}, g_{1}) = (yg_{1}, hh_{1}). Similarly, Iso_{H×K}^{K×H} ∘ (H × Y) is isomorphic to H × Y as (K × H, H × G)-set, where the left action of K × H is given by (k_{1}, h_{1})(h, y) = (h_{1}h, k_{1}y). Hence, it is easy to verify that the map Y × H → H × Y sending (y, h) to (h, y) defines an isomorphism between these two bisets.

To see that CA is closed under the product ×, let a be in CA(G), b be in CA(H) and c be in A(K). We have

a × (b × c) = a × A(Iso_{K×H}^{H×K})(c × b),

which is clearly equal to A(Iso_{G×K×H}^{G×H×K})(a × c × b). Similarly

(a × c) × b = A(Iso_{K×G×H}^{G×K×H})(c × a × b).

Finally, clearly we have

Iso_{G×K×H}^{G×H×K} ∘ Iso_{K×G×H}^{G×K×H} = Iso_{K×G×H}^{G×H×K},

which yields the first equality

(a × b) × c = A(Iso_{K×G×H}^{G×H×K}) (c × (a × b)) .

To finish the proof, it is clear that the identity element ε ∈ A(1) belongs to CA(1).

Corollary 20. Let A be a Green D-biset functor. Then the image of the unique Green biset functor morphism υ_{A} : RB → A is contained in CA.

Proof. Indeed, by uniqueness of υ_{A} and υ_{CA}, the diagram

\xymatrix{ & CA \ar@{^{(}->}[dr] \\ RB \ar@{-->}[ur]^{υ_{CA}} \ar[rr]_{υ_{A}} &  & A}

is commutative.

Definition 21. We will say that a Green D-biset functor is commutative if A = CA.

It is easy to see that CA is commutative. All the examples considered in Example 5 are commutative Green biset functors.

If A is commutative, then clearly A_{G} is commutative for any G. More generally we have the following result.

Proposition 22. Let A be a Green D-biset functor and G ∈ D. Then CA_{G} = (CA)_{G}.

Proof. Observe that CA_{G} and (CA)_{G} are both Green D-biset subfunctors of A_{G}, so to prove they are equal as Green D-biset functors, it suffices to prove that for every group H ∈ D, we have (CA)_{G}(H)=CA_{G}(H).

To prove that (CA)_{G}(H) ⊆ CA_{G}(H), we choose a group K in D, and elements a ∈ (CA)_{G}(H) and b ∈ A_{G}(K). We must prove that

a ×^{d} b = A_{G}(Iso_{K×H}^{H×K})(b ×^{d} a).

We have

a ×^{d} b = A (Res_{H×K×Δ(G)}^{H×G×K×G}) (a × b) and b ×^{d} a = A (Res_{K×H×Δ(G)}^{K×G×H×G}) (b × a).

Now, by definition (CA)_{G}(H) = CA(H × G), so the element a satisfies

a × b = A (Iso_{K×G×H×G}^{H×G×K×G}) (b × a).

Substituting this in the above equation on the left we easily obtain what we wanted.

To prove the reverse inclusion CA_{G}(H) ⊆ (CA)_{G}(H), we now let a ∈ CA_{G}(H) and b ∈ A(K), and consider c = A(Inf_{K}^{K×G})(b). Then we have

a ×^{d} c = A_{G}(Iso_{K×H}^{H×K})(c ×^{d} a),

and clearly

a ×^{d} c = A (Res_{H×K×Δ(G)}^{H×G×K×G} ∘ Inf_{H×G×K}^{H×G×K×G}) (a × b).

But it is easy to see (for example from Section 1.1.3 of [2]) that

Res_{H×K×Δ(G)}^{H×G×K×G} ∘ Inf_{H×G×K}^{H×G×K×G} ≅ Iso_{H×G×K}^{H×K×Δ(G)}.

By doing a similar transformation with c ×^{d} a, and applying the corresponding isomorphisms, we easily obtain what we wanted.

Lemma 23. For any group G in D, the commutant CA(G) is a subring of Z(A(G)).

Proof. Take a ∈ CA(G) and b ∈ A(G), then

a · b = A (Iso_{Δ(G)}^{G} ∘ Res_{Δ(G)}^{G×G}) (a × b)
= A (Iso_{Δ(G)}^{G} ∘ Res_{Δ(G)}^{G×G} ∘ Iso(σ_{G})) (b × a)
= A (Iso_{Δ(G)}^{G} ∘ Res_{Δ(G)}^{G×G}) (b × a) = b · a,

where σ_{G} is the automorphism of G × G switching the components. Since CA(G) and Z(A(G)) have the same ring structure, inherited from the Green D-biset functor structure of A, this shows that CA(G) is a subring of Z(A(G)).

Remark 24. It is not hard to see then that A is a commutative Green biset functor if and only if for every group G, the ring A(G) is a commutative ring.

We now answer the question raised in Remark 11.

Proposition 25. Let G, H, K, L ∈ D. Let α ∈ A(HG) and β ∈ A(LK). Then the square

\xymatrix{G × K \ar[r]^{G×β} \ar[d]_{α×K} & G × L \ar[d]^{α×L} \\ H × K \ar[r]_{H×β} & H × L}

commutes in P_{A} if and only if α and β commute.

Proof. Let u = (α × L) ∘ (G × β). By definition

u = A(Ind_{HGL}^{HLGL}Inf_{HG}^{HGL})(α) ∘ A(Ind_{GLK}^{GLGK}Inf_{LK}^{GLK})(β)

= A(Def_{HLGK}^{HL\underline{GL}GK}Res_{HL\underline{GL}GK}^{HL\underline{GLGL}GK})(A(Ind_{HG\underline{L}}^{H\underline{L}G\underline{L}}Inf_{HG}^{HG\underline{L}})(α) × A(Ind_{\underline{G}LK}^{\underline{G}L\underline{G}K}Inf_{LK}^{\underline{G}LK})(β)),

where the notation Def_{HLGK}^{HL\underline{GL}GK} means the deflation with respect to the underlined normal subgroup, and Res_{HL\underline{GL}GK}^{HL\underline{GLGL}GK} means that the underlined GL in subscript embeds diagonally in the underlined GLGL in superscript. Similarly in Ind_{HG\underline{L}}^{H\underline{L}G\underline{L}}, the group L in subscript embed diagonally in the two underlines copies of L in superscript, and in Inf_{HG}^{HG\underline{L}}, inflation is relative to the underlined L in superscript. Thus

u = A(Def_{HLGK}^{HL\underline{GL}GK}Res_{HL\underline{GL}GK}^{HL\underline{GLGL}GK}Ind_{HG\underline{L}\underline{G}LK}^{H\underline{L}G\underline{L}\underline{G}L\underline{G}K}Inf_{HGLK}^{HG\underline{LG}LK})(α × β)

Standard relations in the composition of bisets (see Section 1.1.3 and Lemma 2.3.26 of [2]) and some tedious but straightforward calculations finally give

u = (α × L) ∘ (G × β) = A(Iso_{HGLK}^{HLGK})(α × β).

Similar calculations show that

(H × β) ∘ (α × K) = A(Iso_{LKHG}^{HLGK})(β × α).

So (H × β) ∘ (α × K) = (α × L) ∘ (G × β) if and only if

β × α = A(Iso_{HLGK}^{LKHG}Iso_{HGLK}^{HLGK})(α × β)
= A(Iso_{HGLK}^{LKHG})(α × β),

that is, if α and β commute.

Corollary 26. The assignment × : P_{A} × P_{A} → P_{A} sending (G, K) to G × K and (α, β) ∈ A(H × G) × A(L × K) to (α × L) ∘ (G × β) ∈ A(H × L × G × K) is a functor if and only if A is commutative. In particular, when A is commutative, this functor × endows P_{A} with a structure of a symmetric monoidal category.

4 The center

Definition 27. Let A be a Green D-biset functor. For a group L in D, we denote by ZA(L) the family of all natural transformations Id → Id_{L} from the identity functor Id : A-Mod → A-Mod to the functor Id_{L}. We call it the center of A at L.

When L is trivial, the functor Id_{L} is isomorphic to the identity functor, hence ZA(1) is the family of natural endotransformations of the identity functor. So our definition is analogous to that of the center of a category (see for example Hoffmann [11] for arbitrary categories, or Section 19 of Butler-Horrocks [5] for abelian categories). Nonetheless, we want to regard this center as a Green D-biset functor, and see its relation with the commutant CA. Our construction is inspired by an analogous construction for Green functors over a fixed finite group in [1] Section 12.2.

4.1 The center as a Green biset functor

Our goal is to show that for each Green D-biset functor A, the assignment L ↦ ZA(L) is itself a Green D-biset functor. For this, we will first give an equivalent description of ZA(L), and then build a Green functor structure on ZA.

Proposition 28. Let A be a Green D-biset functor, and L ∈ D. Then ZA(L) is isomorphic to the family ZA′(L) of natural transformations from the identity functor of P_{A} to ρ_{L}.

Proof. Consider the Yoneda embedding Y_{A} : P_{A} → A-Mod sending L ∈ D to the functor A(−, L). Since Id_{L} preserves the image of Y_{A}, which is a fully faithful functor, we have Id_{L}∘Y_{A} = Y_{A}∘ρ_{L}, and it follows that each element of ZA(L) induces a natural transformation from the identity functor of P_{A}, denoted by ρ_{1}, to ρ_{L}. In this way, we get a linear map f_{L} : ZA(L) → ZA′(L). Conversely, each natural transformation ρ_{1} → ρ_{L} induces a natural transformation Y_{A} → Id_{L} ∘ Y_{A}. Since the image of Y_{A} generates A-Mod, such a natural transformation extends to a natural transformation from the identity functor of A-Mod to Id_{L}. This gives a linear map g_{L} : ZA′(L) → ZA(L). Clearly f_{L} and g_{L} are inverse to one another.

We will now use the previous identification to get a better understanding of ZA(L). Indeed, a natural transformation t from the identity functor of P_{A} to the functor ρ_{L} = − × L = θ_{L}ψ_{L} consists, for each G ∈ D, of a morphism t_{G} : G → G × L in P_{A}, i.e. t_{G} ∈ A(G × L × G), such that for any H ∈ D and any α ∈ A(H × G), the diagram

\xymatrix{G \ar[r]^{t_{G}} \ar[d]_{α} & G × L \ar[d]^{α×L=θ_{L}ψ_{L}(α)} \\ H \ar[r]^{t_{H}} & H × L}  (1)

is commutative in P_{A}.

Lemma 29. Let G, H ∈ D, and α ∈ A(H × G) = Hom_{P_{A}}(G, H). For an element u of A(H × L × G) = Hom_{P_{A}}(G, H × L), let u^{♮} denote the element u, viewed as a morphism from L × G to H in P_{A}. Then for any t ∈ A(G × L × G)

(θ_{L}ψ_{L}(α) ∘ t)^{♮} = α ∘ t^{♮} in A(H × L × G).

Proof. The functor ρ_{L} is a self-adjoint R-linear endofunctor of P_{A}. It follows from the proof of Theorem 13 that for any G, H ∈ P_{A}, the natural bijection given by this adjunction

v ∈ Hom_{P_{A}}(G, ρ_{L}(H)) = A(HLG) → v^{♯} ∈ Hom_{P_{A}}(ρ_{L}(G), H) = A(HGL)

is induced by the isomorphism HLG → HGL switching the components L and G. By adjunction we have commutative diagrams

\xymatrix{G \ar[r]^{t} \ar[dr]_{ρ_{L}(α)∘t} & ρ_{L}(G) \ar[d]^{ρ_{L}(α)} & ρ_{L}(G) \ar[r]^{t^{♯}} \ar[dr]_{(ρ_{L}(α)∘t)^{♯}} & G \ar[d]^{α} \\  & ρ_{L}(H) &  & H}

so (ρ_{L}(α) ∘ t)^{♯} = α ∘ t^{♯}. Since t^{♮} = t^{♯} ∘ τ_{L,G}, where τ_{G,L} : LG → GL is the isomorphism switching G and L, the lemma follows by right composition of the previous equality with τ_{G,L}.

Since v and v^{♮} are actually the same element of A(HLG), for any v ∈ A(HLG), the commutativity in Diagram (1) can be simply written as

α ∘_{G} t_{G} = t_{H} ∘_{H} α,  (2)

where ∘_{G} is the composition A(HG)×A(GLG) → A(HLG), and ∘_{H} is the composition A(HLH) × A(HG) → A(HLG). Thus:

Proposition 30. Let A be a D-green biset functor, and L ∈ D. Then an element t of ZA(L) consists of a family of elements t_{G} ∈ A(GLG), for every G ∈ D, such that α ∘_{G} t_{G} = t_{H} ∘_{H} α, for any G, H in D and α ∈ A(HG). In particular ZA(L) is a set.

Proof. It remains to see that ZA(L) is a set. This is clear, since an element t of ZA(L) is determined by its components t_{G}, where G runs trough our chosen set D of representatives of isomorphism classes of groups in D. More precisely ZA(L) is in one to one correspondence with the set Cr_{A}(L) of sequences of elements (t_{G})_{G∈D} ∈ ∏_{G∈D} A(GLG)

such that the above condition (2) holds for any G, H ∈ D and any α ∈ A(HG).

Proposition 31. 1. Let K, L ∈ D, and β ∈ CA(LK). Then the family of morphisms λ_{G}(β) = G × β : G × K → G × L, for G ∈ D, define a natural transformation of functors ρ_{β} from ρ_{K} to ρ_{L}.

2. Let End_{R}(P_{A}) denote the category of R-linear endofunctors of P_{A}, where morphisms are natural transformations of functors. Then the assignment

\{ \begin{matrix}K ∈ D & ↦ & ρ_{K} ∈ End_{R}(P_{A}) \\ β ∈ CA(LK) & ↦ & (ρ_{β} : ρ_{K} → ρ_{L}) \end{matrix}

is a faithful R-linear functor ρ_{CA} from P_{CA} to End_{R}(P_{A}).

Proof. (1) This follows from Proposition 25.

(2) We have to check that if G, J, K, L ∈ D, if α ∈ A(KJ) and β ∈ A(LK), then (G × β) ∘ (G × α) = G × (β ∘ α) in A(GLGJ), and that if β is the identity element of CA(KK), then G × β is the identity morphism of G × K in P_{A}. This follows from the fact that λ_{G} is a functor.

So we get a functor ρ_{CA} : P_{CA} → End_{R}(P_{A}). Seing that this functor is faithful amounts to seing that if β ∈ CA(LK), then ρ_{β} = 0 if and only if β = 0. But the component 1 × β of ρ_{β} is clearly equal to β, after identification of 1 × K with K and 1 × L with L.

Remark 32. In particular, it follows from Assertion 2 that an isomorphism of groups K → K′ induces an isomorphism of functors ρ_{K} → ρ′_{K}: indeed a group isomorphism φ : K → K′ is represented by a (K′, K)-biset U_{φ} ∈ RB(K′K), hence by an element β_{φ} = υ_{K′K}(U_{φ}) ∈ CA(K′K), by Corollary 20. The corresponding natural transformation ρ_{β_{φ}} is an isomorphism ρ_{K} → ρ′_{K}, with inverse ρ_{β_{φ^{−1}}}.

Lemma 33. Let A be a Green D-biset functor and K, L ∈ D.

1. The endofunctors functor ρ_{L} ∘ ρ_{K} and ρ_{KL} of P_{A} are naturally isomorphic.

2. Let s ∈ ZA(LK), given by the family of elements s_{G} ∈ A(GLKG), for G ∈ D. Then the natural transformation s^{o} : ρ_{K} → ρ_{L} deduced from s : Id → ρ_{K}ρ_{L} by adjunction, is defined by the family of morphisms

s_{G}^{o} = Iso_{GLKG}^{GLGK}(s_{G}) ∈ A(GLGK) = Hom_{P_{A}}(GK, GL).

3. The map s ↦ s^{o} is an isomorphism of R-modules

ZA(LK) → Hom_{End_{R}(P_{A})}(ρ_{K}, ρ_{L}).

Proof. (1) This follows from a straightforward verification.

(2) Indeed, by the proof of Theorem 13, for each G ∈ D, the morphism s_{G} ∈ A(GLKG)

s_{G} : G → GLK = ρ_{K}ρ_{L}(G) = θ_{K}ψ_{K}ρ_{L}(G)

in P_{A} gives by adjunction the morphism

u : ψ_{K}(G) → ψ_{K}ρ_{L}(G),

in P_{A_{K}}, defined as the element u = A(Iso_{GLKG}^{GLGK})(s_{G}) ∈ A_{K}(GLG) = A(GLGK). This element u gives in turn the morphism

v : θ_{K}ψ_{K}(G) = ρ_{K}(G) → ρ_{L}(G)

equal to u ∈ A(GLGK), but viewed as a morphism in P_{A} from GK to GL.

(3) This is clear, by adjunction.

Proposition 34. The center of A is a D-biset functor.

Proof. First ZA(L) is obviously an R-module, for any L ∈ D. Let K ∈ D and t ∈ ZA(K), i.e. t is a natural transformation Id → ρ_{K} of endofunctors of the category P_{A}. If L ∈ D and u ∈ RB(LK), let u_{A} = υ_{LK}(u) ∈ A(LK) be the image of u under the unique morphism of Green functor υ : RB → A. Since u_{A} ∈ CA(LK), by Corollary 20, we can compose t with the natural transformation ρ_{u_{A}} : ρ_{K} → ρ_{L} from Proposition 31, to get a natural transformation ρ_{u_{A}} ∘ t : Id → ρ_{L}, i.e. an element of ZA(L). Hence we get a linear map

u ∈ RB(LK) ↦ (t ↦ ρ_{u_{A}} ∘ t ∈ Hom_{R}(ZA(K), ZA(L))),

and Assertion 4 of Proposition 31 shows that this endows ZA with a structure of biset functor.

We now build a product on ZA, to make it a Green biset functor. For K, L ∈ D, let s ∈ ZA(K) and t ∈ ZA(L). Since s is a natural transformation Id → ρ_{K}, we get, by adjunction, a natural transformation s^{o} : ρ_{K} → Id. By composition with t : Id → ρ_{L}, we obtain a natural transformation t ∘ s^{o} : ρ_{K} → ρ_{L}, which in turn, by adjunction again, gives a natural transformation ^{o}(t ∘ s^{o}) : Id → (ρ_{L})_{K} ≅ ρ_{LK}, i.e. an element of ZA(LK). So we set

∀s ∈ ZA(K), ∀t ∈ ZA(L), t × s = ^{o}(t ∘ s^{o}) ∈ ZA(LK).  (3)

Translating this in the terms of Proposition 30 gives:

Lemma 35. Let s ∈ ZA(K) and t ∈ ZA(L) be defined respectively by families of elements s_{G} ∈ A(GKG) and t_{G} ∈ A(GLG), for G ∈ D. Then t × s is the element of ZA(LK) defined by the family (t × s)_{G} = t_{G} ∘ s_{G} ∈ A(GLKG), for G ∈ D.

Proof. As the adjunction s ↦ s^{o} amounts to switching the last two components of GKG, the element t × s = ^{o}(t ∘ s^{o}) is defined by the family

(t × s)_{G} = A(Iso_{GLGK}^{GLKG})(t_{G} ∘ A(Iso_{GKG}^{GGK})(s_{G}))
= A(Iso_{GLGK}^{GLKG})A(Def_{GLGK}^{GL\underline{G}GK}Res_{GL\underline{G}GK}^{GL\underline{GG}GK})(t_{G} × A(Iso_{GKG}^{GGK})(s_{G})),

where the notation Def_{GLGK}^{GL\underline{G}GK} means that we take deflation with respect to the un-

derlined factor, and Res_{GL\underline{G}GK}^{GL\underline{GG}GK} means that the underlined G in subscript embeds diagonally in the underlined group GG in superscript. It follows that

(t × s)_{G} = A(Def_{GLKG}^{GL\underline{G}KG}Iso_{GLGGK}^{GLGKG}Res_{GL\underline{G}GK}^{GL\underline{GG}GK}Iso_{GLGGKG}^{GLGGGK})(t_{G} × s_{G})
= A(Def_{GLKG}^{GL\underline{G}KG}Res_{GL\underline{G}KG}^{GL\underline{GG}KG})(t_{G} × s_{G})
= t_{G} ∘ s_{G} ∈ A(GLKG).

Notation 36. Let A be a Green D-biset functor, and G, H, K, L ∈ D. For morphisms in P_{A}, namely α : G → H in A(HG) and β : K → L in A(LK), we denote by α ⊠ β : GK → HL the morphism defined by

α ⊠ β = A(Iso_{HGLK}^{HLGK})(α × β) ∈ A(HLGK).

Proposition 37. Let A be a Green D-biset functor, and G, H, K, L ∈ D. Let moreover α ∈ CA(HG) and β ∈ CA(LK). Then for any s ∈ Z(G) and t ∈ Z(K), and for any X ∈ D

(ρ_{α} ∘ s)_{X} ∘ (ρ_{β} ∘ t)_{X} = (ρ_{α⊠β} ∘ (s × t))_{X}.

Proof. The proof amounts to rather lengthy but straighforward calculations on bisets, similar to those we already did several times above, e.g. in the proof of Theorem 13. We leave it as an exercice.

Theorem 38. Let A be a Green D-biset functor. Then ZA, endowed with the product defined in (3), is a Green D-biset functor.

Proof. It is clear from Lemma 35 and Proposition 30 that the product on ZA is associative. Moreover the identity transformation from the identity functor to ρ_{1} = Id_{P_{A}} is obviously an identity element for the product on ZA. This product is also R-bilinear by construction. Finally, the equality ZA(U)(s) × ZA(V)(t) = ZA(U ⊠ V)(s × t) for bisets U and V is a special case of Proposition 37.

4.2 Relations between the commutant and the center

Proposition 39. Let A be a Green D-biset functor.

1. The maps sending α ∈ CA(L) to ρ_{α} ∈ ZA(L), for L ∈ D, define a morphism of Green biset functors ι_{A} : CA → ZA.

2. The maps sending t ∈ Cr_{A}(L) ≅ ZA(L) to t_{1} ∈ A(L), for L ∈ D, define a morphism of Green biset functors π_{A} : ZA → A. The image of this morphism in the component 1 lies in Z(A(1)), hence there is a morphism of rings π_{A, 1} : ZA(1) → Z(A(1)).

3. The composition

\xymatrix{CA \ar@{^{(}->}[r]^{ι_{A}} & ZA \ar[r]^{π_{A}} & A}

is equal to the inclusion \xymatrix{CA \ar@{^{(}->}[r] & A} . In particular ι_{A} is injective.

Proof. For Assertion 1, let α ∈ CA(K), for K ∈ D. Then the element ρ_{α} of ZA(K) corresponds to the family of elements ρ_{α,G} ∈ A(GKG), for G ∈ D, defined by

ρ_{α,G} = A(Ind_{KG}^{GKG}Inf_{K}^{KG})(α).

Similarly, if L ∈ D and β ∈ CA(L), the element ρ_{β} of ZA(L) corresponds to the family ρ_{β,G} = A(Ind_{LG}^{GLG}Inf_{L}^{LG})(β). By Lemma 35, the product q = ρ_{α} × ρ_{β} in ZA(KL) corresponds to the family

q_{G} = ρ_{α,G} ∘ ρ_{β,G}
= A(Ind_{KG}^{GKG}Inf_{K}^{KG})(α) ∘ A(Ind_{LG}^{GLG}Inf_{L}^{LG})(β)
= A(Def_{GKLG}^{GK\underline{G}LG}Res_{GK\underline{G}LG}^{GK\underline{GG}LG})(A(Ind_{KG}^{GKG}Inf_{K}^{KG})(α) ∘ A(Ind_{LG}^{GLG}Inf_{L}^{LG})(β))
= A(Def_{GKLG}^{GK\underline{G}LG}Res_{GK\underline{G}LG}^{GK\underline{GG}LG}Ind_{KGLG}^{GKGGLG}Inf_{KL}^{KGLG})(α × β).

Standard relations in the composition of bisets then show that

q_{G} = A(Ind_{KLG}^{GKLG}Inf_{KL}^{KLG})(α × β),

and it follows that q = ρ_{α×β}. In other words ι_{A}(α × β) = ι_{A}(α) × ι_{A}(β). Moreover, the identity element ε_{A} ∈ CA(1) is mapped by ι_{A} to the element of ZA(1) defined by the family of elements A(Ind_{G}^{GG}Inf_{1}^{G})(ε_{A}), for G ∈ D, that is the identity element of ZA. So ι_{A} is a morphism of Green D-biset functors.

The first part of Assertion 2 is a consequence of Lemma 35. Indeed, if K, L ∈ D, if s ∈ ZA(K) corresponds to the family s_{G} ∈ Cr_{A}(K), and if t ∈ ZA(L) corresponds to the family β_{G} ∈ Cr_{A}(L), for G ∈ D, then the product u = s × t is the element of ZA(KL) corresponding to the family u_{G} = s_{G} ∘ t_{G}. In particular, for G = 1, we have

u_{1} = s_{1} ∘ t_{1} = s_{1} × t_{1}.

This shows that the maps sending t ∈ ZA(L) to t_{1} ∈ A(L), for L ∈ D, is a morphism of Green functors π : ZA → A.

Since composition ∘ : A(1) × A(1) → A(1) coincides with the product as a ring of A(1), the commutativity property defining the series of Cr_{A}(1) shows that π_{A, 1} has image in Z(A(1)). This completes the proof of Assertion 2.

For Assertion 3, we start with an element α ∈ CA(L), for L ∈ D. It is sent by ι_{A} to the element t ∈ ZA(L) corresponding to the family t_{G} = A(Ind_{LG}^{GLG}Inf_{L}^{LG})(α), for G ∈ D, in Cr_{A}(L). In particular t_{1} = A(Ind_{L}^{L}Inf_{L}^{L})(α) = α, do π_{A} ∘ ι_{A} is equal to the inclusion CA ↪ A.

The morphism ι_{A} of the previous proposition allows us to give a CA-module structure to ZA. With this structure, (the image under ι_{A} of) CA is a CA-submodule of ZA. In the particular case where A is commutative, the previous proposition tells us more.

Corollary 40. If A is a commutative Green D-biset functor, then A is isomorphic to a direct summand of ZA in the category A-Mod.

Proof. This follows from the fact that ι_{A} and π_{A} are morphisms of Green D-biset functors, so in particular morphisms of A-modules. Moreover the composition π_{A} ∘ ι_{A} is equal to the identity when A is commutative.

Proposition 41. Let A be a Green D-biset functor. Let End_{R}(P_{A}) be the category of R-linear endofunctors of P_{A}.

1. The assignment

\{ \begin{matrix}K ∈ D & ↦ & ρ_{K} ∈ End_{R}(P_{A}) \\ t ∈ ZA(LK) & ↦ & t^{o} ∈ Hom_{End_{R}(P_{A})}(ρ_{K}, ρ_{L}) \end{matrix}

is a fully faithful R-linear functor ρ_{ZA} from P_{ZA} to End_{R}(P_{A}).

2. The following assignment μ_{A}

\{ \begin{matrix}K ∈ D & ↦ & K ∈ D \\ α ∈ CA(LK) & ↦ & ^{o}ρ_{α} ∈ ZA(LK) \end{matrix}

is equal to the functor P_{ι_{A}} from P_{CA} to P_{ZA}, induced by ι_{A} : CA → ZA. In particular μ_{A} is faithful, and such that

ρ_{ZA} ∘ μ_{A} = ρ_{CA}.

3. The following assignment ν_{A}

\{ \begin{matrix}K ∈ D & ↦ & K ∈ D \\ s ∈ ZA(LK) & ↦ & s_{1} ∈ A(LK) \end{matrix}

is equal to the functor P_{π_{A}} from P_{ZA} to P_{A} induced by the morphism of Green biset functors π_{A} : ZA → A. The composition π_{A} ∘ μ_{A} is equal to the inclusion functor P_{CA} → P_{A}.

Proof. All the assertions are straightforward consequences of Proposition 39.

To conclude this section, we now show that the isomorphism CA_{L} ≅ (CA)_{L} of Proposition 22 only extends to an injection ZA_{L} ↪ (ZA)_{L}. We first prove a lemma.

Lemma 42. Let A be a Green D-biset functor. For L ∈ D, let ψ_{L}^{A} : P_{A} → P_{A_{L}} be the functor ψ_{L} of Theorem 13. If K ∈ D, let ψ_{K}^{A_{L}} : P_{A_{L}} → P_{(A_{L})_{K}} be the similar functor built from A_{L} and K. Then the diagram

\xymatrix{P_{A} \ar[r]^{ψ_{L}^{A}} \ar[drr]_{ψ_{KL}^{A}} & P_{A_{L}} \ar[r]^{ψ_{K}^{A_{L}}} & P_{(A_{L})_{K}} \ar[d]_{e_{K,L}}^{≅} \\  &  & P_{A_{KL}}}

of categories and functors, is commutative, where e_{K,L} is the natural equivalence of categories P_{(A_{L})_{K}} → P_{A_{KL}} provided by the canonical isomorphism of Green D-biset functors (A_{L})_{K} ≅ A_{KL}.

Proof. Indeed, all the functors involved are the identity on objects. And for a morphism α : G → H in P_{A}, i.e. an element α of A(HG), we have

ψ_{K}^{A_{L}}ψ_{L}^{A}(α) = ψ_{K}^{A_{L}}A(Inf_{HG}^{HGL})(α) = A_{L}(Inf_{HG}^{HGK})A(Inf_{HG}^{HGL})(α)
= A(Inf_{HGL}^{HGKL})A(Inf_{HG}^{HGL})(α)
= A(Inf_{HG}^{HGKL})(α) = ψ_{KL}^{A}(α).

Proposition 43. Let A be a Green biset functor and L be a group. Then there is an injective morphism of Green D-biset functors from ZA_{L} to (ZA)_{L}.

Proof. Let K, L ∈ D, and t ∈ ZA_{L}(K), i.e. a natural transformation

t : Id_{P_{A_{L}}} → ρ_{K}^{A_{L}}

from the identity functor of P_{A_{L}} to the functor ρ_{K}^{A_{L}} = θ_{K}^{A_{L}}ψ_{K}^{A_{L}}, where θ_{K}^{A_{L}} is the functor P_{(A_{L})_{K}} → P_{A_{L}} of Theorem 13 built from A_{L} and K. By precomposition of this natural transformation with the functor ψ_{L}^{A}, we get a natural transformation

ψ_{L}^{A} → θ_{K}^{A_{L}}ψ_{K}^{A_{L}}ψ_{L}^{A},

which by adjunction, gives a natural transformation

Id_{P_{A}} → θ_{L}^{A}θ_{K}^{A_{L}}ψ_{K}^{A_{L}}ψ_{L}^{A}.

By Lemma 42, the functor ψ_{K}^{A_{L}}ψ_{L}^{A} is isomorphic to ψ_{KL}^{A}. By Theorem 13, the functor θ_{K}^{A_{L}} is left and right adjoint to the functor ψ_{K}^{A_{L}}, and θ_{L}^{A} is left and right adjoint to ψ_{L}^{A}.

It follows that the functor θ_{L}^{A}θ_{K}^{A_{L}} is isomorphic to the adjoint θ_{KL}^{A} of ψ_{KL}^{A}. Hence we have a natural transformation

T : Id_{P_{A}} → θ_{KL}^{A}ψ_{KL}^{A} = ρ_{KL}^{A},

that is an element of ZA(KL) = ZA_{L}(K).

So we have a map j_{L,K} : t ∈ ZA_{L}(K) ↦ T ∈ (ZA)_{L}(K), which is obviously Rlinear. Lengthy but straightforward calculations show that the family of these maps, for K ∈ D, form a morphism of Green biset functors from ZA_{L} to (ZA)_{L}.

5 Application: some equivalences of categories

5.1 General setting

We begin by recalling some well known folklore facts on the decomposition of a category F_{P} of functors from a small R-linear category P to R-Mod, using an orthogonal decomposition of the identity in the center ZP of P.

Since P is R-linear, its center ZP is a commutative R-algebra. Suppose we have a family (γ_{i})_{i∈I} of elements of ZP indexed by a set I, with the following properties:

1. For i, j ∈ I, the product γ_{i}γ_{j} is equal to 0 if i ≠ j, and to γ_{i} if i = j.

2. For any object G of P, there is only a finite number of elements i ∈ I such that γ_{i,G} ≠ 0. Then, for each object G ∈ P, we can consider the (finite) sum ∑_{i∈I} γ_{i,G},

which is a well defined endomorphism of G. We assume that this endomorphism is the identity of G, for any G ∈ P.

If F is an R-linear functor from P to R-Mod, and i ∈ I, we denote by F γ_{i} the functor that in an object G of P is defined as the image of F (γ_{i,G}), that is

(F γ_{i})(G) = Im(F (γ_{i,G}) : F (G) → F (G)),

which is an R-submodule of F (G). For a morphism α : G → H, we denote by (F γ_{i})(α) the restriction of F (α) to (F γ_{i})(G). The image of (F γ_{i})(α) is contained in F γ_{i}(H), because the square

\xymatrix{G \ar[r]^{γ_{i,G}} \ar[d]_{α} & G \ar[d]^{α} \\ H \ar[r]_{γ_{i},H} & H}

is commutative in P, hence also its image by F.

It is easy to check that F γ_{i} is an R-linear functor from P to R-Mod, which is a subfunctor of F. Moreover, the assigment F ↦ F γ_{i} is an endofunctor Γ_{i} of the category F_{P}. The image of this functor consists of those functors F ∈ F_{R} such that the subfunctor F γ_{i} is equal to F. Let F_{R}γ_{i} be the full subcategory of F_{R} consisting of such functors. It is an abelian subcategory of F_{R}.

For each G ∈ P, the direct sum ⊕_{i∈I} F γ_{i}(G) is actually finite, and our assump-

tions ensure that is is equal fo F (G). This shows that the functor sending F ∈ F_{R} to the family of functors F γ_{i} is an equivalence between F_{R} and the product of the categories F_{R}γ_{i}.

A particular case of the previous situation is when the identity element ε ∈ A(1) of a Green biset functor A has a decomposition in orthogonal idempotents ε = ∑_{i=1}^{n} e_{i} in the ring CA(1). Each e_{i} induces a natural transformation E^{i} : Id → Id_{1}, defined at an A-module M and a group H ∈ D as

E_{M, H}^{i} : M(H) → M_{1}(H) m ↦ M(Iso_{1×H}^{H×1})(e_{i} × m).

For simplicity, we will think of this natural transformation as sending m simply to e_{i} × m, and we will denote by e_{i}M the A-submodule of M given by the image of E_{M}^{i}.

Since the morphism from CA(1) to ZA(1) is a ring homomorphism, we have that the natural transformations E^{i} satisfy E^{i} ∘ E^{i} = E^{i}, E^{i} ∘ E^{j} = 0 if i ≠ j and that the identity natural transformation, 1, is equal to ∑_{i=1}^{n} E^{i}. By Proposition 28, we have then the hypothesis assumed at the beginning of the section and so we obtain the equivalence of categories mentioned above. In this case we can give a more precise description of this equivalence.

Lemma 44. The A-module e_{i}A is a Green D-biset functor, and for every A-module M, the functor e_{i}M is an e_{i}A-module. Furthermore A ≅ ⨁_{i=1}^{n} e_{i}A as Green D-biset functors.

Proof. As we have said, e_{i}A is an A-module, in particular it is a biset functor. We claim that it is a Green biset functor with the product

e_{i}A(G) × e_{i}A(K) → e_{i}A(G × K) (e_{i} × a) × (e_{i} × b) = e_{i} × a × b.

Observe that since all the × represent the product of A, then (e_{i} × a) × (e_{i} × b) is isomorphic to a × e_{i} × e_{i} × b, because e_{i} ∈ CA(1). But the product × coincides with the ring product in A(1), hence this element is isomorphic to a × e_{i} × b and then to e_{i}×a×b. This implies immediately that the product is associative, the identity element in e_{i}A(1) is of course e_{i} × ε. Next, notice that since E_{A}^{i} is a morphism of A-modules, if L, G ∈ D and X is an (L, G)-biset, then A(X)(e_{i} ×a) ≅ e_{i} ×A(X)(a) for all a ∈ A(G). With this, one can easily show the functoriality of the product.

Similar arguments show that e_{i}M is an e_{i}A-module with the product

e_{i}A(G) × e_{i}M(K) → e_{i}M(G × K) (e_{i} × a) × (e_{i} × m) = e_{i} × a × m.

For the final statement, first it is an easy exercise to verify that given A_{1}, …, A_{r} Green biset functors, then their direct sum ⨁_{i=1}^{r} A_{i} in the category of biset functors is again a Green biset functor, with the product given component-wise. With this, it is straightforward to see that the isomorphism of biset functors A ≅ ⨁_{i=1}^{n} e_{i}A is an isomorphism of Green biset functors.

All these observations give us the following result.

Theorem 45. Let A be a Green D-biset functor as above. Then the category A-Mod is equivalent to the product category

∏_{i=1}^{n} e_{i}A-Mod.

Moreover, for each indecomposable A-module M, there exists only one e_{i} such that e_{i}M ≠ 0, and hence e_{i}M ≅ M.

When considering the shifted functor A_{H}, if we have an idempotent e ∈ CA_{H}(1) as before, then the evaluation of eA_{H} at a group G can be seen as follows. Since eA_{H}(G) = e ×^{d} A_{H}(G), then for a ∈ A_{H}(G) it is easy to see that

e ×^{d} a = A(Res_{G×Δ(H)}^{1×H×G×H})(e × a) = A(Inf_{H}^{G×H})(e) · a,

where the product · indicates the ring structure in A(G×H). The last equality follows from Lemma 3 and the properties of restriction and inflation. So, the evaluation of eA_{H} at a given group depends on how inflation of A acts on the idempotents of CA(H).

5.2 Some examples

5.2.1 p-biset functors

When p is a prime and p is invertible in the ring R, a family of orthogonal idempotents in the center of the Green functor RB_{p} of Example 5 has been introduced in [3]. These idempotents b̂_{L} are indexed by atoric p-groups L up to isomorphism, i.e. finite p-groups which cannot be decomposed as a direct product Q × C_{p} of a finite p-group Q and a group C_{p} of order p.

More precisely, for each such atoric p-group L and each finite p-group P, a specific idempotent b_{L}^{P} of RB_{p}(P, P) is introduced (cf. [3], Theorem 7.4), with the property that

a ∘ b_{L}^{P} = b_{L}^{Q} ∘ a

for any finite p-groups P and Q, and any a ∈ RB(Q, P). In other words, the family b_{L} = (b_{L}^{P})_{P} is an element of the center of the biset category RC_{p} of finite p-groups. The elements b̂_{L} of the center of the category of p-biset functors over R - i.e. the category RB_{p}-Mod - are deduced from the elements b_{L} in [3], Corollary 7.5.

The idempotents b_{L}^{P} have the following additional properties:

1. If L and L′ are isomorphic atoric p-groups, then b_{L}^{P} = b_{L′}^{P}

2. If L and L′ are non isomorphic atoric p-groups, then b_{L}^{P}b_{L′}^{P} = 0. Let [At_{p}] denote a set of representatives of isomorphism classes of atoric p-groups.

3. For a given finite p-group P, there are only a finite number of atoric p-groups L, up to isomorphism, such that b_{L}^{P} ≠ 0.

4. The sum ∑_{L∈[At_{p}]} b_{L}^{P}, which is a finite sum by the previous property, is equal to the

identity element of RB(P, P).

It follows that one can consider the sum ∑_{L∈[At_{p}]} b̂_{L} in Z(RB_{p})(1), and that this sum is

equal to the identity element of Z(RB_{p})(1). So we obtain a locally finite decomposition of the identity element of Z(RB_{p})(1) as a sum of orthogonal idempotents, which allows for a splitting of the category of p-biset functors over R as a direct product of abelian subcategories (cf. [3], Corollary 7.5). As a consequence, for each indecomposable pbiset functor F over R, there is an atoric p-group L, unique up to isomorphism, such that b̂_{L} acts as the identity of F (or equivalently, does not act by zero on F). This group L is called the vertex of F (cf. [3], Definition 9.2).

Remark 46. This example shows in particular that ZA can be much bigger than CA: indeed for A = RB_{p}, when R is a field of characteristic different from p, we see that ZA(1) is an infinite dimensional R-vector space, whereas CA(1) ≅ R is one dimensional.

5.2.2 Shifted representation functors

Now we apply the results of Section 5.1 to some shifted classical representation functors, with coefficients in a field 𝔽 of characteristic 0. In each case we will begin with a commutative Green biset functor C such that for each group H, the 𝔽-algebra C(H) is split semisimple. In particular, taking A = C_{H}, in A(1) = C(H) we will have a family

of orthogonal idempotents {e_{i}^{H}}_{i=1}^{n_{H}} such that ε = ∑_{i=1}^{n_{H}} e_{i}^{H}. As we said in Section 5.1, the

evaluation e_{i}^{H}A(G) is given in the following way

e_{i}^{H} ×^{d} a = A(Inf_{1}^{G})(e_{i}^{H}) · a = C(Inf_{H}^{G×H})(e_{i}^{H}) · a

for a ∈ A(G). Now, since inflation is a ring homomorphism, A(Inf_{1}^{G})(e_{i}^{H}) is equal to ∑_{j∈J} e_{j}^{G×H} for some J ⊆ {1, …, n_{G×H}} depending on e_{i}^{H} and G. On the other hand, we

also have a = ∑_{i=1}^{n_{G×H}} α_{i}(a)e_{i}^{G×H}, for some α_{i}(a) ∈ 𝔽. This implies that the idempotents

appearing in the evaluation e_{i}^{H}A(G) depend only on the set {e_{j}^{G×H}}_{j∈J}.

Shifted Burnside functors.

We consider the Burnside functor 𝔽B over 𝔽. We fix a finite group H, and consider the shifted functor A = 𝔽B_{H}. Then the algebra A(1) is isomorphic to 𝔽B(H), hence it is split semisimple. Its primitive idempotents e_{K}^{H} are indexed by subgroups K of H, up to conjugation, and explicitly given (see. [10], [14]) by

e_{K}^{H} = \frac{1}{|N_{H}(K)|} ∑_{L≤K} |K|μ(L, K) [H/L],

where μ is the Möbius function of the poset of subgroups of H and [H/L] ∈ B(H) is the class of the transitive H-set H/L.

By Theorem 45, we get a decomposition of the category A-Mod as a product ∏_{K∈[s_{H}]} e_{K}^{H}A-Mod, where [s_{H}] is a set of representatives of conjugacy classes of sub-

groups of H. From the action of inflation on the primitive idempotents of Burnside rings (see [2] Theorem 5.2.4), it is easy to see that for K ≤ H, the value e_{K}^{H}A(G) of the Green functor e_{K}^{H}A at a finite group G is equal to the set of linear combinations of idempotents e_{L}^{G×H} of 𝔽B(G × H) indexed by subgroups L of (G × H) for which the second projection p_{2}(L) is conjugate to K in H. Also, for each indecomposable A-module M, there exists a unique K ≤ H, up to conjugation, such that e_{K}^{H}M ≠ 0, and then e_{K}^{H}M = M.

Shifted functors of linear representations.

Next we consider the functor 𝔽R_{𝕂} of linear representations over 𝕂, a field of characteristic 0. As before, we fix a finite group H and consider the shifted functor A = (𝔽R_{𝕂})_{H}. This is a commutative Green biset functor, and A(1) is isomorphic to the split semisimple 𝔽-algebra 𝔽R_{𝕂}(H). If |H| = n, it is shown in Section 3.3.1 of [8] (and in a slightly different way in [9]) that 𝔽R_{𝕂}(H) has a complete family of orthogonal primitive idempotents e_{D}^{H} indexed by the E-conjugacy classes of H, where E is certain subgroup of (ℤ/nℤ)^{×}. By E-conjugacy we mean that two elements x, y ∈ H are E-conjugated if there exist [i] ∈ E such that x =_{H} y^{i}. This defines an equivalence relation on H and the set of E-conjugacy classes is denoted by Cl_{E}(H). The group E is built in the following way: First we fix an algebraically closed field 𝕃, which is an extension of 𝔽 and 𝕂, and then we take the intersection 𝔼 = 𝔽 ∩ 𝕂 in 𝕃. By adding an n-th primitive root of unity, ω, to 𝔼, we obtain E as the group isomorphic to Gal(𝔼[ω]/𝔼) in (ℤ/nℤ)^{×}. Observe that, as a group, E depends only on 𝔽, 𝕂 and n, and not on the choice of 𝕃. Then, by Theorem 45, we get a decomposition of the category A-Mod as a product ∏_{D∈Cl_{E}(H)} e_{D}^{H}A-Mod. Also, for each indecomposable A-module M, there exists

a unique E-conjugacy class D of H such that e_{D}^{H}M ≠ 0 and so e_{D}^{H}M = M. On the other hand, in Corollary 3.3.14 of [8] it is shown that e_{D}^{H}A is a simple A-module and hence that A is a semisimple A-module, since A = ∑_{D} e_{D}^{H}A.

Finally, using Lemma 3.3.10 in [8], we see that the idempotents e_{C}^{G×H}, for C an E-class of G × H, appearing in the evaluation A(Inf_{1}^{G})(e_{D}^{H}) are those for which π_{H}(C), the projection of C on H, is equal to D.

Shifted p-permutation functors.

Let k be an algebraically closed field of positive characteristic p. In this case we assume also that 𝔽 contains all the p′-roots of unity, and consider the functor 𝔽pp_{k}. Then 𝔽pp_{k} is a commutative Green biset functor, and the category 𝔽pp_{k}-Mod has been considered in particular in [7] (when 𝔽 is algebraically closed).

We fix a finite group H, and consider the shifted functor A = (𝔽pp_{k})_{H}. Then the algebra A(1) is isomorphic to the algebra 𝔽pp_{k}(H). This algebra is split semisimple, and its primitive idempotents F_{Q,s}^{H} have been determined in [4]: they are indexed by (conjugacy classes of) pairs (Q, s) consisting of a p-subgroup Q of H, and a p′-element s of N_{H}(Q)/Q. We denote by Q_{H,p} the set of such pairs, and by [Q_{H,p}] a set of representatives of orbits of H for its action on Q_{H,p} by conjugation.

If (Q, s) ∈ Q_{H,p} and u ∈ 𝔽pp_{k}(H), then F_{Q,s}^{H}u = τ_{Q,s}^{H}(u)F_{Q,s}^{H}, where τ_{Q,s}^{H}(u) ∈ 𝔽. The maps u ↦ τ_{Q,s}^{H}(u), for (Q, s) ∈ [Q_{H,p}] are the distinct algebra homomorphisms (the species) from 𝔽pp_{k}(H) to 𝔽 (see e.g. [4] Proposition 2.18). Moreover, the map τ_{Q,s}^{H} is determined by the fact that for any p-permutation kH-module M, the scalar τ_{Q,s}^{H}(M) is equal to the value at s of the Brauer character of the Brauer quotient M[Q] of M at Q.

It follows that if N ⊴ H, and v ∈ 𝔽pp_{k}(H/N), then τ_{Q,s}^{H}(Inf_{H/N}^{H}v) = τ_{\overline{Q},\overline{s}}^{H/N}(v), where \overline{Q} = QN/N, and \overline{s} ∈ N_{H/N}(\overline{Q})/\overline{Q} is the projection of s to H/N. As a consequence, if (R, t) ∈ Q_{H/N,p}, then Inf_{H/N}^{H}(F_{R,t}^{H/N}) is equal to the sum of the idempotents F_{Q,s}^{H} for those elements (Q, s) ∈ [Q_{H,p}] for which (\overline{Q}, \overline{s}) is conjugate to (R, t) in H/N.

Now by Theorem 45, we get a decomposition of the category A-Mod as a product ∏_{(Q,s)∈[Q_{H,p}]} F_{Q,s}^{H}A-Mod. Let G be a finite group. It follows from the previous discus-

sion on inflation that the evaluation F_{Q,s}^{H}A(G) of A at G is equal to the set of linear combinations of primitive idempotents F_{L,t}^{G×H}, for (L, t) ∈ Q_{G×H,p}, such that the pair (p_{2}(L), p_{2}(t)) is conjugate to (Q, s) in H, where p_{2} : G × H → H is the second projection. Also, for each indecomposable A-module M, there exists a unique (Q, s) ∈ [Q_{H,p}] such that F_{Q,s}^{H}M ≠ 0, and then F_{Q,s}^{H}M = M.

References

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[12] Nadia Romero. Funtores de Mackey. Tesis de Doctorado UNAM, Mexico, 2011.

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Serge Bouc, CNRS-LAMFA, Université de Picardie, 33 rue St Leu, 80039, Amiens, France.

serge.bouc@u-picardie.fr

Nadia Romero, DEMAT, UGTO, Jalisco s/n, Mineral de Valenciana, 36240, Guanajuato, Gto., Mexico.

nadia.romero@ugto.mx