Mackey functors, induction from restriction functors and coinduction from transfer functors.

Olcay COŞKUN ^{∗}

2007

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey. e-mail: coskun@fen.bilkent.edu.tr.

Abstract

Boltje’s plus constructions extend two well-known constructions on Mackey functors, the fixed-point functor and the fixed-quotient functor. In this paper, we show that the plus constructions are induction and coinduction functors of general module theory. As an application, we construct simple Mackey functors from simple restriction functors and simple transfer functors. We also give new proofs for the classification theorem for simple Mackey functors and semisimplicity theorem of Mackey functors. 2006 Mathematics Subject Classification. Primary: 20C20 ; Secondary: 19A22, 20J05 . Keywords: Mackey functors, restriction functors, transfer functors, plus constructions, fixed-point functor, induction, coinduction, simple Mackey functors.

1 Introduction

The theory of Mackey functors was introduced by Green to provide a unified treatment of group representation theoretic constructions involving restriction, conjugation and transfer. Thévenaz and Webb improved Green’s definition of a Mackey functor, and they realized Mackey functors as representations of the Mackey algebra μ_{R}(G). Using this identification, Thévenaz and Webb applied methods of module theory to classify the simple Mackey functors [11] and to describe the structure of Mackey functors [12]. Their description of simple Mackey functors used induction and inflation from subgroups and two dual constructions, known as the fixedpoint functor and the fixed-quotient functor.

Applying the notion of Mackey functors to the problem of finding an explicit version of Brauer’s induction theorem, Boltje introduced the theory of canonical induction [6], [7]. In order to solve the problem in this general context, Boltje considered not only the category Mack_{R}(G) of Mackey functors, but also two more categories, namely the category Con_{R}(G) of conjugation functors and the category Res_{R}(G) of restriction functors. His main tools were the lower-plus and the upper-plus constructions, which extend the fixed-quotient and the fixedpoint functors, respectively.

∗The author is partially supported by TÜBÍTAK through PhD award program.

The lower-plus construction, denoted by −_{+}, is defined as a functor Res_{R}(G) → Mack_{R}(G). By introducing the restriction algebra ρ_{R}(G), written ρ when R and G are understood, we realize the restriction functors as representations of the restriction algebra. This leads us to

Theorem 5.1 The functors −_{+} and ind_{ρ}^{μ} are naturally equivalent.

On the other hand, the upper-plus construction, denoted by −^{+}, is defined as a functor Con_{R}(G) → Mack_{R}(G). By introducing the transfer algebra τ_{R}(G), written τ, and its representations, called transfer functors and realizing conjugation functors as representations of the conjugation algebra γ_{R}(G), written γ, we prove

Theorem 5.2 The functors −^{+} and coind_{τ}^{μ} inf_{γ}^{τ} are naturally equivalent.

As a consequence of these identifications, we realize the fixed-point and fixed-quotient functors as coinduced and induced modules, respectively. Given an RG-module V, we denote by FQ_{V} the fixed-quotient functor and by FP_{V} the fixed-point functor.

Proposition 5.4 Let V be an RG-module. Then, the following isomorphisms hold.

(i) FQ_{V} ≃ ind_{ρ}^{μ}inf_{γ}^{ρ} D_{V} and (ii) FP_{V} ≃ coind_{τ}^{μ}inf_{γ}^{τ} D_{V}

where D_{V} denotes the γ-module which is non-zero only at the trivial group and D_{V} (1) = V.

We also prove that the Brauer quotient (also known as the bar construction) is the composition of certain restriction and deflation functors (see Corollary 5.7). Via this identification, we see that Thévenaz’ twin functor is the composition of coinduction, inflation, deflation and restriction functors.

The plus constructions are also used by Bouc [4] and Symonds [9]. To obtain information about projective Mackey functors, Bouc considered restriction functors defined only on psubgroups and also the functor −_{+} (which is denoted by I in [4]). In [9], Symonds constructed induction formulae using the plus constructions described in terms of the zero degree group homology and group cohomology functors.

The subalgebra structure of the Mackey algebra, we describe above, leads us to

Theorem 3.2 (Mackey structure theorem) The τ − ρ-bimodule _{τ} μ_{ρ} is isomorphic to τ ⊗_{γ} ρ.

As a consequence of this theorem, we obtain several equivalences relating the functors between the algebras μ, τ, ρ and γ. Using some of these equivalences, we show that the wellknown mark homomorphism corresponds to the identity map on conjugation functors (see Proposition 5.10).

Our module theoretic approach not only unearths the nature of some known constructions for Mackey functors, but also allows us to understand the classification of simple Mackey functors better. The classification theorem of Thévenaz and Webb [11] asserts that the simple Mackey functors are parameterized by the G-classes of simple pairs (H, V) where H is a subgroup of G and V is a simple RN_{G}(H)/H-module. It is easy to see that the simple conjugation functors are also parameterized by the G-classes of simple pairs (H, V). It is almost as easy that the simple restriction functors and the simple transfer functors are parameterized in the same way. As an application of our characterization of the plus constructions, we show how the classification theorem for simple Mackey functors follows quickly from the classification of the simple restriction functors. Moreover, we obtain two new descriptions of the simple Mackey functors. In the case where |G| is invertible in the base field R, we see that induction from the restriction algebra and coinduction from the transfer algebra respect simple modules. We also give a new proof of the semisimplicity theorem [11], which states that the Mackey functors are semisimple when R is a field of characteristic coprime to |G|.

Let us mention that, in a sequel to this paper, we shall be adapting some of these methods and results to the content of biset functors.

The organization of the paper is as follows. In Section 2, we collect together necessary facts concerning the Mackey functors. In Section 3 we prove the Mackey structure theorem and its consequences. Section 4 contains the duality theorems. Our main results, the description of plus constructions via induction, coinduction and restriction are proved in Section 5. Also in this section, we give alternative descriptions of the fixed-point functor, the fixed-quotient functor, the twin functor and the mark homomorphism. The applications to the classification of simple Mackey functors and to the semisimplicity of Mackey functors are the contents of Sections 6 and 7, respectively.

2 Preliminaries

Let G be a finite group and R be a commutative ring with unity. Consider the free algebra on variables c_{g}^{H}, r_{K}^{H}, t_{K}^{H} where K ≤ H ≤ G and g ∈ G. We define the Mackey algebra μ_{R}(G) for G over R as the quotient of this algebra by the ideal generated by the following six relations, where L ≤ K ≤ H ≤ G and h ∈ H and g, g′ ∈ G:

(1) c_{h}^{H} = r_{H}^{H} = t_{H}^{H}

(2) c_{g′}^{^{g}H}c_{g}^{H} = c_{g′g}^{H} and r_{L}^{K}r_{K}^{H} = r_{L}^{H} and t_{K}^{H}t_{L}^{K} = t_{L}^{H}

(3) c_{g}^{K}r_{K}^{H} = r_{^{g}K}^{^{g}H}c_{g}^{H} and c_{g}^{H}t_{K}^{H} = t_{^{g}K}^{^{g}H}c_{g}^{K}

(4) r_{J}^{H}t_{K}^{H} = ∑_{x∈J\H/K} t_{J∩^{x}K}^{J}c_{x}r_{J^{x}∩K}^{K} for J ≤ H (Mackey Relation)

(5) ∑_{H≤G} r_{H}^{H} = 1

(6) All other products of generators are zero.

It is known that, letting H and K run over the subgroups of G and letting g run over the double coset representatives HgK ⊂ G and letting L run over representatives of the subgroups of H^{g} ∩ K up to conjugacy, the elements t_{^{g}L}^{H} c_{g}^{L} r_{L}^{K} run (without repetitions) over the elements of an R-basis for the Mackey algebra μ_{R}(G) (cf.[12, Section 3]).

We denote by ρ_{R}(G), called the restriction algebra for G over R, the subalgebra of the Mackey algebra generated by c_{g}^{H} and r_{K}^{H} for K ≤ H ≤ G and g ∈ G. We denote by τ_{R}(G) the transfer algebra for G over R the subalgebra generated by c_{g}^{H} and t_{K}^{H} for K ≤ H ≤ G and g ∈ G. The conjugation algebra, denoted γ_{R}(G), is the subalgebra generated by the elements c_{g}^{H}. When there is no ambiguity, we write μ = μ_{R}(G), and ρ = ρ_{R}(G) and τ = τ_{R}(G) and γ = γ_{R}(G). Evidently, the restriction algebra ρ has generators c_{g}^{J} r_{J}^{K}, the transfer algebra τ has generators c_{g}^{K} t_{J}^{K} and the conjugation algebra γ has generators c_{g}^{J}.

We define a Mackey functor for G over R to be a μ_{R}(G)-module. Similarly, we define a restriction functor, a transfer functor and a conjugation functor as a ρ_{R}(G)-module, a τ_{R}(G)-module and a γ_{R}(G)-module, respectively.

We can also define a Mackey functor as a quadruple (M, c, r, t) consisting of a family of R-modules M(K) for each K ≤ G and families of three types of maps:

(i) conjugation maps, c_{g}^{K} : M(K) → M(^{g}K) for each g ∈ G and K ≤ G, (ii) restriction maps, r_{L}^{K} : M(K) → M(L) for each L ≤ K ≤ G and

(iii) transfer maps, t_{L}^{K} : M(L) → M(K) for each L ≤ K ≤ G.

These maps have to satisfy the relations (2), (3) and (4), above and the following relation

(1′) c_{h}^{H} = r_{H}^{H} = t_{H}^{H} = id_{H} for all h ∈ H ≤ G.

We write M for the quadruple (M, c, r, t). Then to pass from the first definition to the second one, we put M(K) = c_{1}^{K} M for each K ≤ G and conversely, we take M = ⨁_{K≤G} M(K). Similar comments apply to restriction and transfer and conjugation functors. (cf. [12] and [8])

Defining a morphism of Mackey functors to be an R-module homomorphism compatible with conjugation, restriction and transfer maps, we obtain the category Mack_{R}(G) of Mackey functors for G over R. Similarly, we have the category Res_{R}(G) of restriction functors, the category Tran_{R}(G) of transfer functors and Con_{R}(G) of conjugation functors.

Remark 2.1 In [4], Bouc introduced an algebra, denoted rμ_{R}(G), which is generated by c_{g}^{G} and r_{K}^{H} where K ≤ H ≤ G, g ∈ G and H is a p-subgroup. He also introduced tμ_{R}(G) as the dual of rμ_{R}(G). Upper and lower plus constructions are also introduced in this settings.

In [7] and [8], Boltje introduced two functors −^{+} : Con_{R}(G) → Mack_{R}(G) and −_{+} : Res_{R}(G) → Mack_{R}(G), called upper-plus and lower-plus constructions, respectively. In Section 5, we show that these functors have descriptions as induction and coinduction functors. We review the constructions of these functors.

To a conjugation functor C, we associate a Mackey functor C^{+} where for H ≤ G, we define the modules as

C^{+}(H) = ( ∏_{L≤H} C(L))^{H} .

Here H acts on the product by coordinate-wise conjugation. We define the maps for K ≤ H ≤ G and g ∈ G and x_{L} ∈ C(L) as follows:

Conjugation:

c_{g}^{+H} : C^{+}(H) → C^{+}(^{g}H) where (x_{L})_{L≤H} ↦ (^{g}x_{L^{g}})_{L≤^{g}H}.

Restriction:

r_{K}^{+H} : C^{+}(H) → C^{+}(K) where (x_{L})_{L≤H} ↦ (x_{L})_{L≤K}.

Transfer:

t_{K}^{+H} : C^{+}(K) → C^{+}(H) where (x_{L})_{L≤K} ↦ ∑_{h∈H/K} c_{h}^{+K}((x_{L})_{L≤K}).

The functor −^{+} is defined on morphisms, in the obvious way, that is, if f : B → C is a morphism of conjugation functors, then f^{+} : B^{+} → C^{+} is defined by f_{H}^{+}((x_{L})_{L≤H}) = (f_{L}(x_{L}))_{L≤H}.

To a restriction functor D, we associate a Mackey functor D_{+} where for H ≤ G, the modules are

D_{+}(H) = ( ⨁_{L≤H} D(L))_{H} .

Here, for an RH-module M, we write M_{H} for the (maximal) H-fixed quotient, that is to say, M_{H} = M/I(RH)M where I(RH) denotes the augmentation ideal of RH. For K ≤ H and a ∈ D(K), we write the image of a in D_{+}(H) as [K, a]_{H}. Clearly, [K, a]_{H} = [^{h}K,^{h}a]_{H} for h ∈ H and as an R-module, D_{+}(H) is generated by the elements [K, a]_{H} for K ≤ H and a ∈ D(K).

The maps are defined for L ≤ H ≤ G and g ∈ G as follows:

Conjugation:

c_{+g}^{H} : D_{+}(H) → D_{+}(^{g}H) where [K, a]_{H} ↦ [^{g}K,^{g}a]_{^{g}H}.

Restriction:

r_{+L}^{H} : D_{+}(H) → D_{+}(L) where [K, a]_{H} ↦ ∑_{h∈L\H/K}[L ∩^{h}K, r_{L∩^{h}K}^{^{h}K}(^{h}a)]_{L}.

Transfer:

t_{+L}^{H} : D_{+}(L) → D_{+}(H) where [N, b]_{L} ↦ [N, b]_{H}.

For a morphism f : D → E of restriction functors, we define f_{+} : D_{+} → E_{+} by f_{+H}([K, a]_{H}) = [K, f_{K}(a)]_{H} for K ≤ H and a ∈ D(K).

The plus constructions are related to each other by a morphism, called the mark homomorphism, denoted by ρ in [7],[8]. We write β for the mark homomorphism. It is defined as follows: Let D be a restriction functor and H ≤ G. Then

β_{H} := (π_{K} ∘ r_{+K}^{H})_{K≤H} : D_{+}(H) → (FD)^{+}(H)

where F : Res_{R}(G) → Con_{R}(G) is the forgetful functor and π_{K} is the projection

π_{K}[L, a]_{K} = a

if L = K and equal to zero otherwise. The mark homomorphism is an isomorphism if |G| is invertible in R and is injective if D_{+}(H) has trivial |H|-torsion for all H ≤ G. (cf. [7, Proposition 1.3.2])

The functors −^{+} and −_{+} have crucial use in constructing canonical induction formulae for Mackey functors. For further details, see [7] and [8], for applications see [4] and [9].

Two other constructions in the theory of Mackey functors that are used frequently are the bar construction and the twin functor. We review the definitions of these constructions.

Definition 2.2 Let M be a Mackey functor. The bar construction of M is the conjugation functor \overline{M} where for K ≤ G, we have

\overline{M}(K) = M(K)/ ∑_{L<K} Im(t_{L}^{K})

and the conjugation maps are inherited from those of M.

The bar construction composed with the functor −^{+} gives the twin functor T M of M (cf. [7, Section 1.1.2]). We have the following morphism between a Mackey functor and its twin. For K ≤ G and m ∈ M(K), we define

β_{K} : M(K) → T M(K)

where β_{K}(m) = (π_{L}(r_{L}^{K} m))_{L≤K} and

π_{K} : M(K) → \overline{M}(K)

is the quotient map. Note that the mark homomorphism is a special case of the morphism β : M → T M where we put M = D_{+} for a restriction functor D.

Let E and G be rings and α : E → G be a unital ring homomorphism. We can regard any G-module as an E-module by α. This induces a functor

res_{α} : G−mod → E−mod

called the generalized restriction. There are two functors in the opposite direction.

Induction: We regard G as a right E-module by fe = fα(e) for e ∈ E and f ∈ G. Then, for any (left) E-module M, we make G ⊗_{E} M a (left) G-module by f(f′ ⊗ m) = ff′ ⊗ m for m ∈ M. Note that, the action is well-defined as the natural action of G on itself commutes with the action of E on G. We call G ⊗_{E} M the induced module, written ind_{α}M, and obtain the generalized induction functor

ind_{α}− := G ⊗_{E} − : E−mod → G−mod

Coinduction: Now we regard G as a left E-module by ef = α(e)f for e ∈ E and f ∈ G. Then, for any (left) E-module M, we make Hom_{E}(G, M) a (left) G-module by (fϕ)(f′) = ϕ(f′f) for f, f′ ∈ G and ϕ ∈ Hom_{E}(G, M). Note that, the natural action of G on itself commutes with the action of E on G. We call Hom_{E}(G, M) the coinduced module , written coind_{α}M, and obtain the generalized coinduction functor

coind_{α} := Hom_{E}(G, −) : E−mod → G−mod

We recall the adjointness properties of these three functors:

Proposition 2.3 The induction functor ind_{α} is right adjoint of the restriction res_{α}. The coinduction functor coind_{α} is the left adjoint of the restriction res_{α}.

The proof of the proposition and further details can be found in [2, Section 3.3]. In all our applications, α will be an inclusion E ↪ G or a projection E → G = E/Δ for some ideal Δ of E. For the first case, we write the induction and coinduction functors as ind_{E}^{G} and coind_{E}^{G}, respectively. For the second case, we write induction and restriction as def_{G}^{E} and inf_{G}^{E}, respectively.

Finally, we recall the following well-known proposition.

Proposition 2.4 [1, Section 2.8] Let E and G be rings. Let M be a left G-module and let A be a G − E-bimodule and let N be a left E-module. Then, there is a natural isomorphism

Hom_{E}(N, Hom_{G}(A, M)) ≅ Hom_{G}(A ⊗_{E} N, M).

3 The Mackey triangle

In this section, we examine the relations between the algebras μ, τ, ρ and γ. Mainly, we explain the following triangle, which we call the Mackey triangle.

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Here the arrows ρ ↪ μ and τ ↪ μ denote the inclusions of algebras and so are γ → ρ and γ → τ. The arrows ρ ↠ γ and τ ↠ γ denote surjections explained in the next lemma, which also describes the identifications at the bottom of the triangle.

Lemma 3.1 Let J (ρ) be the two-sided ideal of the restriction algebra ρ generated by all nontrivial restriction maps. Then, there is an evident identification γ = ρ/J (ρ). Similarly, we make the identification γ = τ/J (τ) where J (τ) is generated by all non-trivial transfer maps.

Proof. Recall that the restriction algebra (resp. transfer algebra) is generated by c_{g} r_{H}^{K} where H ≤ K ≤ G and g ∈ G. As an R-module, J (ρ) is spanned by the elements c_{g} r_{H}^{K} where K < H. It is now clear that the quotient is isomorphic to the conjugation algebra. The last part can be proved similarly. □

The main property of the Mackey triangle is the following.

Theorem 3.2 (Mackey structure theorem) The τ − ρ-bimodule _{τ} μ_{ρ} is isomorphic to τ ⊗_{γ} ρ.

Proof. It is clear that τ ⊗_{γ} ρ is generated by the elements t_{^{g}J}^{H} ⊗c_{g}^{J} r_{J}^{K}. Now we show that τ ⊗_{γ} ρ is freely generated by these elements. To this aim, we decompose the left γ-module ρ as

γρ = ⨁_{K≤G, J≤_{K}K} γ r_{J}^{K}

and the right γ-module τ as

τ_{γ} = ⨁_{H≤G, I≤_{H}H} t_{I}^{H} γ.

Then, the tensor product becomes

τ ⊗_{γ} ρ = ⨁_{I≤_{H}H≤G, J≤_{K}K≤G} t_{I}^{H} γ ⊗_{γ} γ r_{J}^{K}
= ⨁_{I≤_{H}H≤G, J≤_{K}K≤G, L≤_{G}G} R t_{I}^{H} γ c^{[L]} ⊗_{γ} c^{[L]} γ r_{J}^{K}

where c^{[L]} = ∑_{L′=_{G}L} c^{L′}. Here c^{L} is the generator c_{1}^{L}.

To focus on each summand separately, fix H, K, L ≤ G. Then,

c^{[L]} γ r_{J}^{K} = ⨁_{x∈G/K, L^{x}=_{K}J} R c^{L} c_{x}^{J} r_{J}^{K}.

Indeed, the equality holds since J is taken up to K-conjugacy and c_{x}^{L} c^{J} = 0 unless L^{x} =_{K} J. Similarly,

t_{I}^{H} γ c^{[L]} = ⨁_{y∈H\G,^{y}L=_{H}I} R t_{I}^{H} c_{y}^{L} c^{L}.

Hence,

t_{I}^{H} γ c^{[L]} ⊗_{γ} c^{[L]} γ r_{J}^{K} = ⨁_{x, y} R t_{I}^{H} c_{y}^{L} ⊗ c_{x}^{J} r_{J}^{K}.

Therefore,

τ ⊗_{γ} ρ = ⨁_{H,K,I,J,L,x,y} R t_{I}^{H} c_{y}^{L} ⊗ c_{x}^{J} r_{J}^{K}
= ⨁_{H,K,I,J} ⨁_{g∈H\G/K,I^{g}=J} R t_{I}^{H} ⊗ c_{g}^{J} r_{J}^{K}.

Hence, we see that τ ⊗_{γ} ρ is freely generated over R by the elements t_{^{g}J}^{H} ⊗ c_{g}^{J} r_{J}^{K}. It is also clear from the last equation that given t_{^{g}J}^{H} ⊗ c_{g}^{J} r_{J}^{K} and t_{^{f} I}^{H} ⊗ c_{f}^{I} r_{I}^{K} then

t_{^{g}J}^{H} ⊗ c_{g}^{J} r_{J}^{K} = t_{^{f} I}^{H} ⊗ c_{f}^{I} r_{I}^{K}

if and only if HgK = HfK and J and I are H^{g} ∩K-conjugate. But this is equivalent to saying that t_{^{g}J}^{H} c_{g}^{J} r_{J}^{K} is equal to t_{^{f} I}^{H} c_{f}^{I} r_{I}^{K} as elements of the Mackey algebra (cf.[12, Proposition 3.2]). Hence the correspondence

Γ : τ ⊗_{γ} ρ → μ

given by Γ(t_{^{g}J}^{H}⊗c_{g}^{J} r_{J}^{K}) = t_{^{g}J}^{H} c_{g}^{J} r_{J}^{K} extends linearly to an isomorphism of R-modules. Evidently, the map Γ is compatible with the left action of the transfer algebra τ and the right action of the restriction algebra ρ. Thus Γ is an isomorphism of τ − ρ-bimodules from τ ⊗_{γ} ρ to _{τ} μ_{ρ}. □

Now as a result of these relations we obtain several induction, coinduction and restriction functors and some equivalences between them. As we shall see in the next section, some of these functors are also naturally equivalent to some well-known constructions. For the rest of this section, we prove some equivalences as consequences of Theorem 3.2. In the next lemma, which we state without proof, we collect some trivial but necessary observations about some of these functors:

Lemma 3.3 In the Mackey triangle, there are two inflation functors, inf_{γ}^{τ} and inf_{γ}^{ρ}. For a γmodule C, the τ-module inf_{γ}^{τ}C is the module C regarded as a τ-module by letting all non-trivial transfer maps t_{L}^{K} for L < K ≤ G act as zero maps. A similar result holds for the ρ-module inf_{γ}^{ρ}C. Moreover, the compositions res_{γ}^{τ}inf_{γ}^{τ} and res_{γ}^{ρ}inf_{γ}^{ρ} are both naturally equivalent to the identity functor on γ-mod.

For the rest of this section, we prove more equivalences. Most of the equivalences are consequences of the Mackey structure theorem.

Theorem 3.4 The following natural equivalences hold.

(i) ind_{γ}^{τ}res_{γ}^{ρ} ≅ res_{τ}^{μ}ind_{ρ}^{μ}.

(ii) coind_{γ}^{ρ}res_{γ}^{τ} ≅ res_{ρ}^{μ}coind_{τ}^{μ}.

Proof. The first equivalence is induced by the isomorphism Γ of τ − ρ-bimodules μ and τ ⊗_{γ} ρ defined in the proof of Theorem 3.2. Indeed

ind_{γ}^{τ}res_{γ}^{ρ} ≅ τ ⊗_{γ} ρ ⊗_{ρ} and res_{τ}^{μ}ind_{ρ}^{μ} ≅ _{τ} μ_{ρ} ⊗_{ρ} .

The induced equivalence is clearly natural. To prove the second equivalence, note that by the definition of coinduction,

coind_{γ}^{ρ}res_{γ}^{τ} = Hom_{γ}(ρ, Hom_{τ} (τ, −)).

Now applying Proposition 2.4, we obtain a natural equivalence

Υ : Hom_{γ}(ρ, Hom_{τ} (τ, −)) ≅ Hom_{τ} (τ ⊗_{γ} ρ, −)

of functors with values in R-mod. It is easy to check that for any τ-module E, the isomorphism Υ_{E} is compatible with conjugation and restriction maps. But, in that case the right hand-side of the last equation becomes

Hom_{τ} (τ ⊗_{γ} ρ, −) ≅ res_{ρ}^{μ}coind_{τ}^{μ}

since τ ⊗_{γ} ρ ≅ μ as left τ-modules. □

Corollary 3.5 The following equivalences hold.

(i) ind_{γ}^{τ} ≅ res_{τ}^{μ}ind_{ρ}^{μ}inf_{γ}^{ρ}.

(ii) coind_{γ}^{ρ} ≅ res_{ρ}^{μ}coind_{τ}^{μ}inf_{γ}^{τ}.

Proof. This follows from Theorem 3.4 and Lemma 3.3 by composing with the corresponding inflations. □

Finally, we have two more functors that are naturally equivalent to the identity functor on γ-mod. Let us write codef_{γ}^{ρ} for the left adjoint of the inflation inf_{γ}^{ρ}. Explicitly, for a ρ-module D and for K ≤ G, we have

codef_{γ}^{ρ}D(K) = ⋂_{L<K} Ker (r_{L}^{K} : D(K) → D(L)).

and the conjugation maps are obtained from those for the ρ-module D. The other functor is the deflation functor def_{γ}^{τ} induced by the map of Lemma 3.1. Note also that we have a deflation functor def_{γ}^{ρ} and a codeflation functor codef_{γ}^{τ}, but we shall not introduce these as we will not use them.

Proposition 3.6 The following equivalences hold:

def_{γ}^{τ}ind_{γ}^{τ} ≅ id_{γ} ≅ codef_{γ}^{ρ}coind_{γ}^{ρ}.

Proof. The equivalences follows easily from Lemma 3.3, since a left and a right adjoint of the identity functor and the identity functor are naturally equivalent to each other. □

4 Duality Theorems

Theorem 3.4 suggests a duality in the Mackey triangle. In this section, we clarify this duality. Following [11], we denote by −^{op}, the opposite functor, defined by

−^{op} : μ−mod → mod−μ

where for a left μ-module M, the right μ-module M^{op} is the same R-module M with the right Mackey functor structure given by

m(t_{^{g}J}^{H} c_{g} r_{J}^{K}) = (t_{J}^{K} c_{g^{−1}} r_{^{g}J}^{H})m

where t_{^{g}J}^{H} c_{g} r_{J}^{K} ∈ μ and m ∈ M(H).

We have another duality (cf. [11])

D_{μ} : μ−mod → mod−μ

where for a left μ-module M, we let D_{μ}M to be the right μ-module Hom_{R}(M, R) where μ acts on the right as usual. Note that D_{μ}M is the usual duality D^{∗} in module theory. Clearly, these functors can be defined in the reverse direction, and we can compose one with the other to obtain

D_{μ}^{op} : μ−mod → μ−mod.

Note that there is no ambiguity writing D_{μ}^{op} since the functors commute.

The functors D_{μ}M and −^{op} also induce functors on the modules of the subalgebras ρ and τ. Since −^{op} interchanges restriction and transfer maps, we obtain dualities

−^{op} : ρ−mod → mod−τ and −^{op} : τ−mod → mod−ρ.

On the other hand, the functor D_{μ} induces

D_{ρ} : ρ−mod → mod−ρ and D_{τ} : τ−mod → mod−τ.

The following theorem describes induction from right τ-modules and coinduction from right ρ-modules to right μ-modules.

Theorem 4.1 (The first duality theorem) Let D be a ρ-module and E be a τ-module. Then

(i) (ind_{ρ}^{μ}D)^{op} ≅ ind_{τ}^{μ}(D^{op}) where ind_{τ}^{μ} : mod−τ → mod−μ.

(ii) (coind_{τ}^{μ}E)^{op} ≅ coind_{ρ}^{μ}(E^{op}) where coind_{ρ}^{μ} : mod−ρ → mod−μ.

Proof. The first part is clear, since we have

(ind_{ρ}^{μ}D)^{op} = (μ ⊗_{ρ} D)^{op} ≅ D^{op} ⊗_{τ} μ = ind_{τ}^{μ}(D^{op}).

The second part can be proved similarly. □

Combining the above functors, we can define

Definition 4.2 The transfer-restriction duality is the equivalence

D_{ρ}^{op} := D_{ρ}^{−1} ∘ −^{op} : τ−mod → ρ−mod

of categories τ−mod ≅ ρ−mod. We call the inverse equivalence

D_{τ}^{op} := D_{τ}^{−1} ∘ −^{op} : ρ−mod → τ−mod

the restriction-transfer duality.

Finally, note that D_{μ}^{op} induces a duality D_{γ}^{op} on γ-modules. The following theorem describes the duality we promised earlier. Theorem 4.3 (Restriction-transfer duality) Let D be a ρ-module and E be a τ-module. Then

(i) D_{μ}^{op}(ind_{ρ}^{μ}D) ≅ coind_{τ}^{μ}(D_{τ}^{op}D).

(ii) D_{μ}^{op}(coind_{τ}^{μ}E) ≅ ind_{ρ}^{μ}(D_{ρ}^{op}E). Proof. The first part follows from Proposition 2.3 as we have

D_{μ}^{op}(ind_{ρ}^{μ}D) = Hom_{R}((ind_{ρ}^{μ}D)^{op}, R)
= Hom_{R}(ind_{τ}^{μ}(D^{op}), R)
≅ Hom_{τ} (μ, Hom_{R}(D^{op}, R))
= coind_{τ}^{μ}(D_{τ}^{op}D).

Note that although the above isomorphism is an isomorphism of R-modules, it is easily checked that it is an isomorphism of left Mackey functors. The second statement can be proved similarly. □

In the next theorem, we collect together some more dualities relating induction, coinduction and restriction. The theorem and any other duality can be proved in the same way. Theorem 4.4 Let M be a μ-module, E be a τ-module and C be a γ-module. Then

(i) D_{τ}^{op}(res_{ρ}^{μ}M) ≅ res_{τ}^{μ}(D_{μ}^{op}M).

(ii) D_{ρ}^{op}(res_{τ}^{μ}M) ≅ res_{ρ}^{μ}(D_{μ}^{op}M). (iii) D_{τ}^{op}(inf_{γ}^{ρ}C) ≅ inf_{γ}^{τ}(D_{γ}^{op}C) . (iv) D_{ρ}^{op}(inf_{γ}^{τ}C) ≅ inf_{γ}^{ρ}(D_{γ}^{op}C).

(v) D_{γ}^{op}(def_{γ}^{τ}E) ≅ codef_{γ}^{ρ}(D_{ρ}^{op}E).

Let us end with an abstraction of the above situation. Let Υ be a finitely generated R-algebra such that it has subalgebras Υ_{↑}, Υ_{↓} and Υ_{−} with the following two properties.

(i) The subalgebras together with Υ form the following triangle.

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where the maps are as explained in the previous section. (ii) The structure theorem, _{Υ_{↑}}Υ_{Υ_{↓}} ≃ Υ_{↑} ⊗_{Υ_{−}} Υ_{↓}, holds.

Then the results in Section 3 and Section 4 hold for the modules of the algebras Υ, Υ_{↑}, Υ_{↓} and Υ_{−} and for induction, coinduction and restriction functors. Moreover our classification and description of simple Mackey functors can be modified for the simple modules of the algebra Υ.

There are at least two more algebras having this structure. The first example is the algebra μ_{A} associated to a Green functor A (see [3] for the definition). Note that the Mackey algebra is obtained by taking A = B^{G}, the Burnside Mackey functor [3].

Another occurrence of this structure is in the biset functors, introduced by Bouc [5]. As mentioned in the introduction we shall adopt the methods of this paper to the analogous algebra for biset functors.

5 Plus constructions via induction and coinduction

In this section, we show that under the equivalence of categories μ_{R}(G) ≅ Mack_{R}(G), the plus constructions −_{+} and −^{+} are realizable in terms of generalized restriction, generalized induction and generalized coinduction. Moreover the well-known fixed-point functor and the fixed-quotient functor [11], and the twin functor [10] have similar descriptions. We begin by proving our first identification.

Theorem 5.1 The functors −_{+} and ind_{ρ}^{μ} are naturally equivalent.

Proof. To specify a natural equivalence Φ : ind_{ρ}^{μ} → −_{+}, we must specify a map of μ-modules

ϕ_{D} : ind_{ρ}^{μ} D → D_{+}

for any ρ-module D and show that it is natural in D. To do that, we must specify an isomorphism of R-modules

ϕ_{D,H} : ind_{ρ}^{μ} D(H) → D_{+}(H)

for any subgroup H ≤ G which is compatible with the actions of transfer, restriction and conjugation. Now

ind_{ρ}^{μ} D(H) = ⨁_{K≤_{H}H}{t_{K}^{H} ⊗_{ρ} a : a ∈ D(K)}

where the notation indicates that K runs over representatives of the conjugacy classes of subgroups of H. Also,

D_{+}(H) = ⨁_{K≤_{H}H}{[K, a]_{H} : a ∈ D(K)}.

We have t_{K}^{H} ⊗a_{K} = 0 if and only if a_{K} ∈ I(N_{H}(K))D(K), where I(N_{G}(H)) is the augmentation ideal as before. But this is equivalent to the condition that [K, a]_{H} = 0. So we can define Φ_{D,H} by

Φ_{D,H}(t_{K}^{H} ⊗_{ρ} a) = [K, a]_{H}.

Thus, we have defined an R-module isomorphism Φ_{D} = (Φ_{D,H})_{H≤G} from ind_{ρ}^{μ} D to D_{+}. Now we show that Φ_{D} is compatible with the actions of conjugation, restriction and transfer. We must also check that Φ is natural.

Given L ≤ G and a ∈ D(K), then

Φ_{D,L}(r_{L}^{H}(t_{K}^{H} ⊗ a)) = Φ_{D,L}( ∑_{h∈L\H/K} t_{L∩^{h}K}^{L} r_{L∩^{h}K}^{^{h}K} c_{h}^{K} ⊗ a)
= ∑_{h∈L\H/K} Φ_{D,L}(t_{L∩^{h}K}^{L} ⊗ r_{L∩^{h}K}^{^{h}K} c_{h}^{K} a)
= ∑_{h∈L\H/K}[L ∩^{h}K, r_{L∩^{h}K}^{^{h}K} c_{h}^{K} a]_{L}
= r_{+L}^{H}[K, a]_{H}
= r_{+L}^{H}Φ_{D,H}(t_{K}^{H} ⊗ a).

We have established compatibility with restriction, Φ_{D}r_{L}^{H} = R_{L}^{H}Φ_{D}. Compatibility with conjugation and transfer can be shown similarly (and more easily).

Finally, for the naturality, consider a map of ρ-modules f : D → D′. The maps of μ-modules

ind_{ρ}^{μ}f : ind_{ρ}^{μ}D → ind_{ρ}^{μ}D′, f_{+} : D_{+} → D′_{+}

are given by (ind_{ρ}^{μ}f)_{H} (t_{K}^{H} ⊗ a) = t_{K}^{H} ⊗ f_{K}(a) and (f_{+})_{H}([K, a]_{H})) = [K, f_{K}(a)]_{H}. Hence,

Φ_{D′}(ind_{ρ}^{μ}f (t_{K}^{H} ⊗ a)) = ϕ_{D′}(t_{K}^{H} ⊗ f_{K}(a)) = [K, f_{K}(a)]_{H} = f_{+}([K, a]_{H}) = f_{+}(Φ_{D}(t_{K}^{H} ⊗ a)).

So Φ_{D′} ∘ ind_{ρ}^{μ}f = f_{+} ∘ Φ_{D}, in other words, Φ is natural. □

Theorem 5.2 The functors −^{+} and coind_{τ}^{μ} inf_{γ}^{τ} are naturally equivalent.

Proof. As in the previous proof, to specify a natural equivalence Ψ : coind_{τ}^{μ} inf_{γ}^{τ} → −^{+}, we must specify a map of μ-modules

Ψ_{C} : coind_{τ}^{μ} inf_{γ}^{τ}C → C^{+}

for any γ-module C and show that it is natural in C. In order to do that, we must specify an R-module isomorphism

Ψ_{C,H} : coind_{τ}^{μ} inf_{γ}^{τ}C(H) → C^{+}(H)

for any subgroup H ≤ G and show that it is compatible with the action of transfer, restriction and conjugation. Now

coind_{τ}^{μ} inf_{γ}^{τ}C(H) = Hom_{τ} (μ, inf_{γ}^{τ}C)(H) = Hom_{τ} (μ c^{H}, inf_{γ}^{τ})

where inf_{γ}^{τ}C = ⊕_{J≤H}C(J). Recall that any element of μ c^{H} is a linear combination of elements of the form t_{^{g}J}^{K} c_{g} r_{J}^{H} where g ∈ G and J ≤ K^{g} ∩ H. But, for such an element and for a map ϕ : μ c^{H} → inf_{γ}^{τ}C of τ-modules, we have

ϕ(t_{^{g}J}^{K} c_{g} r_{J}^{H}) = t_{^{g}J}^{K} ϕ(c_{g} r_{J}^{H}) = 0

unless K = ^{g}J. Indeed, t_{L}^{K} annihilates the τ-module inf_{γ}^{τ}C if L ≠ K. Also, if K = ^{g}J, then

ϕ(c_{g} r_{J}^{H}) = c_{g} ϕ(r_{J}^{H})

that is, the value of ϕ at c_{g} r_{J}^{H} is determined by the value of ϕ at r_{J}^{H}. Moreover, for any h ∈ H, we have

ϕ(r_{^{h}J}^{H}) = ϕ(r_{^{h}J}^{H} c_{h}^{H}) = ϕ(c_{h}^{J} r_{J}^{H}) = c_{h}^{J}(ϕ(r_{J}^{H})).

Now recall that

C^{+}(H) = ( ∏_{J≤H} C(J))^{H} = {(x_{J})_{J≤H} ∈ ∏_{J≤H} C(J) : ^{h}(x_{J}) = x_{^{h}J} for J ≤ H, h ∈ H}.

So, we can define

Ψ_{C,H}(ϕ) = (ϕ(r_{J}^{H}))_{J≤H}.

The map Ψ_{C,H} is an isomorphism of R-modules from coind_{τ}^{μ} inf_{γ}^{τ}C(H) to C^{+}(H) with the inverse given by

Ψ_{C,H}^{−1}(X) = ϕ_{X}.

Here, X = (x_{J})_{J≤H} and ϕ_{X} is the map defined by ϕ_{X}(c_{g} r_{J}^{H}) = ^{g}(x_{J}). Thus, we have defined an R-module isomorphism Ψ_{C} : coind_{τ}^{μ} inf_{γ}^{τ}C → C^{+}. We must show that Ψ_{C} is compatible with the actions of conjugation, restriction and transfer. Also, we must check that Ψ is natural in C.

Given J ≤ H ≤ K ≤ G and ϕ ∈ coind_{τ}^{μ} inf_{γ}^{τ}C(H), then

Ψ_{C,H}(t_{H}^{K}ϕ) = ((t_{H}^{K}ϕ)(r_{J}^{K}))_{J≤K}
= (ϕ(r_{J}^{K} t_{H}^{K}))_{J≤K}
= ( ∑_{x∈J\K/H}(t_{J∩^{x}H}^{J} c_{x})(ϕ(r_{J^{x}∩H}^{H})))_{J≤K}
= (  ∑_{x∈J\K/H, J∩^{x}H=J} c_{x}(ϕ(r_{J^{x}}^{H})))_{J≤K}
= ( ∑_{x∈K/H, J^{x}≤H} c_{x}(ϕ(r_{J^{x}}^{H})))_{J≤K}
= t_{H}^{+K}(ϕ(r_{J}^{H}))_{J≤H} = t_{H}^{+K}Ψ_{C,H}(ϕ).

We have established compatibility with transfer, Ψ_{C,K} ∘ t_{H}^{K} = t_{H}^{K}Ψ_{C,H}. Compatibility with restriction and conjugation can be proved similarly. Finally, one can check that the transformation Ψ is natural as above. □

By Theorems 3.4 and 5.2, we obtain an explicit description of the functor coind_{τ}^{μ}.

Theorem 5.3 Let E be a transfer functor. Then for H ≤ G, we have

coind_{τ}^{μ} E(H) ≅ ( ∏_{L≤H} E(L))^{H}.

The actions of conjugation and restriction are the same as the actions of conjugation and restriction for the functor E^{+}, respectively, and the transfer map is defined for ϕ ∈ coind_{τ}^{μ} E(H) and K ≥ H as

(t_{H}^{K}ϕ)(r_{J}^{K}) = ∑_{k∈J\K/H}(t_{J∩^{k}H}^{J} c_{k})(ϕ(r_{J^{k}∩H}^{H})).

Proof. By Theorem 3.4, there is an isomorphism

res_{ρ}^{μ} coind_{τ}^{μ} E ≅ coind_{γ}^{ρ} res_{γ}^{τ} E

of ρ-modules. Now by Corollary 3.5, we obtain

res_{ρ}^{μ} coind_{τ}^{μ} E ≅ res_{ρ}^{μ} coind_{τ}^{μ} inf_{γ}^{τ} res_{γ}^{τ} E.

Now by Theorem 5.2, the right hand side is (res_{γ}^{τ} E)^{+} regarded as a restriction functor. Hence the isomorphism

coind_{τ}^{μ} E(H) ≅ ( ∏_{L≤H} E(L))^{H}

holds. Evidently, the actions of conjugation and restriction are the same as those for the right hand side. Finally it is clear that the action of transfer is given as above. □

Given an RG-module V, we denote by D_{V} the conjugation functor where D_{V} (1) = V and D_{V} (H) = 0 for 1 ≠ H ≤ G.

Proposition 5.4 The following isomorphisms hold.

(i) FQ_{V} ≅ ind_{ρ}^{μ}inf_{γ}^{ρ} D_{V}.

(ii) FP_{V} ≅ coind_{τ}^{μ}inf_{γ}^{τ} D_{V}.

Proof. It is clear from the construction of the fixed-point functor and the fixed-quotient functor that we have the following isomorphisms (cf. [11, Section 6]):

FP_{V} ≅ (D_{V})^{+} and FQ_{V} ≅ (inf_{γ}^{ρ}D_{V})_{+}.

Now the result follows from Theorem 5.1 and Theorem 5.2. □

Corollary 5.5 [11, Proposition 6.1] The functor ind_{ρ}^{μ}inf_{γ}^{ρ} D_{V} is left adjoint to the functor F : μ−mod → RG−mod which sends a Mackey functor M to the RG-module c_{1}^{1} M = M(1). The right adjoint of F is coind_{τ}^{μ}inf_{γ}^{τ} D_{V}.

Proof. We have inf_{γ}^{ρ} D_{V} (K) = 0 for each subgroup 1 < K ≤ G and inf_{γ}^{ρ} D_{V} (1) = V. Therefore

Hom_{μ}(ind_{ρ}^{μ}inf_{γ}^{ρ} D_{V} , M) ≅ Hom_{ρ}(inf_{γ}^{ρ} D_{V} , res_{ρ}^{μ}M)
≅ Hom_{RG}(inf_{γ}^{ρ} D_{V} (1), res_{ρ}^{μ}M(1))
≅ Hom_{RG}(V, M(1))
≅ Hom_{RG}(V, F M).

The second statement is proved similarly. □

Remark 5.6 It is possible to define the fixed-point functor and the fixed-quotient functor for the right μ-modules, as well as the other constructions. For example, by the Duality Theorem 4.1, we see that

D_{μ}^{op}(ind_{ρ}^{μ}inf_{γ}^{ρ} D_{V}) = coind_{τ}^{μ}inf_{γ}^{τ}(D_{γ}^{op}D_{V})

which is the part (iii) of Proposition 4.1 in [12]. Also, note that we can define a fixed-point functor and a fixed-quotient functor for the right μ-modules using the functor −^{op}. In that case, for a right RG-module V, we define

V FQ := ind_{τ}^{μ} inf_{γ}^{τ}_{V} D.

By the Duality Theorem 4.1, we obtain

V FQ = (ind_{ρ}^{μ}inf_{γ}^{ρ} D_{V^{op}})^{op} = ind_{τ}^{μ}inf_{γ}^{τ}D_{V}

We can define _{V} FP similarly.

Finally, we have the following proposition.

Proposition 5.7 The bar construction \overline{?}, defined in Definition 2.2 is naturally equivalent to def_{γ}^{τ} res_{τ}^{μ}.

Proof. This is immediate from the equality

(J (τ)M)(H) = ∑_{L<H} t_{L}^{H}M(L)

for H ≤ G. □

Corollary 5.8 The twin functor T is naturally equivalent to coind_{τ}^{μ} inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ}. □

The morphism β between a Mackey functor and its twin can be expressed in terms of the above equivalence.

Proposition 5.9 Let M be a Mackey functor. The morphism

β : M → coind_{τ}^{μ} inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} M

as an element in Hom_{μ}(M, coind_{τ}^{μ} inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} M) is induced by the identity endomorphism id_{def_{γ}^{τ} res_{τ}^{μ} M} in Hom_{γ}(def_{γ}^{τ} res_{τ}^{μ} M, def_{γ}^{τ} res_{τ}^{μ} M).

Proof. By Proposition 2.3

Hom_{μ}(M, coind_{τ}^{μ} inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} M) ≅ Hom_{τ} (res_{τ}^{μ}M, inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} M)
≅ Hom_{γ}(def_{γ}^{τ} res_{τ}^{μ} M, def_{γ}^{τ} res_{τ}^{μ} M).

Now the counit of the adjunction

Hom_{τ} (res_{τ}^{μ}M, inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} M) ≅ Hom_{γ}(def_{γ}^{τ} res_{τ}^{μ} M, def_{γ}^{τ} res_{τ}^{μ} M)

is given by composition with the quotient map. That is, for ϕ : def_{γ}^{τ} res_{τ}^{μ} M → def_{γ}^{τ} res_{τ}^{μ} M, the corresponding morphism \overline{ϕ} : res_{τ}^{μ} M → inf_{γ}^{τ}def_{γ}^{τ} res_{τ}^{μ} M is given by

\overline{ϕ}_{H}(m) = ϕ_{H}(π_{H}(m))

where m ∈ M(H) and π : res_{τ}^{μ} M → def_{γ}^{τ} res_{τ}^{μ} M is the quotient map.

On the other hand, the counit of the adjunction

Hom_{μ}(M, coind_{τ}^{μ} inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} M) ≅ Hom_{τ} (res_{τ}^{μ}M, inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} M)

is given by composition with the restriction maps. Explicitly, for ψ : res_{τ}^{μ} M → inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} M, the corresponding morphism \overline{ψ} : M → coind_{τ}^{μ}inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} M is given by

\overline{ψ}_{H}(m) = (ψ_{K}(r_{K}^{H}m))_{K≤H}

where m ∈ M(H).

Now put ϕ = id. Then \overline{ϕ}_{H}(m) = π_{H}(m) is the quotient map. Then, put ψ = \overline{ϕ} and get \overline{ψ}_{H}(m) = (ψ_{K}(r_{K}^{H}m))_{K≤H} = (π_{K}(r_{K}^{H}m))_{K≤H}, which coincides with the definition of the morphism β defined in Section 2. □

Since the mark homomorphism is a special case of the morphism β, we have the following corollary.

Corollary 5.10 Let D be a ρ-module. The mark homomorphism

β : ind_{ρ}^{μ}D → coind_{τ}^{μ}inf_{γ}^{τ}res_{γ}^{ρ}D

is induced by the identity endomorphism id_{res_{γ}^{ρ}D} of the γ-module res_{γ}^{ρ}D.

Proof. Let us put M = ind_{ρ}^{μ}D for some ρ-module D. Then by part (i) of Theorem 3.4 and by Proposition 3.6

coind_{τ}^{μ} inf_{γ}^{τ} def_{γ}^{τ} res_{τ}^{μ} ind_{ρ}^{μ}D ≅ coind_{τ}^{μ} inf_{γ}^{τ} res_{γ}^{ρ} D

Also the quotient map π above coincides with the projection map π since def_{γ}^{τ} res_{τ}^{μ} M = D. □

6 Simple Mackey functors

Throughout this section, we assume that R is a field. In [11], Thévenaz and Webb established a bijective correspondence between the G-classes of the simple pairs (H, V) where H ≤ G and V a simple R\overline{N}_{G}(H)-module where \overline{N}_{G}(H) := N_{G}(H)/H and the isomorphism classes of the simple Mackey functors. We denote by S_{H,V} the simple Mackey functor corresponding to the pair (H, V), under this correspondence.

To illustrate the usefulness of our module-theoretic approach we give an alternative proof to this result by realizing simple Mackey functors as quotients of induced simple restriction functors. As we shall see below, the classification of simple restriction functors is trivial. Then we give another new description of the simple Mackey functor S_{H,V} as the unique minimal subfunctor of the Mackey functor coind_{τ}^{μ} S_{H,V}^{τ}, where S_{H,V}^{τ} is a simple transfer functor, introduced below.

Throughout this section, let H ≤ G and V be a simple R\overline{N}_{G}(H)-module. We write S_{H,V}^{γ} for the conjugation functor defined for K ≤ G by

S_{H,V}^{γ} (K) = ^{g}V if K = ^{g}H

and zero otherwise. We also write S_{H,V}^{ρ} = inf_{γ}^{ρ} S_{H,V}^{γ} and S_{H,V}^{τ} = inf_{γ}^{τ} S_{H,V}^{γ}.

Proposition 6.1 [7, Remark 1.6.6], [4, Proposition 3.2] The followings hold.

(i) The conjugation functor S_{H,V}^{γ} is simple. Moreover, any simple conjugation functor is isomorphic to S_{H,V}^{γ} for some simple pair (H, V).

(ii) The restriction functor S_{H,V}^{ρ} is simple. Moreover, any simple restriction functor is isomorphic to S_{H,V}^{ρ} for some simple pair (H, V).

(iii) The transfer functor S_{H,V}^{τ} is simple. Moreover, any simple transfer functor is isomorphic to S_{H,V}^{τ} for some simple pair (H, V).

We recall, without proof, the description of the simple Mackey functors S_{H,V} := S_{H,V}^{G} from [11]. Since the Mackey algebra μ_{R}(H) is a (non-unital) subalgebra of the Mackey algebra μ_{R}(G) for H ≤ G and the Mackey algebra μ_{R}(G/H) is a quotient of the Mackey algebra μ_{R}(G) for H ⊴G, we obtain an induction functor ind_{μ_{R}(H)}^{μ_{R}(G)} and an inflation functor inf_{μ_{R}(G/H)}^{μ_{R}(G)}. Explicit descriptions of these functors are given in [11, Section 4,5].

Lemma 6.2 [11, Lemma 8.1]Let H be a subgroup of G and V be a simple RN̄_{G}(H). Then

(i) The functor M = ind_{N_{G}(H)}^{G} inf_{N̄_{G}(H)}^{N_{G}(H)}F P_{V} has a unique minimal subfunctor S_{H,V}^{G} generated by M(H) = V.

(ii) The functor ind_{N_{G}(H)}^{G} inf_{N̄_{G}(H)}^{N_{G}(H)}F Q_{V} has a unique maximal subfunctor. Moreover, the quotient is isomorphic to S_{H,V}^{G}.

Now we want to state the main result of this section. For this we need the following notation. Let M be a Mackey functor and H ≤ G be a minimal subgroup for M, that is, M(L) = 0 for L < H and M(H) ≠ 0. After [11], we define two subfunctors of M as follows:

I_{M(H)}(K) = ∑_{L≤K: L=_{G}H} Im(t_{L}^{K} : M(L) → M(K))

and

K_{M(H)}(K) = ⋂_{L≤K: L=_{G}H} Ker(r_{L}^{K} : M(K) → M(L)).

Theorem 6.3 We have the following isomorphisms of Mackey functors

S_{H,V}^{G} ≃ ind_{ρ}^{μ}S_{H,V}^{ρ} /K_{ind_{ρ}^{μ}S_{H,V}^{ρ} (H)} ≃ I_{coind_{τ}^{μ} S_{H,V}^{τ} (H)}.

We prove the theorem in several steps. The first step is the following lemma.

Lemma 6.4 The subfunctor K = K_{ind_{ρ}^{μ}S_{H,V}^{ρ} (H)} of the Mackey functor ind_{ρ}^{μ}S_{H,V}^{ρ} is the unique maximal subfunctor of ind_{ρ}^{μ}S_{H,V}^{ρ}.

Proof. Let T be a proper subfunctor of ind_{ρ}^{μ}S_{H,V}^{ρ}. We are to show that T ≤ K, that is

T (K) ⊂ ⋂_{L≤K: L=_{G}H} Ker(r_{L}^{K})

for any K ≤ G. So, we must show that for each K ≤ G and any x ∈ T (K), we have r_{L}^{K}x = 0 for all H =_{G} L ≤ K. But since ind_{ρ}^{μ}S_{H,V}^{ρ} (H) = V, it is evident that T (L) = 0 for any L =_{G} H. Indeed, otherwise T (H) = V as V is a simple RN̄_{G}(H)-module. But, by definition of the action of t_{L}^{K}, the functor ind_{ρ}^{μ}S_{H,V}^{ρ} is generated by the images of the transfer maps t_{L}^{K} for H =_{G} L ≤ K, that is, we have I_{ind_{ρ}^{μ}S_{H,V}^{ρ} (H)} = ind_{ρ}^{μ}S_{H,V}^{ρ}. Hence the subfunctor T containing the subfunctor generated by T (H) = V is not proper, contradicting our assumption. Thus, T ≤ K as required. □

We denote the simple quotient of ind_{ρ}^{μ}S_{H,V}^{ρ} by

S̃_{H,V} = ind_{ρ}^{μ}S_{H,V}^{ρ} /K.

Note that if (K, W) is another simple pair, then S̃_{K,W} is not isomorphic to S̃_{H,V}. Indeed, since K(H) = 0, the subgroup H is still a minimal subgroup of the quotient S̃_{H,V} and similarly, K is a minimal subgroup of S̃_{K,W}. Hence for K ≠ H, the simple modules S̃_{H,V} and S̃_{K,W} are non-isomorphic. Also, for K = H, any morphism S̃_{H,V} → S̃_{K,W} of Mackey functors induces a map V → W of RN̄_{G}(H)-modules. But, by the Schur’s lemma, any such map is either an isomorphism or the zero map. Thus S̃_{H,V} is not isomorphic to S̃_{K,W} unless H = K and V ≅ W.

Having the above description, we get another proof of Thévenaz and Webb’s classification theorem:

Theorem 6.5 Any simple Mackey functor is isomorphic to S̃_{H,V} for some simple pair (H, V).

Proof. Let S be a simple Mackey functor with a minimal subgroup H and S(H) = V. It suffices to show that there is a non-zero morphism of Mackey functors S̃_{H,V} → S. We show that there is a morphism of Mackey functors F : ind_{ρ}^{μ} S_{H,V}^{ρ} → S such that F_{H} ≠ 0. By Proposition 2.4, we have

Hom_{μ}(ind_{ρ}^{μ} S_{H,V}^{ρ} , S) ≃ Hom_{ρ}(S_{H,V}^{ρ} , res_{ρ}^{μ}S).

But, S_{H,V}^{ρ} (K) = 0 unless K =_{G} H. So the identity map id_{V} : V → V of RN̄_{G}(H)-modules induces a non-zero map f : S_{H,V}^{ρ} → res_{ρ}^{μ}S of ρ-modules. Hence the corresponding map F ∈ Hom_{μ}(ind_{ρ}^{μ} S_{H,V}^{ρ} , S) is non-zero. Moreover, since ind_{ρ}^{μ}S_{H,V}^{ρ} (H) = V, we have F_{H} = f_{H} = id ≠ 0. Thus, the induced morphism F̃ : S̃_{H,V} → S is non-zero, as required. □

Hereafter, we identify S̃_{H,V} with S_{H,V}^{G} and write S_{H,V} when the group G is understood. We complete the proof of Theorem 6.3 by the following lemma.

Lemma 6.6 The subfunctor I = I_{coind_{τ}^{μ} S_{H,V}^{τ} (H)} generated by I(H) = V is the unique minimal subfunctor. Moreover, the subfunctor I is isomorphic to S_{H,V}.

Proof. Let T be a non-zero subfunctor of coind_{τ}^{μ} S_{H,V}^{τ}. We must show that I ≤ T. It suffices to show that T (H) ≠ 0. Indeed, in that case, since coind_{τ}^{μ} S_{H,V}^{τ} (H) = V is simple, T (H) = V and hence I ≤ T. But K_{coind_{τ}^{μ} S_{H,V}^{τ} (H)} = 0 by the definition of the map r_{H}^{K} and the diagram

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commutes for all g ∈ G. That is to say, r_{H}^{K}T (K) ≠ 0, or T (H) ≠ 0. The last claim follows from the classification theorem 6.5. □

To find the modules S_{H,V} (K) for K ≤ G, we need to know the subfunctor K of ind_{ρ}^{μ} S_{H,V}^{ρ}. In particular, when K is zero, we get a more explicit description. The following is a characterization of the subfunctor K.

Lemma 6.7 The subfunctor K of ind_{ρ}^{μ}S_{H,V}^{ρ} (K) coincides with the kernel of the mark homomorphism β : ind_{ρ}^{μ}S_{H,V}^{ρ} → coind_{τ}^{μ}S_{H,V}^{τ}. Moreover, the subfunctor I of coind_{τ}^{μ} S_{H,V}^{τ} is the image of β.

Proof. As K is the unique maximal subfunctor of ind_{ρ}^{μ}S_{H,V}^{ρ} (K), we have ker β ⊂ K. So, it suffices to show the inverse inclusion. Given K ≤ G and x ∈ K(K), then

β_{K}(x) = (η_{L}(r_{L}^{K} x))_{L≤K,L=_{G}H} = 0

since r_{L}^{K} x = 0 by definition of K(K). Therefore, K ⊂ ker β. The second claim is easy since β_{H} is identical. □

Now, using the next proposition from [7], and the above identification of the subfunctor K, we describe K, in some cases.

Proposition 6.8 (cf. [7, Proposition 1.3.2], [10, Section 3]) The mark homomorphism β_{K} is injective if ind_{ρ}^{μ}S_{H,V}^{ρ} (K) has trivial |K|-torsion. It is an isomorphism if |K| is invertible in R.

Corollary 6.9 (i) If ind_{ρ}^{μ}S_{H,V}^{ρ} (K) has trivial |K|-torsion, then K(K) = 0.

(ii) If ind_{ρ}^{μ}S_{H,V}^{ρ} (K) has trivial |K|-torsion for all K ≤ G, then ind_{ρ}^{μ} S_{H,V}^{ρ} is simple.

(iii) If |G| is invertible in R, then ind_{ρ}^{μ} S_{H,V}^{ρ} ≅ coind_{τ}^{μ} S_{H,V}^{τ} is simple for any simple pair (H, V).

Remark 6.10 In the case that |G| is invertible in R, we get two different descriptions of S_{H,V} (K) for K ≤ G. By Corollary 6.9 and proof of Lemma 6.7, we have

S_{H,V} (K) = ( ⨁_{L≤K: L=^{g}H} ^{g}V )_{K}

with the maps t_{K}^{N} and r_{K}^{N} given explicitly in Section 2. Also, by Corollary 6.9, we have

S_{H,V} (K) = ( ∏_{L≤K: L=^{g}H} ^{g}V )^{K}

with the maps t_{K}^{N} and r_{K}^{N} given explicitly in Section 2.

7 Semisimplicity

Throughout this section, suppose R is a field in which |G| is a unit. It is well-known that the Mackey algebra over R is semisimple (see [11], [12] and [7]). The first proof by Thévenaz and Webb [11] is constructive and uses the semisimplicity of the twin functor. In this section we reprove this result by giving a shorter proof of the fact that, in this case, the twin functor of a Mackey functor is isomorphic to itself.

Definition 7.1 (Thévenaz [10]) Let M be a Mackey functor. A subgroup H ≤ G is called a primordial subgroup for M if def_{γ}^{τ}res_{τ}^{μ}M(H) ≠ 0.

Recall, without proof, the following lemma.

Lemma 7.2 [11, Lemma 9.4] Let M be a Mackey functor and χ be a subconjugacy closed family of subgroups of G. Then,

M = Ker r_{χ} ⊕ Im t_{χ}

where

Ker r_{χ}(K) = ⋂_{L≤K, L∈χ} Ker r_{L}^{K} and Im t_{χ}(K) = ∑_{L≤K, L∈χ} Im t_{L}^{K}

are Mackey subfunctors.

As a consequence of this lemma, we obtain the following decomposition.

Lemma 7.3 Let P = {H_{0}, H_{1}, …, H_{n}} be the set of all primordial subgroups of a Mackey functor M taken up to conjugacy and indexed such that for i < j, no G-conjugate of H_{j} is contained in H_{i}. Let T_{i} denotes the subfunctor of M generated by def_{γ}^{τ}res_{τ}^{μ}M(H_{i}). Then

M ≅ ⨁_{H_{i}∈P} T_{i}

as Mackey functors.

Proof. By Lemma 7.2, we have

M = T_{0} ⊕ Ker r_{[H_{0}]}

where

Ker r_{[H_{0}]} = Ker r_{χ} and T_{0} = Im t_{χ}.

Here [H_{0}] is the set of all G-conjugates of H_{0} and χ is the subconjugacy closure of [H_{0}]. Indeed, we have the equalities since H_{0} is a minimal subgroup for M. We denote Ker r_{[H_{0}]} by N_{0}. Then, clearly, N_{0}(H_{0}) = 0 and N_{0}(H_{1}) = (def_{γ}^{τ}res_{τ}^{μ}M)(H_{1}). Therefore, by Lemma 7.2 we obtain

N_{0} = T_{1} ⊕ N_{1}

where N_{1} = Ker r_{[H_{1}]}. Note that H_{1} is a minimal subgroup for N_{0}. Applying the same procedure, we obtain

M = ⨁_{H_{i}∈P} T_{i}

as required. □

Let M_{i} denote the conjugation functor generated by M_{i}(H_{i}) = def_{γ}^{τ}res_{τ}^{μ}M(H_{i}).

Lemma 7.4 There is an isomorphism of Mackey functors

T_{i} ≅ coind_{τ}^{μ} inf_{γ}^{τ}M_{i}.

Proof. Decomposing M_{i} into simple summands and applying Corollary 6.9 to each summand, we obtain

ind_{ρ}^{μ} inf_{γ}^{ρ}M_{i} ≅ coind_{τ}^{μ} inf_{γ}^{τ}M_{i}

where the isomorphism is given by the mark homomorphism. Note that we can decompose M_{i} into simple summands since it is clear that the conjugation algebra for G over R is semisimple when |G| is invertible in R.

Now consider the following triangle.

indμ ρ infρ γM^{i} coindμ τ infτγM^{i} T_{i} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ϕ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ψ ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} β

where β is the mark homomorphism and ψ is the induction morphism defined by ψ(t_{L}^{K} ⊗ v) = t_{L}^{K}v for H =_{G} L ≤ K ≤ G and v ∈ M_{i}(L) . The map ψ is a morphism of Mackey functors since M_{i} is a minimal subgroup both for T_{i} and for ind_{ρ}^{μ} inf_{γ}^{ρ}M_{i} (thus ψ commutes with restriction). The map ϕ is given by

ϕ_{K}(w) = (r_{L}^{K} w)_{L≤K,L=_{G}H}

where K ≤ G and w ∈ T_{i}(K). Note that ψ is surjective since T_{i} is generated by its value on the conjugacy class of H_{i}. Also, since t_{L}^{K} acts as the zero map on inf_{γ}^{τ}M_{i} for L ≠ K, the composition ϕ ∘ ψ is the mark homomorphism, that is, the triangle commutes. Moreover since β is injective, the map ψ is also injective. Hence it is an isomorphism. Now it follows that ϕ = β ψ^{−1} is also an isomorphism, as required. □

Finally we are ready to prove the semisimplicity theorem.

Theorem 7.5 [11, Theorem 9.1] The Mackey algebra μ_{R}(G) is semisimple if R is a field of characteristic coprime to |G|.

Proof. Assume the notation of the section. By Lemma 7.3 and Lemma 7.4, we have

M ≅ ⨁_{H_{i}∈P} coind_{τ}^{μ} inf_{γ}^{τ}M_{i}.

Inflation and coinduction functors are additive. So decomposing M_{i}(H_{i}) into simple R\overline{N}_{G}(H_{i})- modules, we obtain a decomposition of the Mackey functor T_{i}. But, by Corollary 6.9, the Mackey functor coind_{τ}^{μ} inf_{γ}^{τ}M_{i} is simple if M_{i}(H_{i}) is a simple R\overline{N}_{G}(H_{i})-module. □

Corollary 7.6 Let M and N be Mackey functors such that

def_{γ}^{τ}res_{τ}^{μ} M ≅ def_{γ}^{τ}res_{τ}^{μ} N

as conjugation functors. Then M ≅ N as Mackey functors. In particular,

M ≅ coind_{τ}^{μ} inf_{γ}^{τ} def_{γ}^{τ}res_{τ}^{μ} M.

Proof. This follows from Theorem 7.5 since the simple summands of a Mackey functor M are determined by the γ-module def_{γ}^{τ}res_{τ}^{μ} M. Note that the second statement is Corollary 4.4 in [10] and it holds for Mackey functors by Theorem 12.3 of that paper. □

Acknowledgment

I wish to thank my supervisor Laurence J. Barker and Ergün Yalçın for their guidance . I am also grateful to the referee for some important corrections and many detailed suggestions.

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