Inducing native Mackey functors to biset functors

Olcay Coşkun^{1}

Boğaziçi Üniversitesi, Matematik Bölümü 80815 Bebek, İstanbul, Turkey

Abstract

In this paper, we describe the induction functor from the category of native Mackey functors to the category of biset functors for a finite group G over an algebraically closed field k of characteristic zero. We prove two applications of this description. As the first application, we exhibit that any projective biset functor over k is induced from a (rational) virtual native Mackey functor. The second application is the explicit description of the projective indecomposable biset functors parameterized by simple groups.

Keywords: Native Mackey functor, projective biset functor, simple group. 2000 MSC: 19A22, 16E20

1. Introduction

In this paper, we consider the induction functor from the category of native Mackey functors for a finite group G over a field k to the category of biset functors for G over k. The theory of biset functors is introduced by Serge Bouc [2] as a unified theory of the classical operations in finite group representation theory. A basic result in this theory is the classification of the simple objects, which is done by Bouc. According to the classification, the simple functors are parameterized by the pairs (H, V) where H runs over all finite groups and V runs over all simple kOut(H)-modules, both up to isomorphism. Here Out(H) is the group of all outer automorphisms of H. On the other hand, the theory of global Mackey functors is an older theory which only deals with induction, restriction and transport of structure maps. We call a global Mackey functor defined only on the group G a native Mackey functor, see Section 3 for details.

Email address: olcay.coskun@boun.edu.tr (Olcay Coşkun)

1The author is supported by Tubitak-1001-113F240.

Preprint submitted to Elsevier August 13, 2014

In the context of biset functors, native Mackey functors can be identified as functors over bi-free bisets and hence there is a forgetful functor from the category of biset functors to that of native Mackey functors. The left adjoint of this forgetful functor is the functor that we are interested in this paper.

More precisely, we let G be a finite group and let k be an algebraically closed field of characteristic zero and consider all biset functors defined only on the set of subquotients of G together with the native Mackey functors for G. With this restriction on the characteristic of k, it is well-known that the category of global Mackey functors is semisimple, see [1] or [2]. Similarly, in this case, the category of native Mackey functors is also semisimple. Moreover, as we show below, the induction functor is also easier to describe.

As an application of this explicit description of the induction functor, we are able to prove an induction theorem for projective biset functors. According to Theorem 5.1, any projective biset functor is a rational combination of induced native Mackey functor.

We also consider some special cases of this induction functor. We explicitly describe the induction of a simple native Mackey functor parameterized by a cyclic group. Also when P is a p-group for some prime p of order at least p^{3} and 1 is the trivial module, we show that the induction of the simple native Mackey functor parameterized by the pair (P, 1) contains at least two summands, one of them being the projective cover of the torsion-free Dade group functor. The remaining case of the elementary abelian p-group of rank 2 is also considered.

The last case we consider is the case where we induce a simple functor parameterized by a simple non-cyclic group. In this case, we show that the induced functor is indecomposable, and hence obtain a, fairly explicit, description of this indecomposable projective functor. Finally, remark that, we have no other example of a non-simple group with the property that the corresponding induced simple native Mackey functor is indecomposable and leave the question whether the induced functor being indecomposable characterizes the simplicity of the group open.

Acknowledgement

I would like to thank to the referee for his/her comments ard corrections.

2. Preliminaries on biset functors

In this section, we recall basics of biset functors and introduce our notation. We refer to [2] and [6] for details. Let G be a finite group and let k be an algebraically closed field of characteristic zero. We denote by Sq(G) the set of all subquotients of G, that is,

Sq(G) = {H/N : N ⊴ H ≤ G}.

Following Bouc [2], we define the biset category C_{G} := C_{k,G} for G over k as the category whose set of objects are the groups in Sq(G). Given H, K ∈ Sq(G), we put

Mor_{C_{G}}(H, K) = k ⊗ B(K × H)

where B(K × H) is the Grothendieck group of (K, H)-bisets. The composition of morphisms is the linear extension of the well-known amalgamated product of bisets.

Now a biset functor for G over k is a k-linear functor from the biset category C_{G} for G to the category _{k}vect of finite dimensional k-vector spaces. (Note that one can define biset functors for G over any commutative ring with unity, but in this paper, we concentrate only on this special case.) Further we denote by F_{G} := F_{k,G} the category of biset functors for G over k, with the morphisms given by the natural transformations of biset functors. The category F_{G} is abelian. The parametrization of its simple objects follows from Bouc’s parametrization of biset functors [2]. According to [2, Theorem 4.3.10], there is a bijective correspondence between the set of isomorphism classes of simple biset functors for G over k and the set of pairs (H, V) where H is a subquotient of G and V is a simple kOut(H)-module, both taken up to isomorphism. We denote a representative of the isomorphism class of the simple biset functor corresponding to the pair (H, V) by S_{H,V}^{G}.

Remark 2.1. The category C_{G} is a full subcategory of the biset category C, defined in [2]. By the general theory of induction-restriction, it is straightforward to conclude that the simple biset functors S_{H,V}^{G} for G are restrictions to C_{G} of the simple biset functors S_{H,V} defined on the category C.

Another way to define a biset functor for G is to consider the category algebra of the category C_{G}. Precisely, we let Γ(G) := Γ_{k}(G) be the algebra generated by all morphisms in C_{G}, with the multiplication induced by the composition of morphisms. Following Barker [1], we call Γ(G) the alchemic algebra for G over k. By Bouc’s decomposition formula, [2, Lemma 2.3.26], the alchemic algebra is generated by the set of all bisets given by the following list.

1. Tin_{K/N}^{H} := Ind_{K}^{H}Inf_{K/N}^{K} for all N ⊴ K ≤ H ∈ Sq(G),

2. Des_{K/N}^{H} := Def_{K/N}^{K}Res_{K}^{H} for all N ⊴ K ≤ H ∈ Sq(G),

3. Iso_{H,H′}^{λ} for all isomorphisms λ : H′ → H with H, H′ ∈ Sq(G).

These generators are subject to a list of conditions determined by the Mackey product formula [2, Lemma 2.3.24]. With this definition, the category of finitely generated Γ(G) modules is equivalent to the category F_{G} of biset functors for G. The proof of the equivalence is standard. We refer to [6] for further details on this approach. We only recover a special subalgebra structure of the alchemic algebra which will be useful later.

Let Δ := Δ(G) := Δ_{k}(G) (resp. ∇ := ∇(G) := ∇_{k}(G)) be the subalgebra of Γ_{k}(G) generated by the bisets Tin_{K}^{H} and Iso_{H,H′}^{λ} (resp. Des_{K}^{H} and Iso_{H,H′}^{λ}). We also denote by Ω := Ω(G) := Ω_{k}(G) the subalgebra generated by the bisets Iso_{H,H′}^{λ}. Then we have the following triangle summarizing this subalgebra structure of the alchemic algebra.

Γ Δ  ∇ Ω Ω Ω  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}

In the above diagram, all arrows going upward are inclusions, and the down arrows in the last row are quotient maps from the corresponding algebra to its quotient by the ideal generated by all non-iso generators. These algebra morphisms induce several induction, coinduction and restriction functors between the corresponding categories of modules. We also have several natural isomorphisms between these functors and natural equivalences between module categories. The key result to prove such equivalences is the following theorem. Note that this is an alchemic version of Theorem 3.2 in [5] where the author uses a similar triangle in the context of ordinary Mackey functors.

Theorem 2.2. Assume the above notation. Then there is an isomorphism

ΔΓ_{∇} ≅ Δ ⊗_{Ω} ∇

of (Δ, ∇)-bimodules, where we regard Γ as a (Δ, ∇)-bimodule via left and right multiplication.

The proof of this theorem follows the same lines of the proof of Theorem 3.2 in [5]. Now the following theorem, which is a version of Theorem 3.4 in [5], also holds for in the case of the alchemic algebra. The proof is again almost the same as the proof in [5].

Theorem 2.3. The following equivalences hold.

1. Res_{Δ}^{Γ}Ind_{∇}^{Γ} ≅ Ind_{Ω}^{Δ}Res_{Ω}^{∇}.

2. Res_{∇}^{Γ}Coind_{Δ}^{Γ} ≅ Coind_{Ω}^{∇}Res_{Ω}^{Δ}.

3. Native Mackey functors

Other than the biset functors, there are several other natural constructions of functors associated to bisets, like global Mackey functors, inflation functors and restriction functors. In this paper, we are mainly interested in a local version of global Mackey functors, called native Mackey functors. By definition, a native Mackey functor is a kind of global Mackey functor which is defined only on the subquotients of a fixed finite group G. In other words, it is a biset functor for G without inflation and deflation maps. In this section, we introduce the formal definition and basic properties of native Mackey functors. We shall use the module theoretic approach, however it is possible to define them in a way similar to the first definition of a biset functor.

We denote by μ := μ(G) := μ_{k}(G) the subalgebra of Γ(G) generated by the bi-free bisets. In other words, it is the subalgebra of Γ(G) generated by the following list of generators.

1. Tin_{K}^{H} = Ind_{K}^{H} for all K ≤ H ∈ Sq(G),

2. Des_{K}^{H} = Res_{K}^{H} for all K ≤ H ∈ Sq(G),

3. Iso_{H,H′}^{λ} for all isomorphisms λ : H′ → H with H, H′ ∈ Sq(G).

A module over μ_{k}(G) is called a native Mackey functor for G over k. We denote the category of all native Mackey functors for G by _{μ(G)}mod and call the algebra μ(G) the native Mackey algebra.

Notice that a native Mackey functor is a structure very similar to the global Mackey functors, considered by Webb in [8]. Indeed, if M is a global Mackey functor, then the restriction of this functor to the set of all subquotients of G would give a native Mackey functor. On the other hand, an ordinary Mackey functor is not necessarily a native Mackey functor, since in the case of ordinary Mackey functors, one is only allowed to consider isomorphisms induced by the conjugation action of G on the subgroups, whereas in our case, all isomorphisms between subquotients of G are allowed.

Moreover, the native Mackey algebra is an idempotent truncation

μ_{k}(G) = e_{G}μ_{k}^{1,1}e_{G}

of the global Mackey algebra μ_{k}^{1,1} of [8]. Here the idempotent e_{G} is given by

e_{G} = ∑_{H∈Sq(G)} Iso_{H,H}^{1}.

Therefore we can specialize results on the global Mackey functors in [2] and [8] to the case of the native Mackey functors, easily. We give a couple of these results below. (cf. [8, Theorem 9.5, ] and [2, Theorem 4.3.10])

Theorem 3.1. Let G be a finite group and k be an algebraically closed field of characteristic zero. Then

1. The native Mackey algebra μ_{k}(G) is semisimple.

2. The simple μ_{k}(G)-modules are parameterized by the set of pairs (H, V) where H runs over the isomorphism classes of subquotients of G and V runs over the isomorphism classes of simple kOut(H)-modules. The simple functor corresponding to the pair (H, V) is denoted by S_{H,V}^{μ}. It is characterized by the property that H is the group of minimal order such that S_{H,V} (H) ≠ 0 and V = S_{H,V} (H).

The proof of this theorem follows from the general results on modules of the truncated algebras. A construction of the simple native Mackey functors follows from the construction of simple global Mackey functors. However, in the rest of the paper, another construction, similar to the construction of the simple ordinary Mackey functors given in [5] will be more useful. For the rest of the section, we adapt certain results from [5] to the native case, without formal proofs.

First, we need to introduce a subalgebra structure of μ(G). This structure is very similar to the structure introduced above for the alchemic algebra and hence to the structure introduced in [5] for the ordinary Mackey algebra. Therefore the results that are recalled above for the alchemic algebra will also hold for the native Mackey algebra, as explained below.

We denote by τ := τ(G) := τ_{k}(G) (resp. ρ := ρ(G) := ρ_{k}(G)) the subalgebra of μ generated by the bisets Ind_{K}^{H} and Iso_{H,H′}^{λ} (resp. Res_{K}^{H} and Iso_{H,H′}^{λ}). We still denote by Ω the subalgebra generated by the bisets Iso_{H,H′}^{λ}.

Together with the alchemic algebra, we can represent this subalgebra structure of the alchemic algebra by the following diagram.

Γ μ τ  ρ Ω Ω Ω  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}

In this diagram, as in the previous one, all arrows going upward are inclusions, and the down arrows in the last row are quotient maps from the corresponding algebra to its quotient by the ideal generated by all non-iso generators. Again, there are several natural equivalences between functors induced by these maps. We refer to [5] and [6] for the full descriptions of such equivalences. We only note that the equivalences in [5] that are stated in the case of the ordinary Mackey algebra are also valid for the native Mackey algebra. In this paper, we only need several of these equivalences. For the readers convenience, we shall state the native versions of the results that we use in this paper. Finally, note that there is an ambiguity in our notation since the ordinary Mackey algebra is also denoted by μ and the ordinary versions of the algebras τ and ρ above are also denoted by the same symbol. However, we will not use the ordinary versions of these algebras in this paper. So the ambiguity should not be a problem.

The first result determines the (τ, ρ)-bimodule structure of the native Mackey algebra μ(G) and it is actually the key for the rest of the equivalences.

Theorem 3.2. Assume the above notation. Then there is an isomorphism

τμ_{ρ} ≅ τ ⊗_{Ω} ρ

of (τ, ρ)-bimodules, where we regard μ as a (τ, ρ)-bimodule via left and right multiplication.

As we have remarked in the case of the alchemic algebra, the proof of this theorem follows the same steps of the proof of [5, Theorem 3.2] with the only difference that this time we are working with isomorphisms instead of conjugation maps. Using this result, one gets the following natural equivalences (cf. [5, Theorem 3.4 and Proposition 3.6]).

Theorem 3.3. The following equivalences hold.

1. Res_{τ}^{μ}Ind_{ρ}^{μ} ≅ Ind_{Ω}^{τ}Res_{Ω}^{ρ}.

2. Res_{ρ}^{μ}Coind_{τ}^{μ} ≅ Coind_{Ω}^{ρ}Res_{Ω}^{τ}.

3. Def_{Ω}^{τ}Ind_{Ω}^{τ} ≅ Id_{Ω} ≅ Codef_{Ω}^{ρ}Coind_{Ω}^{ρ}.

4. Def_{Ω}^{τ}Inf_{Ω}^{τ} ≅ Id_{Ω} ≅ Codef_{Ω}^{ρ}Inf_{Ω}^{ρ}.

The proof of this theorem is again very similar to the proof of [5, Theorem 3.4]. Indeed, the proof of the cited theorem only uses general results together with [5, Theorem 3.2], whose native version is given above.

Now the construction of simple functors can be carried out exactly as the construction of simple Mackey functors explained in Section 6 of [5]. Indeed, it is easy to see that any simple Ω_{k}(G)-module is of the form S_{H,V}^{Ω} where S_{H,V}^{Ω} (H) = V and its evaluation is equal to zero at any other subquotient not isomorphic to H. Here H is a subquotient of G and V is a simple kOut(H)-module. Then as in [5, Proposition 6.1], the simple τ-modules and the simple ρ-modules are just the corresponding inflations of the simple Ω-modules, that is, any simple ρ-module is of the form

S_{H,V}^{ρ} := Inf_{Ω}^{ρ}S_{H,V}^{Ω}  (1)

as S_{H,V}^{Ω} runs over all simple Ω-modules. Similar remark holds for simple τ-modules. Finally, we have isomorphisms

S_{H,V}^{μ} ≅ Ind_{ρ}^{μ}Inf_{Ω}^{ρ}S_{H,V}^{Ω} ≅ Coind_{τ}^{μ}Inf_{Ω}^{τ}S_{H,V}^{Ω} .  (2)

Here the simplicity of the induced and the coinduced functors follows easily since they are indecomposable and the native Mackey algebra is semisimple. They are also isomorphic by Schur’s Lemma since there is a non-zero map between them induced by the identity homomorphism V → V, see Corollary 5.10 in [5] for the derivation of this map. Finally, the first isomorphism is just the identification coming from the parametrization of the simple functors.

Note also that, since the native Mackey algebra is semisimple, the first isomorphism above actually gives an equivalence of categories

Ind_{ρ}^{μ}Inf_{Ω}^{ρ} : _{Ω}mod → _{μ}mod

with the inverse equivalence

Def_{Ω}^{τ}Res_{τ}^{μ} : _{μ}mod → _{Ω}mod.

Indeed, the composition Def_{Ω}^{τ}Res_{τ}^{μ}Ind_{ρ}^{μ}Inf_{Ω}^{ρ} is naturally isomorphic to the identity functor on Ω-mod by Theorem 3.3. To see that the other composition is naturally isomorphic to the identity functor on μ-mod, one needs to show that there is an isomorphism

M ≅ Ind_{ρ}^{μ}Inf_{Ω}^{ρ}Def_{Ω}^{τ}Res_{τ}^{μ}M  (3)

of native Mackey functors for any native Mackey functor M. Since μ is semisimple, it is sufficient to prove the isomorphism only for the simple native Mackey functors. So let M = S_{H,V}^{μ} for some (H, V). Then by the definition of the deflation functor, for any subquotient K of G, the module Def_{Ω}^{τ}Res_{τ}^{μ}M(K) is the quotient of M(K) by the sum of images of all induction maps to K. But since M is simple, this quotient will be non-zero only if K ≅ H in which case, it will be equal to M(K). In particular, there is an isomorphism

Def_{Ω}^{τ}Res_{τ}^{μ}S_{H,V}^{μ} ≅ S_{H,V}^{Ω} .

Now the result follows form the above construction of the simple native Mackey functors.

4. The induction functor Ind_{μ(G)}^{Γ(G)}

We are interested in the induction functor Ind_{μ}^{Γ} := Ind_{μ(G)}^{Γ(G)} = Γ(G)⊗_{μ(G)}−. Note that, in general, the right μ(G)-module structure of Γ(G) is very complicated and hence it is difficult to describe the induction functor. However, in our case, when the coefficients are from a field of characteristic zero, the problem can be simplified by using the equivalences of the previous section. Explicitly, we have the following result.

Theorem 4.1. Let M be a native Mackey functor for G over k. Then there is an isomorphism of biset functors

Ind_{μ}^{Γ} M ≅ Ind_{ρ}^{Γ} Inf_{Ω}^{ρ} Def_{Ω}^{τ} Res_{τ}^{μ} M.

Proof. Note that since the algebra ρ is a subalgebra of the native Mackey algebra μ, we can rewrite the right hand side of the above isomorphism as

Ind_{ρ}^{Γ} Inf_{Ω}^{ρ} Def_{Ω}^{τ} Res_{τ}^{μ} M = Ind_{μ}^{Γ} Ind_{ρ}^{μ} Inf_{Ω}^{ρ} Def_{Ω}^{τ} Res_{τ}^{μ} M.

Therefore it suffices to show that there is an isomorphism of native Mackey functors

M ≅ Ind_{ρ}^{μ} Inf_{Ω}^{ρ} Def_{Ω}^{τ} Res_{τ}^{μ} M

which follows from the category equivalence explained at the end of the previous section.

Now since the native Mackey algebra μ_{k}(G) is semisimple and the induction functor preserves projectivity, we obtain the following corollary on projectivity of the induced functors.

Corollary 4.2. Let M be a native Mackey functor for G over k. Then the biset functor Ind_{μ}^{Γ} M is projective.

Remark 4.3. To use the above identification more efficiently, one can consider the induction functor Ind_{ρ}^{Γ} as a composition Ind_{∇}^{Γ}Ind_{ρ}^{∇} where ∇ is the subalgebra of Γ generated by all morphisms of the form Iso_{K′,K/N}^{λ}Des_{K/N}^{H} for all appropriate choices of the subquotients H, K/N and K′ and isomorphisms λ. Now the description of Ind_{∇}^{Γ} can be found in [6] and we include a description of the other functor in the appendix.

5. Projective biset functors

In this section, we prove that any projective biset functor for G over k is obtained by inducing a virtual native Mackey functor. We first introduce our notation. Let R_{μ,k}(G) denote the Grothendieck group of the category of native Mackey functors for G over k and let P_{Γ,k}(G) denote the Grothendieck group of the category of projective biset functors for G over k. Note that both of these groups are free of the same rank with basis parameterized by the pairs (H, V) described in the previous sections. In the first case, the basis consists of the set of isomorphism classes of simple native Mackey functors S_{H,V}^{μ}, whereas, in the second case, the basis consists of the set of isomorphism classes of indecomposable projective biset functors P_{H,V}^{Γ} .

Now we have the following theorem.

Theorem 5.1. The induction functor

Ind_{μ(G)}^{Γ(G)} : _{μ}mod → _{Γ}mod

induces an isomorphism

Ind_{μ(G)}^{Γ(G)} : ℚR_{μ,k}(G) → ℚP_{Γ,k}(G).

of ℚ-vector spaces.

Proof. For simplicity, we write Γ and μ, instead of Γ(G) and μ(G). Evidently it suffices to prove that any projective indecomposable biset functor is a rational combination of induced native Mackey functors. For this aim, let H be a subquotient of G and let V be a simple kOut(H)-module. Then, by Corollary 4.2, the biset functor Ind_{μ}^{Γ}S_{H,V}^{μ} is projective, so we have

Ind_{μ}^{Γ} S_{H,V}^{μ} ≅ ⨁_{(K,W)} n_{K,W}^{H,V} P_{K,W}^{Γ}

where the sum is over the set of isomorphism classes of simple biset functors for G over k and P_{K,W}^{Γ} denotes the projective cover of the simple functor S_{K,W}^{Γ} . Since k is algebraically closed, we further have that

n_{K,W}^{H,V} = dim_{k} Hom_{Γ}(Ind_{μ}^{Γ} S_{H,V}^{μ} , S_{K,W}^{Γ} ).

Now, by Equation 1 and Equation 2, there is an isomorphism of native Mackey functors

S_{H,V}^{μ} ≅ Ind_{ρ}^{μ} S_{H,V}^{ρ} .  (4)

Here the simple ρ-module S_{H,V}^{ρ} is the functor which takes the value zero for all subquotients of G except for the isomorphism class of H and S_{H,V}^{ρ} (H) = V. Substituting this into the above equality, we get

n_{K,W}^{H,V} = dim_{k} Hom_{Γ}(Ind_{ρ}^{Γ}S_{H,V}^{ρ} , S_{K,W}^{Γ} ).

On the other hand, since the induction functor is the left adjoint of the restriction functor, we get

n_{K,W}^{H,V} = dim_{k} Hom_{ρ}(S_{H,V}^{ρ} , Res_{ρ}^{Γ}S_{K,W}^{Γ} ).

Note that, to finish the proof, it is sufficient to show that the matrix

N_{G} = (n_{K,W}^{H,V} )_{((K,W),(H,V))}

is invertible. We proceed to show that, with a correct ordering, the matrix N_{G} is lower triangular with diagonal entries equal to 1.

Indeed let H := {1 = H_{0}, H_{1}, ⋯, H_{n} = G} be a set of representatives of the isomorphism classes of subquotients of G ordered in a way that if i < j then |H_{i}| ≤ |H_{j}|. Also let

S := {(H_{i}, V_{ij})|H_{i} ∈ H, V_{ij} ∈ Irr(kOut(H_{i}))}.

Then we claim that

n_{H_{i},V_{ij}}^{H_{k},V_{kl}} = 0

if i > k. Indeed, since the ρ-module S_{H_{k},V_{kl}}^{ρ} is non-zero only on the isomorphism class of the group H_{k}, any ρ-module homomorphism

ϕ : S_{H_{k},V_{kl}}^{ρ} → Res_{ρ}^{Γ}S_{H_{i},V_{ij}}^{Γ}

is uniquely determined by its component

ϕ_{H_{k}} : S_{H_{k},V_{kl}}^{ρ}(H_{k}) → Res_{ρ}^{Γ}S_{H_{i},V_{ij}}^{Γ}(H_{k}).

Moreover the map ϕ_{H_{k}} should be a homomorphism of kOut(H_{k})-modules by its definition. Thus there is an inclusion

Hom_{ρ}(S_{H_{k},V_{kl}}^{ρ}, Res_{ρ}^{Γ}S_{H_{i},V_{ij}}^{Γ}) ⊆ Hom_{kOut(H_{k})}(V_{kl}, S_{H_{i},V_{ij}}^{Γ}(H_{k}))

of k-vector spaces. Therefore we get

n_{H_{i},V_{ij}}^{H_{k},V_{kl}} = dim_{k} Hom_{ρ}(S_{H_{k},V_{kl}}^{ρ}, Res_{ρ}^{Γ}S_{H_{i},V_{ij}}^{Γ}) ≤ dim_{k} Hom_{kOut(H_{k})}(V_{kl}, S_{H_{i},V_{ij}}^{Γ}(H_{k})). (5)

But H_{i} is a minimal group for the simple functor S_{H_{i},V_{ij}}^{Γ}. Thus we have S_{H_{i},V_{ij}}^{Γ}(H_{k}) = 0 which implies that the multiplicity n_{H_{i},V_{ij}}^{H_{k},V_{kl}} is also zero, as claimed.

On the other hand, if i = k, then there are two cases. If j ≠ l, then by the inequality (5), we get

n_{H_{i},V_{ij}}^{H_{i},V_{il}} = dim_{k} Hom_{ρ}(S_{H_{i},V_{il}}^{ρ}, Res_{ρ}^{Γ}S_{H_{i},V_{ij}}^{Γ})
≤ dim_{k} Hom_{kOut(H_{i})}(V_{il}, V_{ij}).

But since j ≠ l, the module V_{ij} is not isomorphic to V_{il}, so the latter dimension is equal to zero by Schur’s Lemma. Finally, if j = l, then we have

n_{H_{i},V_{ij}}^{H_{i},V_{ij}} = dim_{k} Hom_{ρ}(S_{H_{i},V_{ij}}^{ρ}, Res_{ρ}^{Γ}S_{H_{i},V_{ij}}^{Γ})
≤ dim_{k} Hom_{kOut(H_{i})}(V_{ij}, V_{ij}).

This time, by Schur’s Lemma, the later dimension is equal to 1. Now since the identity morphism ι : V_{ij} → V_{ij} induces a non-zero homomorphism

ι^{∗} : S_{H_{i},V_{ij}}^{ρ} → Res_{ρ}^{Γ}S_{H_{i},V_{ij}}^{Γ}

of ρ-modules, the set Hom_{ρ}(S_{H_{i},V_{ij}}^{ρ}, Res_{ρ}^{Γ}S_{H_{i},V_{ij}}^{Γ}) is non-zero, and hence has dimension 1. Therefore, we obtain that

n_{H_{i},V_{ij}}^{H_{i},V_{il}} = δ_{jl}

and hence combining this result with the previous case, we conclude that the matrix

N_{G} = (n_{H_{i},V_{ij}}^{H_{k},V_{kl}})_{(H_{i},V_{ij}),(H_{k},V_{kl})∈S}

is lower triangular having ones in the diagonal, as required.

The following corollary is immediate from the proof of the above theorem.

Corollary 5.2. Let H, K ∈ Sq(G) and V be a simple kOut(H)-module and W be a simple kOut(K)-module. Then the multiplicity of the projective indecomposable biset functor P_{K,W}^{Γ} as a direct summand in the biset functor Ind_{μ}^{Γ}S_{H,V}^{μ} is non-zero only if K is a subquotient of H.

In the next section, a better description of this coefficient will be obtained by considering the off-diagonal entries of the above matrix. However, when H is equal to K in the above corollary, we get the following precise result.

Corollary 5.3. Let H ∈ Sq(G) and V, W be simple kOut(H)-modules. Then the multiplicity of the projective indecomposable functor P_{H,W}^{Γ} as a direct summand in the biset functor Ind_{μ}^{Γ}S_{H,V}^{μ} is non-zero if and only if V ≅ W. When the coefficient is non-zero, it is equal to 1.

Thus for any simple native Mackey functor S_{H,V}^{μ}, we have

Ind_{μ}^{Γ}S_{H,V}^{μ} ≅ P_{H,V}^{Γ} ⊕ P

where P is a (possibly zero) projective biset functor having no summands of the form P_{H,W}^{Γ} .

6. Off-diagonal entries of N_{G}

For the rest of the paper, we study the summand P defined above, in some special cases. The following characterization and estimation of the off-diagonal entries will be useful.

In the proof of Theorem 5.1, we have seen the following equality.

n_{K,W}^{H,V} = dim_{k}Hom_{ρ}(S_{H,V}^{ρ} , Res_{ρ}^{Γ}S_{K,W}^{Γ} ).

Now, as remark in the previous section, we have that S_{H,V}^{ρ} = Inf_{Ω}^{ρ}S_{H,V}^{Ω} . Therefore, we have

n_{K,W}^{H,V} = dim_{k}Hom_{ρ}(Inf_{Ω}^{ρ}S_{H,V}^{Ω} , Res_{ρ}^{Γ}S_{K,W}^{Γ} ).

If we denote the right adjoint of Inf_{Ω}^{ρ} by Codef_{Ω}^{ρ}, the above equality becomes

n_{K,W}^{H,V} = dim_{k}Hom_{Ω}(S_{H,V}^{Ω} , Codef_{Ω}^{ρ}Res_{ρ}^{Γ}S_{K,W}^{Γ} )
= dim_{k}Hom_{kOut(H)}(V, Codef_{Ω}^{ρ}Res_{ρ}^{Γ}S_{K,W}^{Γ} (H)).

Here we use the Morita equivalence, described in [6, Section 4] of the algebras Ω(G) and ∏_{H} kOut(H) where the product is over all subquotients of G, up to isomorphism.

Since the algebra Ω is the quotient of the algebra ρ by the ideal generated by all proper restriction bisets, the evaluation of Codef_{Ω}^{ρ}D for a ρ-module D at a subquotient K of G is given by

Codef_{Ω}^{ρ}D(K) = ⋂_{L<K} ker(Res_{L}^{K} : D(K) → D(L)).

Now the following proposition is immediate since the group algebra kOutH is semisimple.

Proposition 6.1. The multiplicity n_{K,W}^{H,V} of P_{K,W}^{Γ} in the biset functor Ind_{μ}^{Γ}S_{H,V}^{μ} is equal to the multiplicity of the kOut(H)-module V in the kOut(H)-module Codef_{Ω}^{ρ}Res_{ρ}^{Γ}S_{K,W}^{Γ} (H).

Note that, in general, the evaluations of simple biset functors are not easy to determine. Therefore, the above numbers are not easy to calculate. Next, we derive an upper bound for this number which will be useful to determine if the number is zero or not.

To start with, note that

n_{K,W}^{H,V} = dim_{k} Hom_{Γ}(Ind_{μ}^{Γ}S_{H,V}^{μ} , S_{K,W}^{Γ} ) = dim_{k}Hom_{Γ}(Ind_{ρ}^{Γ}S_{H,V}^{ρ} , S_{K,W}^{Γ} )

by the isomorphism (4) of the previous section. On the other hand, by [6, Corollary 4.7], the simple functor S_{K,W}^{Γ} is the unique minimal subfunctor of the functor Coind_{Δ}^{Γ}S_{K,W}^{Δ} . Here Δ denotes the subalgebra of Γ generated by all bisets of the form Tin_{L/N}^{H} and Iso_{H,H′}^{λ}. Thus any morphism of Γ-modules Ind_{ρ}^{Γ}S_{H,V}^{ρ} → S_{K,W}^{Γ} can be regarded as a morphism Ind_{ρ}^{Γ}S_{H,V}^{ρ} → Coind_{Δ}^{Γ}S_{K,W}^{Δ} .Thus we get

n_{K,W}^{H,V} ≤ dim_{k} Hom_{Γ}(Ind_{ρ}^{Γ}S_{H,V}^{ρ} , Coind_{Δ}^{Γ}S_{K,W}^{Δ} ).

The right hand side of the above inequality can be calculated as follows.

Hom_{Γ}(Ind_{ρ}^{Γ}S_{H,V}^{ρ} , Coind_{Δ}^{Γ}S_{K,W}^{Δ} ) ≅ Hom_{∇}(Ind_{ρ}^{∇}S_{H,V}^{ρ} , Res_{∇}^{Γ}Coind_{Δ}^{Γ}S_{K,W}^{Δ} )
≅ Hom_{∇}(Ind_{ρ}^{∇}S_{H,V}^{ρ} , Coind_{Ω}^{∇}Res_{Ω}^{Δ}S_{K,W}^{Δ} )
≅ Hom_{Ω}(Res_{Ω}^{∇}Ind_{ρ}^{Γ}S_{H,V}^{ρ} , Res_{Ω}^{Δ}S_{K,W}^{Δ} ).

Here we use Theorem 2.3 to obtain the second line, and then use the well-known adjointness properties of restriction and induction-coinduction functors to get the last line.

Now since S_{K,W}^{Δ} is non-zero only on the isomorphism class of K, the last term in the above isomorphisms gives the following isomorphism.

Hom_{Γ}(Ind_{ρ}^{Γ}S_{H,V}^{ρ} , Coind_{Δ}^{Γ}S_{K,W}^{Δ} ) ≅ Hom_{kOut(K)}(Ind_{ρ}^{∇}S_{H,V}^{ρ} (K), W).

Thus the coefficient n_{K,W}^{H,V} is non-zero only if W appears in the kOut(K)-module Ind_{ρ}^{∇}S_{H,V}^{ρ} (K) as a direct summand. To determine this number, we need a description of the induction functor Ind_{ρ}^{∇} which is given in the appendix. Now using the description in Theorem Appendix A.1, we have

Hom_{Γ}(Ind_{ρ}^{Γ}S_{H,V}^{ρ} , Coind_{Δ}^{Γ}S_{K,W}^{Δ} ) ≅

⨁_{R∈Sq(G):R≅H} Hom_{kOut(K)}(kX(K, R) ⊗_{kOut(R)} V (R), W)

where V (R) = S_{H,V}^{ρ} (R) and X(K, R) is basically the set of isomorphisms from quotients of R to K, see the appendix for the precise definition. As a result, we have the following proposition.

Proposition 6.2. Assume the above notation. Then the projective indecomposable biset functor P_{K,W}^{Γ} occurs as a direct summand of Ind_{μ}^{Γ}S_{H,V}^{μ} only if W occurs as a direct summand of the kOut(K)-module kX(K, R) ⊗_{kOut(R)} V (R) for some subquotient R of G isomorphic to H.

In particular, we have the following corollary.

Corollary 6.3. Assume the above notation. Then the projective indecomposable biset functor P_{K,W}^{Γ} occurs as a direct summand of Ind_{μ}^{Γ}S_{H,V}^{μ} only if there is a normal subgroup N of H such that K ≅ H/N.

7. Induced functors for cyclic groups

In this section, we describe the biset functor Ind_{μ}^{Γ}S_{C,V}^{μ} where C is a cyclic subquotient of G. First, by Corollary 6.3, any summand of this functor is determined by a quotient of C. Thus we can write

Ind_{μ}^{Γ}S_{C,V}^{μ} = ⨁_{(L,W)} n_{L,W}^{C,V} P_{L,W}^{Γ}  (6)

where the sum is over all pairs (L, W) such that L is a quotient of C and W is a simple kOut(L)-module. Moreover, by Proposition 6.1, to evaluate the above coefficients, we do not need the evaluations of the involved functors at groups larger than C. Thus we can only consider the evaluations at subquotients of C.

On the other hand, for any pair (H, V), we have the following isomorphism

Res_{Γ(H)}^{Γ(G)}Ind_{μ(G)}^{Γ(G)}S_{H,V}^{μ(G)} ≅ Ind_{μ(H)}^{Γ(H)}Res_{μ(H)}^{μ(G)}S_{H,V}^{μ(G)}

of biset functors for H. Indeed, by Theorem 4.1, we have

Ind_{μ(G)}^{Γ(G)}S_{H,V}^{μ(G)} ≅ Ind_{ρ(G)}^{Γ(G)}S_{H,V}^{ρ(G)}.

Now denote the identity element of Γ(H) by 1_{H} and use the definition of the restriction functor to write

Res_{Γ(H)}^{Γ(G)}Ind_{μ(G)}^{Γ(G)}S_{H,V}^{μ(G)} ≅ Res_{Γ(H)}^{Γ(G)}Ind_{ρ(G)}^{Γ(G)}S_{H,V}^{ρ(G)} = 1_{H}Ind_{ρ(G)}^{Γ(G)}S_{H,V}^{ρ(G)} = 1_{H}Γ(G) ⊗_{ρ(G)} S_{H,V}^{ρ(G)}.

Then since 1_{H} acts on the simple functor S_{H,V}^{ρ(G)} trivially, we further have

1_{H}Γ(G) ⊗_{ρ(G)} S_{H,V}^{ρ(G)} = 1_{H}Γ(G) ⊗_{ρ(G)} 1_{H}S_{H,V}^{ρ(G)}
= 1_{H}Γ(G)1_{H} ⊗_{ρ(G)} 1_{H}S_{H,V}^{ρ(G)}
= Γ(H) ⊗_{ρ(G)} 1_{H}S_{H,V}^{ρ(G)}

where the equality Γ(H) = 1_{H}Γ(G)1_{H} holds by the definition of the alchemic algebra Γ(H). In the above tensor product, we can change ρ(G) by ρ(H) since the elements of the algebra ρ(G) which are not contained in ρ(H) are redundant in the tensor product. Thus we get

Res_{Γ(H)}^{Γ(G)}Ind_{μ(G)}^{Γ(G)}S_{H,V}^{μ} ≅ Γ(H) ⊗_{ρ(H)} 1_{H}S_{H,V}^{ρ(H)}
= Ind_{ρ(H)}^{Γ(H)}Res_{ρ(H)}^{ρ(G)}S_{H,V}^{ρ(G)}.

Moreover since S_{H,V}^{ρ(H)} = Res_{ρ(H)}^{ρ(G)}S_{H,V}^{ρ(G)} holds by the definition of this simple functor, we get

Res_{Γ(H)}^{Γ(G)}Ind_{μ(G)}^{Γ(G)}S_{H,V}^{μ} ≅ Ind_{ρ(H)}^{Γ(H)}S_{H,V}^{ρ(H)}
≅ Ind_{μ(H)}^{Γ(H)}S_{H,V}^{μ(H)}.

Finally, by [2, Proposition 4.2.2], we have S_{H,V}^{μ(H)} = Res_{μ(H)}^{μ(G)}S_{H,V}^{μ(G)}, which completes the proof of the following lemma.

Lemma 7.1. Let G be a finite group and H be a subquotient of G. Then for any simple kOut(H)-module V, there is an isomorphism

Res_{Γ(H)}^{Γ(G)}Ind_{μ(G)}^{Γ(G)}S_{H,V}^{μ(G)} ≅ Ind_{μ(H)}^{Γ(H)}Res_{μ(H)}^{μ(G)}S_{H,V}^{μ(G)}

of biset functors for H.

In particular, if we apply the restriction functor Res_{Γ(C)}^{Γ(G)} to Equation 6 and use the above lemma, we get

⨁_{(L,W)} n_{L,W}^{C,V} Res_{Γ(C)}^{Γ(G)}P_{L,W}^{Γ} = Res_{Γ(C)}^{Γ(G)}Ind_{μ}^{Γ}S_{C,V}^{μ} = Ind_{μ(C)}^{Γ(C)}Res_{μ(C)}^{μ(G)}S_{C,V}^{μ}  (7)

On the other hand, by [8, Proposition 3.3], we have

P_{L,W}^{Γ(C)} ≅ Res_{Γ(C)}^{Γ(G)}P_{L,W}^{Γ} .

Furthermore, since, by [1] and [2], the alchemic algebra Γ(C) is semisimple, we have P_{L,W}^{Γ(C)} = S_{L,W}^{Γ(C)} for any pair (L, W). Finally, by [2, Proposition 4.2.2], we have Res_{μ(C)}^{μ(G)}S_{C,V}^{μ} =

S_{C,V}^{μ(C)}. Thus Equation 7 becomes

Ind_{μ(C)}^{Γ(C)}S_{C,V}^{μ(C)} = ⨁_{(L,W)} n_{L,W}^{C,V} S_{L,W}^{Γ(C)}  (8)

where the coefficients n_{L,W}^{C,V} are the ones determined by Equation 6. However, these coefficients are also given by

n_{L,W}^{C,V} = dim_{k} Hom_{Γ(C)}(Ind_{μ(C)}^{Γ(C)}S_{C,V}^{μ(C)}, S_{L,W}^{Γ(C)}).

To evaluate the right hand side, note that by [6, Theorem 6.7], there is an isomorphism

Ind_{∇(C)}^{Γ(C)}S_{L,W}^{∇(C)} ≅ Coind_{Δ(C)}^{Γ(C)}S_{L,W}^{Δ(C)}

of biset functors, which implies, by [6, Proposition 6.2] that there is an isomorphism

S_{L,W}^{Γ(C)} ≅ Coind_{Δ(C)}^{Γ(C)}S_{L,W}^{Δ(C)}

of biset functors. Therefore, we obtain

n_{L,W}^{C,V} = dim_{k} Hom_{Γ(C)}(Ind_{μ(C)}^{Γ(C)}S_{C,V}^{μ(C)}, Coind_{Δ(C)}^{Γ(C)}S_{L,W}^{Δ(C)}).

Now using adjointness properties of the restriction and induction-coinduction functors together with the equivalences given in Theorem 2.3, we can evaluate the right hand side of the above equality as follows.

Hom_{Γ(C)}(Ind_{μ(C)}^{Γ(C)}S_{C,V}^{μ(C)}, Coind_{Δ(C)}^{Γ(C)}S_{L,W}^{Δ(C)}) ≅ Hom_{Γ(C)}(Ind_{ρ(C)}^{Γ(C)}S_{C,V}^{ρ(C)}, Coind_{Δ(C)}^{Γ(C)}S_{L,W}^{Δ(C)})
≅ Hom_{∇(C)}(Ind_{ρ(C)}^{∇(C)}S_{C,V}^{ρ(C)}, Res_{∇(C)}^{Γ(C)}Coind_{Δ(C)}^{Γ(C)}S_{L,W}^{Δ(C)})
≅ Hom_{∇(C)}(Ind_{ρ(C)}^{∇(C)}S_{C,V}^{ρ(C)}, Coind_{Ω(C)}^{∇(C)}Res_{Ω(C)}^{Δ(C)}S_{L,W}^{Δ(C)})
≅ Hom_{Ω(C)}(Res_{Ω(C)}^{∇(C)}Ind_{ρ(C)}^{∇(C)}S_{C,V}^{ρ(C)}, Res_{Ω(C)}^{Δ(C)}S_{L,W}^{Δ(C)})
≅ Hom_{Ω(C)}(Res_{Ω(C)}^{∇(C)}Ind_{ρ(C)}^{∇(C)}S_{C,V}^{ρ(C)}, S_{L,W}^{Ω(C)})
≅ Hom_{kOut(L)}(Ind_{ρ(C)}^{∇(C)}S_{C,V}^{ρ(C)}(L), W).

With the above isomorphisms, we have proved the following theorem.

Theorem 7.2. Let G be a finite group and C be a cyclic subquotient of G. Let V be a simple kOut(C)-module. Then there is an isomorphism

Ind_{μ}^{Γ}S_{C,V}^{μ} ≅ ⨁_{L,W} n_{L,W}^{C,V} P_{L,W}^{Γ} .

of biset functors where n_{L,W}^{C,V} is the multiplicity of W in the kOut(L)-module Ind_{ρ}^{∇}S_{C,V}^{ρ} (L).

8. Induced functors for p-groups

Let p be a prime number and P be a non-cyclic p-group. In this section, we prove that the biset functor Ind_{μ}^{Γ}S_{P,1}^{μ} is never indecomposable. By Corollary 5.3, we have

Ind_{μ}^{Γ}S_{P,1}^{μ} ≅ P_{P,1}^{Γ} ⊕ F

where F is a projective biset functor. Our first aim is to show that if P has order at least p^{3}, then P_{E_{2},1}^{Γ} is a summand of F, where E_{2} is an elementary abelian group of rank 2.

By Proposition 6.1, we need to prove that the trivial kOut(P)-module k is a direct summand of the kOut(P)-module

Codef_{Ω}^{ρ}Res_{ρ}^{Γ}S_{E_{2},1}^{Γ}(P) = ⋂_{L<P} ker(Res_{L}^{P} : S_{E_{2},1}^{Γ}(P) → S_{E_{2},1}Γ(L)).

In other words, we need to show that there is an element ζ ∈ S_{E_{2},1}^{Γ}(P) such that

1. ζ is Out(P)-invariant, that is, for any ϕ ∈ Out(P), we have Iso_{P,P}^{ϕ} (ζ) = ζ, and 2. for any proper subgroup L of P, the restriction Res_{L}^{P}ζ is equal to zero.

Now by [4, Theorem 10.1], we have an isomorphism

S_{E_{2},1}^{Γ} ≅ kD

of biset functors, where kD is the biset functor of the torsion-free Dade group. The element ζ with the above stated properties exists in the Dade group. Indeed, let

ζ = Ω_{M(P)}

be the relative syzygy with respect to the P -set M(P), where M(P) is the disjoint union of the sets P/Q as Q runs over all maximal subgroups of P, as defined by Bouc in [3, Notation 6.2.1]. Then by [3, Remark 6.2.2], the restriction of ζ to any proper subgroup of P is zero. Moreover by [3, Corollary 4.1.2], for any ϕ ∈ Out(P), we have

Iso_{P,P}^{ϕ} ζ = Iso_{P,P}^{ϕ} (Ω_{M(P)}) = Ω_{^{ϕ}M(P)} = ζ

since any automorphism of P only permutes the maximal subgroups of P. In particular, ζ is Out(P)-invariant. Therefore the kOut(P)-module Codef_{Ω}^{ρ}Res_{ρ}^{Γ}kD(P) contains the trivial kOut(P)-module k as a direct summand. Hence we have proved the following theorem.

Theorem 8.1. Let p be a prime number and P be a non-cyclic p-group of order at least p^{3}. Then

Ind_{μ}^{Γ}S_{P,1}^{μ} ≅ P_{P,1}^{Γ} ⊕ P_{E_{2},1}^{Γ} ⊕ Z

where E_{2} is an elementary abelian group of rank 2 and Z is a (possibly zero) projective biset functor. In particular, the biset functor Ind_{μ}^{Γ}S_{P,1}^{μ} is not indecomposable.

Next we consider the remaining case of p-groups, namely the case where P = E_{2} is an elementary abelian p-group of rank 2. In this case, we claim that the following decomposition holds.

Ind_{μ}^{Γ}S_{E_{2},1} ≅ P_{E_{2},1}^{Γ} ⊕ P_{C_{p},1}^{Γ} ⊕ P

where P is a (possibly-zero) projective biset functor whose only indecomposable summands are of the form P_{C_{p},W}^{Γ} for some non-trivial simple kOut(C_{p})-module W. To obtain this decomposition, note that by [4, Theorem 10.1], we have kD(E_{2}) ≅ k. Moreover since E_{2} is minimal for the functor kD, we have kD(H) = 0 if the order of H is less than p^{2}. Therefore, we have

kD(E_{2}) = Codef_{Ω}^{ρ}Res_{ρ}^{Γ}kD(E_{2}) ≅ k.

Thus P_{E_{2},1}^{Γ} appears in Ind_{μ}^{Γ}S_{E_{2},1}^{μ} only once. We also know, by Lemma 9.3, that P_{1,1}^{Γ} does not appear. Thus, by Corollary 6.3, the only other possibility is P_{C_{p},W}^{Γ} , where C_{p} is a cyclic group of order p and W is a simple kOut(C_{p})-module. We claim that the multiplicity n_{p} of P_{C_{p},1}^{Γ} is equal to 1. By the above considerations, the number n_{p} is equal to the dimension of the subspace of Out(E_{2})-invariant elements in the kOut(E_{2})-module Codef_{Ω}^{ρ}Res_{ρ}^{Γ}S_{C_{p},1}^{Γ}(E_{2}). In other words, we need to show, as in the previous case, that there is an element ζ ∈ S_{C_{p},1}^{Γ}(E_{2}) such that

1. ζ is Out(E_{2})-invariant, that is, for any ϕ ∈ Out(E_{2}), we have Iso_{E_{2},E_{2}}^{ϕ}(ζ) = ζ, and 2. for any proper subgroup L of E_{2}, the restriction Res_{L}^{E_{2}}ζ is equal to zero.

In order to show that such an element exists, we first describe the evaluation of S_{C_{p},1}^{Γ} at E_{2}. By [6, Theorem 4.4], we have

S_{C_{p},1}^{Γ}(E_{2}) ≅ \frac{Ind_{∇}^{Γ}S_{C_{p},1}^{∇}(E_{2})}{K_{C_{p},1}(E_{2})}

where K_{C_{p},1}(E_{2}) is the intersection of kernels of all morphisms from E_{2} to a group of smaller order. We claim that this submodule is zero and hence there is an isomorphism

S_{C_{p},1}^{Γ}(E_{2}) ≅ Ind_{∇}^{Γ}S_{C_{p},1}^{∇}(E_{2})

of kOut(E_{2})-modules. Now by [6, Section 5], we have

Ind_{∇}^{Γ}S_{C_{p},1}^{∇}(E_{2}) = ( ⨁_{\begin{matrix}H∈Sq(E_{2}), \\ H≅C_{p} \end{matrix}} k)_{E_{2}}

where for any kG-module M, we write M_{G} for the largest quotient of M on which G acts trivially. Here note that, although the sum in [6] should be over all subquotients of E_{2}, in our case, we can restrict to the subquotients of order p since the functor S_{C_{p},1}^{∇} is non-zero only on the isomorphism class of the group C_{p}.

Moreover, by Section 5 of [6], the group Out(E_{2}) acts on the sum by permuting the components. Since the E_{2}-action on its subquotients is trivial, we have

Ind_{∇}^{Γ}S_{C_{p},1}^{∇}(E_{2}) = ⨁_{\begin{matrix}H∈Sq(E_{2}) \\ H≅C_{p} \end{matrix}} k.

Now the group E_{2} has p + 1 subgroups of order p, say A_{0}, A_{1}, …, A_{p} and p + 1 corresponding quotients Q_{0} := E_{2}/A_{0}, …, Q_{p} := E_{2}/A_{p}, each of order p. Thus we have

Ind_{∇}^{Γ}S_{C_{p},1}^{∇}(E_{2}) ≅ ⨁_{i=0}^{p} k ⊕ ⨁_{j=0}^{p} k

as permutation kOut(E_{2})-modules. Furthermore, by the proof of Theorem 5.1 in [6], the module Ind_{∇}^{Γ}S_{C_{p},1}^{∇}(E_{2}) is generated by the elements of the form Tin_{H}^{E_{2}} ⊗ a where H is a subquotient of E_{2} and a ∈ S_{C_{p},1}^{∇}(H). Since S_{C_{p},1}^{∇}(H) is one-dimensional when H is isomorphic to C_{p}, we can rewrite the above equality in the following form

Ind_{∇}^{Γ}S_{C_{p},1}^{∇}(E_{2}) ≅ ⨁_{i=0}^{p} kTin_{A_{i}}^{E_{2}} ⊗ 1 ⊕ ⨁_{j=0}^{p} kTin_{Q_{j}}^{E_{2}} ⊗ 1

still as permutation kOut(E_{2})-modules. Now to determine the submodule K_{C_{p},1}(E_{2}), we need to evaluate the restrictions and deflations of the above basis elements. By the descriptions of these maps given in Section 5 of [6], we have

Res_{A_{i}}^{E_{2}}Tin_{A_{j}}^{E_{2}} ⊗ 1 = |A_{i}\E_{2}/A_{j}|Tin_{A_{i}∩A_{j}}^{A_{i}}Res_{A_{i}∩A_{j}}^{A_{j}} ⊗ 1 = \{ \begin{matrix}p(Iso_{A_{i}} ⊗ 1) & if i = j \\ 0 & otherwise. \end{matrix}

Here to get the second row of the above equality, note that, when i ≠ j, we have A_{i}∩A_{j} = 1 and by the definition of the simple functor S_{C_{p},1}^{∇}, the restriction of any element to a group of order smaller than p is zero. In the first row, we put Iso_{A_{i}} := IsoA_{i},A_{i}^{id}. Similarly, we have the following equalities.

Res_{A_{i}}^{E_{2}}Tin_{E_{2}/A_{j}}^{E_{2}} ⊗ 1 = \{ \begin{matrix}Iso_{A_{i}} ⊗ 1 & if i ≠ j \\ 0 & otherwise. \end{matrix}

Def_{E_{2}/A_{i}}^{E_{2}}Tin_{A_{j}}^{E_{2}} ⊗ 1 = \{ \begin{matrix}Iso_{A_{i}} ⊗ 1 & if i ≠ j \\ 0 & otherwise. \end{matrix}

Def_{E_{2}/A_{i}}^{E_{2}}Tin_{E_{2}/A_{j}}^{E_{2}} ⊗ 1 = \{ \begin{matrix}Iso_{A_{i}} ⊗ 1 & if i = j \\ 0 & otherwise. \end{matrix}

Now let

x = ∑_{i} a_{i}Tin_{A_{i}}^{E_{2}} ⊗ 1 + ∑_{j} b_{j}Tin_{E_{2}/A_{j}}^{E_{2}} ⊗ 1

be an element of Ind_{∇}^{Γ}S_{C_{p},1}^{∇}(E_{2}). Then by the above calculations, we have

Res_{A_{k}}^{E_{2}}(x) = p · a_{k} + ∑_{j≠k} b_{j}

and

Def_{E_{2}/A_{k}}^{E_{2}}(x) = ∑_{i≠k} a_{i} + b_{k}.

Therefore x ∈ K_{C_{p},1}(E_{2}) if and only if the equations

p · a_{k} + ∑_{j≠k} b_{j} = 0 and ∑_{i≠k} a_{i} + b_{k} = 0

are satisfied for all k. Note that if we order the subquotients of order p as A_{0}, A_{1}, …, A_{p}, Q_{0}, …, Q_{p}, the corresponding coefficient matrix is of the form

C = [ \begin{matrix}p · I & A \\ A & I \end{matrix} ]

where I is the identity matrix of size p + 1 and A is the matrix

[ \begin{matrix}0 & 1 & ⋯ & 1 \\ 1 & 0 & ⋯ & 1 \\ ⋮ & & ... & ⋮ \\ 1 & 1 & ⋯ & 0 \end{matrix} ]

which has zeros on the diagonal and 1 at any other entry. Straightforward calculations show that the (2p + 2) × (2p + 2)-matrix C is non-singular (actually it has determinant (−1)^{p}p(1 − p)^{p+1}). Therefore the equation Cx = 0 has no non-zero solution, hence K_{C_{p},1}(E_{2}) = 0, as required, and we have

S_{C_{p},1}^{Γ}(E_{2}) ≅ Ind_{∇}^{Γ}S_{C_{p},1}^{∇}(E_{2}) = ⨁_{H∈Sq(E_{2}),H≅C_{p}} k  (9)

as permutation kOut(E_{2})-modules. This completes the description of the evaluation of the simple functor S_{C_{p},1}^{Γ} at E_{2}. Next we proceed to show that an element ζ with the above stated properties exists in the evaluation in (9).

We first determine the intersection of kernels of restrictions to subgroups A_{i}. Let x be as above. Then, by the above calculations, we have

Res_{A_{k}}^{E_{2}}(x) = 0 for all k if and only if p · a_{k} + ∑_{j≠k} b_{j} = 0 for all k.

Thus if the restriction to all subgroups A_{i} of x is equal to zero, then the coefficients a_{i} are uniquely determined by the rest of the coefficients b_{j} by the formula

a_{k} = \frac{1}{p} ∑_{j≠k} b_{j}.

In particular, we are free to choose the coefficients b_{j} and hence

dim_{k} Codef_{Ω}^{ρ}Res_{ρ}^{Γ}S_{C_{p},1}^{Γ}(E_{2}) = p + 1.

Next we look for Out(E_{2})-invariant elements in this kernel since the element ζ whose existence in claimed above is Out(E_{2})-invariant. Now it is clear that x is Out(E_{2})- invariant if and only if b_{i} = b_{j} for all i and j since Out(E_{2}) permutes these coefficients. Thus in the kOut(E_{2})-module Codef_{Ω}^{ρ}Res_{ρ}^{Γ}S_{C_{p},1}^{Γ}(E_{2}), there is a unique Out(E_{2})-invariant element, up to a constant multiple, and hence we have completed the proof of the following result.

Proposition 8.2. Let p be a prime number. Then there is an isomorphism

Ind_{μ}^{Γ}S_{E_{2},1}^{μ} ≅ P_{E_{2},1}^{Γ} ⊕ P_{C_{p},1}^{Γ} ⊕ P

of biset functors where P is a (possible zero) projective biset functor whose only indecomposable summands are of the form P_{C_{p},W}^{Γ} for some simple kOut(C_{p})-module W.

With this proposition, Lemma 9.3 and Theorem 8.1, the following corollary is immediate.

Corollary 8.3. Let p be a prime number and P be a p-group. Then the biset functor Ind_{μ}^{Γ}S_{P,1}^{μ} is not indecomposable.

9. Induced functors for simple groups

In this section, we describe the induced simple native Mackey functors that are indexed by simple groups. We start with the trivial case of the trivial group. In this case, by the classification of the simple native Mackey functors, there is a unique simple functor which is S_{1,1}^{μ}. Note that, by the isomorphism in (2), this functor is the constant functor \underline{k} which takes the value k at any subquotient of G and maps any morphism to the identity homomorphism k → k.

By Corollary 6.3, we immediately get the isomorphism

Ind_{μ}^{Γ}S_{1,1}^{μ} ≅ P_{1,1}^{Γ}

of biset functors. Moreover by [2, Remark 5.1.3], the projective indecomposable biset functor P_{1,1}^{Γ} is isomorphic to the Burnside functor kB^{G}. Thus we obtain the following result.

Proposition 9.1. There is an isomorphism

Ind_{μ}^{Γ}S_{1,1}^{μ} ≅ kB^{G}

of biset functors.

Note that, in this example, the trivial group has no non-trivial proper quotients. This property is also shared by all simple groups. Our next aim is to determine the induction of the simple native Mackey functors parameterized by simple groups, that is, the biset functor Ind_{μ}^{Γ}S_{H,V}^{μ} where H is a simple group. By Corollary 6.3, this functor has at most two summands, namely P_{H,V}^{Γ} and P_{1,1}^{Γ} with the first one of multiplicity one. Thus we only need to determine the multiplicity of kB in the induced functor.

Next we determine the multiplicity m_{B} of the Burnside functor kB^{G} = P_{1,1}^{Γ} in the induced functor Ind_{μ}^{Γ}S_{H,V}^{μ} for any pair (H, V) and then specialize to the case of simple groups. First, by Proposition 6.1, we have

m_{B} = dim_{k} Hom_{kOut(H)}(V, Codef_{Ω}^{ρ}Res_{ρ}^{Γ}S_{1,1}^{Γ}(H)).

Also, by [2, Proposition 4.4.8], we have

S_{1,1}^{Γ} ≅ kR_{ℚ}

where R_{ℚ} is the functor of rational characters of finite groups. As in the previous sections, we need to determine the Out(H)-invariant elements in Codef_{Ω}^{ρ}Res_{ρ}^{Γ}S_{1,1}^{Γ}(H). For this aim, we first determine the restriction of S_{1,1}^{Γ} to the category of native Mackey functors. By the above identification, this amounts to determine the simple summands of the native Mackey functor kR_{ℚ} of rational representations. Note further that, by the isomorphism given in (3), we have

kR_{ℚ} ≅ Ind_{ρ}^{μ}Inf_{Ω}^{ρ}Def_{Ω}^{τ}Res_{τ}^{μ}kR_{ℚ}.

On the other hand, by the definition of the deflation functor Def_{Ω}^{τ}, for any H ∈ Sq(G), we have

Def_{Ω}^{τ}kR_{ℚ}(H) = kR_{ℚ}(H)/ ∑_{L<H} Im(Ind_{L}^{H}).

Now, by Artin’s Induction Theorem, the above quotient is non-zero only if the group H is cyclic. Moreover, when the group H is cyclic, the quotient is isomorphic to k as kOut(H)-modules. Indeed, in this case, by [7, Theorem 30], the set of characters Ind_{L}^{H}1 as L runs over all subgroups of H generates kR_{ℚ}(H) and moreover, as indicated in Exercise 13.8 in [7], this set is actually a basis of kR_{ℚ}(H). It is now clear that the quotient is one dimensional. Moreover the Out(H)-action on this vector space is trivial since kOut(H) fixes the character 1 = Ind_{H}^{H}1.

With this observation, we conclude that there is an isomorphism

Def_{Ω}^{τ}Res_{τ}^{Γ}S_{1,1}^{Γ} ≅ ⨁_{H∈Sq(G):H:cyclic up to isomorphism} S_{H,1}^{Ω}

of Ω-modules. Thus by the above isomorphism and the isomorphism given in Equation (2), we get that following result.

Proposition 9.2. As a native Mackey functor for G over k, the functor of rational representations decomposes as

kR_{ℚ} ≅ ⨁_{H∈Sq(G):H:cyclic up to isomorphism} S_{H,1}^{μ}.

Recall that we are looking for elements ζ in kR_{ℚ}(H) such that

1. ζ is Out(H)-invariant, that is, for any ϕ ∈ Out(H), we have Iso_{H,H}^{ϕ}(ζ) = ζ, and 2. for any proper subgroup L of H, the restriction Res_{L}^{H}ζ is equal to zero.

The elements satisfying the second property lies in

Codef_{Ω}^{ρ}Res_{ρ}^{Γ}kR_{ℚ}(H) = ⋂_{K<H} ker(Res_{K}^{H} : kR_{ℚ}(H) → kR_{ℚ}(K)).

Now by Equation (2) and by Proposition 9.2, we have an isomorphism

kR_{ℚ} ≅ Coind_{τ}^{μ}Inf_{Ω}^{τ}Def_{Ω}^{τ}Res_{τ}^{μ}kR_{ℚ}

of native Mackey functors. Therefore, we have

Codef_{Ω}^{ρ}Res_{ρ}^{μ}kR_{ℚ} ≅ Codef_{Ω}^{ρ}Res_{ρ}^{μ}Coind_{τ}^{μ}Inf_{Ω}^{τ}Def_{Ω}^{τ}Res_{τ}^{μ}kR_{ℚ}
≅ Codef_{Ω}^{ρ}Coind_{Ω}^{ρ}Res_{Ω}^{τ}Inf_{Ω}^{τ}Def_{Ω}^{τ}Res_{τ}^{μ}kR_{ℚ}

where we use Theorem 3.3 to interchange the coinduction and the restriction functors. Then, again by Theorem 3.3, the composition Codef_{Ω}^{ρ}Coind_{Ω}^{ρ} and the composition Res_{Ω}^{τ}Inf_{Ω}^{τ} are naturally equivalent to the identity functor. Therefore, we get that

Codef_{Ω}^{ρ}Res_{ρ}^{μ}kR_{ℚ} ≅ Def_{Ω}^{τ}Res_{τ}^{μ}kR_{ℚ}.

In particular,

Codef_{Ω}^{ρ}Res_{ρ}^{μ}kR_{ℚ}(H) = 0

unless H is cyclic and hence no element ζ with the above prescribed properties exists in kR_{ℚ}(H). Therefore when H is not cyclic, the Burnside functor is not a direct summand of the biset functor Ind_{μ}^{Γ}S_{H,V}.

On the other hand, if H is cyclic, then by the above results,

Codef_{Ω}^{ρ}Res_{ρ}^{μ}kR_{ℚ}(H) ≅ k

as kOut(H)-modules. Therefore all elements in this module are Out(H)-fixed and this completes the proof of the following lemma.

Lemma 9.3. The multiplicity m_{B} of the Burnside functor kB^{G} as a direct summand in the biset functor Ind_{μ}^{Γ}S_{H,V}^{μ} is given by

m_{B} = \{ \begin{matrix}1 & if H is cyclic and V is trivial. \\ 0 & otherwise. \end{matrix}

Combining the above lemma with Corollary 6.3, we get the following theorem.

Theorem 9.4. Let H be a simple group.

1. If H is of prime order, and 1 is the trivial kOut(H)-module, then

Ind_{μ}^{Γ}S_{H,1}^{μ} ≅ P_{H,1}^{Γ} ⊕ P_{1,1}^{Γ}.

2. Otherwise, if H is cyclic but V is non-trivial, or if H is not cyclic, then

Ind_{μ}^{Γ}S_{H,V}^{μ} ≅ P_{H,V}^{Γ} .

Note that when H is a simple, non-cyclic group and V is non-trivial, the functor Ind_{μ}^{Γ}S_{H,V}^{μ} gets easier. Indeed, in this case, we have Ind_{ρ}^{∇}S_{H,V}^{ρ} ≅ S_{H,V}^{∇} by Theorem Appendix A.1. To prove this isomorphism, note that, for any K ∈ Sq(G), we have

Ind_{ρ}^{∇}S_{H,V}^{ρ} (K) = 0

if K is not isomorphic to a quotient of H. But since H is simple this means that the above evaluation is equal to zero unless K = H or K = 1. Now the evaluation at H is equal to V by the definition of the induction functor. Also, we have

Ind_{ρ}^{Δ}S′_{H,V} ρ(1) = kX(1, H) ⊗_{kOut(H)} V ≅ k ⊗_{kOut(H)} V = 0.

Therefore we get

Ind_{ρ}^{∇}S_{H,V}^{ρ} (1) = 0

and hence the isomorphism

Ind_{ρ}^{∇}S_{H,V}^{ρ} ≅ S_{H,V}^{∇}

of ∇-modules hold. Note that, when V = 1 is the trivial kOut(H)-module, we have k ⊗_{kOut(H)} k = k, and hence Ind_{ρ}^{∇}S_{H,V} (1) = k. In particular, the above isomorphism does not hold when V is trivial. We can summarize this result as follows.

Corollary 9.5. Assume the notation of the above theorem and suppose that H is not cyclic and V is non-trivial. Then there is an isomorphism

P_{H,V}^{Γ} ≅ Ind_{μ}^{Γ}S_{H,V}^{μ} ≅ Ind_{∇}^{Γ}S_{H,V}^{∇}

of biset functors.

Note that the induction functor Ind_{∇}^{Γ} can be evaluated using [6, Theorem 5.1].

Problem 9.6. By Corollary 8.3 and Lemma 9.3, the biset functor Ind_{μ}^{Γ}S_{H,1}^{μ} is not indecomposable if H is a p-group for some prime p or a cyclic group. Also in some other groups of small order, not covered by these results, we showed, by hand, that the above functor has at least two direct summands. We do not include these calculations. However we could not answer whether Ind_{μ}^{Γ}S_{H,V}^{μ} being indecomposable (and hence isomorphic with P_{H,V}^{Γ} ) characterizes simple groups, and leave it as open.

Appendix A. The functor Ind_{ρ}^{∇}

In this section, we describe the functor Ind_{ρ(G)}^{∇(G)} = ∇(G)⊗_{ρ(G)}. For simplicity, we write ∇ = ∇(G) and ρ = ρ(G). For our aims, it is sufficient to describe the functor only for atomic functors, that is, for functors which are non-zero only on one isomorphism class of subquotients. Thus, let D_{H,V} be the ρ-module such that D_{H,V} (K) = 0 if K is not isomorphic to H and D_{H,V} (H) = V. Note that, here, the module V is not necessarily simple. Further let K be a subquotient of G. We want to describe

M := Ind_{ρ}^{∇}D_{H,V} (K) = Iso_{K,K}^{id}∇ ⊗_{ρ} D_{H,V}

where Iso_{K,K}^{id} is the (K, K)-biset K where the group K acts on both sides via multiplication. By [6, Theorem 3.3], the algebra ∇ has a k-basis consisting of all triples [R, L, ϕ : L → K] where R runs over all subquotients of G, L runs over all subquotients of R, up to conjugation, K runs over all subquotients of G and ϕ : L → K is an isomorphism, taken up to (K, R)- conjugacy, in the sense of [6, Section 3]. Thus we can write

M = Iso_{K,K}^{id}∇ ⊗_{ρ} e_{H}D_{H,V}
= ⨁_{[R,L,ϕ:L→K],R≅H} kIso_{K,L}^{ϕ}Des_{L}^{R}e_{H} ⊗_{ρ} e_{H}D_{H,V} .

where e_{H} is the sum of Iso_{R,R}^{id} where R ≅ H runs over all subquotients of G isomorphic to H. Now if L is a proper subgroup of R, then the corresponding summand will be equal to zero since the restriction map Des_{L}^{R} annihilates D_{H,V}. Thus the sum becomes

M = ⨁_{\begin{matrix}[R,R/N,ϕ:R/N→K], \\ R≅H \end{matrix}} kIso_{K,R/N}^{ϕ}Des_{R/N}^{R}e_{H} ⊗_{ρ} e_{H}D_{H,V} .

Let Iso(L, K) denote the set of all isomorphisms from L to K. Then the above sum can be written as follows.

M = ⨁_{\begin{matrix}R∈Sq(G),N⊴R \\ R≅H,R/N≅K \\ ϕ∈Iso(R/N,K) \end{matrix}} kIso_{K,R/N}^{ϕ}Des_{R/N}^{R}e_{H} ⊗_{ρ} e_{H}D_{H,V} .

Now there is a left K-action and a right R action on the set Iso(R/N, K) which are trivial on the tensor product. Thus the above sum reduces to

M = ⨁_{\begin{matrix}R∈Sq(G),N⊴R \\ R≅H,R/N≅K \\ ϕ∈K\Iso(R/N,K)/R \end{matrix}} kIso_{K,R/N}^{ϕ}Des_{R/N}^{R}e_{H} ⊗_{ρ} e_{H}D_{H,V}

= ⨁_{\begin{matrix}R∈Sq(G),N⊴R \\ R≅H,R/N≅K \\ ϕ∈K\Iso(R/N,K)/R \end{matrix}} kIso_{K,R/N}^{ϕ}Des_{R/N}^{R} ⊗_{kOut(R)} V (R)

where V (R) := D_{H,V} (R). To simplify the above sum, let

X(K, R) = ⨆_{N⊴R:R/N≅K} K\Iso(R/N, K)/R.

The set X(K, R) is an (Out(K), Out(R))-biset, via pre and post compositions. Now we have

M = ⨁_{R∈Sq(G):R≅H} kX(K, R) ⊗_{kOut(R)} V (R).

This completes the proof of the following result.

Theorem Appendix A.1. Let V be a kOut(H)-module. Denote by D_{H,V} the ρ-module which is zero on all subquotients of G not isomorphic to H and D_{H,V} (H) = V. Let K be a subquotient of G. Then

Ind_{ρ}^{∇}D_{H,V} (K) = ⨁_{R∈Sq(G):R≅H} kX(K, R) ⊗_{kOut(R)} V (R)

where X(K, R) is the (Out(K), Out(R))-biset defined above.

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