Alcahestic subalgebras of the alchemic algebra and a correspondence of simple modules

Olcay COŞKUN^{∗}

5 March 2008

Bilkent Universitesi, Matematik Bölümü, 06800 Bilkent, Ankara, Türkiye (Turkey). e-mail: coskun@fen.bilkent.edu.tr.

Abstract

The unified treatment of the five module-theoretic notions, transfer, inflation, transport of structure by an isomorphism, deflation and restriction, is given by the theory of biset functors, introduced by Bouc. In this paper, we construct the algebra realizing biset functors as representations. The algebra has a presentation similar to the well-known Mackey algebra. We adopt some natural constructions from the theory of Mackey functors and give two new constructions of simple biset functors. We also obtain a criterion for semisimplicity in terms of the biset functor version of the mark homomorphism. The criterion has an elementary generalization to arbitrary finite-dimensional algebras over a field. 2000 Mathematics Subject Classification. Primary: 20C20; Secondary: 19A22. Keywords: biset functor, globally-defined Mackey functor, inflation functor, simple functor, simple module, coinduction, semisimplicity, mark homomorphism.

1 Introduction

In [3] and [4], Bouc introduced two different notions of functors. The first notion, now known as a biset functor, is an R-linear functor from a category whose objects are finite groups to the category of modules over the commutative ring R with unity. In this case, the morphisms between finite groups are given by finite bisets and the composition product of bisets is given by the usual amalgamated product of bisets, called the Mackey product in [3]. In his paper, Bouc proved that any biset is a composition of five types of bisets, namely composition of a transfer biset, an inflation biset, an isomorphism biset, a deflation biset and a restriction biset. The names of these special bisets comes from their actions on the representation ring functor and the Burnside ring functor where the actions coincide with the transfer map, the inflation map, the transport by an isomorphism, the deflation map and the restriction map respectively. Hence, this approach gives a unified treatment of these five module-theoretic notions which extend the operations available with Mackey functors. An application of the biset functors is to the Dade group of a p-group (see [9], [10]).

∗The author is supported by TÜBİTAK (Turkish Scientific and Technological Research Council)

through a PhD award program (BDP).

In [4], Bouc used the category of finite groups with morphisms given by the bisets to classify the functors between the categories of finite G-sets. He proved that any such functor preserving disjoint unions and cartesian products corresponds to a biset, and conversely, any biset induces a functor with these properties. However, the usual amalgamated product of bisets does not correspond to the composition of these functors. Therefore, he obtains a restricted product of bisets and a category of functors defined as above with different composition of morphisms.

Both of these categories of functors are abelian and the corresponding simple functors are classified by Bouc in [3] and [4]. The description of simple objects in both of the categories are given by the same construction but the Mackey product of the first category makes the construction complicated while the restricted product of the second category makes the description explicit.

In [17], Webb considered two other kinds of functors by allowing only certain types of bisets as morphisms. For the first one, he allows only the bisets that are free on both sides. These functors are well-known as global Mackey functors. The other case is where the right-free bisets are allowed. He called this kind inflation functors. The main example of the inflation functors is the functor of group cohomology. He described simple functors for both of these cases. As we shall see below, his construction is a special case of one of our main results. A general framework for these functors were also introduced by Bouc in [3]. Given two classes X and Y of finite groups having certain properties, Bouc considered R-linear functors C_{R}^{X ,Y} → R-mod, called globally-defined Mackey functor in [18]. Here the category C_{R}^{X ,Y} consists of all finite groups with morphisms given by all finite bisets with left-point stabilizers in X and right point stabilizers in Y. The product is still the amalgamated product of bisets. Now global Mackey functors are obtained by letting both X and Y consist of the trivial group and the biset functors are obtained by letting them contain all finite groups, and inflation functors are obtained by letting X consist of all finite groups and Y consist of the trivial group.

In this paper, we only consider the biset functors defined over the category restricted to a fixed finite group. That is, we consider the category C_{G} of subquotients of a fixed finite group G with morphisms given by the Grothendieck group of bisets between the subquotients. The composition is still given by the Mackey product. Here by a subquotient, we mean a pair (H^{∗}, H_{∗}) of subgroups of G such that H_{∗} ⊴ H^{∗}. We always write H for the pair (H^{∗}, H_{∗}). Now a biset functor for G (over a commutative ring R with unity) is an R-linear functor C_{G} → R-mod. Note that although we consider a special case of Bouc’s biset functors, it is easy to generalize our results from this finite category to the infinite case since the Grothendieck group of bisets for any two finite groups is independent of their inclusions to any larger group as subquotients.

It follows from a well-known result that the category of biset functors for G over R is equivalent to the module category of the algebra generated by finite bisets appearing as morphisms in C_{G}. Following Barker [1], we call this algebra the alchemic algebra^{1}. Description of the alchemic algebra using Bouc’s decomposition theorem for bisets produces five types of generators and so many relations that cannot be specified in a tractable way. Our approach to the alchemic algebra is to amalgamate some of the variables in a way that the relations become tractable. In order to do this, we introduce two new variables, namely tinflation which is the composition of transfer and inflation, and destriction, composition of deflation and restriction. The fifth element remains the same but we call it isogation. In this way, the relations become not only tractable but also similar to the relations between the generators of the Mackey algebra.

1The name refers to the five elements of nature in alchemy; air, fire, water, earth and the fifth element known as quintessence or aether. Two elements, transfer and inflation, go upwards as the two elements fire and air. Similarly deflation and restriction go downwards as water and earth. See [1].

Note that in [18], Webb specified a list of relations that works for globally-defined Mackey functors for any choice of the classes X and Y. Our list of relations, although obtained using the Mackey product, can also be obtained from these relations with some straightforward calculations. Also in [1], Barker constructed the alchemic algebra, without specifying the relations explicitly, for a family of finite groups closed under taking subquotients and isomorphisms. This covers our case when the family is the family of subquotients of G.

Having the above description of the alchemic algebra, we are able to adapt some natural constructions from the context of the Mackey functors to the context of biset functors. Our first main result in this direction concerns the construction of simple biset functors using the techniques in [12]. In fact our result extends to classification and description of simple modules of certain unital subalgebras. The study of subalgebras of the alchemic algebra is crucial since some natural constructions, such as group homology, group cohomology and modular representation rings, are only modules over some subalgebras of the alchemic algebra. For instance, Webb’s inflation functors are representations of such a subalgebra of the alchemic algebra. This subalgebra realizes the modular character ring as well as the group cohomology. Group homology is a module of the opposite algebra of this subalgebra. In Section 4, we shall prove the following classification theorem which extends both Bouc’s and Webb’s classification of simple functors.

Theorem 1.1 Let Π be an alcahestic subalgebra of the alchemic algebra. There is a bijective correspondence between

(i) The isomorphism classes S^{Π} of simple Π-modules.

(ii) The isomorphism classes S of simple Ω_{Π}-modules.

Here by an alcahestic subalgebra, we mean a subalgebra of the alchemic algebra containing a certain set of orthogonal idempotents summing up to the identity, see Section 4 for a precise definition. The algebra Ω_{Π} is the subalgebra of Π generated by isogations in Π. Our theorem generalizes Bouc’s and Webb’s classification theorems because in both cases Ω_{Π} is Morita equivalent to the algebra ∏_{H} ROut(H) where the product is over subquotients of G up to isomorphism. Note also that globally-defined Mackey functors are always modules of a suitably chosen alcahestic subalgebra.

This classification theorem follows easily from the correspondence theorem below. Note further that a version of the following theorem, Corollary 4.7, gives the bijection of Theorem 1.1, explicitly.

Theorem 1.2 Let Π ⊂ Θ be alcahestic subalgebras of the alchemic algebra such that all destrictions in Θ are contained in Π. Let S be a simple Π-module. Then the induced Θ-module ind_{Π}^{Θ} S := Θ ⊗_{Π} S has a unique maximal submodule.

In Section 5, we apply this result to give two descriptions of the simple biset functors. These descriptions are both alternatives to Bouc’s construction. Moreover our construction has an advantage that the induced module in Theorem 1.2 is smaller than a similar one used by Bouc in the sense that it is a quotient of the latter. Also one of our descriptions explicitly gives a formula for the action of the tinflation on the simple module. The action is similar to the action of transfer on a simple Mackey functor, which is given by the relative trace map.

Another natural construction that we adapt from the Mackey functors is the mark morphism. Originally the mark morphism is a morphism from the Burnside ring to its ghost ring. Boltje [2] generalized this concept to a morphism connecting his plus constructions. It is shown in [12] that the plus constructions are usual induction and coinduction functors and the mark morphism can be constructed by applying a series of adjunctions. In Section 6, we further generalize the mark morphism to biset functors in a similar way. In particular, we have a mark morphism β^{S} associated to any simple biset functor S. Here we see another advantage of our alternative descriptions that this mark morphism can only be constructed between the induced and coinduced modules described in Section 5.

Our final main result is a characterization of semisimplicity of the alchemic algebra in terms of the mark morphism. Note that the equivalence of the first two statements below is proved by Barker [1] and Bouc, independently. Our proof is less technical than the previous two proofs since Barker and Bouc compared the dimension of the alchemic algebra with the dimensions of simple functors. Instead we have the following result.

Theorem 1.3 Let G be a finite group and R be a field of characteristic zero. The followings are equivalent.

(i) The alchemic algebra for G over R is semisimple.

(ii) The group G is cyclic.

(iii) The mark morphism β^{S} is an isomorphism for any simple biset functor S.

We end the paper by a generalization of this criterion to an arbitrary finite-dimensional algebra over a field. This is an elementary result and it completes a well-known result on semisimplicity of such algebras.

Theorem 1.4 Let A be a finite dimensional algebra over a field and e be an idempotent of A. Let f = 1 − e. Then the following are equivalent

1. The algebra A is semisimple.

2. (a) The algebras eAe and fAf are semisimple.

(b) For any simple gAg-module V, for g ∈ {e, f}, there is an isomorphism of A-modules Ag ⊗_{gAg} V ≅ Hom_{gAg}(gA, V).

Finally note that our results still hold if we change the composition product of the category C_{G} with the restricted product in [4] and also for globally-defined Mackey functors, with appropriate changes.

Acknowledgement. I wish to thank to the referee for several suggestions. I am also grateful to Robert Boltje for his valuable remarks.

2 Biset functors, an overview

In this section, we summarize some basic definitions and constructions concerning biset functors. First, we introduce the category of bisets. This is the standard theory of bisets and can be found in [3]. Then we review Bouc’s definition of biset functors and Bouc’s construction of simple functors [3].

2.1 Bisets

Let H and K be two finite groups. An (H, K)-biset is a set with a left H-action and a right K-action such that

h(xk) = (hx)k

for all elements h ∈ H and k ∈ K.

An (H, K)-biset X is called transitive if for any elements x, y ∈ X there exists an element h ∈ H and an element k ∈ K such that hxk is equal to y.

We can regard any (H, K)-biset as a right H × K-set with the action given by

x.(h, k) = h^{−1}xk

for all h ∈ H and k ∈ K. Clearly, X is a transitive (H, K)-biset if and only if X is a transitive H × K-set. Hence there is a bijective correspondence between

(i) isomorphism classes [X] of transitive (H, K)-bisets,

(ii) conjugacy classes [L] of subgroups of H × K

where the correspondence is given by [X] ↔ [L] if and only if the stabilizer of some point x ∈ X is in [L].

Hence we can denote a transitive biset by (H × K)/L. Given finite groups H, K, M and transitive bisets (H × K)/L and (K × M)/N, we define the product of these bisets by the Mackey product [3], given by

(H × K)/L ×_{K} (K × M)/N = ∑_{x∈p_{2}(L)\K/p_{1}(N)}(H × M)/L ∗^{(x,1)}N

where the subgroup L ∗ N of H × M is defined by

L ∗ N = {(h, m) ∈ H × M : (h, k) ∈ L and (k, m) ∈ N for some k ∈ K}

and the subgroups p_{1}(N) and p_{2}(L) of K are projections of N and L to K, respectively, that is,

p_{1}(L) = {l ∈ H : (l, k) ∈ L for some k ∈ K} and p_{2}(L) = {k ∈ K : (l, k) ∈ L for some l ∈ L}.

In [3], Bouc proved that any transitive biset is a Mackey product of the following five types of bisets: Let H be a finite group and N ⊴ J be subgroups of H and let L, M be two isomorphic finite groups with a fixed isomorphism ϕ : L → M, then the five bisets are given as follows.

1. Induction biset: ind_{J}^{H} := (H × J)/T where T = {(j, j) : j ∈ J}.

2. Inflation biset: inf_{J/N}^{J} := (J × J/N)/I where I = {(j, jN) : j ∈ J}.

3. Isomorphism biset: c_{M,L}^{ϕ} = (M × L)/C^{ϕ} where C^{ϕ} = {(ϕ(l), l) : l ∈ L}.

4. Deflation biset: def_{J/N}^{J} = (J/N × J)/D where D = {(jN, j) : j ∈ J}.

5. Restriction biset: res_{J}^{H} = (J × H)/R where R = {(j, j) : j ∈ J}.

The following theorem explicitly shows the decomposition of any transitive biset in terms of these special bisets.

Theorem 2.1 [3] Let L be any subgroup of H × K. Then

(H × K)/L = ind_{p_{1}(L)}^{H} × inf_{p_{1}(L)/k_{1}(L)}^{p_{1}(L)} × c_{p_{1}(L)/k_{1}(L),p_{2}(L)/k_{2}(L)}^{ϕ} × def_{p_{2}(L)/k_{2}(L)}^{p_{2}(L)} × res_{p_{2}(L)}^{K}

where the subgroup k_{1}(L) of H and the subgroup k_{2}(L) of K are given by

k_{1}(L) = {h ∈ H : (h, 1) ∈ L} and k_{2}(L) = {x ∈ K : (1, x) ∈ L}.

The isomorphism

ϕ : p_{2}(L)/k_{2}(L) → p_{1}(L)/k_{1}(L)

is the one given by associating lk_{2}(L) to mk_{1}(L) where for a given element l ∈ p_{2}(L) we let m be the unique element in p_{1}(L) be such that (m, l) ∈ L.

2.2 Biset functors

Let C be the category whose objects are finite groups and let the set of morphisms between two finite groups H and K be given by

Hom_{C}(H, K) = R ⊗_{ℤ} Γ(K, H) =: RΓ(K, H)

where Γ(K, H) denotes the Grothendieck group of the isomorphism classes of finite (K, H)- bisets with addition as the disjoint union and where R is a field. The composition of the morphisms in C is given by the Mackey product of bisets.

Now a biset functor F over R is an R-linear functor C → R-mod. Defining a morphism of biset functors as a natural transformation of functors, we obtain the category Biset_{R} of biset functors. Since the category R-mod is abelian, the category Biset_{R} is also abelian. Simple objects of this category are described by Bouc [3]. We shall review his construction.

Let H be a finite group. We denote by E_{H} the endomorphism algebra End_{C}(H) of H in the category C. It is easy to show that E_{H} decomposes as an R-module as

E_{H} = I_{H} ⊕ ROut(H)

where Out(H) = Aut(H)/Inn(H) is the group of outer automorphisms of H and ROut(H) is the group algebra of Out(H) and I_{H} is a two-sided ideal of E_{H} (see [3] for explicit description of I_{H}). Therefore, we obtain an epimorphism

E_{H} ↠ ROut(H)

of algebras. In particular, we can lift any simple ROut(H)-module V to a simple E_{H}-module, still denoted by V.

Denote by e_{H} the evaluation at H functor, that is, let e_{H} : Biset_{R} → E_{H}-mod be the functor sending a biset functor to its value at the group H. Now let L_{H,V} denote the left adjoint of the functor e_{H}. Explicitly, for a finite group K, we get

L_{H,V} (K) = Hom_{C}(H, K) ⊗_{E_{H}} V.

The action of a biset on L_{H,V} is given by composition of morphisms. The functor L_{H,V} has a unique maximal subfunctor

J_{H,V} (K) = {∑_{i} ϕ_{i} ⊗ v_{i}|∀ψ ∈ Hom_{C_{G}}(K, H), ∑_{i}(ψϕ_{i})v_{i} = 0}.

Hence taking the quotient of L_{H,V} with this maximal ideal, we obtain a simple biset functor

S_{H,V} := L_{H,V} /J_{H,V} .

Moreover, we have

Theorem 2.2 (Bouc [3]) Any simple biset functor is of the form S_{H,V} for some finite group H and a simple ROut(H)-module V.

As mentioned in the introduction, the main examples of the biset functors are the functor of the Burnside ring and the functor of the representation ring. Further, over a field of characteristic zero, the rational representation ring is an example of a simple biset functor. More precisely, Bouc proved in [3] that the biset functor of rational representation ring ℚR_{ℚ} over ℚ is isomorphic to the simple biset functor S_{1,ℚ}. Another interesting example of a simple biset functor is the functor S_{C_{p}×C_{p},ℚ} defined only over p-groups where p is a prime number. In [10], it is shown that this functor is isomorphic to the functor of the Dade group ℚD with coefficients extended to the rational numbers ℚ and there is an exact sequence of biset functors

0  ℚD  ℚB ℚR^{ℚ}  0.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

Here the map ℚB → ℚR_{ℚ} can be chosen as the natural map sending a P -set X to the permutation module ℚX. For an improvement of this result to ℤ, see [7] and for more exact sequences relating these functors, see [11]. Some other well-known examples of biset functors are the functor of units of the Burnside ring [5] and the functor of the group of relative syzygies [6]. For further details also see [3], [9], [10], [8] and [1].

For the rest of the paper, we concentrate on the biset functors defined only for subquotients of a fixed finite group G. We introduce some notations that will be used throughout the paper. Recall that a subquotient of G is a pair (H^{∗}, H_{∗}) where H_{∗} ⊴ H^{∗} ≤ G. We write the pair (H^{∗}, H_{∗}) as H and denote the subquotient relation by H ⪯ G. Here, and afterwards, we regard any group L as the subquotient (L, 1). We write H ⪯_{G} G to mean that H is taken up to G-conjugacy and write H ⪯_{∗} G to mean that H is taken up to isomorphism. Note that we always consider H as the quotient group H^{∗}/H_{∗}. Clearly the relation ⪯ extends to a relation on the set of subquotients of G in the following way. Let J and H be two subquotients of G. Then we write J ⪯ H if and only if H_{∗} ≤ J_{∗} and H^{∗} ≥ J^{∗}. In this case the pair (J^{∗}/H_{∗}, J_{∗}/H_{∗}) is a subquotient of H. Finally we say that two subquotients H and K of G are isomorphic if and only if they are isomorphic as groups, that is, if H^{∗}/H_{∗} ≅ K^{∗}/K_{∗}.

In this case, we have the category C_{G} of subquotients of G with objects the groups H as H runs over the set of subquotients of G and with the same morphisms and the same composition product as C. Note that since the set Hom_{C}(H, K) of (H, K)-bisets depends only on the subquotients of the groups H and K, it is easy to generalize the results from the finite category C_{G} to the infinite case.

Now we define a biset functor for G over R as an R-linear functor C_{G} → R-mod. We also denote by Biset_{R}(G) the category of biset functors for G over R.

Finally let us introduce a notation that we will use throughout the paper. Let H, K ⪯ G. We define the intersection H ⊓ K of the subquotients of H and K as

H ⊓ K = \frac{(H^{∗} ∩ K^{∗})H_{∗}}{(H^{∗} ∩ K_{∗})H_{∗}} .

Note that, in general, this intersection is neither commutative nor associative. But, there is an isomorphism of groups

λ : H ⊓ K → K ⊓ H

which we call the canonical isomorphism between the groups H ⊓ K and K ⊓ H. The isomorphism is the one that comes from the Zassenhaus-Butterfly Lemma.

3 Alchemic algebra

It is clear that the category of Biset_{R}(G) of biset functors for G over R is equivalent to the category of modules of the algebra Γ_{R}(G) defined by

Γ_{R}(G) = ⨁_{H,K⪯G} Hom_{C}(H, K).

It is evident that this algebra has a basis consisting of the isomorphism classes of transitive bisets. Hence by Theorem 2.1, it is generated by the five special types of bisets, namely by transfer, inflation, isomorphism, deflation and restriction bisets. Following [1], we call this algebra the alchemic algebra for G over R, written shortly as Γ when G and R are understood.

It is possible to define the alchemic algebra by forgetting the bisets altogether. In order to do this, we can consider the algebra generated freely over R by the five types of variables corresponding to the five special types of bisets. Then the alchemic algebra is the quotient of this algebra by the ideal generated by relations between the variables induced by the Mackey product of bisets. But the relations obtained in this way are not tractable. To get the relations in a tractable way, we introduce two new amalgamated variables. Instead of the five variables, we consider the composition of transfer and inflation as the first variable, which we call tinflation, and the composition of deflation and restriction as the second one, called destriction. The third and final variable is the transport of structure by an isomorphism, which we call isogation. In this way, we obtain a set of relations that is very similar to the defining relations of the well-known Mackey algebra. Explicitly, consider the algebra freely generated over R by the following three types of variables.

V1. tin_{J}^{H} for each J ⪯ H ⪯ G,

V2. des_{J}^{H} for each J ⪯ H ⪯ G,

V3. c_{M,L}^{ϕ} for each M, L ⪯ G such that M ≅ L and for each isomorphism ϕ : M → L.

Then we let Γ̃_{R}(G), written Γ̃, be the quotient of this algebra by the ideal generated by the following relations.

R1. Let h : H → H denotes the inner automorphism of H induced by conjugation by h ∈ H. Then c_{H,H}^{h} = tin_{H}^{H} = des_{H}^{H}.

R2. Let L ⪯ J and ψ : M → S be an isomorphism. Then

(i) c_{S,M}^{ψ} c_{M,L}^{ϕ} = c_{S,L}^{ψ∘ϕ},

(ii) tin_{J}^{H} tin_{L}^{J} = tin_{L}^{H},

(iii) des_{L}^{J} des_{J}^{H} = des_{L}^{H}.

R3. Let K ⪯ G and let α : H → K be an isomorphism and let ^{α}J denote α(J^{∗})/α(J_{∗}), then

(i) c_{K,H}^{α} tin_{J}^{H} = tin_{^{α}J}^{K} c_{^{α}J,J}^{α},

(ii) des_{I}^{K} c_{K,H}^{α} = c_{I,^{α^{−1}}I}^{α}des_{^{α^{−1}}I}^{H}.

R4. (Mackey relation) Let I ⪯ H. Then

des_{I}^{H} tin_{J}^{H} = ∑_{x∈I^{∗}\H/J^{∗}} tin_{I⊓^{x}J}^{I} c^{x∘λ} des_{J⊓I^{x}}^{J}

Here c^{x∘λ} := c_{I⊓^{x}J,J⊓I^{x}}^{x∘λ} and λ is the canonical isomorphism introduced in the previous section.

R5. 1 = ∑_{H⪯G} c_{H} where c_{H} := c_{H,H}^{1}.

R6. All other products of the generators are zero.

Remark 3.1 In [9], the amalgamated variables tinflation and destriction are abbreviated as indinf and defres, respectively.

Even it is clear from the construction, the following theorem formally shows that the algebra Γ̃ is isomorphic to the alchemic algebra Γ.

Theorem 3.2 The algebras Γ and Γ̃ are isomorphic.

Proof. The correspondence

tin_{J}^{H}c_{J,I}^{ϕ}des_{I}^{K} ⟶ (H × K)/A

where A = {(h, k) ∈ J^{∗} × I^{∗} : hJ_{∗} = ϕ(kI_{∗})} extends linearly to a map α : Γ̃ → Γ. We must show that α is an algebra isomorphism. Indeed, α is an isomorphism of R-modules by Theorem 2.1. Thus, it suffices to check that it respects the multiplication. We shall only check the Mackey relation. The others can be checked similarly. First note that the images of tinflation and destriction are

α(tin_{J}^{H}) = ind_{J^{∗}/H_{∗}}^{H} inf_{(J^{∗}/H_{∗})/(J_{∗}/H_{∗})}^{J^{∗}/H_{∗}} c_{(J^{∗}/H_{∗})/(J_{∗}/H_{∗}),J}^{λ} =: (H × J)/T′

and

α(des_{J}^{H}) = c_{J,(J/H_{∗})/(J_{∗}/H_{∗})}^{λ^{−1}}def_{(J/H_{∗})/(J_{∗}/H_{∗})}^{J^{∗}/H_{∗}} res_{J^{∗}/H_{∗}}^{H} =: (J × H)/R′

where λ is the canonical map J → (J^{∗}/H_{∗})/(J_{∗}/H_{∗}) and

T′ = {(jH_{∗}, j′J_{∗}) ∈ J^{∗}/H_{∗} × J : (jH_{∗})J_{∗}/H_{∗} = λ(j′J_{∗})}

and

R′ = {(jJ_{∗}, j′H_{∗}) ∈ J × J^{∗}/H_{∗} : (j′H_{∗})J_{∗}/H_{∗} = λ(jJ_{∗})}.

Hence, we must show that

α(des_{I}^{H} tin_{J}^{H}) = (I × H)/R′ ×_{H} (H × J)/T′.

By the Mackey product formula, we have

(I × H)/R′ ×_{H} (H × J)/T′ = ∑_{x∈p_{2}(R′)\H/p_{1}(T′)}(I × J)/R′ ∗^{(x,1)}T′

where

R ∗ T = {(iI_{∗}, jJ_{∗}) ∈ I × J : (iI_{∗}, hH_{∗}) ∈ R and (hH_{∗}, jJ_{∗}) ∈ T for some hH_{∗} ∈ H}.

Straightforward calculations show that p_{1}(R ∗ T)/k_{1}(R ∗ T) = I ⊓ J and p_{2}(R ∗ T)/k_{2}(R ∗ T) = J ⊓ I, and hence the Mackey relation. □

Hereafter, we shall identify Γ and Γ̃ via the above isomorphism α. Now let us describe the free basis of the alchemic algebra consisting of the isomorphism classes of transitive bisets in terms of the new variables. Clearly any transitive biset corresponds to a product of tinflation, isogation and destriction, in this order. We are aiming to find an equivalence relation on the set B = {tin_{J}^{H}c_{J,I}^{ϕ}des_{I}^{K} : J ⪯ H, I ⪯ K, ϕ : I → J} such that under α, the equivalence classes of the relation correspond to the isomorphism classes of transitive bisets.

Given two subquotients H and K of G. Also given subquotients J, A of H and subquotients I, C of K such that there are isomorphisms ϕ : I → J and ψ : C → A. We say that the triples (J, I, ϕ) and (A, C, ψ) are (H, K)−conjugate if there exist k ∈ K and h ∈ H such that

1. the equalities ^{h}J = A and ^{k}I = C hold and

2. (compatibility of ϕ and ψ) the following diagram commutes.

C I  A J^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}k^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ψ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ϕ  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}h

We denote by [J, I, ϕ] the (H, K)−conjugacy class of (J, I, ϕ). Then we obtain

Theorem 3.3 Letting H and K run over the subquotients of G and [J, I, ϕ] run over the (H, K)−conjugacy classes of triples (J ⪯ H, I ⪯ K, ϕ : I → J), the elements tin_{J}^{H} c_{J,I}^{ϕ} des_{I}^{K} run, without repetitions, over a free R−basis of the alchemic algebra Γ.

Proof. We are to show that (H, K)-conjugacy classes of the triples (J, I, ϕ) are in one-to-one correspondence with the isomorphism classes of transitive (H, K)-bisets. This follows from the following lemma. □

Lemma 3.4 Let H, K ⪯ G. Then there is a one-to-one correspondence between

(i) The (H, K)-conjugacy classes [J, I, ϕ] of triples (J, I, ϕ),

(ii) The isomorphism classes [X] of transitive (H, K)-bisets

where the correspondence is given by associating [J, I, ϕ] to the isomorphism class of the biset α(tin_{J}^{H} c_{J,I}^{ϕ} des_{I}^{K}).

Proof. Let (J, I, ϕ) and (A, C, ψ) be two (H, K)-conjugate triples. Then we have to show that the transitive bisets α(tin_{J}^{H} c_{J,I}^{ϕ} des_{I}^{K}) and α(tin_{A}^{H} c_{A,C}^{ψ} des_{C}^{K}) are isomorphic. Let us write

α(tin_{J}^{H} c_{J,I}^{ϕ} des_{I}^{K}) = (H × K)/𝔞

and

α(tin_{A}^{H} c_{A,C}^{ψ} des_{C}^{K}) = (H × K)/𝔟

for some subgroups 𝔞, 𝔟 ∈ H × K given explicitly in the proof of Theorem 3.2. Let h ∈ H and k ∈ K such that

hJ = A and ^{k}I = C.

We shall show that ^{(h,k)}𝔞 = 𝔟. Let (j, i) ∈ 𝔞. Then by the definition of α, we have jJ_{∗} = ϕ(iI_{∗}). Clearly, (^{h}j,^{k}i) ∈ A × C. So it suffices to show ^{h}jA_{∗} = ψ(^{k}iC_{∗}). But,

h(jJ_{∗}) = hjJ_{∗}h^{−1} = hjh^{−1}hJ_{∗}h^{−1} = ^{h}jA_{∗},

and

hϕ(iI_{∗}) = hϕ(k^{−1}(kik^{−1})(kI_{∗}k^{−1})k)h^{−1} = ψ(^{k}iC_{∗})

by the compatibility of ϕ and ψ. Hence ^{h}jA_{∗} = ψ(^{k}iC_{∗}), as required.

Conversely, let 𝔞, 𝔟 ∈ H×K be two conjugate subgroups of H×K. Then we are to show that the triples (p_{1}(𝔞)/k_{1}(𝔞), p_{2}(𝔞)/k_{2}(𝔞), ϕ) and (p_{1}(𝔟)/k_{1}(𝔟), p_{2}(𝔟)/k_{2}(𝔟), ψ) are (H, K)-conjugate. Here ϕ and ψ are the canonical isomorphisms introduced in Theorem 2.1.

Now let ^{(h,k)}𝔞 = 𝔟 for some h ∈ H and k ∈ K. Then, clearly,

h(p_{1}(𝔞), k_{1}(𝔞)) = (p_{1}(𝔟), k_{1}(𝔟))

and

k(p_{2}(𝔞), k_{2}(𝔞)) = (p_{2}(𝔟), k_{2}(𝔟)).

So, it suffices to show that the diagram

p_{2}(𝔟)/k_{2}(𝔟) p_{2}(𝔞)/k_{2}(𝔞)  p_{1}(𝔟)/k_{1}(𝔟) p_{1}(𝔞)/k_{1}(𝔞)^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}k ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ψ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ϕ  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}h

commutes.

Let (a, c) ∈ 𝔟. Then by the definition of ψ, we have

ψ(a k_{2}(𝔟)) = c k_{1}(𝔟).

But writing i = a^{k} and j for the unique element j = c^{k}, the left hand side becomes

ψ(a k_{2}(𝔟)) = ψ(kik^{−1}kk_{2}(𝔞)k^{−1}) = ψ(k(ik_{2}(𝔞))k^{−1})

and the right hand side becomes

c k_{1}(𝔟) = h(jk_{1}(𝔞))h^{−1} = h(ϕ(jk_{2}(𝔞)))h^{−1}.

Thus combining these two equality we get ψ(k(ik_{2}(𝔞))k^{−1}) = hϕ(jk_{2}(𝔞))h^{−1}, as required. □

Note that the unit

1_{Γ} = ∑_{H⪯G} c_{H}

of the alchemic algebra Γ induces a decomposition

F = ⨁_{H⪯G} c_{H}F

of any biset functor F into R-submodules. We call F (H) := c_{H}F the coordinate module of F at H.

Clearly the coordinate module F (H) at the subquotient H ⪯ G is a module for the truncated subalgebra c_{H} Γ c_{H} of the alchemic algebra Γ. Now the actions of the generators can be seen as maps between the coordinate modules. Explicitly, given J ⪯ H ⪯ G and M, L ⪯ G such that there is an isomorphism ϕ : M → L, we have the following maps.

M1. Tinflation map tin_{J}^{H} : F (J) → F (H),

M2. Destriction map des_{J}^{H} : F (H) → F (J),

M3. Isogation map c_{M,L}^{ϕ} : F (L) → F (M).

In this case, the maps are subject to the relations (R2) − (R6) of the alchemic algebra together with the following relation.

R1. The maps c_{H,H}^{h}, tin_{H}^{H} and des_{H}^{H} where h : H → H is conjugation by h ∈ H are all equal to the identity map for all subquotients H ⪯ G.

Hence we have defined a biset functor F as a quadruple (F, tin, des, c) where F is a family consisting of R-modules F (H) for each H ⪯ G and there are three families of maps between these modules given as above. It is straightforward to prove that this definition is equivalent to any of the other two.

4 Alcahestic subalgebras and simple modules

In this section, we explore the simple modules of the alchemic algebra together with its certain unital subalgebras, each called an alcahestic subalgebra. In particular, we describe the simple modules of these subalgebras in terms of the head (or the socle) of induced (or coinduced) simple modules and using some particular choice of these subalgebras, we prove a classification theorem for the simple modules. Note that Bouc’s classification of simple biset functors (Theorem 2.2) follows from our classification theorem. Further, each of these subalgebras has two special types of alcahestic subalgebras. These special subalgebras allow us to introduce a triangle having similar properties as the Mackey triangle introduced in [12]. We shall not introduce this triangle-structure in this paper. But all of the results in [12, Section 3] hold in this case with some modifications. Furthermore it is easy to describe the coordinate modules of the functors induced (or coinduced) from these subalgebras. We shall describe these functors for the alchemic algebra in the next section.

To begin with, let Π be a subalgebra of the alchemic algebra Γ. We call Π an alcahestic subalgebra^{2} of Γ if it contains c_{H} for each subquotient H of G. Clearly the alchemic algebra is an alcahestic subalgebra. Actually the subalgebras of immediate interest, for example the subalgebras realizing cohomology functors or homology functors or Brauer character ring functor, are all alcahestic. However there is a more basic example of such algebras, defined below, which allows us to parameterize the simple modules.

Let Π be an alcahestic subalgebra of Γ. We denote by Ω_{Π} the subalgebra of Π generated by all isogations in Π. We call Ω_{Π} the isogation algebra associated to Π. We write Ω for the isogation algebra associated to the alchemic algebra Γ. Clearly, Ω_{Π} is alcahestic since Π is. The structure of the isogation algebra Ω_{Π} is easy to describe. We examine the structure since it is crucial in proving our main results.

It is evident that the isogation algebra Ω_{Π} associated to Π has the following decomposition

Ω_{Π} = ⨁_{I,J⪯G} c_{I} Ω_{Π} c_{J}.

Now for a fixed subquotient H ⪯ G, the following isomorphism holds.

⨁_{I,J≅H} c_{I} Ω_{Π} c_{J} ≅ Mat_{n}(c_{H} Ω_{Π} c_{H})

where n is the number of subquotients of G isomorphic to H. In particular, we see that the isogation algebra Ω_{Π} is Morita equivalent to the algebra ⨁_{H⪯_{⋆}G} c_{H}Ω_{Π}c_{H}. Here the sum is over the representatives of the isomorphism classes of subquotients of G. Note that if Π is the alchemic algebra, there is an isomorphism c_{H}Ωc_{H} ≅ ROut(H) of algebras. Recall that Out(H) is the group of outer automorphisms of H. In general c_{H}Ω_{Π}c_{H} is a subgroup of Out(H), which we denote by Out_{Π}(H).

Now simple modules of the algebra Mat_{n}(c_{H} Ω_{Π} c_{H}) correspond to the simple c_{H}Ω_{Π}c_{H}- modules. Hence the simple modules of the isogation algebra Ω_{Π} are parameterized by the pairs (H, V) where H is a subquotient of G and V is a simple c_{H}Ω_{Π}c_{H}-module. We call (H, V) a simple pair for Ω_{Π} and denote by S_{H,V}^{Ω} the corresponding simple Ω_{Π}-module. It is clear that S_{H,V}^{Ω} is the Ω_{Π}-module defined for any subquotient K of G by

S_{H,V}^{Ω} (K) = ^{ϕ}V if there exists an isomorphism ϕ : K → H

and zero otherwise. Note that the definition does not depend on the choice of the isomorphism ϕ : K → H since any two such isomorphisms differ by an inner automorphism of H and the group Inn(H) acts trivially on V.

More generally, for any alcahestic subalgebra Π, the following theorem holds.

Theorem 4.1 Let Π be an alcahestic subalgebra of the alchemic algebra Γ. There is a bijective correspondence between (i) the isomorphism classes of simple Π-modules S^{Π},

2In alchemy, alcahest is the universal solvent. The decomposition of unity into a sum of the elements c_{H} for H ⪯ G allows us to decompose any module into coordinate modules as in Section 3.

(ii) the isomorphism classes of simple Ω_{Π}-modules S_{H,V}^{Ω}

given by S^{Π} ↔ S_{H,V}^{Ω} if and only if H is minimal such that S(H) ≠ 0 and S(H) = V.

In other words, the above theorem asserts that given an alcahestic subalgebra Ω′ of the isogation algebra Ω and given any subalgebra Π of the alchemic algebra such that Ω_{Π} ≅ Ω′, the simple modules of Π are parameterized by simple Ω′-modules.

We prove this theorem in several steps. The first step is to characterize the simple modules in terms of images of tinflation maps and kernels of destriction maps. This characterization is an adaptation of a similar result of Thévenaz and Webb [15] for Mackey functors. In particular, this characterization implies that any simple Π-module has a unique minimal subquotient, up to isomorphism, and the minimal coordinate module is simple. In order to do this, we introduce two submodules of a Π-module F, as follows (cf. [15]). Let H be a minimal subquotient for F, that is, H is a subquotient of G minimal subject to the condition that F (H) ≠ 0. Define two R-submodules of F by

I_{F,H}(J) = ∑_{I⪯J, I≅H} Im(tin_{I}^{J} : F (I) → F (J))

and

K_{F,H}(J) = ⋂_{I⪯J, I≅H} Ker(des_{I}^{J} : F (J) → F (I)).

Proposition 4.2 The R-modules I_{F,H} and K_{F,H} are Π-submodules of F, via the induced actions.

Proof. Let us prove that I_{F,H} is a submodule of F. The other claim can be proved similarly. Clearly, I_{F,H} is closed under isogation. It is also clear that I_{F,H} is closed under tinflation, because of the transitivity of tinflation. So it suffices to show that I_{F,H} is closed under destriction, which is basically an application of the Mackey relation. Let A ⪯ K be subquotients of G and let f be an element of I_{F,H}(K). We are to show that des_{A}^{K} f is an element of I_{F,H}(A). Write

f = ∑_{I} tin_{I}^{K} f_{I}

for some f_{I} ∈ F (I). Here the sum is over all subquotients of K isomorphic to H. Applying the Mackey relation, we get

des_{A}^{K} f = ∑_{I} des_{A}^{K}tin_{I}^{K} f_{I}
= ∑_{I} ∑_{y∈A^{∗}\K/I^{∗}} tin_{A⊓^{y}I}^{A} c^{yλ} des_{I⊓A^{y}}^{I} f_{I}.

Since H is minimal for F, the last sum contains only the terms tin_{A⊓^{y}I}^{A} c^{yλ} where A ⊓ ^{y}I is isomorphic to H. Therefore des_{A}^{K} f ∈ I_{F,H}(A), as required. □

The characterization of simple Π-modules via these subfunctors is as follows. (cf. [15] and [17])

Proposition 4.3 Let Π be an alcahestic subalgebra of the alchemic algebra Γ. Let S be a Π-module. Let H be a minimal subquotient for S and let V denotes the coordinate module of S at H. Then S is simple if and only if

(i) I_{S,H} = S,

(ii) K_{S,H} = 0,

(iii) V is a simple c_{H} Ω_{Π} c_{H}-module.

Proof. It is clear that if S is simple then the conditions (i) and (ii) hold. Also since H is minimal such that S(H) ≠ 0, any map that decomposes through a smaller subquotient is a zero map. Thus S(H) is a module of the algebra c_{H} Ω_{Π} c_{H}. But it has to be simple since any decomposition of the minimal coordinate module gives a decomposition of S. Now it remains to show the reverse implication. Suppose the conditions hold. Let T be a subfunctor of S. Since S(H) = V is simple, T (H) is either 0 or V. If T (H) = V then by condition (i), it is equal to S. So, let T (H) = 0. Then for any K ⪯ G, the module T (K) is a submodule of K_{S,H}, because for any x ∈ T (K) and H ≅ L ⪯ K, we have des_{L}^{K} x ∈ T (L) = 0. Thus by condition (ii), T (K) = 0, that is, T = 0. Thus, any subfunctor of S is either zero or S itself. In other words, S is simple. □

Now it is clear that a simple Π-module S has a unique, up to isomorphism, minimal subquotient, say H. Moreover the coordinate module at H is a simple c_{H} Ω_{Π} c_{H}-module. That is to saying that there is a map from the set of isomorphism classes of simple Π-modules to the set of isomorphism classes of the simple pairs (H, V) for Ω_{Π}, justifying the existence of the correspondence of Theorem 4.1.

The second step in proving Theorem 4.1 is to describe the behavior of simple modules under induction and coinduction to certain alcahestic subalgebras. We need two more definitions.

Let Π still denote an alcahestic subalgebra. Define the destriction algebra ∇_{Π} associated to Π as the subalgebra of Π generated by all destriction maps and isogation maps in the algebra Π. Similarly define Δ_{Π}, the tinflation algebra^{3} associated to Π. We write ∇ and Δ for the destriction algebra and the tinflation algebra associated to the alchemic algebra Γ, respectively. Clearly, both ∇_{Π} and Δ_{Π} are alcahestic subalgebras since Π is.

Now we are ready to state our main theorem. This theorem is a precise statement of Theorem 1.2.

Theorem 4.4 (Correspondence Theorem) Let Π ⊂ Θ be two alcahestic subalgebras of the alchemic algebra and S^{Π} be a simple Π-module with minimal subquotient H and denote by V the coordinate module of S at H.

(i) The Θ-module ind_{Π}^{Θ} S^{Π} has a unique maximal submodule provided that ∇_{Π} = ∇_{Θ}. Moreover the minimal subquotient for the simple quotient is H and the coordinate module of the simple quotient at H is isomorphic to V.

(ii) The Θ-module coind_{Π}^{Θ} S^{Π} has a unique minimal submodule provided that Δ_{Π} = Δ_{Θ}. Moreover the minimal subquotient for the minimal submodule is H and the coordinate module of the simple submodule at H is isomorphic to V.

3Our notation is consistent with that of ancient alchemists. In alchemy, the symbols of fire and water are Δ and ∇, respectively. The symbols of air and earth are the same as the symbols of fire and water, respectively, with an extra horizontal line dividing the symbol into two. Moreover quintessence is also known as spirit which has the symbol Ω.

Proof. We only prove part (i). The second part follows from a dual argument. First, observe that the subquotient H is minimal for the induced module F := ind_{Π}^{Θ} S_{H,V}^{Π} since ∇_{Π} = ∇_{Θ}. Moreover observe that there is an isomorphism F (H) ≅ V. Therefore the submodule K_{F,H} is defined. We claim that K_{F,H} is the unique maximal submodule of F.

To prove this, let T be a proper submodule of F. We are to show that T ≤ K_{F,H}. Since S is generated by its coordinate module at H, the Θ-module F is generated by its coordinate module at H which is simple. So T (H) must be the zero module. Now let K ⪯ G be such that T (K) ≠ 0. Then clearly, T (K) ≤ K_{F,H}(K) as des_{L}^{K}f ∈ T (L) = 0 for any f ∈ T (K) and L ≅ H. Thus T ≤ K_{F,H}, as required. □

To prove Theorem 4.1, we examine a special case of the Correspondence Theorem. This special case also initiates the process of describing simple Θ-module via induction or coinduction using Theorem 4.4. First we describe simple destriction and tinflation modules. For completeness, we include the description of simple Ω_{Π}-modules.

Proposition 4.5 Let Π be an alcahestic subalgebra and (H, V) be a simple pair for the isogation algebra Ω_{Π} associated to Π. Then

(i) The Ω_{Π}−module S_{H,V}^{Ω} is simple. Moreover any simple Ω_{Π}−module is of this form for some simple pair (H, V).

(ii) The ∇_{Π}−module S_{H,V}^{∇} := inf_{Ω_{Π}}^{∇_{Π}}S_{H,V}^{Ω} is simple. Moreover any simple ∇_{Π}−module is of this form for some simple pair (H, V) for Ω_{Π}.

(iii) The Δ_{Π}−module S_{H,V}^{Δ} := inf_{Ω_{Π}}^{Δ_{Π}}S_{H,V}^{Ω} is simple. Moreover any simple Δ_{Π}−module is of this form for some simple pair (H, V) for Ω_{Π}.

Here the inflation functor inf_{Ω_{Π}}^{∇_{Π}} is the inflation induced by the quotient map ∇_{Π} → ∇_{Π}/J (∇_{Π}) where J (∇_{Π}) is the ideal generated by proper destriction maps. We identify the quotient with the isogation algebra Ω_{Π} in the obvious way. Proof. The first part follows from the above discussion of simple isogation modules. Moreover it is clear that the module S_{H,V}^{∇} is simple. To see that any simple is of this form, notice that if a ∇-module D has non-zero coordinates at two non-isomorphic subquotient, then D has a non-zero submodule generated by the coordinate module at the smaller subquotient. So any simple ∇_{Π}−module has a unique, up to isomorphism, non-zero coordinate module. Clearly, this coordinate module should be simple. The same argument applies to the second part. □

Remark 4.6 Alternatively, one can apply the correspondence theorem to obtain simple ∇_{Π}- modules and simple Δ_{Π}-modules. In the first case, to identify the submodule K, one should identify des_{I}^{J} with c_{I} des_{I}^{J} c_{J}. Similar modification is also needed to identify the submodule I.

Evidently, for any alcahestic subalgebra Π, the destriction algebra ∇_{Π} and the tinflation algebra Δ_{Π} associated to Π are the minimal examples of the subalgebras satisfying the conditions of the first and the second part of the Correspondence Theorem, respectively. The following corollary restates the Correspondence Theorem for these special cases. We shall refer to this corollary in proving Theorem 4.1 and also in describing the simple biset functors in the next section.

Corollary 4.7 Let Π be an alcahestic subalgebra of the alchemic algebra such that the destriction algebra ∇_{Π} and the tinflation algebra Δ_{Π} are proper subalgebras. Let (H, V) be a simple pair for the isogation algebra Ω_{Π} associated to Π.

(i) The Π-module ind_{∇}^{Π}S_{H,V}^{∇} has a unique maximal subfunctor. Moreover H is a minimal subquotient for the simple quotient and the coordinate module of S at H is isomorphic to V.

(ii) The Π-module coind_{Δ}^{Π}S_{H,V}^{Δ} has a unique minimal subfunctor. Moreover H is a minimal subquotient for the simple subfunctor and the coordinate module of S at H is isomorphic to V.

Now, we are ready to prove Theorem 4.1. By Corollary 4.7, we associated a simple module S_{H,V}^{Π} to each simple pair (H, V) for the isogation algebra Ω_{Π}. Clearly this is an inverse to the correspondence described above. So it suffices to show that the correspondence is injective. This is equivalent to show that any simple Π-module with minimal subquotient H and S(H) = V is isomorphic to S_{H,V}^{Π} . So let S be a simple Π-module with this property. Then we are to exhibit a non-zero morphism S_{H,V}^{Π} → S. By our construction of S_{H,V}^{Π} , it suffices to exhibit a morphism ϕ : ind_{∇}^{Γ} S_{H,V}^{∇} → S such that ϕ_{H} is non-zero. The morphism exists since

Hom_{Π}(ind_{∇}^{Π}S_{H,V}^{∇} , S) ≅ Hom_{∇}(S_{H,V}^{∇} , res_{∇}^{Π} S) ≅ Hom_{ROut_{Π}(H)}(V, V) ≠ 0.

Here the first isomorphism holds because induction is left adjoint of restriction. On the other hand, the second isomorphism holds since S_{H,V}^{∇} is non-zero only on the isomorphism class of H. Now the identity morphism V → V induces a morphism ϕ : ind_{∇}^{Π}S_{H,V}^{∇} → S. Clearly, ϕ_{H} is non-zero, as required. Therefore we have established the injectivity, as required.

Having proved the classification theorem, we can restate the Correspondence Theorem more precisely. Let Π ⊂ Θ be two alcahestic subalgebras of the alchemic algebra Γ and (H, V) be a simple pair for the isogation algebra Ω_{Π}.

(i) The Θ-module ind_{Π}^{Θ} S_{H,V}^{Π} has a unique maximal submodule provided that ∇_{Π} = ∇_{Θ}. Moreover the simple quotient is isomorphic to S_{H,V}^{Θ} .

(ii) The Θ-module coind_{Π}^{Θ} S^{Π} has a unique minimal submodule provided that Δ_{Π} = Δ_{Θ}. Moreover the simple submodule is isomorphic to S_{H,V}^{Θ} .

Finally the next result shows that in both cases of the Correspondence Theorem, the inverse correspondence is given by restriction.

Theorem 4.8 Let Π ⊂ Θ be alcahestic subalgebras of the alchemic algebra. Then

(i) The Π-module res_{Π}^{Θ} S_{H,V}^{Θ} has a unique maximal submodule, provided that Δ_{Π} = Δ_{Θ}. Moreover the simple quotient is isomorphic to S_{H,V}^{Π} .

(ii) The Π-module res_{Π}^{Θ} S_{H,V}^{Θ} has a unique minimal submodule, provided that ∇_{Π} = ∇_{Θ}. Moreover this submodule is isomorphic to S_{H,V}^{Π} .

Proof of the first part of (i) is similar to the proof of Theorem 4.4. Indeed the maximal subfunctor of the restricted module res_{Π}^{Θ} S_{H,V}^{Θ} is generated by intersection of kernels of the destriction maps having range isomorphic to H. Note that this subfunctor is non-zero, in general. The second part of (i) follows from Theorem 4.1. Part (ii) can be proved by a dual argument.

In particular, this theorem shows that restriction of a simple Θ-module to a subalgebra Π is indecomposable provided that Π contains either all destriction maps or tinflation maps in Θ. On the other hand, if none of these conditions holds then the restricted module can be zero, semisimple or indecomposable.

5 An application: Simple biset functors

As an application of Theorem 4.7, we shall present two descriptions of the simple biset functors by describing induction (and coinduction) from destriction (and tinflation) algebra. Similar descriptions can be made for other alcahestic subalgebras. In [17], Webb constructed simple functors for the alcahestic subalgebra generated by all tinflation and restriction maps. He called the modules of this algebra inflation functors. His construction is equivalent to the construction by coinduction from the tinflation algebra associated to this subalgebra. In his paper, Webb also constructed simple global Mackey functors. In our terminology, global Mackey functors are modules of the alcahestic subalgebra generated by all transfer, restriction and isogation maps and his construction is, again, equivalent to the construction by coinduction from transfer algebra. Note that another description, from restriction algebra, is also possible.

Turning to the construction of simple biset functors, we see that in general, our construction is more efficient than Bouc’s construction in the sense that the biset functor in Theorem 4.7 is smaller than the one in Section 2.2. It is also advantageous to have explicit descriptions of the coordinate modules of the induced (or coinduced) modules.

In the following theorem, we characterize the coordinate modules of the induced module ind_{∇}^{Γ} D where D is a ∇-module. The proof is similar to the proof of Theorem 5.1 in [12] but we include the proof to introduce our notation.

Theorem 5.1 Let D be a ∇−module and H be a subquotient of G. Then there is an isomorphism of R−modules

(ind_{∇}^{Γ} D)(H) ≅ ( ⨁_{J⪯H} D(J))_{H}

where the right hand side is the maximal H−fixed quotient of the direct sum.

Proof. Let

D_{+}(H) := ( ⨁_{J⪯H} D(J))_{H}.

We write [J, a]_{H} for the image of a ∈ D(J) in D_{+}(H). Since H acts trivially, it is clear that [J, a]_{H} = ^{h}[J, a]_{H} for all h ∈ H. Moreover, D_{+}H is generated as an R-module by [J, a]_{H} for J ⪯_{H} H and a ∈ D(J). Here ⪯_{H} means that we take J up to H-conjugacy. In other words,

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

On the other hand,

(ind_{∇}^{Γ} D)(H) = ⨁_{J⪯_{H}H}{tin_{J}^{H} ⊗ a : a ∈ D(J)}.

Now tin_{J}^{H} ⊗ a = 0 if and only if a ∈ I(Out(H))D(J) where I(Out(H)) is the augmentation ideal of ROut(H). Therefore, the correspondence tin_{J}^{H} ⊗ a ⟷ [J, a]_{H} extends linearly to an isomorphism of R−modules (ind_{∇}^{Γ} D)(H) ≅ D_{+}(H). Evidently, this is an ROut(H)−modules isomorphism. □

Let us describe the action of tinflation, destriction and isogation on the generating elements tin_{J}^{H} ⊗ a of the biset functor ind_{∇}^{Γ} D. Note that we obtain these formulae by multiplying from the left with the corresponding generator and use the defining relations of the alchemic algebra. Let J ⪯ H ⪯ G and T ⪯ H ⪯ K and a ∈ D(J). Finally let A ⪯ G such that ϕ : H ≅ A. Then Tinflation

tin_{H}^{K}(tin_{J}^{H} ⊗ a) = tin_{J}^{K} ⊗ a.

Destriction

des_{T}^{H}(tin_{J}^{H} ⊗ a) = ∑_{x∈T^{∗}\H/J^{∗}} tin_{T ⊓^{x}J}^{T} ⊗ c^{xλ} des_{J⊓T^{x}}^{J} a.

Isogation

c_{A,H}^{ϕ}(tin_{J}^{H} ⊗ a) = tin_{ϕ(J)}^{A} ⊗ c_{ϕ(J),J}^{ϕ} a.

The other functor, that we will make use of, is coinduction from the tinflation algebra Δ to the alchemic algebra Γ. We can describe the coordinate modules in terms of fixed-points as follows.

Theorem 5.2 Let E be a Δ−module and H ⪯ G. Then

(coind_{Δ}^{Γ} E)(H) ≅ ( ∏_{J⪯H} E(J))^{H}

where the right hand side is the H−fixed points of the direct product.

The proof of this theorem is similar to the proof of the above theorem. We shall only describe the actions of tinflation, destriction and isogation on the tuples (x_{J})_{J⪯H}. Let J ⪯ H ⪯ G and T ⪯ H ⪯ K and A ⪯ G such that ϕ : H ≅ A. Then

Tinflation

(tin_{H}^{K}((x_{J})_{J⪯H}))_{I} = ∑_{y∈I^{∗}\K/H^{∗}} tin_{I⊓^{y}H}^{I} c^{yλ} x_{H⊓I^{y}}.

Here, we write x_{L} for the L-th coordinate of an element x ∈ coind_{Δ}^{Γ} E.

Destriction

des_{T}^{H}((x_{J})_{J⪯H})) = (x_{J})_{J⪯T} .

Isogation

c_{A,H}^{ϕ}((x_{J})_{J⪯H})) = (^{ϕ}x_{ϕ^{−1}(J)})_{J⪯A})

where ^{ϕ}x = c_{J,ϕ^{−1}(J)}^{ϕ}x.

By these theorems together with Theorem 4.7, we obtain the following two descriptions of the coordinate modules of the simple biset functors. For the first description, recall that S_{H,V}^{∇} denotes the simple ∇-module corresponding to the simple pair (H, V) where H ⪯ G and V is a simple ROut(H)-module. (Note that ROut(H) ≅ c_{H} Ω c_{H}.) By Theorem 5.1, the coordinate module at K ⪯ G of the induced module ind_{∇}^{Γ}S_{H,V}^{∇} is given by

ind_{∇}^{Γ}S_{H,V}^{∇} (K) = ( ⨁_{L⪯K,ϕ:L≅H} ^{ϕ}V)_{K}.

To obtain the coordinate module at K of the simple biset functor S_{H,V}, we take the quotient of the above module by the submodule K(K), defined in the proof of Theorem 4.4. Explicitly, K(K) is generated by the elements x ∈ ind_{∇}^{Γ}S_{H,V}^{∇} (K) such that des_{L}^{K} x = 0 for any L ⪯ K with L ≅ H. More explicit conditions on x can be obtained using the generating set {tin_{L}^{K} ⊗ v : L ⪯ K, L ≅ H, v ∈ S_{H,V}^{∇} (L)}.

The second description provides the description of the coordinate modules of S_{H,V} in terms of the images of tinflation maps.

Corollary 5.3 Let H and K be subquotients of G and suppose that K contains H. Also let V be a simple ROut(H)-module. Then the K-th coordinate of S_{H,V}^{Γ} is

S_{H,V}^{Γ} (K) = ∑_{I⪯K, ϕ:H≅I} Im (tin_{I}^{K})

where for v ∈ S_{H,V}^{Γ} (I) ≅ V, the J-th coordinate of tin_{I}^{H} v is given by

(tin_{I}^{K} v)_{J} = ∑ c_{J,I}^{yλ} v

where the sum is over the representatives of the double cosets J^{∗}\K/I^{∗} such that I = I ⊓ J^{y} and J = J ⊓^{y}I.

6 The mark morphism and semisimplicity

In this section, we introduce the mark morphism for biset functors. We also show that it connects the two constructions of simple biset functors in an exact sequence. Furthermore we prove a characterization of semisimplicity of the alchemic algebra in terms of the mark morphism. As a corollary to this theorem, we give an alternative proof of the semisimplicity theorem for the alchemic algebra proved independently by Barker [1] and Bouc. Note further that it is straightforward to generalize this construction and the semisimplicity criterion to the alcahestic subalgebras. The characterization is also valid in more general cases where a mark morphism exists, for example for Mackey functors. Finally we shall show that our criterion has an adaptation to any finite-dimensional algebra over a field.

To introduce the mark morphism, let D be a ∇-module. We denote by π^{D} the morphism

π^{D} : res_{Δ}^{Γ} ind_{∇}^{Γ} D → inf_{Ω}^{Δ} res_{Ω}^{∇} D

that forgets the tensor product. That is, given subquotients H ⪯ K of G and an element a of D(H), then π_{K}^{D}(tin_{H}^{K} ⊗ a) = tin_{H}^{K} a. Notice that since non-trivial tinflation maps of the Δ-module inf_{Ω}^{Δ} res_{Ω}^{∇} D are the zero maps, the morphism π_{K}^{D} is actually the projection ind_{∇}^{Γ} D(K) → D(K). Now via the following adjunction

Hom_{Γ}(ind_{∇}^{Γ} D, coind_{Δ}^{Γ} inf_{Ω}^{Δ} res_{Ω}^{∇} D) ≅ Hom_{Δ}(res_{Δ}^{Γ} ind_{∇}^{Γ} D, inf_{Ω}^{Δ} res_{Ω}^{∇} D)

we obtain a map

β^{D} : ind_{∇}^{Γ} D → coind_{Δ}^{Γ} inf_{Ω}^{Δ} res_{Ω}^{∇} D.

Explicitly, given a subquotient H of K and an element tin_{H}^{K} ⊗ a in ind_{∇}^{Γ} D(K), the morphism β^{D} is given by

β_{K}^{D}(tin_{H}^{K} ⊗ a) = (π_{I}(des_{I}^{K}tin_{H}^{K} ⊗ a))_{I⪯K}.

Following Boltje [2], we call β^{D} the mark morphism for D. Using the Mackey relation, we can calculate β_{K}^{D} more explicitly, as follows (cf. [2]).

β_{K}^{D}(tin_{H}^{K} ⊗ a) = (π_{I}( ∑_{x∈I^{∗}\K/H^{∗}} tin_{I⊓^{x}H}^{I} c^{x∘λ}des_{H⊓I^{x}⊗a}^{H}))_{I⪯K}
= (  ∑_{x∈I^{∗}\K/H^{∗},I=I⊓^{x}H} c^{x∘λ}des_{H⊓I^{x}}^{H} a)_{I⪯K}.

Remark 6.1 The mark morphism defined as above is a generalization of the well-known mark homomorphism

β_{G} : B(G) → B(G)

where B(G) is the Burnside ring of G and B(G) is the ghost ring of the Burnside ring. The ghost ring B(G) is defined as the dual of the Burnside ring and is isomorphic to the space of ℤ-valued functions constant on conjugacy classes of subgroups of G, i.e. B(G) = (∏_{H≤G} ℤ)^{G}. The mark morphism is now given by associating a finite G-set X to the function (|X^{H}|)_{H≤G}. For appropriate choices, this morphism becomes a special case of the morphism between Boltje’s plus constructions. For a detailed explanation, see [2].

Now if we put D = S_{H,V}^{∇}, the simple ∇−module associated to the simple pair (H, V), then clearly inf_{Ω}^{Δ} res_{Ω}^{∇} S_{H,V}^{∇} = S_{H,V}^{Δ} . Hence the mark morphism is a morphism between the two dual constructions

β^{H,V} : ind_{∇}^{Γ} S_{H,V}^{∇} → coind_{Δ}^{Γ} S_{H,V}^{Δ} .

Moreover, we have the following exact sequence.

Proposition 6.2 The following spliced sequence is exact.

0  K_{ind_{∇}^{Γ} S_{H,V}^{∇} ,H}  ind_{∇}^{Γ} S_{H,V}^{∇} S_{H,V}^{Π} coind_{Δ}^{Γ} S_{H,V}^{Δ}  C_{H,V}  0 0 0  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} β^{H,V}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

where C_{H,V} = coind_{Δ}^{Γ} S_{H,V}^{Δ} /I_{coind_{Δ}^{Γ} S_{H,V}^{Δ} ,H}.

Proof. It suffices to show that the kernel of the mark morphism is the unique maximal subfunctor of ind_{∇}^{Γ} S_{H,V}^{∇}. The inclusion ker β^{H,V} ⊂ K_{ind_{∇}^{Γ} S_{H,V}^{∇} ,H} is trivial, as the right hand side is max-

imal and β^{H,V} is non-zero. To show the reverse inclusion, let K ⪯ G and a ∈ K_{ind_{∇}^{Γ} S_{H,V}^{∇} ,H}(K). Then

β_{K}^{H,V} (a) = (π_{I}(des_{I}^{K} a))_{I⪯K,I≅H} = 0

as by definition of K_{ind_{∇}^{Γ} S_{H,V}^{∇} ,H}(K), we have des_{I}^{K} a = 0 for any I ≅ H. □

The following is immediate from the exactness of the above sequence.

Corollary 6.3 Let (H, V) be a simple pair for Ω over a field R and K ⪯ G. Then

dim_{R} S_{H,V}^{Γ} (K) = rank β_{K}^{H,V} .

Note that when G is a p-group for some prime p and V = R is the trivial ROut(H)-module, the matrix for the mark morphism is the same as the matrix for the bilinear form introduced by Bouc in [3].

A special case of the mark morphism is the well-known natural morphism, called the linearization morphism

ℚlin_{G} : ℚB(G) → ℚR_{ℚ}(G)

where B(G) is the Burnside group, as above, and R_{ℚ}(G) is the Grothendieck group of rational representation of G. The morphism is given by associating a G-set X to the permutation module ℚX. To see that this morphism is a mark morphism, we prove the following identification of the Burnside biset functor B^{G}.

Proposition 6.4 There is an isomorphism of biset functors B^{G} ≅ ind_{∇}^{Γ}S_{1,1}^{∇} where S_{1,1}^{∇} is the simple ∇-module having one copy of the trivial module at each trivial coordinate.

Proof. We need to specify an isomorphism Φ : B^{G} → ind_{∇}^{Γ}S_{1,1}^{∇} of biset functors. To do this, we specify an isomorphism Φ_{H} : B^{G}(H) → ind_{∇}^{Γ}S_{1,1}^{∇}(H) of R-modules for each H ⪯ G which is compatible with the actions of tinflation, destriction and isogation. By Theorem 5.1, ind_{∇}^{Γ}S_{1,1}^{G}(H) is generated by {tin_{L/L}^{H} ⊗ 1 : L ≤ H}. Now we define Φ_{H} by associating [H/L] to tin_{L/L}^{H} ⊗ 1. Straightforward calculations show that Φ is an isomorphism of biset functors. □

Now the image of the mark morphism β^{1,1} is the simple biset functor S_{1,1} by Proposition 6.2. It is shown by Bouc [3] that over a field of characteristic zero, the functor of rational representations is simple and isomorphic to S_{1,1}. Hence, over a field of characteristic zero, the linearization morphism lin_{G} and the mark morphism β_{G}^{1,1} coincide since they coincide on the trivial subgroup. Note that the mark morphism is given explicitly by associating tin_{L/L}^{H} ⊗ 1 to the function (|K\H/L|)_{K/K⪯H} where |K\H/L| is the number of double coset representatives of K and L in H. Notice that the mark morphism for the Burnside biset functor is different than the usual mark morphism, which indeed is not a morphism of biset functors. Finally, in [9], Bouc describe the biset functor structure of the kernel of the linearization morphism for p-groups where p is a prime number. Over a field of characteristic zero and for a p-group P, the kernel is known to be isomorphic to the rational Dade group ℚ ⊗ D(P) as mentioned in Section 2.

Another corollary of Proposition 6.4 is the following well-known result, see [3] and [1].

Corollary 6.5 Let R be a field. Then the Burnside biset functor B^{G} is the projective cover of the simple biset functor S_{1,1}.

Indeed it is clear that the simple destriction functor S_{1,1}^{∇} is projective, which implies that B^{G} is projective. Also it is indecomposable since the adjunction isomorphism

Hom_{Γ}(ind_{∇}^{Γ}S_{1,1}^{∇}, ind_{∇}^{Γ}S_{1,1}^{∇}) ≅ Hom_{∇}(S_{1,1}^{∇}, res_{∇}^{Γ}ind_{∇}^{Γ}S_{1,1}^{∇})

is actually an isomorphism of rings. Further to this, the following dual statement also holds.

Proposition 6.6 There are isomorphisms of biset functors B^{G} ≅ coind_{Δ}^{Γ}S_{1,1}^{Δ} ≅ I_{1,1} where B^{G} denotes the functor of the ghost ring of Burnside ring and I_{1,1} denotes the injective hull of the simple biset functor S_{1,1}. □

In particular, S_{1,1} is both projective and injective if and only if the mark morphism β^{1,1} : B^{G} → B^{G} is an isomorphism. This observation inspires the following criterion of semisimplicity, which is a precise statement of Theorem 1.3. Note that the equivalence of the first two statements is proved independently by Barker [1] and Bouc.

Theorem 6.7 Let G be a finite group and R be a field of characteristic zero. The following statements are equivalent.

(i) The alchemic algebra Γ_{R}(G) for G over R is semisimple.

(ii) The group G is cyclic.

(iii) The mark morphism β^{H,V} : ind_{∇}^{Γ} S_{H,V}^{∇} → coind_{Δ}^{Γ} S_{H,V}^{Δ} is an isomorphism for any simple pair (H, V) for Ω.

We prove the theorem in two steps. The first step is to prove the equivalence of (i) and (iii). This follows from the following criterion of projectiveness and injectivity of the simple biset functors.

Theorem 6.8 Let (H, V) be a simple pair for Ω such that V is projective. Then the following statements are equivalent.

(i) The simple biset functor S_{H,V}^{Γ} is both projective and injective.

(ii) The mark morphism β^{H,V} : ind_{∇}^{Γ}S_{H,V}^{∇} → coind_{Δ}^{Γ}S_{H,V}^{Δ} is an isomorphism.

Proof. First, suppose that β is an isomorphism. We are to show that the simple biset functor S_{H,V}^{Γ} is both injective and projective. By the exact sequence in Proposition 6.2, we get

ind_{Δ}^{Γ}S_{H,V}^{∇} ≅ S_{H,V}^{Γ} ≅ coind_{∇}^{Γ}S_{H,V}^{Δ} .

Now consider any short exact sequence

0  K  F  S{Γ \atop H,V}  0^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ϕ

of biset functors. By the second isomorphism above we get ϕ ∈ Hom_{Γ}(F, coind_{∇}^{Γ}S_{H,V}^{Δ} ). (Note that since ϕ is a morphism of biset functors, preimage of any v ∈ S_{H,V}^{Γ} (X) is in the kernel of destriction maps to the isomorphism class of H). By adjointness of coinduction and restriction we get a morphism ϕ̄ ∈ Hom_{∇}(res_{∇}^{Γ}F, S_{H,V}^{Δ} ). Since V is projective as an ROut(H)-module, the morphism ϕ̄_{H} : F (H) → V of ROut(H)-modules splits. That is to saying that V is a direct summand of the ROut(H)-module F (H). Therefore, there is a morphism ψ̄ : S_{H,V}^{∇} → res_{Δ}^{Γ}F of Δ-modules given by sending V to the direct summand of F (H) which is isomorphic to V. Now by adjointness of induction and restriction the map ψ̄ induces a map ψ ∈ Hom_{Γ}(ind_{Δ}^{Γ}S_{H,V}^{∇} , F). Clearly the composition ϕ ∘ ψ is the mark homomorphism, which is an isomorphism by our assumption. Hence the composition ψ ∘ β^{−1} is the required splitting for the above exact sequence.

Similarly it can be shown that any short exact sequence of biset functors

0  S{Γ \atop H,V}  F  C  0^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} α^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

splits, that is, S_{H,V}^{Γ} is injective.

Conversely, suppose the simple biset functor S_{H,V}^{Γ} is both injective and surjective. Then both of the sequences

0  K  indΓ ΔS{∇ \atop H,V}  S_{H,V}^{Γ}  0^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

and

0  S_{H,V}^{Γ}  coindΓ ∇S{Δ \atop H,V}  C  0^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

splits. But ind_{Δ}^{Γ}S_{H,V}^{∇} and coind_{∇}^{Γ}S_{H,V}^{Δ} are indecomposable. Hence the result follows. □

Now the equivalence of the statements (i) and (iii) of Theorem 6.7 is clear. Incidentally, the approach of the proof of this theorem, applied in the context of Mackey functors, taken together with [2, Proposition 2.4], can be used to give a proof of the following corollary.

Corollary 6.9 ([16]) The Mackey algebra μ_{R}(G) is semisimple if and only if |G| is invertible in R.

Indeed in [12], it is shown that the simple Mackey functors can be constructed using the methods of the present paper. In particular, we have a mark morphism associated to each simple Mackey functor S_{H,V}^{G} where H is a subgroup of G and V is a simple RN_{G}(H)/Hmodule. Now one can modify the above proof to get the above corollary.

However, the mark morphism for the alchemic algebra for G over a field R of characteristic zero is not an isomorphism, in general. In the following we find necessary and sufficient condition for the mark morphism to be an isomorphism, completing the proof of Theorem 6.7. Note that the result is inspired by the semisimplicity theorem proved by Barker and Bouc, independently.

Theorem 6.10 Let R be a field of characteristic zero. The mark morphism β^{H,V} : ind_{∇}^{Γ} S_{H,V}^{∇} → coind_{Δ}^{Γ} S_{H,V}^{Δ} is an isomorphism for any simple pair (H, V) for G if and only if the group G is cyclic.

Proof. First, suppose G is not cyclic. We are to show that the mark morphism is not an isomorphism for some pair (H, V). But we have ind_{∇}^{Γ}S_{1,1}^{∇} ≅ B^{G} by Lemma 6.4 and it is wellknown by a theorem of Ritter and Segal that R_{ℚ} ≅ S_{1,1} is not isomorphic to B^{G}. So the mark morphism for the simple pair (1, 1) is not an isomorphism.

Conversely, suppose that the group G is cyclic. Then we are to show that the mark morphism is an isomorphism. But this is equivalent to showing that coind_{Δ}^{Γ} S_{H,V}^{Δ} and ind_{∇}^{Γ} S_{H,V}^{∇} are simple, that is, to showing that the subfunctor I_{coind_{Δ}^{Γ} S_{H,V}^{Δ} ,H} of coind_{Δ}^{Γ} S_{H,V}^{Δ} is not proper and the subfunctor K_{ind_{∇}^{Γ} S_{H,V}^{∇} ,H} is zero. We shall show the first claim. The second one follows from a similar argument. Without loss of generality, we shall prove the claim for the top coordinate module, that is we shall prove that coind_{Δ}^{Γ} S_{H,V}^{Δ} (G) = I_{coind_{Δ}^{Γ} S_{H,V}^{Δ} ,H}(G). For simplicity, write I(G) for I_{coind_{Δ}^{Γ} S_{H,V}^{Δ} ,H}(G). Now any element of I(G) is a sum of tinflated

elements tin_{I}^{G} v_{I} for some I ≅ H and where we write v_{I} for the generator of the one-dimensional ROut(H)-module S_{H,V}^{Δ} (I). By Corollary 5.3, we have

(tin_{I}^{G} v_{I})_{J} = ∑ c_{J,I}^{yλ} v_{I}

where the sum is over the representatives of the double cosets J^{∗}\G/I^{∗} such that I = I ⊓ J^{y} and J = J ⊓^{y}I. But G is cyclic, hence J^{y} = J for any y ∈ G. Therefore the J-th coordinate (tin_{I}^{G} v_{I})_{J} of tin_{I}^{G} v_{I} is non-zero only if the equalities I = I ⊓ J and J = J ⊓ I hold. Moreover when the equalities hold we have

(tin_{I}^{G} v_{I})_{J} = |J^{∗}\G/I^{∗}|c_{J,I}^{λ} v_{I}.

Clearly c_{J,I}^{λ} v_{I} = v_{J} and hence

(tin_{I}^{G} v_{I})_{J} = |J^{∗}\G/I^{∗}|v_{J}.

Finally we have

tin_{I}^{G} v_{I} = (|J^{∗}\G/I^{∗}|v_{J})_{J∈S_{I}}

where S_{I} = {J ⪯ G : J ≅ H, I = I ⊓ J, J = J ⊓ I}. Now our aim is to show that the set B = {tin_{I}^{G} v_{I} : I ⪯ G, I ≅ H} forms a basis for the coordinate module coind_{Δ}^{Γ} S_{H,V}^{Δ} (G). In order to show this, we first determine the set S_{I} for any I and then show that the set B is linearly independent. Since the order of the set B is equal to the order of the canonical basis of coind_{Δ}^{Γ} S_{H,V}^{Δ} (G), consisting of characteristic functions of subquotients of G, the proof would be established.

Now write G = C_{1} × C_{2} × … C_{n} as a direct product of (cyclic) Sylow subgroups. For simplicity, we shall prove our claim for the case that H ⪯ C_{1}. The general case follows from the same argument.

Let I ⪯ G be a subquotient isomorphic to H. Then it is easy to see that J ⪯ G is in S_{I} if and only if the equalities J^{∗} = I^{∗} ×Z and J_{∗} = I_{∗} ×Z hold for some Z ⪯ G with (|J^{∗}|, |Z|) = 1. Indeed the converse statement is trivial and the first implication follows from the equalities

\frac{J^{∗}}{J_{∗}} = \frac{(J^{∗} ∩ I^{∗})J_{∗}}{(J^{∗} ∩ I_{∗})J_{∗}} and \frac{I^{∗}}{I_{∗}} = \frac{(I^{∗} ∩ J^{∗})I_{∗}}{(I^{∗} ∩ J_{∗})I_{∗}}

using the fact that all groups involved are cyclic.

Hence we obtain a description of the set S_{I}. However we can make it more explicit. Let J ∈ S_{J} be the subquotient with J^{∗} has minimal order among all J′ ∈ S_{I}. Then clearly J ⪯ C_{1} and S_{J} = S_{I}. Now we can identify the set S_{J} with the set of subgroups of G_{1} = G/C_{1}. Note further that given L, N ∈ S_{J} we have the equality

|L^{∗}\G/N^{∗}| = |C_{1}/J^{∗}| |L_{1}^{∗}\G_{1}/N_{1}^{∗}|

where L^{∗} = J^{∗} × L_{1}^{∗} and N^{∗} = J^{∗} × N_{1}^{∗}. Indeed we have

|L^{∗}\G/N^{∗}| = \frac{|G|}{|L^{∗} N^{∗}|} = \frac{|G| |L^{∗} ∩ N^{∗}|}{|L^{∗}| |N^{∗}|}.

Now both the denominator and the numerator of the last quotient is divisible by the square of the order of J^{∗}. Cancelling these, we get the above equality.

Hence it remains to show that the elements of the set B are linearly independent. It is clear that the sets S_{I} gives a partition of the set of subquotients of G isomorphic to H. Hence it suffices to show that the elements of the set B_{J} = {tin_{I}^{G} v_{I} : I ∈ S_{J}} are linearly independent where J is the representative of the set S_{J} with minimal order of its numerator. But this is clear since for any I ⪯ J, we have

tin_{I}^{G} v_{I} = (|C_{1}/J^{∗}||J_{1}^{∗}\G_{1}/I_{1}^{∗}|v_{I})_{I∈S_{J}}

and any linear relation among these elements gives a linear relation between the columns of the double coset matrix [|L\G_{1}/N|]_{L,N≤G_{1}} which is well-known to be invertible for cyclic groups. Thus we have proved that coind_{Δ}^{Γ} S_{H,V}^{Δ} (G) = I(G) proving that coind_{Δ}^{Γ} S_{H,V}^{Δ} is simple, as required. □

Now by Corollary 5.3, we can describe the simple biset functors for a cyclic group G, explicitly.

Corollary 6.11 Suppose the group G is cyclic. Let H ⪯ G and V be a simple ROut(H)- module. Then

S_{H,V}^{Γ} (K) = ⨁_{C⪯K,ϕ:C≅H} ^{ϕ}V.

The isogation, destriction and tinflation maps are as in the Corollary 5.3.

We end with a generalization of Theorem 6.7 to the finite-dimensional algebras over a field. Let A be a finite-dimensional algebra over a field k. Let e be an idempotent of the algebra A. We denote by A_{e} the truncated subalgebra eAe.

Given a simple A-module V, it is well-known that the restriction eV of V to the subalgebra A_{e} is either zero or a simple A_{e}-module. Conversely, given a simple A_{e}-module W, there exists unique, up to isomorphism, simple A-module S_{e,W} such that eS_{e,W} ≅ W. Indeed we can construct the simple module S_{e,W} in two ways, as follows. See [14] for further details.

For the first construction, consider the A-module

F = ind_{A_{e}}^{A} W := Ae ⊗_{A_{e}} W.

It is clear that the A-module F satisfies eF ≅ W. However the A-module F is not necessarily simple. We let K be the sum of all submodules of F annihilated by the idempotent e. Then it is easy to show that the A-submodule K is the unique maximal submodule of the A-module F and it intersects eF trivially. Therefore the quotient F/K is simple satisfying e(F/K) ≅ W. We leave it as an easy exercise to show that the simple module F/K is unique, up to isomorphism. Note further that the induced module F is indecomposable. To see this consider the adjunction isomorphism

Hom_{A}(F, F) ≅ Hom_{A_{e}}(W, eF) ≅ Hom_{A_{e}}(W, W)

which is indeed an isomorphism of rings.

The second way of constructing the simple modules is to consider the dual construction. Let W be as above. Put

E = coind_{A_{e}}^{A} W := Hom_{A_{e}}(eA, W).

Again we have

eE = e Hom_{A_{e}}(A, W) ≅ Hom_{A_{e}}(eAe, W) ≅ W

since e is the identity element of A_{e} and we only consider the A_{e}-invariant maps. As above, the A-module E is not simple, in general. But it is indecomposable, that is its socle is simple with the property that e soc(E) ≅ W. Therefore by the uniqueness of the simple module S_{e,W}, we have soc(E) ≅ S_{e,W}.

Now we are ready to prove our generalization, Theorem 1.4.

Proof.[of Theorem 1.4] It is clear that (1) implies (2). We shall prove that (2) implies (1). Given a simple A-module V, we have g V ≠ 0 for some g ∈ {e, f}. Then by construction, we have S_{g,gV} ≅ V. But by the assumption we have

S_{g,gV} ≅ ind_{A_{g}}^{A} gV.

Moreover the simple module gV is projective since the algebra A_{g} is semisimple. Hence the Amodule V is projective. Similarly V is also injective. But V was arbitrary, thus A is semisimple. □

References

[1] L. Barker, Rhetorical Biset Functors, Rational p-biset Functors and Their Semisimplicity in Characteristic Zero, J. Algebra (to appear).

[2] R. Boltje, A General Theory of Canonical Induction Formulae, J. Algebra, 206 (1998), 293-343.

[3] S. Bouc, Foncteurs d’ensembles munis d’une double action, J. Algebra, 183 (1996), 664-736.

[4] S. Bouc, Construction de foncteurs entre catégories de G-ensembles, J. Algebra, 183 (1996), 737-825.

[5] S. Bouc, The functor of units of Burnside rings for p-groups, Comm. Math. Helv., 82 (2007), 583-615.

[6] S. Bouc, Tensor induction of relative syzygies, J. reine angew. Math., 523 (2000), 113-171.

[7] S. Bouc, A Remark on the Dade Group and the Burnside Group, J.Algebra, 279 (2004), 180-190.

[8] S. Bouc, The functor of rational representations for p-groups, Adv. Maths, 186 (2004), 267-306.

[9] S. Bouc, Dade group of a p-group, Invent. Math., 164 (2006), 189-231.

[10] S. Bouc, J. Thévenaz The Group of Endo-permutation Modules, Invent. Math., 139 (2000), 275-349.

[11] S. Bouc, E. Yalçın Borel-Smith Functions and the Dade Group, J.Algebra, 311 (2007), 821-839.

[12] O. Coşkun, Mackey functors, induction from restriction functors and coinduction from transfer functors, J.Algebra, 315 (2007), 224-248.

[13] C.W. Curtis, I. Reiner, Methods of representation theory: with applications to finite group theory, Vol. I, Wiley, New York (1987).

[14] J.A. Green, Polynomial Representations of GL_{n}, Lecture Notes in Mathematics No: 830, Berlin, New York, Springer-Verlag, (1980).

[15] J. Thévenaz and P. Webb, Simple Mackey functors, Proc. of 2nd international group theory conference, Bressonone (1989), Supplement to Rendiconti del Circulo Matematico di Palermo 23 (1990) 299-319.

[16] J. Thévenaz, P. Webb, The Structure of Mackey Functors, Trans. Amer. Math. Soc., 347 (1995), 1865-1961.

[17] P. Webb, Two classifications of simple Mackey functors with applications to group cohomology and decomposition of classifiying spaces, J. Pure and Appl. Algebra, 88 (1993), 265-304.

[18] P. Webb, A Guide to Mackey Functors,Handbook of Algebra vol 2 (ed M. Hazewinkel), Elsevier 2000.