Vanishing evaluations of simple functors

Serge Bouc, Radu Stancu, and Jacques Thévenaz

Abstract : The classification of simple biset functors is known, but the evaluation of a simple biset functor at a finite group G may be zero. We investigate various situations where this happens, as well as cases where this does not occur. We also prove a closed formula for such an evaluation under some restrictive conditions on G.

AMS Subject Classification : 19A22, 20C20. Key words : biset, Burnside ring, simple module.

1. Introduction

Let k be a field. The biset category kC is the k-linear category whose objects are finite groups, with morphisms Hom_{kC}(H, G) = kB(G, H), where B(G, H) is the Burnside group of (G, H)-bisets and kB(G, H) = k ⊗_{ℤ} B(G, H). A biset functor is a k-linear functor from kC to the category k-Mod of k-vector spaces. The category of biset functors is an abelian category and is used in various ways in representation theory, see [Bo2].

The classification of simple biset functors was obtained in [Bo1]. They are parametrized by equivalence classes of pairs (H, V), where H is a finite group and V is a simple kOut(H)-module. We write S_{H,V} for the simple functor associated to the pair (H, V). However, the problem of describing the evaluation of simple functors at specific finite groups is much harder. In the present paper, we consider the question of the possible vanishing of such evaluations. We are interested in both questions of vanishing and non-vanishing.

This question is related to the problem of describing all simple modules for the double Burnside ring kB(G, G), because any simple kB(G, G)-module has the form S_{H,V} (G) for some (H, V), and conversely any evaluation S_{H,V} (G) is either zero or a simple kB(G, G)-module. We refer to [BST] and [BD] for this related question.

For other types of functors, there are explicit formulas for the evaluations of simple functors. This holds in particular for Mackey functors for a fixed finite group G (see Proposition 8.8 in [TW]) and for global Mackey functors and inflation functors (see Theorem 2.6 in [We]). Thus the vanishing of evaluations of such simple functors can be checked, at least in principle, by watching directly the formula. The situation is much more complicated for general biset functors, whenever both inflation and deflation are present (as well as restriction and induction). No closed formula for the evaluation of a simple functor is known. The most general known results are, on the one hand, a description of S_{H,V} (G) as the image of a suitable linear map (see Theorem 4.3.20 in [Bo2]), and on the other hand, a formula for its dimension in terms of the rank of a suitable bilinear form (see Theorem 7.1 in [BST]). But neither of those results allows for an easy way to determine whether or not the evaluation is zero. The essential purpose of the present paper is to give some answers to this question.

In Section 3, we give some easy conditions which guarantee the non-vanishing S_{H,V} (G) ≠ 0. Then we prove in Section 4 a general criterion, which has the disadvantage of being difficult to apply. In Section 5, we describe a suitable subquotient of S_{H,V} (G), which implies a non-vanishing condition. Finally in Section 6, we prove that a closed formula for the evaluation at G of a simple functor exists under some restrictive conditions on G and we immediately deduce a criterion for the vanishing of this evaluation. Various special cases can then be handled, as shown in Section 7.

2. Preliminaries

We review some known facts about biset functors. For more details, we refer to [Bo1] and [Bo2]. Given two finite groups G and H, the Burnside group B(G, H) is the Grothendieck group of the category of finite (G, H)-bisets and kB(G, H) = k ⊗_{ℤ} B(G, H). In particular, kB(G, G) is a finite dimensional k-algebra, called the double Burnside ring of G.

A section of a finite group G is a pair (T, S) of subgroups of G such that S is a normal subgroup of T. In that case, the group T/S is called a subquotient of G. We write H ⊑ G when the group H is isomorphic to a subquotient of G and we write H ⊏ G if H ⊑ G and H ≅ G̸ (hence |H| < |G|). We also write N_{G}(T, S) for the normalizer of the section, that is, the set of all g ∈ G such that gT g^{−1} = T and gSg^{−1} = S. If (T, S) is a section of G, then there are elementary bisets Res_{T}^{G}, Def_{T/S}^{T}, Ind_{T}^{G}, Inf_{T/S}^{T}, and their composites Defres_{T/S}^{G} and Indinf_{T/S}^{G} (see Section 2.3 in [Bo2]). Also any group isomorphism σ : G → G′ defines a (G′, G)- biset Iso_{σ}. Given finite groups G and H, any transitive (G, H)-biset has the form Indinf_{B/A}^{G} Iso_{σ} Defres_{T/S}^{H}, where (B, A) is a section of G, (T, S) is a section of H, and σ : T/S → B/A is a group isomorphism (see Lemma 3 in [Bo1] or Lemma 2.3.26 in [Bo2]).

If kI(G, G) is the ideal of kB(G, G) generated by all (G, G)-bisets which factorize through a proper subquotient of G, then kB(G, G)/kI(G, G) ≅ kOut(G), where Out(G) = Aut(G)/Inn(G) is the group of outer automorphisms of G. In particular, any kOut(G)-module can be viewed as a kB(G, G)-module, with kI(G, G) acting by zero.

The biset category kC is the k-linear category whose objects are finite groups, with morphisms Hom_{kC}(H, G) = kB(G, H) (note that a (G, H)-biset is a morphism from H to G). The composition of morphisms, which we often write ∘, is the k-linear extension of the usual products of bisets U ×_{H} V. Recall that, if U is a (G, H)-biset and V is an (H, L)-biset, then U ×_{H} V is a (G, L)-biset in the obvious way.

A biset functor is a k-linear functor from kC to the category k-Mod of kvector spaces. The category of all such biset functors is abelian. A biset functor is called simple if it is non-zero and has no proper non-zero subfunctor. Recall the classification of simple functors (see Section 4 in [Bo1] or Section 4.3 in [Bo2]).

2.1. Proposition. Let S be a simple biset functor, let H be a group of minimal order such that S(H) ≠ 0, and let V = S(H).

1. H is unique up to isomorphism.

2. The ideal kI(H, H) acts by zero on V and V is a kOut(H)-module.

3. V is a simple kOut(H)-module.

4. If S(G) ≠ 0 for some finite group G, then H ⊑ G.

This provides a parametrization of simple functors by (equivalence classes of) pairs (H, V) where H is a finite group and V is a simple kOut(H)-module. We write S_{H,V} for the simple functor as in the statement, so that H is its minimal group and S_{H,V} (H) = V.

We shall need a direct description of simple functors as quotients of suitable standard functors and we now recall this construction, which appears in [BST]. Let us fix a finite group H and consider the representable functor kB(−, H). For every finite group G, define

kI(G, H) := ∑_{J⊏H} kB(G, J)B(J, H) .

Then kI(−, H) is a subfunctor of kB(−, H) and we define

k\overline{B}(−, H) = kB(−, H)/kI(−, H) .

For any finite group G, the evaluation k\overline{B}(G, H) has a natural structure of right kOut(H)-module, because the right action of kI(H, H) is zero. This structure depends on G and we need to describe it more precisely.

We let Σ_{H}(G) be the set of all sections (T, S) of G such that T/S ≅ H, and we let [Σ_{H}(G)/G] be a set of representatives of G-orbits in Σ_{H}(G). For every (T, S) ∈ Σ_{H}(G), we choose an isomorphism σ_{T,S} : H → T/S. The group N_{G}(S, T) acts by conjugation on T/S and therefore \overline{N}_{G}(S, T) = N_{G}(T, S)/T maps into the group Out(T/S). We use the isomorphism σ_{T,S} to transport the image of this map to a subgroup of Out(H), that is, we define Γ_{G}(T, S) to be the subgroup of Out(H) consisting of all elements induced by automorphisms σ_{T,S}^{−1} Conj_{g} σ_{T,S}, where g ∈ N_{G}(T, S). If σ_{T,S} is replaced by σ_{T,S} α where α ∈ Aut(H), then Γ_{G}(T, S) is replaced by \overline{α}^{−1} Γ_{G}(T, S) \overline{α}, where \overline{α} ∈ Out(H) is the class of α. Thus the conjugacy class of Γ_{G}(T, S) only depends on (T, S).

2.2. Lemma. The k-space k\overline{B}(G, H) is a permutation right kOut(H)-module decomposing as follows :

k\overline{B}(G, H) = ⨁_{(T,S)∈[Σ_{H}(G)/G]} \overline{Indinf_{T/S}^{G}} ∘ k\overline{B}(T/S, H)
≅ ⨁_{(T,S)∈[Σ_{H}(G)/G]} k[Γ_{G}(T, S)\Out(H)] ,

where \overline{Indinf_{T/S}^{G}} denotes the image of Indinf_{T/S}^{G} in k\overline{B}(G, H).

Proof : By Lemma 7.2 in [BST], k\overline{B}(G, H) has a basis consisting of the set of elements of the form Indinf_{T/S}^{G} ∘ Iso_{σ}, where (T, S) ∈ [Σ_{H}(G)/G], and where σ : H → T/S runs over all isomorphisms, up to left composition by conjugation by elements of N_{G}(T, S) (because Indinf_{T/S}^{G} = Indinf_{T/S}^{G} ∘ Conj_{g} whenever g ∈ N_{G}(T, S)). This provides the first decomposition of the statement. Now for any fixed section (T, S), we have a fixed isomorphism σ_{T,S} : H → T/S and we obtain a permutation right kOut(H)-module

\overline{Indinf_{T/S}^{G}} ∘ k\overline{B}(T/S, H) = \overline{Indinf_{T/S}^{G}} ∘ \overline{Iso_{σ_{T,S}}} ∘ kOut(H) .

The generator \overline{Indinf_{T/S}^{G}} ∘ \overline{Iso_{σ_{T,S}}} has Γ_{G}(T, S) as a stabilizer in Out(H). The result follows.

For any left kOut(H)-module V, we define the functor

\overline{L}_{H,V} = k\overline{B}(−, H) ⊗_{kOut(H)} V .

This has a subfunctor \overline{J}_{H,V} defined as follows (see Remark 4.5 in [BST]) :

\overline{J}_{H,V} (X) = \{ ∑_{i} \overline{ϕ}_{i} ⊗ v_{i} ∈ \overline{L}_{H,V} (X) | ∀ψ ∈ kB(H, X), ∑_{i}(\overline{ψ ∘ ϕ}_{i}) · v_{i} = 0\} ,

where ϕ_{i} ∈ kB(X, H) and \overline{ϕ}_{i} denotes its image in k\overline{B}(X, H). When V is a simple kOut(H)-module, we obtain the following result.

2.3. Proposition. Suppose that V is a simple kOut(H)-module.

1. \overline{J}_{H,V} is the unique maximal subfunctor of \overline{L}_{H,V}.

2. \overline{L}_{H,V} /\overline{J}_{H,V} ≅ S_{H,V}.

3. For every finite group G and for any fixed non-zero element v ∈ V,

\overline{J}_{H,V} (G) = \{\overline{ϕ} ⊗ v ∈ \overline{L}_{H,V} (G) | ∀ψ ∈ kB(H, G), (\overline{ψ ∘ ϕ}) · v = 0\} .

4. For every finite group G, \overline{L}_{H,V} (G) is generated by all elements of the form \overline{Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}}} ⊗ v, where v ∈ V and where (T, S) is a section of G such that T/S ≅ H and σ_{T/S} : H → S/T is a fixed isomorphism.

Proof : (1) and (2) follow from Proposition 4.4 in [BST]. Since V is generated by any of its non-zero elements v, (3) is a consequence of the description of \overline{J}_{H,V} (X) given above. Finally (4) is a consequence of Lemma 2.2 above.

3. Some easy cases

We have seen that S_{H,V} (G) vanishes if H is not isomorphic to a subquotient of G. Also, S_{H,V} (H) = V ≠ 0. This is of course the starting point in our investigation of vanishing or non-vanishing of evaluations. The following is another elementary result.

3.1. Lemma. Let S_{H,V} be a simple biset functor. If H is isomorphic to a quotient group G/N of G, then S_{H,V} (G) ≠ 0.

Proof : We have bisets Inf_{G/N}^{G} and Def_{G/N}^{G} which satisfy Def_{G/N}^{G} ∘ Inf_{G/N}^{G} = id_{G/N}. Thus we have maps

S_{H,V} (G/N) ⟶^{Inf_{G/N}^{G}} S_{H,V} (G) ⟶^{Def_{G/N}^{G}} S_{H,V} (G/N)

whose composite is the identity. Since S_{H,V} (G/N) ≠ 0 (because G/N ≅ H by assumption), we must have S_{H,V} (G) ≠ 0.

The lemma suffices to obtain the following result for the evaluation at an abelian group.

3.2. Proposition. Let S_{H,V} be a simple biset functor. If H ⊑ G/[G, G], then S_{H,V} (G) ≠ 0. In particular, if H ⊑ G and if G is abelian (hence H too), then S_{H,V} (G) ≠ 0.

Proof : In view of the structure theorem for finite abelian groups, any subquotient of the finite abelian group G/[G, G] is isomorphic to a quotient of G/[G, G], hence to a quotient of G. Then the result follows from Lemma 3.1.

Our purpose is to generalize Lemma 3.1 and we need the following notions. The set of all sections of G is partially ordered by the relation ⪯ defined as follows : (V, U) ⪯ (T, S) if and only if V ≤ T and the inclusion α : V → T induces an isomorphism \overline{α} : V/U → T/S (or in other words V S = T and V ∩ S = U).

Two sections (B, A) and (T, S) are said to be linked if (B∩T, A∩S) ⪯ (B, A) and (B ∩ T, A ∩ S) ⪯ (T, S) (see 4.3.11 in [Bo2] or Section 2 in [BT2]). In that case, the composition of the canonical isomorphisms

ϕ_{B/A,T/S} : T/S ⟶^{∼}(B ∩ T)/(A ∩ S) ⟶^{∼} B/A

maps xS to xA for every x ∈ B ∩ T and is called the isomorphism induced by the linking. We write (B, A)−(T, S) whenever (B, A) and (T, S) are linked.

3.3. Lemma. Let (B, A) and (T, S) be two sections of G. The following are equivalent :

a) (B, A)−(T, S),

b) |B/A| = |T/S|, S(B ∩ T) = T, and S(A ∩ T) = S.

Of course, the last equality is equivalent to A ∩ T ≤ S, but we shall need below the equality as stated.

Proof : We know that (B, A) and (T, S) generate a butterfly diagram, as in Lemma 2.3 of [BT2], and the two sections

(A(B ∩ S), A(B ∩ T)) and (S(B ∩ T), S(A ∩ T))

are linked. If now S(B ∩ T) = T and S(A ∩ T) = S, then the second section is (T, S). Thus (T, S) is linked to (A(B ∩ S), A(B ∩ T)), which is a section of B/A. If moreover, |B/A| = |T/S|, then this section of B/A cannot be proper and must be (B, A). It follows that (T, S) is linked to (B, A) (i.e. the butterfly diagram collapses to a linking).

If conversely (B, A)−(T, S), then B/A ≅ T/S, hence |B/A| = |T/S|. Moreover, the linking implies that S(B ∩ T) = T, and also that A ∩ T = A ∩ S, so that S(A ∩ T) = S.

3.4. Proposition. Let S_{H,V} be a simple biset functor. Suppose that H is isomorphic to T/S, where (T, S) is a section of G such that, for every g ∈ G with g ∉ T, the conjugate section (^{g}T , ^{g}S) is not linked to (T, S). Then S_{H,V} (G) ≠ 0.

It is clear that if S is a normal subgroup S of G, then the section (G, S) satisfies the assumption (because in that case there is no g ∉ G), so Proposition 3.4 actually generalizes Lemma 3.1.

Proof : As in the proof of Lemma 3.1, we consider the maps

S_{H,V} (T/S) ⟶^{Indinf_{T /S}^{G}} S_{H,V} (G) ⟶^{Defres_{T /S}^{G}} S_{H,V} (T/S)

and we want to prove that the composite is the identity. This will then force S_{H,V} (G) to be non-zero since S_{H,V} (T/S) ≅ S_{H,V} (H) ≠ 0.

By the generalized Mackey formula (see Proposition A.1 in [BT1]), the composite above decomposes as a sum indexed by double cosets representatives g ∈ [T \G/T]. There is one double coset, indexed by an element of T which can be chosen to be 1_{G}, and the corresponding term is the identity. We show that all the other terms vanish. Any such term has the form

Indinf_{X/Y}^{T/S} Iso_{ϕ} Conj_{g} Defres_{S(^{g}T ∩T)/S(^{g}S∩T)}^{T/S}

for some subquotient X/Y of T/S and some group isomorphism ϕ. Since (^{g}T , ^{g}S) is not linked to (T, S) by assumption, and since |^{g}T /^{g}S| = |T/S|, Lemma 3.3 tells us that the section (S(^{g}T ∩ T), S(^{g}S ∩ T)) must be a proper section of T/S. Therefore the group S(^{g}T ∩ T)/S(^{g}S ∩ T) has order strictly smaller than the order of T/S ≅ H. The functor S_{H,V} vanishes on such a group and so the term above factors through zero.

The special case where S = 1 is worth mentioning.

3.5. Corollary. Suppose that H is isomorphic to a subgroup T of G such that T = N_{G}(T). Then S_{H,V} (G) ≠ 0.

We now show that the assumption of Proposition 3.4 holds in particular for a section (N_{G}(S), S) where S is an expansive subgroup of G. Recall that a subgroup S of G is called expansive in G if, for every g ∉ N_{G}(S), the subgroup S(^{g}S ∩ N_{G}(S))/S has a non-trivial core in the group N_{G}(S)/S, in other words, there exists a normal subgroup M of N_{G}(S) contained in S(^{g}S ∩ N_{G}(S)) and containing S properly. This notion is defined and used in [Bo2] and in [BT2]. In particular, the subgroup S(^{g}S ∩ N_{G}(S)) contains S properly, and this implies, by Lemma 3.3, that the section (^{g}N_{G}(S), ^{g}S) is not linked to (N_{G}(S), S), so that Proposition 3.4 applies. This proves the following corollary.

3.6. Corollary. Let S_{H,V} be a simple biset functor. If H is isomorphic to N_{G}(S)/S, where S is an expansive subgroup of G, then S_{H,V} (G) ≠ 0.

It is clear that any normal subgroup S of G is expansive in G (because in that case there is no g ∉ N_{G}(S)), so again Lemma 3.1 is a special case of Corollary 3.6.

For example, the Mathieu group G = M_{11} has a subgroup S, isomorphic to A_{6}, which is expansive in G and such that N_{G}(S)/S has order 2 (in fact N_{G}(S) = M_{10}). It follows that S_{C_{2},k}(M_{11}) ≠ 0, independently of the characteristic of k.

4. A general criterion

Let S_{H,V} be a simple biset functor and let G be a finite group. The analysis of the evaluation S_{H,V} (G) involves the set Σ_{H}(G) of all sections (T, S) of G such that T/S ≅ H, because S_{H,V} is a quotient of \overline{L}_{H,V} and the evaluation \overline{L}_{H,V} (G) involves those sections (see Proposition 2.3). We may assume that H ⊑ G, that is, Σ_{H}(G) ≠ ∅.

Now we come to a criterion for the vanishing of the evaluation of a simple functor. It gives a general answer to our main question, although it is rather hard to use it in practice. A similar result appears in Theorem 7.1 of [BD].

4.1. Theorem. Let S_{H,V} be a simple biset functor and let G be a finite group. For every (T, S) ∈ Σ_{H}(G), fix an isomorphism σ_{T/S} : H → T/S. The following are equivalent :

1. S_{H,V} (G) = 0.

2. For any (B, A), (T, S) ∈ Σ_{H}(G), the action on V of the automorphism

∑_{{g∈[B\G/T] \atop (B,A)−^{g}(T,S)}} σ_{B/A}^{−1} ϕ_{B/A,^{g}T /^{g}S} Conj_{g} σ_{T/S}

is zero, where ϕ_{B/A,^{g}T /^{g}S} : ^{g}T /^{g}S → B/A denotes the isomorphism in-

duced by the linking (B, A)−^{g}(T, S).

Proof : By Proposition 2.3, we have S_{H,V} (G) = 0 if and only if \overline{L}_{H,V} (G) = \overline{J}_{H,V} (G). By Proposition 2.3 again, the latter equality holds if and only if (\overline{ψ ∘ ϕ}) acts by zero on V for all ϕ ∈ kB(G, H) and all ψ ∈ kB(H, G). Since we have passed to the quotient by all morphisms factorizing below H, we can assume that

ϕ = Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}} and ψ = Iso_{σ_{B/A}}^{−1} ∘ Defres_{B/A}^{G}

where (B, A), (T, S) ∈ Σ_{H}(G). Then the generalized Mackey formula applies to Defres_{B/A}^{G} ∘ Indinf_{T/S}^{G} (see Proposition A.1 in [BT1]), indexed by double cosets representatives g ∈ [B\G/T]. But we are interested in the image \overline{ψ ∘ ϕ} in

kB(H, H)/kI(H, H) ≅ kOut(H)

and all terms in the formula factorize through a group isomorphic to a proper subquotient of H, except those indexed by an element g ∈ [B\G/T] such that (B, A)−^{g}(T, S), where ^{g}(T, S) = (^{g}T , ^{g}S) denotes the conjugate section. For such an element g, we are left with the (B/A, T/S)-biset Iso_{ϕ_{B/A,^{g}T /^{g}S}} ∘Iso_{Conj_{g}}. Composing with σ_{T/S} and σ_{B/A}^{−1}, we see that the action on V of (\overline{ψ ∘ ϕ}) is equal to the action of the automorphism

∑_{{g∈[B\G/T] \atop (B,A)−^{g}(T,S)}} σ_{B/A}^{−1} ϕ_{B/A,^{g}T /^{g}S} Conj_{g} σ_{T/S} .

Thus the condition is that this sum must act by zero on V.

Note that if (B, A)−^{g}(T, S), then the isomorphism ϕ_{B/A,^{g}T /^{g}S} Conj_{g} induced by the linking is given by the (B/A, T/S)-biset A\BgT/S. This appears explicitly in the proof of the generalized Mackey formula in [BT1], but we do not need this here.

5. Minimal sections

Given finite groups H and G, we let again Σ_{H}(G) be the set of all sections (T, S) of G such that T/S ≅ H. A section (T, S) ∈ Σ_{H}(G) will be called minimal if it is minimal with respect to the partial order ⪯ defined in Section 3. In that case, if (T, S) is linked to (B, A), then (B ∩ T, A ∩ S) = (T, S), that is, T ≤ B and S ≤ A (and also T A = B and T ∩ A = S because of the linking). We write Σ_{H}(G)^{min} for the subset of Σ_{H}(G) consisting of minimal sections. Clearly G acts by conjugation on Σ_{H}(G) and Σ_{H}(G)^{min} and we let [Σ_{H}(G)^{min}/G] denote a set of representatives of G-orbits in Σ_{H}(G)^{min}.

5.1. Lemma. Let (T, S) ∈ Σ_{H}(G), let f : T → H be a surjective group homomorphism with kernel S, and let Φ(T) be the Frattini subgroup of T (that is, the intersection of all maximal subgroups of T). The following are equivalent.

1. (T, S) is minimal.

2. S ≤ Φ(T).

3. f induces an isomorphism T/Φ(T) ⟶^{∼} H/Φ(H).

Proof : If H = 1, then the only minimal section in Σ_{1}(G) is (1, 1) and the result follows easily. Assume that H ≠ 1, that is S < T. Suppose (T, S) is not minimal and let (B, A) ∈ Σ_{H}(G) such that (B, A) ≺ (T, S). Then B < T and there is some maximal subgroup M of T containing B. It follows that T = BS = MS, so S ≰ Φ(T). Conversely, if S ≰ Φ(T), there is some maximal subgroup M of T which does not contain S. Then MS = T, so (M, M ∩ S) ≺ (T, S) and (T, S) is not minimal. The proof of the equivalence of (2) and (3) is easy and is left to the reader.

Recall that, for every section (T, S) ∈ Σ_{H}(G), we have set \overline{N}_{G}(S, T) = N_{G}(S, T)/T and we have fixed an isomorphism σ_{T/S} : H → T/S. This allows us to view any kOut(H)-module V as a k[\overline{N}_{G}(T, S)]-module, as follows :

\overline{g} · v = \overline{σ_{T/S}^{−1} Conj_{g} σ_{T/S}} · v , g ∈ N_{G}(T, S) , v ∈ V ,

where the bar denotes the class in Out(H) of the automorphism in Aut(H). By Proposition 2.3, we know that \overline{L}_{H,V} (G) is generated as a k-vector space by all the elements of the form

\overline{Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}}} ⊗ v , (T, S) ∈ Σ_{H}(G), v ∈ V .

Let \overline{L}_{H,V} (G)^{min} be the subspace of \overline{L}_{H,V} (G) generated by all the elements of the form

\overline{Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}}} ⊗ v , (T, S) ∈ Σ_{H}(G)^{min}, v ∈ V .

Let also S_{H,V} (G)^{min} be the image of \overline{L}_{H,V} (G)^{min} under the canonical surjection

π : \overline{L}_{H,V} (G) ⟶ \overline{L}_{H,V} (G)/\overline{J}_{H,V} (G) ≅ S_{H,V} (G) .

For any finite group X and any kX-module W, we let Tr_{1}^{X} : W → W be the klinear map defined by Tr_{1}^{X}(w) = ∑_{x∈X} x·w (relative trace), and we let Tr_{1}^{X}(W) denote its image.

5.2. Theorem. Let S_{H,V} be a simple biset functor and let G be a finite group. With the notation above, there is a surjective k-linear map

\overline{τ} : S_{H,V} (G)^{min} ⟶ ⨁_{(T,S)∈[Σ_{H}(G)^{min}/G]} Tr_{1}^{\overline{N}_{G}(T,S)}(V) .

Hence the right hand side is isomorphic to a subquotient of S_{H,V} (G).

Proof : We have \overline{L}_{H,V} (G) = k\overline{B}(G, H)⊗_{kOut(H)}V and we know how k\overline{B}(G, H) decomposes, by Lemma 2.2. By tensoring with V this decomposition, we obtain

\overline{L}_{H,V} (G) = ⨁_{(T,S)∈[Σ_{H}(G)/G]} \overline{Indinf_{T/S}^{G}} ∘ k\overline{B}(T/S, H) ⊗_{kOut(H)} V
≅ ⨁_{(T,S)∈[Σ_{H}(G)/G]} k[Γ_{G}(T, S)\Out(H)] ⊗_{kOut(H)} V
≅ ⨁_{(T,S)∈[Σ_{H}(G)/G]} V_{Γ_{G}(T,S)} ,

where V_{Γ_{G}(T,S)} denotes the k-space of coinvariants for the group Γ_{G}(T, S) (i.e. the quotient of V by all elements of the form (γ−1)v, γ ∈ Γ_{G}(T, S), v ∈ V). By the proof of Lemma 2.2, we see that a generator \overline{Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}}} ⊗ v of \overline{L}_{H,V} (G) is mapped to 1 ⊗ v ∈ k[Γ_{G}(T, S)\Out(H)] ⊗_{kOut(H)} V, hence to the class of v in V_{Γ_{G}(T,S)}.

By definition, Γ_{G}(T, S) is the image of \overline{N}_{G}(T, S) in Out(H) (using the isomorphism σ_{T/S} : H → T/S). Therefore V_{Γ_{G}(T,S)} = V_{\overline{N}_{G}(T,S)} and we obtain

\overline{L}_{H,V} (G) = ⨁_{(T,S)∈[Σ_{H}(G)/G]} V_{\overline{N}_{G}(T,S)} .

Restricting to minimal sections, we obtain

\overline{L}_{H,V} (G)^{min} ≅ ⨁_{(T,S)∈[Σ_{H}(G)^{min}/G]} V_{\overline{N}_{G}(T,S)} .

Now the relative trace map Tr_{1}^{\overline{N}_{G}(T,S)} induces a surjective k-linear map

Tr_{1}^{\overline{N}_{G}(T,S)} : V_{\overline{N}_{G}(T,S)} ⟶ Tr_{1}^{\overline{N}_{G}(T,S)}(V) ,

and the direct sum of these maps defines a surjective k-linear map

τ : \overline{L}_{H,V} (G)^{min} ⟶ ⨁_{(T,S)∈[Σ_{H}(G)^{min}/G]} Tr_{1}^{\overline{N}_{G}(T,S)}(V) ,

mapping a generator \overline{Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}}} ⊗ v to Tr_{1}^{\overline{N}_{G}(T,S)}(v).

Recall the canonical surjection

π : \overline{L}_{H,V} (G) ⟶ \overline{L}_{H,V} (G)/\overline{J}_{H,V} (G) ≅ S_{H,V} (G) .

We claim that \overline{J}_{H,V} (G) ∩ \overline{L}_{H,V} (G)^{min} ⊆ Ker(τ). It will follow that τ induces a surjective k-linear map

\overline{τ} : S_{H,V} (G)^{min} ⟶ ⨁_{(T,S)∈[Σ_{H}(G)^{min}/G]} Tr_{1}^{\overline{N}_{G}(T,S)}(V) ,

proving the theorem.

In order to prove the claim, we let x ∈ \overline{J}_{H,V} (G) ∩ \overline{L}_{H,V} (G)^{min} and we write

x = ∑_{(T,S)∈[Σ_{H}(G)^{min}/G]}(\overline{Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}}}) ⊗ v_{T,S} ,

where v_{T,S} ∈ V for every (T, S). By the description of J_{H,V} (G) in Proposition 2.3, we have

∑_{(T,S)∈[Σ_{H}(G)^{min}/G]}(\overline{ψ} ∘ \overline{Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}}}) · v_{T,S} = 0

for all ψ ∈ kB(H, G). Fix (B, A) ∈ [Σ_{H}(G)^{min}/G] and choose

ψ = Iso_{σ_{B/A}}^{−1} ∘ Defres_{B/A}^{G} .

As in the proof of Theorem 4.1, Defres_{B/A}^{G} ∘ Indinf_{T/S}^{G} decomposes according to the generalized Mackey formula (see Proposition A.1 in [BT1]), indexed by double cosets representatives g ∈ [B\G/T], and all terms in the formula factorize through a group isomorphic to a proper subquotient of H, except those indexed by an element g ∈ [B\G/T] such that (B, A)−^{g}(T, S). Since both (B, A) and g(T, S) are minimal, the only possible linking is the identity, hence (B, A) = g(T, S). But we have chosen orbit representatives in Σ_{H}(G)^{min}, so (B, A) = (T, S). Thus the sum over (T, S) reduces to a single term, indexed by (B, A). Moreover, in the Mackey formula, we are left with the sum over all g ∈ [B\G/B] such that ^{g}(B, A) = (B, A), that is, g ∈ [N_{G}(B, A)/B]. Therefore we obtain

∑_{g∈[N_{G}(B,A)/B]}(Iso_{σ_{B/A}}^{−1} ∘ Iso_{Conj_{g}} ∘ Iso_{σ_{B/A}}) · v_{B,A} = 0 .

This is the action of the automorphism σ_{B/A}^{−1} Conj_{g} σ_{B/A}, so by definition of the action of \overline{N}_{G}(B, A), we obtain

Tr_{1}^{\overline{N}_{G}(B,A)}(v_{B,A}) = 0 .

This holds for all (B, A) ∈ Σ_{H}(G)^{min} and therefore

τ(x) = τ(  ∑_{(T,S)∈[Σ_{H}(G)^{min}/G]} \overline{Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}}} ⊗ v_{T,S})

= ∑_{(T,S)∈[Σ_{H}(G)^{min}/G]} Tr_{1}^{\overline{N}_{G}(T,S)}(v_{T,S}) = 0 .

Thus x ∈ Ker(τ), proving the claim.

Theorem 5.2 immediately implies the following result about vanishing evaluations.

5.3. Corollary. Let S_{H,V} be a simple biset functor and let G be a finite group. If Tr_{1}^{\overline{N}_{G}(T,S)}(V) ≠ 0 for some (T, S) ∈ Σ_{H}(G)^{min}, then S_{H,V} (G) ≠ 0.

This can be applied for instance in the following situation.

5.4. Corollary. Suppose that there exists a minimal section (T, S) ∈ Σ_{G}(H)^{min} such that N_{G}(T, S) acts by inner automorphisms on T/S. Suppose also that |\overline{N}_{G}(T, S)| ≠ 0 in k. Then S_{H,V} (G) ≠ 0.

Proof : By assumption, the image of \overline{N}_{G}(T, S) in Out(T/S) is trivial. Therefore Tr_{1}^{\overline{N}_{G}(T,S)}(V) = |\overline{N}_{G}(T, S)| · V ≠ 0 and the result follows from Corollary 5.3.

Note that the first assumption holds in particular if N_{G}(T, S) is equal to the centralizer C_{G}(T/S) of T/S, because N_{G}(T, S) acts trivially on T/S in this case. This applies in particular if N_{G}(T, S) is abelian, improving the first statement of Proposition 3.2 in the case where k has characteristic not dividing |G|.

6. A closed formula for some evaluations

As already mentioned in the introduction, there is no known closed formula for the evaluation S_{H,V} (G) of a simple biset functor S_{H,V}. However, with suitable assumptions, such a formula exists.

For instance, if the kOut(H)-module V is primitive, in the sense defined on page 721 of [Bo1], and if k has characteristic zero, then

S_{H,V} (G) ≅ ⨁_{(T,S)} V^{\overline{N}_{G}(T,S)} ,

where (T, S) runs over a suitable subset of Σ_{H}(G). We refer to Proposition 20 in [Bo1] for more details. Of course, a criterion for the vanishing of S_{H,V} (G) is immediately deduced in this case.

The purpose of this section is to prove a closed formula for the evaluation S_{H,V} (G) under a suitable assumption on the structure of G, more precisely when Σ_{H}(G)^{min} = Σ_{H}(G). Then this can be used to give a criterion for the vanishing of S_{H,V} (G).

6.1. Theorem. Let S_{H,V} be a simple biset functor and let G be a finite group. If every section (T, S) ∈ Σ_{H}(G) is minimal, the map \overline{τ} of Theorem 5.2 is an isomorphism and

S_{H,V} (G) ≅ ⨁_{(T,S)∈[Σ_{H}(G)/G]} Tr_{1}^{\overline{N}_{G}(T,S)}(V) .

Proof : By Theorem 5.2, we already know that the map \overline{τ} is surjective, so we need to prove that it is injective. In other words, we have to show that

Ker(τ) ⊆ \overline{J}_{H,V} (G) ∩ \overline{L}_{H,V} (G)^{min} ,

where τ is the map defined in the proof of Theorem 5.2, namely

τ : \overline{L}_{H,V} (G)^{min} ⟶ ⨁_{(T,S)∈[Σ_{H}(G)^{min}/G]} Tr_{1}^{\overline{N}_{G}(T,S)}(V) .

Let x ∈ Ker(τ) and write

x = ∑_{(T,S)∈[Σ_{H}(G)^{min}/G]} \overline{Indinf_{T/S}^{G} ∘ Iso_{σ_{T /S}}} ⊗ v_{T,S} ,

where v_{T,S} ∈ V for every (T, S). Since τ(x) = 0, we have Tr_{1}^{\overline{N}_{G}(T,S)}(v_{T,S}) = 0 for every (T, S) ∈ Σ_{H}(G)^{min}. Fix (B, A) ∈ [Σ_{H}(G)^{min}/G] and let

ψ_{B,A} = Iso_{σ_{B/A}}^{−1} ∘ Defres_{B/A}^{G} .

Exactly the same computation as in the proof of Theorem 5.2 above shows that

\overline{ψ_{B,A}} · x = ∑_{(T,S)∈[Σ_{H}(G)^{min}/G]}(\overline{ψ_{B,A}} ∘ \overline{Indinf_{T/S}^{G} Iso_{σ_{T /S}}}) · v_{T,S}

= Tr_{1}^{\overline{N}_{G}(B,A)}(v_{B,A}) = 0 .

But now, since Σ_{H}(G)^{min} = Σ_{H}(G) by assumption, k\overline{B}(H, G) is generated by elements of the form \overline{Iso_{α} ∘ ψ_{B,A}} where ψ_{B,A} is as above and α ∈ Aut(H). Therefore \overline{ψ} · x = 0 for all ψ ∈ kB(H, G). But this means that x ∈ \overline{J}_{H,V} (G), as was to be shown.

6.2. Corollary. Let S_{H,V} be a simple biset functor and let G be a finite group. Assume that Σ_{H}(G)^{min} = Σ_{H}(G). Then S_{H,V} (G) = 0 if and only if Tr_{1}^{\overline{N}_{G}(T,S)}(V) = 0 for every (T, S) ∈ Σ_{H}(G).

Our next result gives a first application of Theorem 6.1. 6.3. Proposition. Suppose that G and H are p-groups with the same sectional rank. Then

S_{H,V} (G) ≅ ⨁_{(T,S)∈[Σ_{H}(G)/G]} Tr_{1}^{\overline{N}_{G}(T,S)}(V) .

In particular S_{H,V} (G) = 0 if and only if the action of ∑_{g∈[N_{G}(T,S)/T]} Conj_{g} on V is zero, for every (T, S) ∈ Σ_{H}(G).

Proof : Let (T, S) ∈ Σ_{H}(G) and let f : T → H be a surjective group homomorphism with kernel S. Let r be the sectional rank of H and let 1 ≤ H_{2} ◁ H_{1} ≤ H such that H_{1}/H_{2} is elementary abelian of rank r. Let T_{i} = f^{−1}(H_{i}). Then T_{1}/T_{2} is elementary abelian of rank r, and this must be the largest possible rank of an elementary abelian quotient of T_{1}, because the sectional rank of G is also r. It follows that Φ(T_{1}) = T_{2}. Since Φ(T_{1}) ≤ Φ(T) (because T_{1} ≤ T and T is a p-group), we deduce that S ≤ T_{2} = Φ(T_{1}) ≤ Φ(T). By Lemma 5.1, this proves that the section (T, S) is minimal. Thus Theorem 6.1 applies and yields the result.

7. The case of a single section

Theorem 6.1 can be applied to various cases to obtain a closed formula for the evaluation S_{H,V} (G), hence a criterion for its vanishing. We concentrate here on cases where the set Σ_{H}(G) reduces to a single conjugacy class, so that clearly every section (T, S) ∈ Σ_{H}(G) is minimal. As before, S_{H,V} denotes a simple biset functor and G a finite group. We can assume that H ⊑ G, since otherwise S_{H,V} (G) = 0.

7.1. Proposition. Suppose that Σ_{H}(G) contains a unique section (T, S) up to conjugation. Then

S_{H,V} (G) ≅ Tr_{1}^{\overline{N}_{G}(T,S)}(V) .

In particular S_{H,V} (G) = 0 if and only if the action of ∑_{g∈[N_{G}(T,S)/T]} Conj_{g} on V is zero.

Proof : This is a special case of Theorem 6.1.

There are many instances where G has a subgroup H such that (H, 1) is the only section in Σ_{H}(G) up to conjugation. Here are a few such cases.

7.2. Corollary. Suppose that H is a normal Hall subgroup of G. By the Schur-Zassenhaus theorem, we know that G = H ⋊ Y for some subgroup Y. Assume that Y acts faithfully on H. Then

S_{H,V} (G) ≅ Tr_{1}^{Y} (V) .

In particular S_{H,V} (G) = 0 if and only if the action of ∑_{y∈Y} Conj_{y} on V is zero.

Proof : Since (|H|, |Y |) = 1 and Y acts faithfully on H, it is easy to see that H cannot normalize a non-trivial subgroup of Y. Therefore there is no section in Σ_{H}(G) apart from (H, 1) and Proposition 7.1 applies. Moreover, N_{G}(H)/H = G/H ≅ Y.

7.3. Corollary. Suppose that H is a Sylow p-subgroup of G and that H does not normalize any non-trivial q-subgroup, for every prime q ≠ p. Then

S_{H,V} (G) ≅ Tr_{1}^{N_{G}(H)/H}(V) .

In particular S_{H,V} (G) = 0 if and only if the action of ∑_{g∈[N_{G}(H)/H]} Conj_{g} on V is zero.

Proof : Let (T, S) be a section of G such that T/S ≅ H. Then a Sylow psubgroup of T is isomorphic to H, hence conjugate to H, and we may assume that it is equal to H. Then H normalizes S, hence also a Sylow q-subgroup Q of S because (|H|, |S|) = 1. Therefore Q = 1 by assumption. Since this holds for every prime divisor q of |S|, it follows that S = 1 and that T = H. Thus Σ_{H}(G) reduces to the conjugacy class of (H, 1) and Proposition 7.1 applies.

7.4. Corollary. Let H be a non-abelian simple group and let G = Aut(H). Suppose that V is a non-trivial kOut(H)-module (so in particular G/H ≅ Out(H) is a non-trivial group). Then S_{H,V} (G) = 0.

Proof : Let (T, S) be a section of G such that T/S ≅ H. Then HT/HS is isomorphic to a quotient of T/S ≅ H and to a subquotient of G/H ≅ Out(H). Since Out(H) is solvable (the Schreier conjecture) and H is non-abelian simple, we must have HT/HS = 1, hence HT = HS. It follows that T = (H ∩ T)S, hence T/S ≅ (H ∩ T)/(H ∩ S). But T/S ≅ H, while (H ∩ T)/(H ∩ S) is a subquotient of H. Therefore H ∩ T = H and H ∩ S = 1. But then H and S are normal subgroups of T with H ∩ S = 1, so [S, H] = 1, that is, S acts trivially on H. Since S is isomorphic to a subgroup of Out(H), we must have S = 1. This proves that (H, 1) is the only section in Σ_{H}(G) and therefore

S_{H,V} (G) ≅ Tr_{1}^{G/H}(V) = Tr_{1}^{Out(H)}(V) ⊆ V^{Out(H)} = 0 ,

because the non-trivial simple module V has no non-zero fixed point.

If k has characteristic zero, then under the assumptions of Corollary 7.4, the module V is in fact primitive, in the sense mentioned at the beginning of Section 6, and this provides another approach of the result.

7.5. Corollary. Let H be a non-abelian simple group and let G be its universal central extension. Then S_{H,V} (G) = V (hence non-zero).

Proof : Let Z be the centre of G, so that G/Z = H. Let (T, S) be a section of G such that T/S ≅ H. Then ZT/ZS is isomorphic to a quotient of T/S ≅ H and is a subquotient of G/Z = H, so it is either trivial of the whole of G/Z. If ZT/ZS = 1, then ZT = ZS and T = (Z ∩ T)S, so that the group

H ≅ T/S ≅ (Z ∩ T)/(Z ∩ S)

is abelian, contrary to our assumption. Thus ZT/ZS = G/Z, hence ZT = G and S ≤ Z. It follows that G/S ≅ Z/S × T/S. Therefore G has an abelian quotient Z/S, so Z/S = 1 since G is perfect. So we have S = Z and T = G. This proves that (G, Z) is the only section in Σ_{H}(G) and Proposition 7.1 applies. Therefore

S_{H,V} (G) ≅ Tr_{1}^{G/G}(V) = V ,

as was to be shown.

7.6. Examples. Let H = A_{5} and let ε be the sign representation of the group Out(A_{5}) of order 2 (over a field k of characteristic different from 2). By Corollaries 7.4 and 7.5, we have S_{A_{5},ε}(S_{5}) = 0 and S_{A_{5},ε}(Ã_{5}) ≠ 0. We also have S_{A_{5},ε}(S_{7}) = 0, because it is easy to check that (A_{5}, 1) is the only section in Σ_{A_{5}}(A_{7}) up to conjugation.

References

[BD] R. Boltje, S. Danz. A ghost algebra of the double Burnside algebra in characteristic zero, J. Pure Appl. Alg., to appear.

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[Bo2] S. Bouc. Biset functors for finite groups, Springer Lecture Notes in Mathematics no. 1990 (2010).

[BST] S. Bouc, R. Stancu, J. Thévenaz. Simple biset functors and double Burnside ring, J. Pure Appl. Alg. 217 (2013), 546–566.

[BT1] S. Bouc, J. Thévenaz. Gluing torsion endo-permutation modules, J. London Math. Soc. 78 (2008), 477–501.

[BT2] S. Bouc, J. Thévenaz. Stabilizing bisets, Adv. in Math. 229 (2012), 1610–1639.

[TW] J. Thévenaz, P. Webb. The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), 1865–1961.

[We] P. Webb. Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces, J. Pure Appl. Alg. 88 (1993), 265–304.

Serge Bouc, CNRS-LAMFA, Université de Picardie - Jules Verne, 33, rue St Leu, F-80039 Amiens Cedex 1, France.

serge.bouc@u-picardie.fr

Radu Stancu, CNRS-LAMFA, Université de Picardie - Jules Verne, 33, rue St Leu, F-80039 Amiens Cedex 1, France.

radu.stancu@u-picardie.fr

Jacques Thévenaz, Section de mathématiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland.

Jacques.Thevenaz@epfl.ch