On Alperin’s conjecture and functorial equivalence of blocks

Robert Boltje, Serge Bouc, and Deniz Yılmaz

February 11, 2026

Abstract

Let k be an algebraically closed field of positive characteristic p and let 𝔽 be an algebraically closed field of characteristic 0. We consider Alperin’s weight conjecture (over k) from the point of view of (stable) functorial equivalence of blocks over 𝔽. We formulate a functorial version of Alperin’s blockwise weight conjecture, and show that it is equivalent to the original one. We also show that this conjecture holds stably, i.e. in the category of stable diagonal p-permutation functors over 𝔽.

MSC2020: 20C20, 20J15, 19A22.

Keywords: blocks of group algebras, Alperin’s weight conjecture, diagonal p-permutation functor, functorial equivalence.

1 Introduction

Let k be an algebraically closed field of positive characteristic p. The first step in studying the representation theory of a finite group G over k consists in splitting the group algebra kG as a direct product of the block algebras kGb, where b is a block idempotent of kG, that is, a primitive idempotent of the center of kG. In this paper, such a pair (G, b) of a finite group G and a block idempotent b of kG is called a group-block pair (over k).

To speak very loosely, block theory is the study of such group-block pairs (G, b), and of the numerous invariants and structures attached to them: The block algebra kGb itself, in the first place, and its various - derived or stable - categories of modules. But also invariants of a more combinatorial nature, such as defect groups, source algebras or fusion systems. For each of these invariants, there is a corresponding notion of equivalence of group-block pairs: Morita equivalence, splendid Rickard equivalence, Puig equivalence, stable equivalence of Morita type, p-permutation equivalence, isotypy, perfect isometry ([9], [8], [2], [6])… All these equivalences have been the object of intense research, and a huge amount of literature is now devoted to them.

These invariants also led to important conjectures, of various kinds. Some of them, called finiteness conjectures, predict that there are only finitely many group-block pairs up to some particular equivalence, once a specific invariant is given: For example, Donovan’s conjecture ([8, Conjecture 6.1.9]) states that there are only finitely many block algebras of group-block pairs with a given defect group, up to Morita equivalence. Other types of conjectures - called local-global or counting conjectures - assert that some invariants for a group-block pair (G, b) can be computed from the invariants attached to “smaller” pairs and “local data” - generally obtained by applying the Brauer morphism. This is in particular the case of Alperin’s weight conjecture, in its blockwise form ([1]). Finally, some structural conjectures, like Broué’s abelian defect group conjecture, predict the existence of some specific equivalence between derived categories of modules associated to group-block pairs related by the Brauer correspondence ([6]). All these conjectures have been verified in numerous cases, and proved for some particular classes of group-block pairs, but they remain essentially open.

Further new invariants were attached recently ([4], [5]) to group-block pairs, in the form of (stable) diagonal p-permutation functors over some commutative ring R (see Section 2 for details), leading to a corresponding notion of (stable) functorial equivalence of group-block pairs over R. These invariants lead in particular to a finiteness theorem ([4], Theorem 10.6), in the spirit of Donovan’s conjecture: If 𝔽 is an algebraically closed field of characteristic 0, and D is a finite p-group, then there is only a finite number of group-block pairs (G, b) with defect groups isomorphic to D, up to functorial equivalence over 𝔽.

In the present paper, we consider Alperin’s weight conjecture from the point of view of (stable) diagonal p-permutation functors. In Section 3, we first formulate (Conjecture 3.3) a functorial form FAwc(G, b) of Alperin’s conjecture Awc(G, b), for each groupblock pair (G, b). The main theorem of this section is Theorem 3.5, stating that Conjecture FAwc(G, b) holds stably, i.e. in the Grothendieck group of the category of stable diagonal p-permutation functors over 𝔽. As a consequence, our conjecture FAwc(G, b) is in fact equivalent to Awc(G, b).

It has been suspected by many experts that the original formulation of Alperin’s weight conjecture, namely the equality of two numbers associated to (G, b), is only the shadow of a more structural (yet hidden) phenomenon. We hope that FAwc(G, b) is a first step towards a more structural explanation.

1.1 Notation Let G be a group. By i_{g}: G → G, we denote the conjugation map x ↦ gxg^{−1}. Moreover, for x ∈ G and H ⩽ G we set ^{g}x := i_{g}(x) and ^{g}H := i_{g}(H). If X is a left G-set, the stabilizer of an element x ∈ X is denoted by G_{x}. By G\X we denote the set of G-orbits of X, and by [G\X] we denote a set of representatives of the G-orbits of X.

2 Review of (stable) functorial equivalence of blocks

2.1. Let R be a commutative ring (with 1), and k be an algebraically closed field of characteristic p > 0. We denote by Rpp_{k}^{Δ} the category introduced in [3], where objects are finite groups, and the set of morphisms from a group G to a group H is RT^{Δ}(H, G) = R ⊗_{ℤ} T^{Δ}(H, G), where T^{Δ}(H, G) is the Grothendieck group of diagonal p-permutation (kH, kG)-bimodules.

A diagonal p-permutation functor is by definition (see [3]) an R-linear functor from Rpp_{k}^{Δ} to the category R-Mod of all R-modules. These functors, together with their natural transformations, form an abelian category which we simply denote by F_{R,k} (instead of F_{Rpp_{k}}^{Δ} as in [3]).

2.2. We denote by \overline{Rpp_{k}^{Δ}} the quotient category of Rpp_{k}^{Δ} by the morphisms that factor through the trivial group. A stable diagonal p-permutation functor (see [5]) is an R-linear functor from \overline{Rpp_{k}^{Δ}} to R-Mod, or equivalently, a diagonal p-permutation functor which vanishes at the trivial group. Stable diagonal p-permutation functors also form an abelian category, that we simply denote by \overline{F_{R,k}}. If F is a diagonal p-permutation functor, we denote by \overline{F} its largest stable quotient, i.e. the quotient of F by the subfunctor generated by F (1). In other words \overline{F} (G) = F (G)/RT^{Δ}(G, 1)F (1) for any finite group G ([5], Remark 3.4).

2.3. Let (G, b) be a pair of a finite group G and a central idempotent b of kG (recall that when b is moreover primitive, the pair (G, b) is called a group-block pair). Then the (isomorphism class of the) (kG, kG)-bimodule kGb is an idempotent endomorphism of G in Rpp_{k}^{Δ}. The diagonal p-permutation functor RT_{G,b}^{Δ} associated to (G, b) is the corresponding direct summand RT^{Δ}(−, G) ∘ kGb of the representable functor at G obtained by precomposition with kGb.

We say that two group-block pairs (G, b) and (H, c) are functorially equivalent over R ([4], Definition 10.1) if the functors RT_{G,b}^{Δ} and RT_{H,c}^{Δ} are isomorphic in F_{R,k}. Similarly, we say that (G, b) and (H, c) are stably functorially equivalent over R if the functors \overline{RT_{G,b}^{Δ}} and \overline{RT_{H,c}^{Δ}} are isomorphic in \overline{F_{R,k}}.

2.4. We denote by RBℓ_{k} the (partial^{1}) idempotent completion of Rpp_{k}^{Δ} constructed from blocks of group algebras, i.e. the category defined as follows:

• The objects of RBℓ_{k} are pairs (G, b), where G is a finite group and b is a central idempotent of kG.

• The set of morphisms from (G, b) to (H, c) in RBℓ_{k} is the subset of RT^{Δ}(H, G) obtained by precomposition with kGb and postcomposition with kHc, in other words the set kHc ∘ RT^{Δ}(H, G) ∘ kGb.

• The composition of morphisms u : (G, b) → (H, c) and v : (H, c) → (K, d) in RBℓ_{k} is induced by the tensor product of bimodules over kH.

1In the classical definition of the idempotent completion of a category C, the objects are all the pairs (X, e) of an object X of C and an idempotent endomorphism e of X in C. Here we consider only some of these pairs. This does not affect the main properties of the idempotent completion.

• The identity morphism of (G, b) in RBℓ_{k} is (the isomorphism class of) the (kG, kG)- bimodule kGb.

Note that the objects (G, 0) in RBℓ_{k} are 0-objects. Similarly, we denote by \overline{RBℓ_{k}} the quotient category of RBℓ_{k} by the morphisms which factor through the trivial pair (1, 1_{k}). Equivalently \overline{RBℓ_{k}} is the partial idempotent completion built as above from the category \overline{Rpp_{k}^{Δ}}.

We denote by FBℓ_{R,k} (resp. \overline{FBℓ_{R,k}}) the category of R-linear functors from RBℓ_{k} (resp. \overline{RBℓ_{k}}) to R-Mod. By standard results, the inclusion functor Rpp_{k}^{Δ} → RBℓ_{k} (resp. \overline{Rpp_{k}^{Δ}} → \overline{RBℓ_{k}}) sending a group G to the pair (G, 1_{kG}) induces by composition an equivalence of categories from FBℓ_{R,k} to F_{R,k} (resp. from \overline{FBℓ_{R,k}} to \overline{F_{R,k}}). We observe that if a central idempotent b of kG is an orthogonal sum of two central idempotents b_{1} and b_{2}, the functor RT_{G,b}^{Δ} is naturally isomorphic to the direct sum RT_{G,b_{1}}^{Δ} ⊕ RT_{G,b_{2}}^{Δ} in F_{R,k}. We also observe that two group-block pairs (G, b) and (H, c) are functorially equivalent (resp. stably functorially equivalent) over R if and only if (G, b) and (H, c) are isomorphic in RBℓ_{k} (resp. in \overline{RBℓ_{k}}).

2.5 Let 𝔽 be an algebraically closed field of characteristic 0. It was shown in [4] and [5] that the categories F_{𝔽,k} and \overline{F_{𝔽,k}} are semisimple, and it follows that the categories FBℓ_{𝔽,k} and \overline{FBℓ_{𝔽,k}} are also semisimple. We denote by K_{0}(FBℓ_{𝔽,k}) and K_{0}(\overline{FBℓ_{𝔽,k}}) their respective Grothendieck groups.

The simple diagonal p-permutation functors over 𝔽 are parametrized ([4], Corollary 6.15) by means of D^{Δ}-pairs: By definition ([4], Definition 3.2), this is a pair (L, u) of a finite p-group L and a generator u of a p′-group acting faithfully on L - in other words u is not only a p′-element acting on L, but a p′-automorphism of L. An isomorphism φ : (L, u) → (M, v) of D^{Δ}-pairs is a group isomorphism φ : L⋊⟨u⟩ → M⋊⟨v⟩ between the corresponding semidirect products, which sends u to a conjugate of v. We denote by Aut(L, u) the group of automorphisms of the pair (L, u), and by Out(L, u) the quotient of Aut(L, u) by the subgroup Inn(L⋊⟨u⟩) of inner automorphisms of L⋊⟨u⟩.

The simple diagonal p-permutation functors S_{L,u,V} over 𝔽 (up to isomorphism) are then parametrized by triples (L, u, V) (up to isomorphism), where (L, u) is a D^{Δ}-pair, and V is a simple 𝔽Out(L, u)-module. The simple stable diagonal p-permutation functors are the functors S_{L,u,V}, where L is a nontrivial p-group. So the group K_{0}(FBℓ_{𝔽,k}) has a ℤ-basis consisting of the isomorphism classes [S_{L,u,V}] of simple functors S_{L,u,V}, and K_{0}(\overline{FBℓ_{𝔽,k}}) has a ℤ-basis consisting of the classes [S_{L,u,V}] with L ≠ 1. Any diagonal p-permutation functor F has an image in K_{0}(FBℓ_{𝔽,k}) (resp. in K_{0}(\overline{FBℓ_{𝔽,k}})), and two diagonal p-permutation functors are isomorphic (resp. stably isomorphic) if and only if they have the same image, i.e. if the multiplicity of each simple functor S_{L,u,V} (resp. S_{L,u,V} with L ≠ 1) in both of them is the same. In particular, two group-block pairs (G, b) and (H, c) are functorially equivalent (resp. stably functorially equivalent) if and only if the multiplicities of each S_{L,u,V} (resp. each S_{L,u,V} with L ≠ 1) in 𝔽T_{G,b}^{Δ} and 𝔽T_{H,c}^{Δ} are equal.

When G is a finite group, and b is a central idempotent of kG, we simply denote by [[G, b]]_{𝔽} the image of the functor 𝔽T_{G,b}^{Δ} in K_{0}(FBℓ_{𝔽,k}), and by \overline{[[G, b]]}_{𝔽} its image in K_{0}(\overline{FBℓ_{𝔽,k}}).

3 Alperin’s weight conjecture and stable functorial equivalence

By Knörr-Robinson, see [7, Theorem 3.8], Alperin’s blockwise weight conjecture, that we refer to as AWC, is equivalent to saying that the following conjecture Awc(G, b) holds for all group-block pairs (G, b).

3.1 Conjecture Let (G, b) be a group-block pair over k. Then

(Awc(G, b))  ∑_{σ∈[G\S_{p}(G)]}(−1)^{|σ|} l(kG_{σ}b_{σ}) = \begin{cases}1 & if d(b) = 0; \\ 0 & if d(b) > 0. \end{cases}

Here:

• S_{p}(G) denotes the set of strictly ascending chains (1 = P_{0} < P_{1} < ⋯ < P_{n}) of p-subgroups of G.

• G_{σ} is the stabilizer in G of σ.

• |σ| = n if σ = (1 = P_{0} < P_{1} < ⋯ < P_{n}) ∈ S_{p}(G).

• For σ as above, b_{σ} := Br_{P_{n}}(b), where Br_{P_{n}} is the Brauer homomorphism with respect to P_{n}.

• l associates to a finite-dimensional k-algebra the number of its simple modules (up to isomorphism).

• d(b) denotes the defect of b.

3.2 Remark One can show that b_{σ} is actually a sum of block idempotents of kG_{σ} and that it does not depend on the chain σ but only on the stabilizer G_{σ} (see Lemma 3.1 and the following Remark in [7]). If b is a block of kG with trivial defect group, Conjecture Awc(G, b) holds trivially.

For a group-block pair (G, b), we propose the following conjecture, denoted by FAwc(G, b) (with “F” standing for functorial), using the notation of Conjecture 3.1:

3.3 Conjecture Let (G, b) be a group-block pair over k. Then, in K_{0}(FBℓ_{𝔽,k}), we have

(FAwc(G, b))  ∑_{σ∈[G\S_{p}(G)]}(−1)^{|σ|} [[G_{σ}, b_{σ}]]_{𝔽} = \begin{cases}[S_{1,1,𝔽}] & if d(b) = 0; \\ 0 & if d(b) > 0. \end{cases}

Again, if b has trivial defect group, it is easy to show that Conjecture FAwc(G, b) holds, see [4, Corollary 8.23]. The statement that FAwc(G, b) holds for all group-block pairs (G, b) is abbreviated by FAWC.

It is straightforward to see that for a group-block pair (G, b), Conjecture FAwc(G, b) implies Conjecture Awc(G, b):

3.4 Theorem 1. Let (G, b) be a group-block pair over k. If Conjecture FAwc(G, b) holds, then Conjecture Awc(G, b) holds.

2. In particular, Conjecture FAWC implies Conjecture AWC.

Proof Since 𝔽T_{G,b}^{Δ}(1) is isomorphic to the 𝔽-vector space spanned by the indecomposable projective kGb-modules, one has l(kGb) = dim_{𝔽} 𝔽T_{G,b}^{Δ}(1), and by [4, Corollary 8.23] the latter is equal to the multiplicity of the simple functor S_{1,1,𝔽} in 𝔽T_{G,b}^{Δ}. Assertion 1 is now immediate by considering the multiplicity of [S_{1,1,𝔽}] in both sides of FAwc(G, b), and then Assertion 2 follows.

In the following theorem, the notation is the same as in Conjecture 3.1:

3.5 Theorem 1. Let (G, b) be a group-block pair over k. Then there exists an integer n_{G,b} such that, in K_{0}(FBℓ_{𝔽,k}),

∑_{σ∈[G\S_{p}(G)]}(−1)^{|σ|}[[G_{σ}, b_{σ}]]_{𝔽} = n_{G,b}[S_{1,1,𝔽}].

2. In particular, Conjecture FAWC is equivalent to AWC, and for any group-block pair (G, b) over k, Conjecture FAwc(G,b) is equivalent to Awc(G,b).

Proof 1. Let Σ_{G,b} ∈ K_{0}(FBℓ_{𝔽,k}) denote the alternating sum in the left hand side of FAwc(G, b), that is,

Σ_{G,b} := ∑_{σ∈[G\S_{p}(G)]}(−1)^{|σ|}[[G_{σ}, b_{σ}]]_{𝔽} .

We want to show that the multiplicity of a simple functor S_{L,u,V} in Σ_{G,b} is equal to 0 if L ≠ 1. By [4, Theorem 8.22], we know that the multiplicity m_{L,u,V} (G_{σ}, b_{σ}) of S_{L,u,V} as a composition factor of the functor 𝔽T_{G,b}^{Δ} is given by

m_{L,u,V} (G_{σ}, b_{σ}) = ∑_{(P_{γ},π)∈[G_{σ}\L_{bσ} (G_{σ},L,u)/Aut(L,u)]} dim_{𝔽} V^{Aut(L,u)\overline{(Pγ ,π)}},

where the notation is as follows:

• L_{b_{σ}}(G_{σ}, L, u) is the set of pairs (P_{γ}, π) of a local point P_{γ} on kG_{σ}b_{σ}, i.e., a psubgroup P of G_{σ} and a conjugacy class γ of primitive idempotents of (kG_{σ}b_{σ})^{P}, and π : L → P is a group isomorphism such that πuπ^{−1} ∈ N_{G_{σ}}(P_{γ}), i.e., such that there exists g ∈ N_{G}(P_{γ}) with i_{g}π = πu.

• The set L_{b_{σ}}(G_{σ}, L, u) is a (G_{σ}, Aut(L, u))-biset via the action defined by

g · (P_{γ}, π) · φ = (^{g}P_{^{g}γ}, i_{g}πφ) ,

for (g, φ) ∈ G_{σ} × Aut(L, u) and (P_{γ}, π) ∈ L_{b_{σ}}(G_{σ}, L, u). For (P_{γ}, π) ∈ L_{b_{σ}}(G_{σ}, L, u), we denote by \overline{(P_{γ}, π)} the left orbit G_{σ}(P_{γ}, π), and by Aut(L, u)\overline{(P_{γ},π)} the stabilizer of this orbit in Aut(L, u), namely

Aut(L, u)\overline{(P_{γ},π)} = \{φ ∈ Aut(L, u) | ∃g ∈ N_{G_{σ}}(P_{γ}), i_{g}π = πφ\}.

So we want to show that

∑_{σ∈[G\S_{p}(G)]}(−1)^{|σ|}m_{L,u,V} (G_{σ}, b_{σ}) = 0

whenever L ≠ 1. We observe that

dim_{𝔽} V^{Aut(L,u)\overline{(Pγ ,π)}} = (Ind_{Aut(L,u)\overline{(Pγ ,π)}}^{Aut(L,u)}𝔽, V )_{Aut(L,u)} ,

where (−, −)_{Aut(L,u)} denotes the Schur inner product on the Grothendieck group (character ring) R_{𝔽}(Aut(L, u)) of finite dimensional 𝔽Aut(L, u)-modules. We set

W (G, L, u) = ∑_{σ∈[G\S_{p}(G)]}(−1)^{|σ|} ∑_{(P_{γ},π)∈[G_{σ}\L_{bσ} (G_{σ},L,u)/Aut(L,u)]} Ind_{Aut(L,u)\overline{(Pγ ,π)}}^{Aut(L,u)}𝔽,

which we view as an element in R_{𝔽}(Aut(L, u)). We want to show that

(W (G, L, u), V )_{Aut(L,u)} = 0 ,

for all L ≠ 1 and all simple 𝔽Aut(L, u)-modules V. But this amounts to saying that the virtual character W (G, L, u) is equal to 0. Since W (G, L, u) is a (virtual) permutation character its value at φ ∈ Aut(L, u) is equal to

|W (G, L, u)^{φ}| = ∑_{σ∈[G\S_{p}(G)]}(−1)^{|σ|}|(G_{σ}\L_{b_{σ}}(G_{σ}, L, u))^{φ}| .

So, we want to show that this number is equal to 0 if L ≠ 1.

Summing over all σ ∈ S_{p}(G) rather than representatives of G-orbits, we have that

|W (G, L, u)^{φ}| = ∑_{σ∈S_{p}(G)} \frac{|G_{σ}|}{|G|}(−1)^{|σ|}|(G_{σ}\L_{b_{σ}}(G_{σ}, L, u))^{φ}|

= ∑_{\begin{matrix}σ∈S_{p}(G) \\ (P_{γ},π)∈L_{bσ} (G_{σ},L,u) \\ G_{σ}(P_{γ},π)·φ=G_{σ}(P_{γ},π) \end{matrix}} \frac{|G_{σ}|}{|G|}(−1)^{|σ|} \frac{|G_{σ} ∩ G_{(P_{γ},π)}|}{|G_{σ}|}

= ∑_{\begin{matrix}σ∈S_{p}(G) \\ (P_{γ},π)∈L_{bσ} (G_{σ},L,u) \\ G_{σ}(P_{γ},π)·φ=G_{σ}(P_{γ},π) \end{matrix}}(−1)^{|σ|} \frac{|G_{σ} ∩ G_{(P_{γ},π)}|}{|G|}

where G_{(P_{γ},π)} is the left stabilizer of (P_{γ}, π), i.e.,

G_{(P_{γ},π)} = {g ∈ G | ^{g}P = P, ^{g}γ = γ, i_{g}π = π} = C_{G}(P) ∩ N_{G}(P_{γ}).

Now G_{σ}(P_{γ}, π) · φ = G_{σ}(P_{γ}, π) if and only if there exists g ∈ G_{σ} such that (^{g}P_{^{g}γ}, i_{g}π) = (P_{γ}, πφ), and in this case, the number of such elements g ∈ G_{σ} is equal to |G_{σ} ∩ G_{(P_{γ},π)}|. It follows that

|W (G, L, u)^{φ}| = \frac{1}{|G|} ∑_{\begin{matrix}(σ,P_{γ},π,g) \\ σ∈S_{p}(G) \\ P_{γ}∈L_{bσ} (G_{σ},L,u) \\ g∈G_{σ} \\ (^{g}P_{^{g}γ},i_{g}π)=(P_{γ},πφ) \end{matrix}}(−1)^{|σ|} .

We can rewrite this as

|W (G, L, u)^{φ}| = \frac{1}{|G|} ∑_{(σ,P,γ,π,g)∈𝕊}(−1)^{|σ|},

where 𝕊 is the set of quintuples (σ, P, γ, π, g) such that:

• σ ∈ S_{p}(G),

• P ⩽ G_{σ},

• γ is a local point of (kG_{σ}b_{σ})^{P},

• π : L →^{≅} P is a group isomorphism such that πuπ^{−1} ∈ N_{G_{σ}}(P_{γ}),

• g ∈ G_{σ} is such that (^{g}P_{^{g}γ}, i_{g}π) = (P_{γ}, πφ), in other words g ∈ N_{G_{σ}}(P_{γ}) and i_{g}π = πφ.

The proof that |W (G, L, u)^{φ}| = 0 is inspired by the proof of Lemma 4.1 of [7]. We will build an involution (σ, P, γ, π, g) ↦ (σ′, P, γ′, π, g) of 𝕊 such that |σ′| = |σ| ± 1.

Let (σ, P, γ, π, g) ∈ 𝕊, with σ = (1 = P_{0} < P_{1} < … < P_{n}). Since P ≅ L ≠ 1 = P_{0}, there is a largest integer i ∈ {0, …, n} such that P ⩽̸ P_{i}. There are two cases:

• Either i = n or P P_{i} < P_{i+1}, then set σ′ = σ ⊔ {P P_{i}}, i.e.,

σ′ = (P_{0} < … < P_{n} < P P_{n}) or σ′ = (P_{0} < P_{1} < … P_{i} < P P_{i} < P_{i+1} < … < P_{n}) ,

respectively.

• Or P P_{i} = P_{i+1}, and then set σ′ = σ ∖ {P_{i+1}}, i.e.,

σ′ = (P_{0} < P_{1} < … P_{i} < P_{i+2} < … < P_{n}).

One checks easily that (σ′)′ = σ, and it is clear that |σ′| = |σ| ± 1. Moreover P ⩽ G_{σ′} if and only if P ⩽ G_{σ}, and N_{G}(P) ∩ G_{σ} = N_{G}(P) ∩ G_{σ′}, i.e., N_{G_{σ}}(P) = N_{G_{σ′}} (P).

Now by [10, Corollary 37.6], the Brauer morphism Br_{P}^{G_{σ}} induces a bijection between the local points of (kG_{σ}b_{σ})^{P} and the points of kC_{G_{σ}}(P)Br_{P}^{G_{σ}}(b_{σ}). Since N_{G_{σ}}(P) = N_{G_{σ′}} (P), we also have C_{G_{σ}}(P) = C_{G_{σ′}} (P). Moreover, Br_{P}^{G_{σ}}(b_{σ}) = Br_{P}^{G_{σ′}} (b_{σ′}). In fact, Br_{P}^{G_{σ}}(b_{σ}) is the truncation to kC_{G_{σ}}(P) of b_{σ} ∈ kC_{G_{σ}}(P_{n}), so Br_{P}^{G_{σ}}(b_{σ}) is the truncation of b to kC_{G_{σ}}(P P_{n}) = kC_{G_{σ′}} (P P_{n}), which is equal to Br_{P}^{G_{σ′}} (b_{σ′}).

It follows that the Brauer morphism at P induces a bijection γ ↦ γ′ between the local points of (kG_{σ}b_{σ})^{P} and those of (kG_{σ′}b_{σ′})^{P}. This bijection is N_{G_{σ}}(P)-equivariant, so N_{G_{σ}}(P_{γ}) = N_{G_{σ′}} (P_{γ′}), and it follows that (σ′, P, γ′, π, g) ∈ 𝕊. Now (σ, P, γ, π, g) ↦ (σ′, P, γ′, π, g) is an involution of the set 𝕊, with the property that |σ′| = |σ| ± 1. Hence |W (G, L, u)^{φ}| = 0 for any φ ∈ Aut(L, u) if L ≠ 1. This completes the proof of Assertion 1.

2. It follows from Assertion 1 that FAwc(G, b) and Awc(G, b) are both equivalent to n_{G,b} being equal to 1 if d(b) = 0, and to 0 otherwise. Both parts of Assertion 2 follow.

3.6 Corollary Let (G, b) be a group-block pair over k. Then the following stable version of Conjecture FAwc(G, b) holds:

∑_{σ∈[G\S_{p}(G)]}(−1)^{|σ|} \overline{[[G_{σ}, b_{σ}]]}_{𝔽} = 0 in K_{0}(\overline{FBℓ_{𝔽,k}}) .

Proof Indeed, the simple functor S_{1,1,𝔽} becomes zero in the category \overline{FBℓ_{𝔽,k}} and the result follows from Theorem 3.5.

Acknowledgment The first two authors are very grateful for the hospitality they experienced during their visits at the Mathematics Department at Bilkent University. The third author is supported by the Scientific and Technological Research Council of Türkiye (TÜBİTAK) under the 3501 Career Development Program with Project No. 123F456.

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Robert Boltje, Department of Mathematics, University of California, Santa Cruz, 95064, California, USA.

boltje@ucsc.edu

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

serge.bouc@u-picardie.fr

Deniz Yılmaz, Department of Mathematics, Bilkent University, 06800 Ankara, Turkey.

d.yilmaz@bilkent.edu.tr