k*-Fusions and Alperin’s Conjecture†

by

LLUIS PUIG

CNRS, Université Paris 7, 6 Av Bizet, 94340 JOINVILLE-LE-PONT, FRANCE, puig@math.jussieu.fr

1. In a lecture with almost the same title — at the 2002 Durham Symposium — Markus Linckelmann made a claim for the existence of a suitable k^{∗}-extension of the full subcategory of the Brauer category of a block over the set of all its selfcentralizing Brauer pairs, relating the k-algebra associated with this extension to Alperin’s Weight Conjecture [1] on the number of modular irreducible characters in the block.

2. The purpose of this Note is to explicit our comment at the end of this lecture, in the sense that the existence of such a k^{∗}-extension depends on the existence of the k^{∗}-fusions of the local pointed groups — introduced in a series of lectures we gave in the École Normale Supérieure des Jeunes Filles in 1985-86 — and is equivalent to the existence — here proved — of the localizing sheaf of k^{∗}-localizers we have arisen in our talks at the MRSI in 1990 [11]††, and that the relationship with Alperin’s Conjecture does not depend on the particular form of this k^{∗}-extension but only on the translation in terms of k^{∗}-fusions of this conjecture, that we have several times explained — in particular in the Claude Chevalley Seminar.

3. Let p be a prime, k an algebraically closed field of characteristic p, G a finite group and b a block of G. Recall that the objects of the Brauer category of b [2] are the Brauer b-pairs, namely the pairs (Q, f) where Q is a p-subgroup of G and f a block of C_{G}(Q) such that Br_{Q}(b)f = f , whereas the objects of the local category of b [8] are the local pointed groups on kGb , namely the pairs Q_{δ} where Q is a p-subgroup of G and δ is a conjugacy class of primitive idempotents in the k-algebra (kGb)^{Q} such that Br_{Q}(δ) ≠ {0}; we denote by (Q, b(δ)) the Brauer b-pair determined by Q_{δ} .

4. It is clear that G acts on both sets and the morphisms in both categories are the group homomorphisms induced by the G-conjugation and by the corresponding inclusion relation. We say that a local pointed group R_{ε} on kGb is contained in Q_{δ} if Q contains R and, for any i ∈ δ , there is j ∈ ε such that ij = j = ji , whereas (Q, f) contains a Brauer b-pair (R, g) if for some local pointed groups Q_{δ} and R_{ε} on kGb we have

4.1  b(δ) = f , b(ε) = g and R_{ε} ⊂ Q_{δ} .

† This is a revised version of a previous Note, where the “obvious bijection” 12.1 was obviously false.

†† It is not a question of priority but historical facts, as it is the attendance of Markus Linckelmann to both series of lectures.

5. As in any noncommutative algebraic structure, in both categories we have an obvious notion of inner automorphisms of the objects, and we consider the corresponding exterior quotients of these categories; then, we respectively denote by E_{G}((R, g), (Q, f)) and E_{G}(R_{ε}, Q_{δ}) the sets of morphisms in the exterior quotients; thus, we have

5.1  \begin{aligned}E_{G}(Q, f) = E_{G}((Q, f), (Q, f)) & ≅ N_{G}(Q, f)/Q·C_{G}(Q) \\ E_{G}(Q_{δ}) = E_{G}(Q_{δ}, Q_{δ}) & ≅ N_{G}(Q_{δ})/Q·C_{G}(Q) \end{aligned} .

6. We say that Q_{δ} or (Q, f) is selfcentralizing if, setting C̄_{G}(Q) = Q·C_{G}(Q)/Q , the image in kC̄_{G}(Q) of b(δ) or f — which is a block of C̄_{G}(Q) — has defect zero [8, (41.1)]; note that in this case b(δ) determines δ; that is to say, up to suitable identifications, the full subcategories of the Brauer and the local categories over the selfcentralizing objects coincide [8, (41.2)]. Let us recall that Q_{δ} is selfcentralizing if and only if C_{P} (Q) = Z(Q) for any local pointed group P_{γ} on kGb containing Q_{δ} — mutatis mutandi for (Q, f) — which justifies the terminology [8, (41.3)].

7. Following Alperin [1], a weight of b is a pair (Q, χ) , where Q is a p-subgroup of G and χ is an irreducible character of N_{G}(Q) associated with Br_{Q}(b) , coming from a modular irreducible character χ̄ in a block of defect zero of N̄_{G}(Q) = N_{G}(Q)/Q , and his remarkable conjecture affirms that the number of G-conjugacy classes of weights of b coincides with the number |Irr_{k}(G, b)| of modular irreducible characters in b .

8. But note that χ̄ is projective and unique in its block b(χ̄) of defect zero, and that its restriction to C̄_{G}(Q) is projective too, determining a set of blocks of defect zero of C̄_{G}(Q) which N̄_{G}(Q) acts transitively on; that is to say, a genuine application of the so-called Clifford’s theory [3] yields

8.1  χ̄ = Ind_{N̄_{G}(Q,f)}^{N̄_{G}(Q)}(ζ̂ ⊗ θ̂)

where f is a block of C_{G}(Q) which has an image f̄ of defect zero in kC̄_{G}(Q) fulfilling f̄b(χ̄) = f̄, ζ̂ is the modular irreducible character of the Clifford extension ˆ N̄_{G}(Q, f) — the k^{∗}-extension of N̄_{G}(Q, f) obtained from its action on the simple k-algebra kC̄_{G}(Q)f̄ — acting on a simple kC̄_{G}(Q)f̄-module, and θ̂ is a suitable modular irreducible character of ˆ N̄_{G}(Q, f)^{∘} coming from a block of defect zero of the quotient ˆ N̄_{G}(Q, f)^{∘}/C̄_{G}(Q) .

9. Hence, a weight (Q, χ) of b determines, up to G-conjugacy, a selfcentralizing local pointed group Q_{δ} and, denoting by Ê_{G}(Q_{δ}) the k^{∗}-extension of E_{G}(Q_{δ}) coming from ˆN̄_{G}(Q_{δ})/C̄_{G}(Q) via isomorphism 5.1, a modular irreducible character of Ê_{G}(Q_{δ})^{∘} in a block of defect zero. Conversely, it is not difficult to check that such a pair determines a weight of b via equality 8.1. Thus, denoting by e_{Q_{δ}} the sum of all the blocks of defect zero of Ê_{G}(Q_{δ})^{∘} for any selfcentralizing local pointed group Q_{δ} , with an obvious notation Alperin’s Conjecture can be reformulated as follows

9.1  |Irr_{k}(G, b)| = ∑_{Q_{δ}} |Irr_{k}(Ê_{G}(Q_{δ})^{∘}, e_{Q_{δ}})|

where Q_{δ} runs on a set of representatives for the G-conjugacy classes of selfcentralizing local pointed groups.

Remark 10. Although Ê_{G}(Q_{δ}) determines an element γ_{Q_{δ}} in the 2-cohomology group ℍ^{2}(E_{G}(Q_{δ}), k^{∗}) , it would be misleading to denote Ê_{G}(Q_{δ}) by the pair (E_{G}(Q_{δ}), γ_{Q_{δ}}) since, whenever |Hom(E_{G}(Q_{δ}), k^{∗})| ≠ 1 , both mathematical structures have different automorphism groups.

11. Let us come back to the local category F_{b} of b and denote by F_{b}(Q_{δ}, R_{ε}) the set of morphisms from R_{ε} to Q_{δ} , both being local pointed groups on kGb; that is to say, the set of group homomorphisms from R to Q induced by elements x ∈ G such that R_{ε} ⊂ (Q_{δ})^{x} . Setting F_{b}(Q_{δ}) = F_{b}(Q_{δ}, Q_{δ}) and denoting by F̂_{b}(Q_{δ}) the corresponding k^{∗}-extension and by F_{b}(Q_{δ})_{R} and F̂_{b}(Q_{δ})_{R} the stabilizers of a subgroup R of Q in F_{b}(Q_{δ}) and F̂_{b}(Q_{δ}) respectively, we claim that if R_{ε} is a selfcentralizing local pointed group contained in Q_{δ} then the group homomorphism F_{b}(Q_{δ})_{R} → F_{b}(R_{ε}) induced by the restriction can be lifted to a k^{∗}-group homomorphism

11.1  F̂_{b}(Q_{δ})_{R} ⟶ F̂_{b}(R_{ε}) ;

indeed, since for any subgroup N of Q there is a local point ν of N on kG such that N_{ν} ⊂ Q_{δ} , and ε is uniquely determined by δ , arguing by induction on |Q: R| we may assume that R is normal in Q; in this case, the Brauer homomorphism induces an isomorphism

11.2  ((kG)(R))(Q̄) ≅ (kG)(Q) ,

where Q̄ = Q/R , and, since (kG)(R_{ε}) ≅ k ⊗_{kZ(R)} kC_{G}(R)b(ε) , Q̄ stabilizes a basis of (kG)(R_{ε}) , so that (kG)(R_{ε}) becomes a Dade Q̄-algebra; hence, ((kG)(R_{ε}))(Q̄) is a simple k-algebra (cf. Corollary 5.8 in [9]) and Dade’s Spliting Theorem guarantees the existence of an extension (cf. (e) in [7])

11.3  Br∗ ; Q̄ : N_{(kG)(R_{ε})^{∗}}(Q̄) ⟶ ((kG)(R_{ε}))(Q̄)^{∗}

of the Brauer homomorphism Br(kG)(R_{ε}) ; Q̄ to the normalizer of the canonical image of Q̄ in (kG)(R_{ε})^{∗}; moreover, by Proposition 3.3 and Example 3.9 in [5], we get an embedding (kG)(Q_{δ}) → ((kG)(R_{ε}))(Q̄) (see in 47 below a more detailed argument). Now, from the structural k^{∗}-group homomorphism

11.4  F̂_{b}(Q_{δ}) ⟶ (kG)(Q_{δ})^{∗} ⟶ ((kG)(R_{ε}))(Q̄)^{∗} ,

it is not difficult to obtain a k^{∗}-group isomorphism N_{F̂_{b}(R_{ε})}(I_{Q}(R)) ≅ F̂_{b}(Q_{δ})_{R} , where I_{Q}(R) denotes the image of Q in Aut(R) .

12. A more than fifteen years old problem (cf. 1.20 and 1.21 in [11]) is whether the k^{∗}-group homomorphisms 11.1 can be chosen in a coherent way. Precisely, consider the chains 𝔮 of selfcentralizing local pointed groups on kGb , namely the injective orderpreserving maps from Δ_{n} = {i | 0 ≤ i ≤ n} , where n ∈ ℕ , to the ordered set SC_{b} of selfcentralizing local pointed groups on kGb , and denote by F_{b}(𝔮) the stabilizer of the family {𝔮(i)}_{i∈Δ_{n}} in F_{b}(𝔮(n)) and by F̂_{b}(𝔮) the corresponding k^{∗}-subgroup of F̂_{b}(𝔮(n)); it is clear that G acts by conjugation on the set of all these chains and we say that a chain 𝔯: Δ_{m} → SC_{b} is contained in 𝔮 , and write 𝔯 ⊂ 𝔮 , when Im(𝔯) ⊂ Im(𝔮) , which forces n ≤ m . Let us denote by F_{b}^{sch} the category where the objects are the chains of selfcentralizing local pointed groups and the morphisms from a chain 𝔯: Δ_{m} → SC_{b} to a chain 𝔮: Δ_{n} → SC_{b} are the group homomorphisms from 𝔯(m) to 𝔮(n) induced by conjugation by elements x ∈ G such that 𝔯 ⊂ 𝔮^{x}; then, it is clear that the correspondence sending 𝔮 to F_{b}(𝔮) extends to a contravariant functor from F_{b}^{sch} to the category of groups 𝔊𝔯 and the existence of the k^{∗}-group homomorphism 11.1 guarantees that any group homomorphism F_{b}(𝔮) → F_{b}(𝔯) coming from this functor can be lifted to a k^{∗}-group homomorphism

12.1  F̂_{b}(𝔮) ⟶ F̂_{b}(𝔯) .

The problem is whether the functor itself can be lifted to a functor from F_{b}^{sch} to the category of k^{∗}-groups k^{∗}-𝔊𝔯 mapping any chain 𝔮 on the corresponding k^{∗}-subgroup F̂_{b}(𝔮) of F̂_{b}(𝔮(n)) and the inclusion in 𝔮 of the chain mapping 0 on 𝔮(n) on the inclusion

12.2  F̂_{b}(𝔮) ⊂ F̂_{b}(𝔮(n)) .

13. As a matter of fact, Linckelmann’s problem is equivalent to this one. Indeed, firstly assume that there is a suitable k^{∗}-extension of the full subcategory over SC_{b} of the local category of b; that is to say assume that, for any Q_{δ}, R_{ε} ∈ SC_{b} , we have k^{∗}-sets F̂_{b}(Q_{δ}, R_{ε}) having F_{b}(Q_{δ}, R_{ε}) as k^{∗}-quotient and holding an associative composition which lifts the composition in the local category and centralizes k^{∗} . Then, if 𝔮: Δ_{n} → SC_{b} and 𝔯: Δ_{m} → SC_{b} are chains such that 𝔮(n) = Q_{δ} and 𝔯(m) = R_{ε} , and x ∈ G fulfills 𝔯 ⊂ 𝔮^{x} , it is quite clear that any lifting ˆ x̄ in F̂_{b}(Q_{δ}, R_{ε}) of the element x̄ of F_{b}(Q_{δ}, R_{ε}) determined by x induces a k^{∗}-group homomorphism

13.1  θ̂_{ˆx̄} : F̂_{b}(𝔮) ⟶ F̂_{b}(𝔯)

fulfilling ˆ x̄·θ̂_{ˆx̄}(ˆȳ) = ˆ ȳ·ˆx̄ for any ˆ ȳ ∈ F̂_{b}(𝔮) ; moreover, since we still have λˆx̄·θ̂_{ˆx̄}(ˆȳ) = ˆ ȳ·λˆx̄ for any λ ∈ k^{∗} , θ̂_{ˆx̄} only depends on x̄; finally, it is easily checked that the correspondence sending x̄ to θ̂_{ˆx̄} is compatible with the compositions in F_{b}^{sch} and k^{∗}-𝔊𝔯.

14. Conversely, assume that we have a functor from F_{b}^{sch} to k^{∗}-𝔊𝔯 fulfilling the above condition. In particular, if Q_{δ}, Q′_{δ′} ∈ SCb are G-conjugated each other and x̄ ∈ F_{b}(Q′_{δ′}, Q_{δ}) then, applying the functor, we get a k^{∗}-group isomorphism θ̂x̄ : F̂_{b}(Q_{δ}) ≅ F̂_{b}(Q′_{δ′}) and we denote by F̂_{b}(Q′_{δ′}) ×x̄ F̂_{b}(Q_{δ}) the quotient set of F̂_{b}(Q′_{δ′}) × F̂_{b}(Q_{δ}) by the corresponding θ̂x̄-twisted diagonal; it is clear that F̂_{b}(Q′_{δ′}) acts in this quotient on the left and F̂_{b}(Q_{δ}) on the right, both regularly, and that these actions are compatible. Moreover, if Q′′_{δ′′} ∈ SC_{b} is G-conjugated to Q′_{δ′} and x̄′ ∈ F_{b}(Q′′_{δ′′}, Q′_{δ′}) , it follows from the existence of the functor that we have an evident map

14.1  (F̂_{b}(Q′′_{δ′′}) ×_{}_{x̄′} F̂_{b}(Q′_{δ′})) × (F̂_{b}(Q′_{δ′}) ×x̄ F̂_{b}(Q_{δ})) ⟶ F̂_{b}(Q′′_{δ′′}) ×_{}_{x̄′}x̄ F̂_{b}(Q_{δ}) ;

similarly, it is easily checked that the maps associated with all these triples Q_{δ}, Q′_{δ′}, Q′′_{δ′′} fulfill evident associative properties. A consequence of this associativity is that if ȳ is another element of F_{b}(Q′_{δ′}, Q_{δ}) , the bijections between the sets F̂_{b}(Q′_{δ′}) ×x̄ F̂_{b}(Q_{δ}) and F̂_{b}(Q′_{δ′}) ×ȳ F̂_{b}(Q_{δ}) determined by x̄^{−1}ȳ and ȳx̄^{−1} , together with the corresponding unity elements, coincide and that, denonting by τ_{ȳ,x̄} this unique bijection, for a third z̄ ∈ F_{b}(Q′_{δ′}, Q_{δ}) we have τ_{z̄,ȳ} ∘ τ_{ȳ,x̄} = τ_{z̄,x̄}; that is to say, we have a family of sets and a transitive family of bijections between them

14.2  {F̂_{b}(Q′_{δ′}) ×x̄ F̂_{b}(Q_{δ})}_{x̄∈F_{b}(Q′_{δ′} ,Q_{δ})} and {τ_{ȳ,x̄}}_{x̄,ȳ∈F_{b}(Q′_{δ′} ,Q_{δ})}

and we denote by F̂_{b}(Q′_{δ′}, Q_{δ}) their inverse limit or, equivalently, the subset of the elements of the direct product ∏_{x̄∈F_{b}(Q′_{δ′} ,Q_{δ})} F̂_{b}(Q′_{δ′}) ×x̄ F̂_{b}(Q_{δ}) compatible with the bijec-

tions. Then, we have a map F̂_{b}(Q′_{δ′}, Q_{δ}) → F_{b}(Q′_{δ′}, Q_{δ}) and the maps 14.1 determine an “associative composition” compatible with the composition in F_{b}

14.3  μ_{Q′′_{δ′′} ,Q′_{δ′} ,Q_{δ}} : F̂_{b}(Q′′_{δ′′}, Q′_{δ′}) × F̂_{b}(Q′_{δ′}, Q_{δ}) ⟶ F̂_{b}(Q′′_{δ′′}, Q_{δ}) .

15. Let R_{ε}, R′_{ε′} ∈ SCb be G-conjugated each other and assume that we have R_{ε} ⊂ Q_{δ} and R′_{ε′} ⊂ Q′_{δ′} , and that there is x̄ ∈ F_{b}(Q′_{δ′}, Q_{δ}) mapping R_{ε} on R′_{ε′} ; denote by F̂_{b}(Q_{δ})_{R} and F̂_{b}(Q′_{δ′})_{R′} the corresponding stabilizers, and by ȳ the element of F_{b}(R′_{ε′}, R_{ε}) determined by x̄; it is clear that θ̂x̄ induces a k^{∗}-group isomorphism F̂_{b}(Q_{δ})_{R} ≅ F̂_{b}(Q′_{δ′})_{R′} and that, applying the functor to ȳ and to the inclusions, we get a commutative diagram of k^{∗}-group homomorphisms

15.1  \begin{aligned}F̂_{b}(Q_{δ})_{R} & ≅ & F̂_{b}(Q′_{δ′})_{R′} \\ ↓ & & ↓ \\ F̂_{b}(R_{ε}) & ≅ & F̂_{b}(R′_{ε′}) \end{aligned} ;

consequently, denoting by F̂_{b}(Q′_{δ′})_{R′} ×x̄ F̂_{b}(Q_{δ})_{R} the image of F̂_{b}(Q′_{δ′})_{R′} × F̂_{b}(Q_{δ})_{R} in F̂_{b}(Q′_{δ′}) ×x̄ F̂_{b}(Q_{δ}) , we get a map

15.2  F̂_{b}(Q′_{δ′})_{R′} ×x̄ F̂_{b}(Q_{δ})_{R} ⟶ F̂_{b}(R′_{ε′}) ×ȳ F̂_{b}(R_{ε}) ;

moreover, denoting by F_{b}(Q′_{δ′}, Q_{δ})_{R′,R} the set of homomorphisms in F_{b}(Q′_{δ′}, Q_{δ}) mapping R on R′ , and by F̂_{b}(Q′_{δ′}, Q_{δ})_{R′,R} the converse image of F_{b}(Q′_{δ′}, Q_{δ})_{R′,R} in F̂_{b}(Q′_{δ′}, Q_{δ}), it is easily checked that the canonical bijection F̂_{b}(Q′_{δ′}, Q_{δ}) ≅ F̂_{b}(Q′_{δ′}) ×x̄ F̂_{b}(Q_{δ}) induces a new bijection

15.3  F̂_{b}(Q′_{δ′}, Q_{δ})_{R′,R} ≅ F̂_{b}(Q′_{δ′})_{R′} ×x̄ F̂_{b}(Q_{δ})_{R}

which, together with the map 15.2 and the bijection F̂_{b}(R′_{ε′}, R_{ε}) ≅ F̂_{b}(R′_{ε′}) ×ȳ F̂_{b}(R_{ε}) , determines a map independent of the choice of x̄

15.4  ρ\begin{matrix}Q{′ \atop δ′} ,Q_{δ} \\ R{′ \atop ε′} ,Rε \end{matrix} : F̂_{b}(Q′_{δ′}, Q_{δ})_{R′,R} ⟶ F̂_{b}(R′_{ε′}, R_{ε}) .

16. Further, if Q′′_{δ′′}, R′′_{ε′′} ∈ SC_{b} still fulfill R′′_{ε′′} ⊂ Q′′_{δ′′} and there is x̄′ ∈ F_{b}(Q′′_{δ′′}, Q′_{δ′}) mapping R′_{ε′} on R′′_{ε′′} , the compatibility of the corresponding diagrams 15.1 allows to prove the commutativity of the following diagram

16.1 \begin{matrix}F̂_{b}(Q′′_{δ′′}, Q′_{δ′})_{R′′,R′} × F̂_{b}(Q′_{δ′}, Q_{δ})_{R′,R} & ⟶^{μ_{Q{′′ \atop δ′′} ,Q{′ \atop δ′} ,Q_{δ}}} & F̂_{b}(Q′′_{δ′′}, Q_{δ})_{R′′,R} \\ ρ\begin{matrix}Q{′′ \atop δ′′} ,Q{′ \atop δ′} \\ R{′′ \atop ε′′} ,R{′ \atop ε′} \end{matrix} × ρ\begin{matrix}Q{′ \atop δ′} ,Q_{δ} \\ R{′ \atop ε′} ,Rε \end{matrix} ↓ & & ↓ ρ\begin{matrix}Q{′′ \atop δ′′} ,Q_{δ} \\ R{′′ \atop ε′′} ,Rε \end{matrix} \\ F̂_{b}(R′′_{ε′′}, R′_{ε′}) × F̂_{b}(R′_{ε′}, R_{ε}) & ⟶^{μ_{R{′′ \atop ε′′} ,R{′ \atop ε′} ,Rε}} & F̂_{b}(R′′_{ε′′}, R_{ε}) \end{matrix} .

On the other hand, if T_{ν}, T′_{ν′} ∈ SCb are G-conjugated each other, if we have T_{ν} ⊂ R_{ε} and T′_{ν′} ⊂ R′_{ε′} , and if there is ȳ ∈ F_{b}(R′_{ε′}, R_{ε}) mapping T_{ν} on T′_{ν′} , considering the chains 𝔮 , 𝔮′ , 𝔯 and 𝔯′ fulfilling 16.2 Im(𝔮) = {Q_{δ}, R_{ε}, T_{ν}} , Im(𝔮′) = {Q′_{δ′}, R′_{ε′}, T′_{ν′}} , Im(𝔯) = {R_{ε}, T_{ν}} , Im(𝔯′) = {R′_{ε′}, T′_{ν′}} ,

and applying the functor to the inclusions 𝔯 ⊂ 𝔮 and 𝔯′ ⊂ 𝔮′ , we get k^{∗}-group homomorphisms

16.3  F̂_{b}(Q_{δ})_{𝔯} ⟶ F̂_{b}(R_{ε})_{T} and F̂_{b}(Q′_{δ′})_{𝔯′} ⟶ F̂_{b}(R′_{ε′})_{T′}

which, according to our hypothesis, have to coincide with the restrictions of the corresponding vertical arrows in diagram 15.1. Similarly, considering the chains 𝔱 and 𝔱′ fulfilling Im(𝔱) = {T_{ν}} and Im(𝔱′) = {T′_{ν′}}, and applying the functor to the inclusions 𝔱 ⊂ 𝔮 and 𝔱′ ⊂ 𝔮′ , we get the commutative diagrams

16.4  \begin{aligned}F̂_{b}(Q_{δ})_{𝔯} & ⊂ & F̂_{b}(Q_{δ})_{T} \\ ↓ & ↘ & ↓ \\ F̂_{b}(R_{ε})_{T} & → & F̂_{b}(T_{ν}) \end{aligned} and \begin{aligned}F̂_{b}(Q′_{δ′})_{𝔯′} & ⊂ & F̂_{b}(Q′_{δ′})_{T′} \\ ↓ & ↘ & ↓ \\ F̂_{b}(R′_{ε′})_{T′} & → & F̂_{b}(T′_{ν′}) \end{aligned} .

Finally, for any x̄ ∈ F̂_{b}(Q′_{δ′}, Q_{δ})_{R′,R} ∩ F̂_{b}(Q′_{δ′}, Q_{δ})_{T′,T} we have

16.5  ρ\begin{matrix}R{′ \atop ε′} ,Rε \\ T{′ \atop ν′} ,Tν \end{matrix} (ρ\begin{matrix}Q{′ \atop δ′} ,Q_{δ} \\ R{′ \atop ε′} ,Rε \end{matrix} (x̄)) = ρ\begin{matrix}Q{′ \atop δ′} ,Q_{δ} \\ T{′ \atop ν′} ,Tν \end{matrix} (x̄) .

17. Now, denoting by F_{b}^{sc} the full subcategory of F_{b} over SC_{b} , we are ready to define a k^{∗}-extension F̂_{b}^{sc} ; for any pair Q_{δ}, R_{ε} ∈ SC_{b} we set

17.7  F̂_{b}(Q_{δ}, R_{ε}) = ⨆_{R′_{ε′}} F̂_{b}(R′_{ε′}, R_{ε}) ,

where R′_{ε′} runs over the set of G-conjugate of R_{ε} contained in Q_{δ} , and for any T_{ν} ∈ SC_{b} we define the composition

17.2  μ_{Q_{δ},R_{ε},T_{ν}} : F̂_{b}(Q_{δ}, R_{ε}) × F̂_{b}(R_{ε}, T_{ν}) ⟶ F̂_{b}(Q_{δ}, T_{ν})

as follows: if ˆ x̄ ∈ F̂_{b}(R′_{ε′}, R_{ε}) and ˆ ȳ ∈ F̂_{b}(T′_{ν′}, T_{ν}) , where R′_{ε′} and T′_{ν′} are G-conjugated of R_{ε} and T_{ν} , and contained in Q_{δ} and R_{ε} respectively, we set

17.3  ˆ  x̄ ∘ ˆ ȳ = μ_{Q_{δ},R_{ε},T_{ν}} (ˆx̄, ˆ ȳ) = μ_{T′′_{ν′′} ,T′_{ν′} ,T_{ν}} (ρ\begin{matrix}R{′ \atop ε′} ,Rε \\ T{′′ \atop ν′′} ,T{′ \atop ν′} \end{matrix} (ˆx̄), ˆ ȳ)

where T′′_{ν′′} = xT′_{ν′}x^{−1} for some x ∈ G lifting the image x̄ of ˆ x̄ in F_{b}(R′_{ε′}, R_{ε}) ; in order to prove the associativity, let U_{ω}, U′_{ω′} ∈ SC_{b} be mutually G-conjugate such that U′_{ω′} ⊂ T_{ν} and ˆ z̄ an element of F_{b}(U′_{ω′}, U_{ω}) , and set U′′_{ω′′} = yU′_{ω′}y^{−1} for some y ∈ G lifting the image ȳ of ˆ ȳ in F_{b}(T′_{ν′}, T_{ν}) , and U′′′_{ω′′′} = xU′′_{ω′′}x^{−1} ; then, we have (cf. 16.1 and 16.5)

17.4 \begin{aligned}(ˆx̄ ∘ ˆ ȳ) ∘ ˆ z̄ & = μ_{U′′′_{ω′′′} ,U′_{ω′} ,U_{ω}}(ρ\begin{matrix}T{′′ \atop ν′′} ,Tν \\ U{′′′ \atop ω′′′} ,U{′ \atop ω′} \end{matrix} (μ_{T′′_{ν′′} ,T′_{ν′} ,T_{ν}} (ρ\begin{matrix}R{′ \atop ε′} ,Rε \\ T{′′ \atop ν′′} ,T{′ \atop ν′} \end{matrix} (ˆx̄), ˆ ȳ)), ˆ z̄) \\ & = μ_{U′′′_{ω′′′} ,U′_{ω′} ,U_{ω}}(μ_{U′′′_{ω′′′} ,U′′_{ω′′} ,U′_{ω′}} (ρ\begin{matrix}T{′′ \atop ν′′} ,T{′ \atop ν′} \\ U{′′′ \atop ω′′′} ,U{′′ \atop ω′′} \end{matrix} (ρ\begin{matrix}R{′ \atop ε′} ,Rε \\ T{′′ \atop ν′′} ,T{′ \atop ν′} \end{matrix} (ˆx̄)), ρ\begin{matrix}T{′ \atop ν′} ,Tν \\ U{′′ \atop ω′′} ,U{′ \atop ω′} \end{matrix} (ˆȳ)), ˆ z̄) \\ & = μ_{U′′′_{ω′′′} ,U′′_{ω′′} ,U_{ω}}(ρ\begin{matrix}R{′ \atop ε′} ,Rε \\ U{′′′ \atop ω′′′} ,U{′′ \atop ω′′} \end{matrix} (ˆx̄), μ_{U′′_{ω′′} ,U′_{ω′} ,U_{ω}}(ρ\begin{matrix}T{′ \atop ν′} ,Tν \\ U{′′ \atop ω′′} ,U{′ \atop ω′} \end{matrix} (ˆȳ), ˆ z̄)) \\ & = ˆ x̄ ∘ (ˆȳ ∘ ˆ z̄) . \end{aligned}

18. Before going further on the discussion of this problem, let us consider it in a more general setting. Let A be a G-interior algebra, namely a finite dimensional k-algebra endowed with a group homomorphism G → A^{∗}; let H_{β} and K_{γ} be pointed groups on A , namely H and K are subgroups of G , whereas β and γ are conjugacy classes of primitive idempotents in A^{H} and A^{K} respectively; then, let us call A-fusion from K_{γ} to H_{β} any group homomorphism φ: K → H which, for a suitable a ∈ A^{∗} and some i ∈ β and j ∈ γ , fulfills

18.1  ji^{a} = j = i^{a}j and aj·y = φ(y)·aj

for any y ∈ K (see [8] for a more comprehensive introduction and [15, §8] for a more general point of view); it is quite clear that this definition does not depend on the choice of i and j , and we denote by F_{A}(H_{β}, K_{γ}) ⊂ Hom(K, H) the set of A-fusions from K_{γ} to H_{β} .

19. Since the composition of A-fusions is an A-fusion too, we get a category F_{A} where the objects are the pointed groups on A and the morphisms are the A-fusions. We claim that, if we restrict ourselves to consider only the fusions between pointed groups of same multiplicity, the corresponding subcategory F_{A}^{=} has a canonical k^{∗}-extension; more precisely, for any pointed group H_{β} on A , denoting by 𝔐_{β} the maximal (two-sided) ideal of A^{H} which does not contain β , the quotient A(H_{β}) = A^{H}/𝔐_{β} is a matrix algebra over k of degree m_{β} — the multiplicity of β — and we have an evident group homomorphism

19.1  det_{β}^{A} : (A^{H})^{∗} ⟶ A(H_{β})^{∗} ⟶ k^{∗};

if K_{γ} is a second pointed group on A such that m_{γ} = m_{β} and φ ∈ F_{A}(H_{β}, K_{γ}) , it follows from the very definition of A-fusion that, for any maximal subsets I ⊂ β and J ⊂ γ of pairwise orthogonal idempotents, setting e = ∑_{i∈I} i and f = ∑_{j∈J} j , the set T_{A}^{φ}(H_{e}, K_{f}) of elements af where a ∈ A^{∗} fulfills

19.2  fe^{a} = f = e^{a}f and af·y = φ(y)·af ,

for any y ∈ K , is not empty and it is easily checked that (eA^{φ(K)}e)^{∗} acts transitively by left multiplication on it; moreover, the multiplication by λ ∈ k^{∗} of one element of J induces an action of k^{∗} on T_{A}^{φ}(H_{e}, K_{f}) and we get a bijection

19.3  Ker(det_{φ(γ)}^{eAe})\T_{A}^{φ}(H_{e}, K_{f}) ≅ k^{∗}

where φ(γ) denotes the point of φ(K) on eAe (and on A) containing aJa^{−1} .

20. On the other hand, it is quite clear that Ker(det_{β}^{A}) and Ker(det_{γ}^{A}) act transitively by conjugation on the sets of choices for I and J respectively, and therefore for another choice I′ ⊂ β and J′ ⊂ γ , setting e′ = ∑_{i′∈I′} i′ and f′ = ∑_{j′∈J′} j′ , we get a canonical bijection

20.1  Ker(det_{φ(γ)}^{e′Ae′})\T_{A}^{φ}(H_{e′}, K_{f′}) ≅ Ker(det_{φ(γ)}^{eAe})\T_{A}^{φ}(H_{e}, K_{f})

compatible with the action of k^{∗} . Consequently, we can set

20.2  F̂_{A}(H_{β}, K_{γ}) = ⨆_{φ∈F_{A}(H_{β},K_{γ})} Ker(det_{φ(γ)}^{eAe})\T_{A}^{φ}(H_{e}, K_{f}) ;

moreover, if L_{δ} is a third pointed group on A such that m_{δ} = m_{γ} = m_{β} , ψ ∈ F_{A}(K_{γ}, L_{δ}) and g is the sum of a maximal subset of pairwise orthogonal idempotents in δ , it is easily checked that the product in A determines a map

20.3  T_{A}^{φ}(H_{e}, K_{f}) × T_{A}^{ψ}(K_{f}, L_{g}) ⟶ T_{A}^{φ∘ψ}(H_{e}, L_{g})

which is compatible with the multiplication by k^{∗} and with the left products by the groups Ker(det_{φ(γ)}^{eAe}) , Ker(det_{ψ(δ)}^{fAf}) and Ker(det_{(φ∘ψ)(δ)}^{eAe}) ; hence, it defines a composition

20.4  F̂_{A}(H_{e}, K_{f}) × F̂_{A}(K_{f}, L_{g}) ⟶ F̂_{A}(H_{e}, L_{g}) ;

then, the associativity of this composition comes from the associativity of the product in A . Note that, if i ∈ β , from the embedding iAi → Res_{H}^{G}(A) we get a k^{∗}-group isomorphism between F̂_{A}(H_{β}) = F̂_{A}(H_{β}, H_{β}) and the set of pairs (φ, ā) , where φ ∈ F_{A}(H_{β}) , a is an element of N_{(iAi)^{∗}}(H·i) such that (φ(y)·i)^{a} = y for any y ∈ H , and ā = a(i + J(iA^{H}i)) , endowed with the evident product and multiplication by k^{∗} .

21. Let us come back to the block algebra case A = kGb; the notation is quite coherent since it follows from the main result in [8] that, for any local pointed groups Q_{δ} and R_{ε} on kGb , we have indeed

21.1  F_{kGb}(Q_{δ}, R_{ε}) = F_{b}(Q_{δ}, R_{ε}) ;

but, according to Proposition 6.12 in [10], we have a canonical k^{∗}-group isomorphism

21.2  F̂_{kGb}(Q_{δ}) ≅ F̂_{b}(Q_{δ})^{∘} .

Let P_{γ} be a defect pointed group of b , namely a maximal local pointed group on kGb; we have seen in 11 above that, for any selfcentralizing local pointed group Q_{δ} contained in P_{γ} , since N̄_{P} (Q) = N_{P} (Q)/Q stabilizes δ and acts on the simple k-algebra (kG)(Q_{δ}) stabilizing a basis, it becomes a Dade N̄_{P} (Q)-algebra (cf. 1.3 in [12]); we claim that if we find a Dade P -algebra S — that we may consider as a P -interior algebra —- such that, for such a subgroup Q , the Dade N̄_{P} (Q)-algebras S(Q) and (kG)(Q_{δ}) are similar (cf. 1.5 in [12]) and we have F_{S}(Q) ⊃ F_{kGb}(Q_{δ}) , we have solved our problem. Indeed, consider the local point γ′ of P on the P -interior algebra S^{∘} ⊗_{k} Res_{P}^{G}(kGb) determined by γ and the unique local point of P on S^{∘} (cf. Theorem 5.3 in [9]), and set

21.3  A′ = (S^{∘} ⊗_{k} Res_{P}^{G}(kGb))_{γ′} .

22. First of all, we prove that, for any selfcentralizing local pointed group Q_{δ} contained in P_{γ} , the multiplicity m_{δ′}^{A′} of the corresponding local pointed group Q_{δ′} on A′ is one; we argue by induction on |P: Q| and, according to our choice, we have m_{γ′}^{A′} = 1 . Assume that Q ≠ P ; since A′(Q_{δ′}) is a Dade N̄_{P} (Q)-algebra embedded in S^{∘}(Q) ⊗_{k} (kG)(Q_{δ}) (cf. Proposition 5.6 in [9]), it is similar to k (cf. 1.8.1 in [12]) and therefore N̄_{P} (Q) fixes a primitive idempotent in A′(Q_{δ′}) (cf. 1.4.1 in [12]), so that, by Proposition 3.3 and Example 3.9 in [5], it fixes a primitive idempotent ℓ′ in A′(Q) ; on the other hand, for any nontrivial subgroup R̄ = R/Q of N̄_{P} (Q) , it follows from the induction hypothesis that the unity element is primitive in

22.1  A′(R) ≅ (A′(Q))(R̄)

and therefore, since the primitivity of ℓ′ forces BrR̄(ℓ′) ≠ 0 , we get BrR̄(ℓ′) = 1 ; hence, the idempotent 1 − ℓ′ belongs to (cf. Proposition 1.3 in [6])

22.2  A′(Q)_{1}^{N̄_{P} (Q)} = Br_{Q}(A′_{Q}^{P})

which forces ℓ′ = 1 since A′_{Q}^{P} ⊂ J(A′^{P}) .

23. Consequently, on the one hand, our claim in 19 applies to the set SC_{A′} of local pointed groups on A′ coming from selfcentralizing local pointed groups on kGb contained in P_{γ} and therefore we get a canonical k^{∗}-extension of the full subcategory of F_{A′} over SC_{A′} . On the other hand, for mutually corresponding elements Q_{δ} ∈ SC_{b} contained in P_{γ} and Q_{δ′} ∈ SC_{A′} , since we assume that F_{S}(Q) ⊃ F_{kGb}(Q_{δ}) , we have F_{A′}(Q_{δ′}) = F_{b}(Q_{δ}) (cf. Theorem 5.3 in [9]) and, since A′(Q_{δ′}) is embedded in S^{∘}(Q) ⊗_{k} (kG)(Q_{δ}) , according to Dade’s Splitting Theorem [7], we have a (non-canonical) k^{∗}-group isomorphism (cf. Theorem 5.3 in [9])

23.1  F̂_{A′}(Q_{δ′}) ≅ F̂_{b}(Q_{δ})^{∘} .

Combining both facts, we get the k^{∗}-extension of the full subcategory of F_{b} over SC_{b} we were looking for.

24. Actually, the above arguments can be extended to all the local pointed groups Q_{δ} such that the associated block b(δ) of the centralizer C_{G}(Q) is nilpotent since, according to the main result in [9], the simple k-algebra (kG)(Q_{δ}) still becomes a Dade N̄_{P} (Q)-algebra, and in [13] we prove the existence of a suitable Dade P -algebra S whenever P is abelian — which is of no help in our problem. Yet, in the nonabelian case the existence of a Dade P -algebra S fulfilling the above conditions in full generality is far to be evident.

25. As a matter of fact, the first way to solve our problem we looked at was the possibility of finding “reasonably canonical splittings” for the k^{∗}-fusions associated with the Dade Q-algebras for any finite p-group Q; more precisely, let us denote by 𝔇_{k}(Q) the category where the objects are the Dade Q-algebras and the morphisms between two Dade Q-algebras A and B are the embeddings from A to B , namely the (B^{Q})^{∗}-conjugacy classes of Q-algebra homomorphisms f̃: A → B such that A ≅ f(1)Bf(1) ; then, we have a functor 𝔣̂_{Q} from 𝔇_{k}(Q) to the category of k^{∗}-groups k^{∗}-𝔊𝔯 mapping any Dade Q-algebra A on F̂_{A}(Q_{π^{A}}) , where π^{A} denotes the unique local point of Q on A (cf. Corollary 5.8 in [9]), and any embedding f̃: A → B between two Dade Q-algebras A and B on the canonical isomorphism F̂f̃(Q) : F̂_{A}(Q_{π^{A}}) ≅ F̂_{B}(Q_{π^{B}}) , (cf. Proposition 6.18 in [10]) and we are interested in the natural maps ω_{Q} — called Q-polarizations — from this functor to the trivial one mapping A on k^{∗} and f̃ on the identity map on k^{∗} .

26. Whereas it is not difficult to prove the existence of a Q-polarization for any finite p-group Q , compatible with all the surjective group homomorphisms — called a polarization — it is impossible to get a polarization compatible with the tensor product, namely fulfilling the following condition

26.1 If A and B are Dade Q-algebras, the following diagram is commutative

F̂_{A}(Q_{π^{A}})ˆ∩F̂_{B}(Q_{π^{B}}) ⟶ F̂_{A⊗_{k}B}(Q_{π^{A⊗_{k}B}})
ω_{Q}(A)⊗ω_{Q}(B) ↘  ↙ ω_{Q}(A⊗_{k}B)
k^{∗}

where F̂_{A}(Q_{π^{A}})ˆ∩F̂_{B}(Q_{π^{B}}) denotes the restriction to the intersection F_{A}(Q_{π^{A}})∩F_{B}(Q_{π^{B}}) of the direct product of k^{∗}-groups F̂_{A}(Q_{π^{A}}) ⊗ F̂_{B}(Q_{π^{B}}) (cf. Proposition 5.11 in [9]).

27. For instance, assume that |Q| = p and A = End_{k}(J(kQ)); then, it is quite clear that F_{A}(Q_{π^{A}}) = Aut(Q) ≅ ℤ/(p − 1)ℤ and it is well-known that there is an embedding from k to A ⊗_{k} A; explicitly, choosing a generator u of Q and setting

27.1  M = J(kQ) ⊗_{k} J(kQ) and u_{n,m} = (1 − u)^{n} ⊗ (1 − u)^{m}

it is easily checked that {u_{n,m}}_{1≤n,m≤p−1} is a basis of M , that we have

27.2  u·u_{n,m} = u_{n,m} − u_{n+1,m} − u_{n,m+1} + u_{n+1,m+1}

and that Q fixes v = ∑_{n,m∈ℕ−\{0\}}(−1)^{m}({n \atop p−m}) u_{n,m} . Moreover, Q stabilizes the k-linear form φ: M → k mapping u_{n,m} on (−1)^{n}({p−m \atop n}) and we have φ(v) = 1 , so that kv is a direct summand of M as kQ-modules. On the other hand, Aut(Q) acts naturally on M and denoting by λ_{τ} ∈ k^{∗} the element of the prime subfield fulfilling τ(u) = u^{λ_{τ}} , it is not difficult to check that Q still fixes w = ∑_{τ∈Aut(Q)}(λ_{τ})^{−1}τ(v) , that kw is also a direct summand of M as kQ-modules and that τ(w) = λ_{τ} w for any τ ∈ Aut(Q) . Now, a modification on the choice of the splitting of F̂_{A}(Q_{π^{A}}) via a group homomorphism ρ: Aut(Q) → k^{∗} induces a new action of Aut(Q) on M where τ maps w on λ_{τ} ρ(τ)^{2}w , which is never the identity map.

28. For our problem, the point is that, restricting ourselves to the full subcategory 2𝔇_{k}(Q) of 𝔇_{k}(Q) over the Dade Q-algebras which can be embedded in A ⊗_{k} A for some Dade Q-algebra A , there is a natural map ω_{Q} from the restriction to 2𝔇_{k}(Q) of the functor 𝔣̂_{Q} determined by the associated fusions to the corresponding trivial functor — called Q-semipolarization — compatible with the tensor product; again, there is such a Q-semipolarization for any finite p-group Q , compatible with all the surjective group homomorphisms — called a semipolarization — and still compatible with the transitivity of the Brauer sections in the following sense. If A is a Dade Q-algebra and R a normal subgroup of Q , it is clear that we have the Dade R-algebra B = Res_{R}^{Q}(A) and the Dade Q/R-algebra A(R) (cf. 1.8.1 in [12]), that we simply consider as Q-algebra, and it is wellknown that, considering all the Dade Q-algebras as Q-interior algebras and denoting by F_{A}(Q_{π^{A}})_{R} the stabilizer of R in F_{A}(Q_{π^{A}}) , we have natural group homomorphisms

28.1  F_{B}(R_{π^{B}}) ⟵ F_{A}(Q_{π^{A}})_{R} ⟶ F_{A(R)}(Q_{π^{A(R)}}) .

29. More explicitly, if a group automorphism φ: Q → Q is an A-fusion from Q_{π^{A}} to Q_{π^{A}} such that φ(R) = R and, for some i ∈ π^{A} , an element a ∈ A^{∗} fulfills i^{a} = i and ai·u = φ(u)·ai for any u ∈ Q (cf. 18.1), it is quite clear that ai ∈ (iAi)^{∗} normalizes (iAi)^{R} and acts on (iAi)(R) , so that this action can be lifted to an element of (iAi)(R)^{∗} . That is to say, there is b ∈ (A^{R})^{∗} fulfilling i^{b} = i and such that ab^{−1}i acts trivially on (iAi)(R) ; in particular, choosing j ∈ π^{B} such that ji = j = ij , we may assume that j^{ab^{−1}} = j and then, for any v ∈ R , we have

29.1  ab^{−1}j·v = φ(v)·ab^{−1}j

which proves that the restriction ψ of φ to R is a B-fusion from R_{π^{B}} to R_{π^{B}} . Similarly, for any u ∈ Q , denoting by Br_{R}^{∗}(u·i) the canonical lifting to (iAi)(R)^{∗} of u·i , we get

29.2  Br_{R}^{∗}(u·i) = Br_{R}^{∗}(φ(u)·i)^{ai} = Br_{R}^{∗}(φ(u)·i)^{bi} = Br_{R}(b^{−1}i)Br_{R}^{∗}(φ(u)·i)Br_{R}(bi)

which proves that φ is an A(R)-fusion from Q_{π^{A(R)}} to Q_{π^{A(R)}} since Br_{R}(A^{Q}) = A(R)^{Q} and, in particular, Br_{R}(i) belongs to π^{A(R)} .

30. But we get more than the natural group homomorphisms 28.1; indeed, setting \overline{ai}^{A} = a(i+J(iA^{Q}i)) and similarly considering \overline{ab^{−1}j}^{B} and \overline{Br_{R}(bi)}^{A(R)} , it follows from 20

that the pairs (φ, \overline{ai}^{A}) , (ψ, \overline{ab^{−1}j}^{B}) and (φ, \overline{Br_{R}(bi)}^{A(R)}) respectively belong to F̂_{A}(Q_{π^{A}}) , to F̂_{B}(R_{π^{B}}) and to F̂_{A(R)}(Q_{π^{A(R)}}) ; moreover, it is not difficult to check that a modification

in our choices of a and b just multiplies \overline{ai}^{A} and \overline{Br_{R}(bi)}^{A(R)} by elements λ and μ of k^{∗} respectively, whereas \overline{ab^{−1}j}^{B} becomes multiplied by λμ^{−1} ; that is to say, we get a k^{∗}-group homomorphism

30.1  F̂_{A}(Q_{π^{A}})_{R} ⟶ F̂_{B}(R_{π^{B}}) ⊗ F̂_{A(R)}(Q_{π^{A(R)}})

mapping (φ, \overline{ai}^{A}) on (ψ, \overline{ab^{−1}j}^{B})⊗(φ, \overline{Br_{R}(bi)}^{A(R)}) . Now, we claim that there is a semipolarization ω which is compatible with the tensor product and the surjective group homomorphisms, and fulfills the following condition 30.2 If Q is a finite p-group, A is an object in 2𝔇_{k}(Q) and R a normal subgroup of Q , setting B = Res_{R}^{Q}(A) , the following diagram is commutative

\begin{aligned}F̂_{A}(Q_{π^{A}})_{R} & ⟶ & F̂_{B}(R_{π^{B}}) ⊗ F̂_{A(R)}(Q_{π^{A(R)}}) \\ ω_{Q}(A) ↘ & & ↙ ω_{R}(B)⊗ω_{Q}(A(R)) \\ & k^{∗} \end{aligned} .

31. First of all, we need the coassociativity of the kind of coproduct defined by homomorphism 30.1; explicitly, let Q be a finite p-group, R and T normal subgroups of Q such that T ⊂ R , and A a Dade Q-algebra; setting B = Res_{R}^{Q}(A) and C = Res_{T}^{Q}(A) , considering A(R) and A(T) as Q-algebras, and denoting by F̂_{A}(Q_{π^{A}})_{R,T} the stabilizer of R and T in F̂_{A}(Q_{π^{A}}) , from 30.1 we get the k^{∗}-group homomorphisms

31.1  \begin{matrix}F̂_{A}(Q_{π^{A}})_{R,T} & ⟶ F̂_{B}(R_{π^{B}})_{T} ⊗ F̂_{A(R)}(Q_{π^{A(R)}}) \\ F̂_{A}(Q_{π^{A}})_{R,T} & ⟶ F̂_{C}(T_{π^{C}}) ⊗ F̂_{A(T)}(Q_{π^{A(T)}})_{R} \\ F̂_{B}(R_{π^{B}})_{T} & ⟶ F̂_{C}(T_{π^{C}}) ⊗ F̂_{B(T)}(R_{π^{B(T)}}) \\ F̂_{A(T)}(Q_{π^{A(T)}})_{R} & ⟶ F̂_{B(T)}(R_{π^{B(T)}}) ⊗ F̂_{A(R)}(Q_{π^{A(R)}}) \end{matrix}

where we have identified (A(T))(R) with A(R) ; then, we claim that the following diagram is commutative

31.2 \begin{matrix}F̂_{B}(R_{π^{B}})_{T} ⊗ F̂_{A(R)}(Q_{π^{A(R)}}) \\ α ↗  ↘ γ \\ F̂_{A}(Q_{π^{A}})_{R,T} & F̂_{C}(T_{π^{C}}) ⊗ F̂_{B(T)}(R_{π^{B(T)}}) ⊗ F̂_{A(R)}(Q_{π^{A(R)}}) \\ β ↘  ↗ δ \\ F̂_{C}(T_{π^{C}}) ⊗ F̂_{A(T)}(Q_{π^{A(T)}})_{R} \end{matrix} .

32. Indeed, keeping the notation above, assume that φ(T) = T and denote by η the restriction of φ to T ; then, (φ, \overline{ai}^{A}) and (ψ, \overline{ab^{−1}j}^{B}) respectively belong to F̂_{A}(Q_{π^{A}})_{R,T} and to F̂_{B}(R_{π^{B}})_{T} , and we already know that

32.1  α(φ, \overline{ai}^{A}) = (ψ, \overline{ab^{−1}j}^{B}) ⊗ (φ, \overline{Br_{R}(bi)}^{A(R)}) .

Moreover, it is quite clear that ab^{−1}i normalizes (iAi)^{T} and acts on (iAi)(T) , so that there is c ∈ (A^{T})^{∗} fulfilling i^{c} = i and such that ab^{−1}c^{−1}i acts trivially on (iAi)(T) ; in particular, choosing ℓ ∈ π^{C} such that ℓj = ℓ = jℓ , we may assume that j^{c^{−1}} = j and ℓ^{ab^{−1}c^{−1}} = ℓ , and then, for any w ∈ T , we have

32.2  ab^{−1}c^{−1}ℓ·w = φ(w)·ab^{−1}c^{−1}ℓ

so that (η, \overline{ab^{−1}c^{−1}ℓ}^{C}) belongs to F̂_{C}(T_{π^{C}}) . As above, for any u ∈ Q , denoting by Br_{T}^{∗} (u·i) the canonical lifting to (iAi)(T)^{∗} of u·i , we get

32.3 Br_{T}^{∗} (u·i) = Br_{T}^{∗} (φ(u)·i)^{ai} = Br_{T}^{∗} (φ(u)·i)^{cbi} = Br_{T} (cb^{−1}i)Br_{T}^{∗} (φ(u)·i)Br_{T} (cbi)

and therefore (φ, \overline{Br_{T} (cbi)}^{A(T)}) belongs to F̂_{A(T)}(Q_{π^{A(T)}})_{R} . Consequently, we have

32.4  β(φ, \overline{ai}^{A}) = (η, \overline{ab^{−1}c^{−1}ℓ}^{C}) ⊗ (φ, \overline{Br_{T} (cbi)}^{A(T)})

On the other hand, since ab^{−1} and c^{−1} fixe j , and ab^{−1}c^{−1}j acts trivially on (jBj)(T) , we still have

32.5  γ(α(φ, \overline{ai}^{A})) = (η, \overline{ab^{−1}c^{−1}ℓ}^{C}) ⊗ (ψ, \overline{Br_{T} (cj)}^{B(T)}) ⊗ (φ, \overline{Br_{R}(bi)}^{A(R)})

Finally, we know that Br_{T} (i) belongs to π^{A(T)} , that Br_{T} (cbi) normalizes (iAi)(T)^{R} , that Br_{T} (b) belongs to (A(T)^{R})^{∗} and that Br_{T} (ci) acts trivially on ((iAi)(T))(R) ≅ (iAi)(R) ; hence, we still get

32.6  δ(β(φ, \overline{ai}^{A})) = (η, \overline{ab^{−1}c^{−1}ℓ}^{C}) ⊗ (ψ, \overline{Br_{T} (cj)}^{B(T)}) ⊗ (φ, \overline{Br_{R}(bi)}^{A(R)}) .

33. We need also the distributivity with respect to the tensor product; that is to say, with the notation above, let A′ be another Dade Q-algebra and set B′ = Res_{R}^{Q}(A′) ; then, we claim that the following diagram is commutative 33.1

F̂_{A⊗_{k}A′}(Q_{π^{A⊗_{k}A′}})_{R} →  F̂_{B⊗_{k}B′}(R_{π^{B}}) ⊗ F̂_{(A⊗_{k}A′)(R)}(Q_{π^{(A⊗_{k}A′)(R)}})
↑  ↑
F̂_{A}(Q_{π^{A}})_{R} ˆ∩F̂_{A′}(Q_{π^{A′}})_{R} → (F̂_{B}(R_{π^{B}})ˆ∩F̂_{B′}(R_{π^{B′}}))⊗(F̂_{A(R)}(Q_{π^{A(R)}})ˆ∩F̂_{A′(R)}(Q_{π^{A′(R)}}))

indeed, we already consider the pairs (φ, \overline{ai}^{A}) , (ψ, \overline{ab^{−1}j}^{B}) and (φ, \overline{Br_{R}(bi)}^{A(R)}) respectively belonging to F̂_{A}(Q_{π^{A}}) , to F̂_{B}(R_{π^{B}}) and to F̂_{A(R)}(Q_{π^{A(R)}}) ; mutatis mutandi, for some i′ ∈ π^{A′} , we get an element a′ ∈ A^{′∗} fulfilling i^{′a′} = i′ and a′i′·u = φ(u)·a′i′ for any u ∈ Q (cf. 18.1), and an element b′ ∈ (A^{′R})^{∗} fulfilling i^{′b′} = i′ and such that a′b^{′−1}i′ acts trivially on (i′A′i′)(R) ; in particular, choosing j′ ∈ π^{B′} such that j′i′ = j′ = i′j′ , we

may assume that j^{′a′b^{′−1}} = j′; then, we know that the pairs (φ, \overline{a′i′}^{A′}) , (ψ, \overline{a′b^{′−1}j′}^{B′}) and

(φ, \overline{Br_{R}(b′i′)}^{A′(R)}) respectively belong to F̂_{A′}(Q_{π^{A′}}) , to F̂_{B′}(R_{π^{B′}}) and to F̂_{A′(R)}(Q_{π^{A′(R)}}) ,

and the bottom arrow maps (φ, \overline{ai}^{A}) ⊗ (φ, \overline{a′i′}^{A′}) on the element

33.2  ((ψ, \overline{ab^{−1}j}^{B}) ⊗ (φ, \overline{Br_{R}(bi)}^{A(R)})) ⊗ ((ψ, \overline{a′b^{′−1}j′}^{B′}) ⊗ (φ, \overline{Br_{R}(b′i′)}^{A′(R)})) .

34. On the other hand, choosing i′′ ∈ π^{A⊗A′} fulfilling (i ⊗ i′)i′′ = i′′ = i′′(i ⊗ i′) , we have Br_{Q}^{A⊗A′}(i′′) = Br_{Q}^{A}(i) ⊗ Br_{Q}^{A′}(i′) and therefore there is a′′ ∈ 1_{A⊗A′} + Ker(Br_{Q}^{A⊗A′}) such that i^{′′(a⊗a′)a′′} = i′′; then, it is easily checked that the left vertical arrow maps the element (φ, \overline{ai}^{A}) ⊗ (φ, \overline{a′i′}^{A′}) on

34.1  (φ, \overline{(a ⊗ a′)a′′i′′}^{A⊗A′}) = (φ, \overline{ai ⊗ a′i′}^{A⊗A′}) .

Similarly, ab^{−1}i⊗a′b^{′−1}i′ acts trivially on (iAi⊗_{k} i′A′i′)(R) , so on (i′′(A⊗A′)i′′)(R) , and, choosing j′′ ∈ π^{B⊗B′} fulfilling i′′j′′ = j′′ = j′′i′′ , we may find b′′ ∈ 1_{A⊗A′} + Ker(Br_{R}^{A⊗A′})

such that j^{′′(ab^{−1}⊗a′b^{′−1})b′′} = j′′; thus, the top arrow maps (φ, \overline{ai ⊗ a′i′}^{A⊗A′}) on the element

34.2  (ψ, \overline{(ab^{−1} ⊗ a′b^{′−1})b′′j′′}^{B⊗B′}) ⊗ (φ, \overline{Br_{R}(b^{′′−1}(b ⊗ b′)i′′)}^{(A⊗A′)(R)})

and it is quite clear that (φ, \overline{Br_{R}(b^{′′−1}(b ⊗ b′)i′′)}^{(A⊗A′)(R)}) = (φ, \overline{Br_{R}(bi ⊗ b′i′)}^{(A⊗A′)(R)}) ; moreover, since Br_{R}^{B}(j)⊗Br_{R}^{B′}(j′) and Br_{R}^{B⊗B′}(j′′) are mutually conjugated in (B⊗B′)(R) , up to the indentification in 20.1, we get

34.3  (ψ, \overline{(ab^{−1} ⊗ a′b^{′−1})b′′j′′}^{B⊗B′}) = (ψ, \overline{ab^{−1}j ⊗ a′b^{′−1}j′}^{B⊗B′}) .

Now, it is easily checked that, in diagram 33.1, the right vertical arrow maps the element 33.2 on the element 34.2.

35. In order to define the announced semipolarization ω , we start by discussing the p-elementary abelian case. Assume that Q is p-elementary abelian; by Dade’s Classification [4], we know that any object A in 2𝔇_{k}(Q) can be embedded in a tensor product of End_{k}(M) , where M is a kQ-module with a Q-stable basis with some fixe element, and the tensor product of a unique family of the Dade Q-algebras AQ̄ = End_{k}(MQ̄ ⊗_{k} MQ̄) or (AQ̄)^{∘} , where Q̄ is a quotient of Q of order at least p^{2} and MQ̄ is the kernel of the augmentation map k(Q̄ ⋊ Aut(Q̄)) → k; in particular, if i is a primitive idempotent in A^{Q} such that Br_{Q}(i) ≠ 0 then we have F̂_{iAi}(Q_{{i}}) ≅ F̂_{A}(Q_{π^{A}}) and iAi can be embedded in the tensor product of this unique family; note that for the empty family we have iAi ≅ k and an obvious scission ω_{Q}(k) .

36. For any subgroup T of Q such that |Q: T | ≥ p^{2} , setting Q̄ = Q/T and denoting by Aut(Q)_{T} the stabilizer of T in Aut(Q) , it quite clear that (cf. Proposition 2.18 in [8])

36.1  F_{AQ̄}(Q_{π^{AQ̄}}) = Aut(Q)_{T}

and therefore the subgroup Aut(Q̄)·id_{MQ̄⊗_{k}MQ̄} of N_{(AQ̄)^{∗}}(Q·id_{MQ̄⊗_{k}MQ̄}) determines a scission

36.2  ω_{Q}(AQ̄) : F̂_{AQ̄}(Q_{π^{AQ̄}}) ⟶ k^{∗} .

We claim that these scissions fulfill condition 30.2; indeed, it is clear that we may choose i ∈ π^{AQ̄} fixed by Aut(Q̄)·id_{MQ̄⊗_{k}MQ̄} ; if R is a subgroup of Q not contained in T , we

have AQ̄(R) ≅ k and therefore, if (φ, \overline{ai}^{AQ̄}) is an element of F̂_{AQ̄}(Q_{π^{AQ̄}})_{R} such that a ∈ Aut(Q̄)·id_{MQ̄⊗_{k}MQ̄} then a acts trivially on AQ̄(R) ; moreover, denoting by R̄ the image of R in Q̄ , it is quite clear that in MQ̄ we may find a direct summand isomorphic to MR̄ via the inclusion of Aut(R̄) in Aut(Q̄) obtained from the choice of a complement of R̄ in Q̄; hence, up a modification of a , we may choose j fixed by a , we may assume that aj ∈ Aut(R̄)·j and, with the notation above, we may choose b = 1_{AQ̄} . Consequently,

(φ, \overline{ai}^{AQ̄}) maps on (ψ, \overline{aj}^{BQ̄}) ⊗ (φ, \overline{Br_{R}(i)}^{AQ̄(R)}) , where BQ̄ = Res_{R}^{Q}(AQ̄) , and the pairs

(φ, \overline{ai}^{AQ̄}) , (ψ, \overline{aj}^{BQ̄}) and (φ, \overline{Br_{R}(i)}^{AQ̄(R)}) belong to the kernels of ω_{Q}(AQ̄) , ω_{R}(AR̄) and ω_{Q}(k) respectively. Finally, if R is contained in T then Res_{R}^{Q}(MQ̄) is a trivial kR-module and we have AQ̄(R) = AQ̄; now, the verification is straightforward.

37. In conclusion, if A is an object in 2𝔇_{k}(Q) and i ∈ π^{A} , it follows from Dade’s Classification that there is a unique family {z_{T} }_{T} of rational integers such that iAi can be embedded in the tensor product

37.1  B = ⨂_{T}(A_{Q/T})^{z_{T}}

where T runs over the set of subgroups of Q of index at least p^{2} and, as usual, the 0-power is k whereas the negative powers are the corresponding positive powers of the opposite Q-algebras; it follows from Proposition 2.18 in [8] and from this uniqueness that F_{B}(Q_{π^{B}}) coincides with the stabilizer in Aut(Q) of all the T such that z_{T} ≠ 0 ; consequently we get

37.2  F̂_{A}(Q_{π^{A}}) ≅ ˆ ⋂_{T} (F̂_{A_{Q/T}} (Q_{π^{A_{Q/T}}}))^{ˆ∩z_{T}}

where T runs over the set of subgroups of Q of index at least p^{2} such that z_{T} ≠ 0 and, for any n ∈ ℕ − {0} and any k^{∗}-group Ĥ we set Ĥ^{ˆ∩0} = k^{∗} × H and inductively define

37.3 .  Ĥ^{ˆ∩n} = Ĥ ˆ∩Ĥ^{ˆ∩(n−1)} and Ĥ^{ˆ∩−n} = Ĥ^{∘} ˆ∩(Ĥ^{∘})^{ˆ∩(n−1)}

Then, we can define ω_{Q}(A): F̂_{A}(Q_{π^{A}}) → k^{∗} from condition 26.1 and from the family {ω_{Q}(A_{Q/T})}_{T} where T runs over the set of subgroups of Q of index at least p^{2} ; moreover, the uniqueness of decomposition 37.2 guaranties that this Q-semipolarization fulfills the corresponding condition 26.1. Finally, since all these Q-semipolarizations ω_{Q}(A_{Q/T}) are compatible with surjective group homomorphisms and fulfill condition 30.2, it follows from the commutativity of diagram 33.1 that this correspondence ω fulfills condition 30.2 too.

38. In order to extend this correspondence ω to all the finite p-groups Q , we proceed by induction on |Q|; we already may assume that Q is not p-elementary abelian and we set Z = Ω_{1}(Z(Q))∩Φ(Q) and Q̄ = Q/Z; for any object A of 2𝔇_{k}(Q) , it is clear that F_{A}(Q_{π^{A}}) stabilizes Z and acts on Q̄ and therefore, since we may assume that the Q̄-semipolarization ωQ̄ is already defined, we may define ω_{Q}(A) by the composition

38.1  F̂_{A}(Q_{π^{A}}) ⟶ F̂_{C}(Z_{π^{C}}) ⊗ F̂_{A(Z)}(Q_{π^{A(Z)}}) ; ↓ ; F̂_{C}(Z_{π^{C}}) ⊗ F̂_{A(Z)}(Q̄_{π^{A(Z)}}) ⟶^{ω_{Z}(C)⊗ωQ̄(A(Z))} k^{∗} .

where C = Res_{Z}^{Q}(A) and in the bottom A(Z) is considered as a Dade Q̄-algebra. Then, the compatibility with the tensor product — the corresponding condition 26.1 — follows easily from the commutativity of diagram 33.1 and the induction hypothesis.

39. It remains to prove that ω fulfills condition 30.2; let R be a proper normal subgroup of Q; with the notation above, if Z ⊂ R then the commutative diagram 31.2 implies that the following diagram is still commutative

39.1 \begin{matrix}F̂_{B}(R_{π^{B}})_{Z} ⊗ F̂_{A(R)}(Q_{π^{A(R)}}) \\ α ↗  ↘ γ \\ F̂_{A}(Q_{π^{A}})_{R} & F̂_{C}(Z_{π^{C}}) ⊗ F̂_{B(Z)}(R̄_{π^{B(Z)}}) ⊗ F̂_{A(R)}(Q̄_{π^{A(R)}}) \\ β ↘  ↗ δ \\ F̂_{C}(Z_{π^{C}}) ⊗ F̂_{A(Z)}(Q̄_{π^{A(T)}})R̄ \end{matrix} .

where B = Res_{R}^{Q}(A) and R̄ = Q/R; according to the induction hypothesis we have

39.2  \begin{aligned}ω_{Q}(A) & = (ω_{Z}(C) ⊗ ωQ̄(A(Z))) ∘ β \\ & = (ω_{Z}(C) ⊗ ωR̄(B(Z)) ⊗ ωQ̄(A(R))) ∘ δ ∘ β \\ & = (ω_{Z}(C) ⊗ ωR̄(B(Z)) ⊗ ωQ̄(A(R))) ∘ γ ∘ α \\ & = (ω_{R}(B) ⊗ ωQ̄(A(R))) ∘ α \end{aligned}

If R ⊄ Z then, since we already have the conclusion for the proper subgroup R·Z , it suffices to apply the commutative diagram 31.2 to the chain R ⊂ R·Z and to develop a similar argument.

40. Let us come back to our setting in 21; having obtained a semipolarization ω compatible with the tensor product, with the surjective group homomorphisms and with the transitivity of the Brauer sections, we may weaken our condition on the Dade P -algebra S there, namely it would suffice that, for any selfcentralizing local pointed group Q_{δ} of b , the Dade N̄_{P} (Q)-algebra S(Q)^{∘} ⊗_{k} (kG)(Q_{δ}) was an object of 2𝔇_{k}(Q) . Moreover, since the k^{∗}-extension F̂_{kG}(Q_{δ}) of F_{kG}(Q_{δ}) comes from a finite subgroup of k^{∗} , there is n ∈ ℕ such that

40.1  (F̂_{kG}(Q_{δ}))^{ˆ∩p^{n}} ≅ F̂_{kG}(Q_{δ}) ;

note that, if p = 2 , combining both statements we are done. For p ≠ 2 , working in the Dade group D_{k}(P) of P (cf. 1.5 in [12]), this remark allows us to replace D_{k}(P) by D_{k}(P)_{p} = ℤ_{{p^{n}}_{n∈ℕ}} ⊗_{ℤ}D_{k}(P) which has the advantage of admitting the following canonical decomposition (cf. Proposition 3 in [16])

40.2  ( ⨁_{Q} R^{∘}D_{k}(Q)_{p})^{P} ≅ D_{k}(P)_{p} ≅ ( ∏_{Q} RD_{k}(Q)_{p})^{P}

where Q runs over the set of subgroups of P , and we set

40.3  R^{∘}D_{k}(Q)_{p} = ⋂_{Q} Ker(res_{R}^{Q}) and RD_{k}(Q)_{p} = D_{k}(Q)_{p}/ ⋂_{R} Im(ten_{R}^{Q}) ,

where R runs over the set of proper subgroups of Q and

40.4  res_{R}^{Q}: D_{k}(Q)_{p} → D_{k}(R)_{p} and ten_{R}^{Q} : D_{k}(R)_{p} → D_{k}(Q)_{p}

respectively denote the group homomorphisms induced by the restriction and the tensor induction. In this framework, for any normal subgroup N of Q , setting Q̄ = Q/N , the Brauer sections induce a group homomorphism (cf. 1.8.2 in [12])

40.5  st_{N,Q} : D_{k}(Q)_{p} ⟶ D_{k}(Q̄)_{p}

mapping the class s of any Dade Q-algebra S on the class of the Dade Q̄-algebra S(N) .

41. As we said above, the existence of the semipolarization ω allows us to work in

41.1  D̃_{k}(P)_{p} = D_{k}(P)_{p}/2D_{k}(P)_{p} .

For any subgroup Q of P , we know that there is a unique Brauer b-pair (Q, f) such that (Q, f) ⊂ (P, b(γ)) (cf. [2]) and we simply denote by F_{b}(P, Q) the set of group homomorphisms φ: Q → P induced by some x ∈ G such that (Q, f) ⊂ (P, b(γ))^{x} . For inductive purposes, we consider a nonempty set 𝔛 of subgroups of P such that any subgroup R of P containing φ(Q) , for some Q ∈ 𝔛 and some φ ∈ F_{b}(P, Q) , belongs to 𝔛 , and we assume that, for any Q ∈ 𝔛 , we have an element s(Q) of D_{k}(N̄_{P} (Q))_{p} in such a way that the following conditions are fulfilled (cf. Theorem 14 in [16]) 41.2 For any Q ∈ 𝔛 and any ζ ∈ F_{b}(P, N_{P} (Q)) , the isomorphism between D_{k}(N_{P} (Q))_{p}

and D_{k}(ζ(N_{P} (Q)))_{p} induced by ζ maps s(Q) on res_{ζ̄(N̄_{P} (Q))}^{N̄_{P} (ζ(Q))}(s(ζ(Q))) .

41.3 For any triple N, Q, R ∈ 𝔛 fulfilling R ⊂ Q ⊂ N ⊂ N_{P} (Q) ∩ N_{P} (R) , we have

s̃t_{Q̄,N̄}(r̃esN̄_{P} (R) ; N̄(s̃(R))) = r̃esN̄_{P} (Q) ; ¯N̄(s̃(Q))

where Q̄ , N̄ and ¯ N̄ are the images of Q and N in N̄_{P} (R) and N̄_{P} (Q) respectively.

42. It is clear that the set of subgroups Q of P such that the corresponding Brauer b-pair (Q, f) is selfcentralizing, with s(Q) equal to the class of the Dade N̄_{P} (Q)-alge-bra (kG)(Q_{δ(f)}) fulfills all the conditions above. Assume that 𝔛 contains this set and that the correspondence mapping Q ∈ 𝔛 on s(Q) coincides with this one over it; if 𝔛 does not contain all the nontrivial subgroups of P , let R be a nontrivial subgroup of P maximal such that R ∉ 𝔛; we may choose R in such a way that, if (R, g) ⊂ (P, b(γ)) , N_{P} (R) is a defect group of the block g of N_{G}(R, g) . Setting N̄ = N̄_{P} (R) and s(Q̄) = res_{N̄N̄ (Q̄)}^{N̄_{P} (Q)}(s(Q)) for any nontrivial subgroup Q̄ = Q/R of N̄ , it is clear that the new system inherites the corresponding condition 41.3, and therefore it follows from Corollary 15 in [16] that there exists a unique s̄ ∈ D_{k}(N̄)_{p} fulfilling conditions 15.2, 15.3 and 15.4 in [16] with respect to some element of ℤ/2ℤ .

43. Now, we claim that, for any φ ∈ F_{b}(P, R) and any pair ζ, ζ′ ∈ F_{b}(P, N_{P} (R′)) such that ζ(R′) = R = ζ′(R′) , where R′ = φ(R) , we have resζ̄(s̄) = resζ̄′(s̄) , where ζ̄ and ζ̄′ are the group homomorphisms from N̄′ = N̄_{P} (R′) to N̄ respectively determined by ζ and ζ′ . Indeed, setting M = ζ(N_{P} (R′)) and M′ = ζ′(N_{P} (R′)) , we have ζ′ = σ ∘ ζ for a group isomorphism σ: M ≅ M′ induced by conjugation by some element of N_{G}(R, g) mapping (M, h) ⊂ (P, b(γ)) on (M′, h′) ⊂ (P, b(γ)) , and our claim amounts to say that resN̄ ; M̄(s̄) = resσ̄(resN̄ ; M̄′(s̄)); thus, according to Alperin’s Fusion Theorem for Brauer pairs (cf. §48 in [17]), it suffices to prove that, for any Brauer b-pair (R, g) ◁ (T, ℓ) ⊂ (P, b(γ)) , N_{P} (T, ℓ) ∩ N_{G}(R, g) fixes resN̄ ; T̄(s̄) ; but, it follows from Corollary 15 in [16] that resN̄ ; T̄(s̄) is the unique element of D_{k}(T̄)_{p} fulfilling conditions 15.2, 15.3 and 15.4 in [16] with respect to the restricted system and to some element of ℤ/2ℤ; hence, since N_{P} (T, ℓ) ∩ N_{G}(R, g) stabilizes the restricted system and acts trivially on R^{∘}D_{k}(Ω_{1}(Z(T̄)))_{p} , it fixes resN̄ ; T̄(s̄)

indeed.

44. Consequently, the set 𝔛∪{φ(R)}_{φ∈F_{b}(P,R)} and the correspondence mapping Q ∈ 𝔛

on s(Q) , and φ(R) on resζ̄(s̄) , where ζ ∈ F_{b}(P, N_{P} (φ(R))) fulfills ζ(φ(R)) = R and ζ̄ is

the group homomorphism from N̄_{P} (φ(R)) to N̄ , still fulfill conditions 41.2 and 41.3 above; in particular, we may assume that 𝔛 coincides with the set of all the nontrivial subgroups of P . In this case, once again from Corollary 15 in [16], there exists a unique s ∈ D_{k}(P)_{p} fulfilling conditions 15.2, 15.3 and 15.4 in [16] with respect to some element of ℤ/2ℤ , and moreover, for any nontrivial subgroup P′ of P , res_{P′}^{P}(s) is the unique element of D_{k}(P′)_{p} fulfilling conditions 15.2, 15.3 and 15.4 in [16] with respect to the restricted system and to some element of ℤ/2ℤ; hence, once again according to Alperin’s Fusion Theorem for Brauer pairs (cf. §48 in [17]), for any subgroup Q of P and any φ ∈ F_{b}(P, Q) , we get

44.1  res_{Q}^{P}(s) = res_{φ}(s) .

45. Finally, setting s = 0 whenever p = 2 , in all the cases we choose n ∈ ℕ such that, on the one hand p^{n}s belongs to D_{k}(P) , and on the other hand we have (cf. 40.1)

45.1  (F̂_{kG}(Q_{δ}))^{ˆ∩p^{n}} ≅ F̂_{kG}(Q_{δ})

for any selfcentralizing local pointed group Q_{δ} of b; then, p^{n}s is the class of some Dade P -algebra S and it follows from equalities 44.1 that (cf. Proposition 2.18 in [8])

45.2  F_{b}(P, Q) ⊂ F_{S}(P_{π^{S}}, Q_{π^{T}})

where T = res_{Q}^{P}(S) . In conclusion, considering the P -interior algebra

45.3  A′ = (S^{∘} ⊗_{k} ( ⊗^{p^{n}} Res_{P}^{G}(kGb)))_{γ′}

where γ′ is the local point of P on S^{∘} ⊗_{k} ( ⊗^{p^{n}} Res_{P}^{G}(kGb)) determined by γ (chosen in 21) and the unique local point of P on S (cf. Proposition 5.6 in [9]), for any selfcentralizing local pointed group Q_{δ} of b , there is a unique local point δ′ of Q on A′ and we have (cf. Theorem 3.1 in [8], Theorem 5.3 in [9] and (e) in [7])

45.4  F_{b}(P, Q) ⊂ F_{A′}(P_{γ′}, Q_{δ′}) and F̂_{kG}(Q_{δ}) ⊂ F̂_{A′}(Q_{δ′}) ;

moreover, since we have an embedding of Dade N̄_{P} (Q)-algebras

45.5  A′(Q_{δ′}) ⟶ S(Q)^{∘} ⊗_{k} ( ⊗^{p^{n}} (kG)(Q_{δ})) ,

it follows from condition 41.3 that A′(Q_{δ′}) is an object of the category 2𝔇_{k}(N̄_{P} (Q)) .

46. Let us denote by F_{A′}^{sch} the category where the objects are the chains of local pointed groups Q_{δ′} over A′ coming from selfcentralizing local pointed groups of b and, denoting by SC_{A′} that ordered set of local pointed groups over A′ , the morphisms from a chain 𝔯′: Δ_{m} → SC_{A′} to a chain 𝔮′: Δ_{n} → SC_{A′} are the A′-fusions φ from 𝔯′(m) to 𝔮′(n) such that φ(Im(𝔯′)) ⊂ Im(𝔮′) ; then, it is clear that the correspondence sending 𝔮′ to F_{A′}(𝔮′) — the stabilizer of Im(𝔮′) in F_{A′}(𝔮′(n)) — can be extended to a contravariant functor from F_{A′}^{sch} to the category of groups 𝔊𝔯 . Now, we claim that the semipolarization ω allows us to lift this functor to a contravariant functor from F_{A′}^{sch} to the category of k^{∗}-groups k^{∗}-𝔊𝔯 mapping 𝔮′ on F̂_{A′}(𝔮′) — the restriction of F̂_{A′}(𝔮′(n)) to F_{A′}(𝔮′) — solving our problem.

47. We mimick our argument in 11; let Q_{δ′} and R_{ε′} be elements of SC_{A′} such that R_{ε′} ◁Q_{δ′} , and set Q̄ = Q/R; since A′ admits a P -stable basis, the Brauer homomorphisms in A′ induce an isomorphism

47.1  (A′(R))(Q̄) ≅ A′(Q) ;

actually, we claim that the Q̄-algebra homomorphism A′(R) → A′(R_{ε′}) is a semicovering (cf. §3 in [5]); it suffices to prove that, for any subgroup T̄ = T/R of Q̄ , the k-algebra homomorphism A′(T) ≅ (A′(R))(T̄) → (A′(R_{ε′}))(T̄) is a semicovering (cf. Theorem 3.16 in [5]); since (kG)(R_{ε}) ≅ k ⊗_{kZ(R)} kC_{G}(R)b(ε) , where ε is the corresponding local point of R on kGb , and C̄_{C̄_{G}(R)}(T̄)/C̄_{G}(T) is a p-group, the statement follows from Proposition 2.2 in [14]. Consequently, we get an embedding (cf. 2.1.1 in [14])

47.2  A′(Q_{δ′}) ⟶ (A′(R_{ε′}))(Q̄)

and therefore, since we have structural k^{∗}-group homomorphisms

47.3  F̂_{A′}(Q_{δ′})^{∘}/I_{Q}(Q) ⟶ A′(Q_{δ′})^{∗} and F̂_{A′}(R_{ε′})^{∘}/I_{R}(R) ⟶ A′(R_{ε′})^{∗}

where I_{Q}(Q) and I_{R}(R) denote the respective inner automorphism groups, and A′(R_{ε′}) is an object of 2𝔇_{k}(Q̄) , the scission determined by the semipolarization ω (cf. 35)

47.4  ωQ̄(A′(R_{ε′})) : F̂_{A′(R_{ε′})}(Q̄_{π^{A′(R_{ε′})}}) ⟶ k^{∗}

induces a k^{∗}-group homomorphism

47.5  F̂_{A′}(Q_{δ′})_{R} ⟶ F̂_{A′}(R_{ε′})

where F̂_{A′}(Q_{δ′})_{R} denotes the stabilizer of R in F̂_{A′}(Q_{δ′}) .

48. Moreover, if T_{ν′} is a third element of SC_{A′} , contained and normal in both Q_{δ′} and R_{ε′} , setting ¯ Q̄ = Q/T and ¯ Q̄ = R/T , and denoting by F̂_{A′}(Q_{δ′})_{T} and F̂_{A′}(R_{ε′})_{T} the stabilizers of T in F̂_{A′}(Q_{δ′}) and F̂_{A′}(R_{ε′}) respectively, we get k^{∗}-group homomorphisms

48.1  F̂_{A′}(R_{ε′})_{T} ⟶ F̂_{A′}(T_{ν′}) and F̂_{A′}(Q_{δ′})_{T} ⟶ F̂_{A′}(T_{ν′})

from the scissions 48.2 ω_{¯R̄}(A′(T_{ν′})) : F̂_{A′(T_{ν′})}( ¯R̄_{π^{A′(T_{ν′})}}) ⟶ k^{∗} and ω_{¯Q̄}(A′(T_{ν′})) : F̂_{A′(T_{ν′})}( ¯Q̄_{π^{A′(T_{ν′})}}) ⟶ k^{∗};

but, condition 30.2 applied to the Dade ¯ Q̄-algebra Ā′ = A′(T_{ν′}) and to the normal subgroup ¯ R̄ gives the following commutative diagram

48.3  \begin{aligned}F̂Ā′( ¯Q̄_{πĀ′})_{¯R̄} & ⟶ & F̂B̄′( ¯R̄_{π^{B}}) ⊗ F̂_{Ā′( ¯R̄)}( ¯Q̄_{π^{Ā′( ¯R̄)}}) \\ ω_{¯Q̄}(Ā′) ↘ & & ↙ ω_{¯R̄}(B̄′)⊗ωQ̄(Ā′( ¯R̄)) \\ & k^{∗} \end{aligned}

and the corresponding embedding 47.2 still gives

48.4 F̂_{Ā′( ¯R̄)}( ¯Q̄_{π^{Ā′( ¯R̄)}}) = F̂_{A′(R_{ε′})}(Q̄_{π^{A′(R_{ε′})}}) and ωQ̄(Ā′( ¯R̄)) = ωQ̄(A′(R_{ε′})) .

That is to say, setting F̂_{A′}(Q_{δ′})_{T,R} = F̂_{A′}(Q_{δ′})_{T} ∩ F̂_{A′}(Q_{δ′})_{T} , we get the following commutative diagram

48.5  \begin{aligned}F̂_{A′}(Q_{δ′})_{T,R} & ⟶ & F̂_{A′}(R_{ε′})_{T} \\ ↘ & & ↙ \\ & F̂_{A′}(T_{ν′}) \end{aligned} .

49. Now, if Q_{δ′} and R_{ε′} are elements of SC_{A′} such that R_{ε′} ⊂ Q_{δ′} , we define a k^{∗}-group homomorphism

49.1  F̂_{A′}(Q_{δ′})_{R} ⟶ F̂_{A′}(R_{ε′})

by induction on |Q: R|; setting N = N_{Q}(R) , there is a unique local point η′ of N over A′ and then we have R_{ε′} ⊂ N_{η′} ⊂ Q_{δ′} ; we may assume that N ≠ Q and, from the induction hypothesis and the construction above, we have already defined the k^{∗}-group homomorphisms

49.2  F̂_{A′}(Q_{δ′})_{N} ⟶ F̂_{A′}(N_{η′}) and F̂_{A′}(N_{η′})_{R} ⟶ F̂_{A′}(R_{ε′}) ;

but, it is clear that F̂_{A′}(Q_{δ′})_{R} ⊂ F̂_{A′}(Q_{δ′})_{N} and we may define homomorphism 49.1 by composition. We claim that, if T_{ν′} is a third element of SC_{A′} strictly contained in R_{ε′} , the corresponding diagram 48.5 is commutative too; we argue by induction on (|Q: T |, |Q: R|) and may assume that either R_{ε′} or T_{ν′} is not normal in Q_{δ′} ; if N ≠ Q then it suffices to consider the composition of F̂_{A′}(N_{η′})_{T,R} → F̂_{A′}(R_{ε′})_{T} with F̂_{A′}(R_{ε′})_{T} → F̂_{A′}(T_{ν′}) and after, the composition of F̂_{A′}(Q_{δ′})_{T,R} → F̂_{A′}(N_{η′})_{T,R} with F̂_{A′}(N_{η′})_{T,R} → F̂_{A′}(T_{ν′}) . If N = Q , set M = N_{Q}(T) and denote by μ′ the unique local point of M on A′; if M = R then we just apply our definiton; otherwise, set L = M ∩ R ≠ T , denote by λ′ the unique local point of L on A′ and consider the following diagram

49.3  \begin{aligned}& & F̂_{A′}(R_{ε′})_{T} \\ & ↗ & ↓ & ↘ \\ F̂_{A′}(Q_{δ′})_{T,R} & → & F̂_{A′}(L_{λ′})_{T} & → & F̂_{A′}(T_{ν′}) \\ & ↘ & ↑ & ↗ \\ & & F̂_{A′}(M_{μ′})_{T} \end{aligned}

By the induction hypothesis, the top left and the bottom left triangles are commutative; by definition, the top right triangle is commutative; finally, in the bottom right triangle we get the normal situation discussed above; consequently, all the diagram is commutative and, by definition, F̂_{A′}(Q_{δ′})_{T} → F̂_{A′}(T_{ν′}) is the composition of F̂_{A′}(Q_{δ′})_{T} → F̂_{A′}(M_{μ′})_{T} with F̂_{A′}(M_{μ′})_{T} → F̂_{A′}(T_{ν′}) .

50. In conclusion, for any pair of objects 𝔮′: Δ_{n} → SC_{A′} and 𝔯′: Δ_{m} → SC_{A′} of the category F_{A′}^{sch} , and any morphism φ: 𝔯′ → 𝔮′ , we have defined a k^{∗}-group homomorphism

50.1  \begin{matrix}F̂_{A′}(𝔮′(n))_{R} & ⟶ F̂_{A′}(R_{ε′}) ≅ & F̂_{A′}(𝔯′(m)) \\ ⋃ & & ⋃ \\ F̂_{A′}(𝔮′) & ⟶ & F̂_{A′}(𝔯′) \end{matrix}

where R_{ε′} is the image in 𝔮′(n) of 𝔯′(m) throughout φ; if 𝔱′: Δ_{ℓ} → SC_{A′} is a third object of this category and ψ: 𝔱′ → 𝔯′ a morphism, respectively denoting by T_{ν′} and T′_{ν′′} the images in 𝔮′(n) and in 𝔯′(m) of 𝔱′(ℓ) throughout φ ∘ ψ and ψ , it is clear that T_{ν′} is the image in R_{ε′} of T′_{ν′′} throughout φ , and we claim that the following diagram is commutative

50.2  \begin{matrix}F̂_{A′}(R_{ε′})_{T} & ⟶ & F̂_{A′}(T_{ν′}) \\ ≀|| & & ≀|| \\ F̂_{A′}(𝔯′(m))_{T′} & ⟶ & F̂_{A′}(T′_{ν′′}) \end{matrix} .

51. Arguing by induction on |R: T | , we may assume that T_{ν′} is normal in R_{ε′} , and then T′_{ν′′} is normal in 𝔯′(m) ; set R′_{ε′′} = 𝔯′(m) , choose maximal subsets I′ ⊂ ε′ and I′′ ⊂ ε′′ of pairwise orthogonal idempotents, and consider a′ ∈ T_{A′}^{φ′}(R_{e′}, R′_{e′′}) where e′ = ∑_{i′∈I′} i′ , e′′ = ∑_{i′′∈I′′} i′′ and φ′: R′ ≅ R is the isomorphism determined by φ (cf. 19); that is to say, the “conjugation” by a′ determines an R′-interior algebra isomorphism

51.1  f′ : e′′A′e′′ ≅ Res_{φ′}(e′A′e′)

and, since T′ has a unique local point over each of these R′-interior algebras, this isomorphism induces a R̄′-algebra isomorphism

51.2  (e′′A′e′′)(T′_{ν′′}) ≅ Res_{}_{φ̄′}((e′A′e′)(T_{ν′})) ,

where R̄ = R/T , R̄′ = R′/T′ and φ̄′: R̄′ ≅ R̄ is the isomorphism induced by φ′ , and therefore we get (cf. 25)

51.3  ωR̄((e′A′e′)(T_{ν′})) ∘ F̂f̃′(R̄′) = ωR̄′((e′′A′e′′)(T′_{ν′′}));

since ωR̄((e′A′e′)(T_{ν′})) = ωR̄((A′)(T_{ν′})) and ωR̄′((e′′A′e′′)(T′_{ν′′})) = ωR̄′((A′)(T′_{ν′′})) , we are done.

52. Now, the functoriality follows from the commutativy of the following diagram

52.1  \begin{aligned}F̂_{A′}(𝔮′(n))_{T,R} & ⟶ & F̂_{A′}(R_{ε′})_{T} & ⟶ & F̂_{A′}(T_{ν′}) \\ ⋃ & ↘ & ≀|| & & ≀|| \\ F̂_{A′}(𝔮′) & & F̂_{A′}(𝔯′(m))_{T′} & ⟶ & F̂_{A′}(T′_{ν′′}) \\ & ↘ & ⋃ & ↘ & ≀|| \\ & & F̂_{A′}(𝔯′) & & F̂_{A′}(𝔱′(ℓ)) \\ & & & ↘ & ⋃ \\ & & & & F̂_{A′}(𝔱′) \end{aligned} .

53. Having solved our problem, it follows from 14, 15, 16 and 17 above that we have the k^{∗}-extension F̂_{b}^{sc} of the category F_{b}^{sc} — the k^{∗}-fusion category of b — Linckelmann was looking for. Let us consider the k-algebra — called k^{∗}-fusion algebra to avoid confusion with the fusion algebra introduced in [15, §11] — associated with the k^{∗}-extension F̂_{b}^{sc}

53.1  𝔉̂_{G}(b) = ⨁_{Q_{δ},R_{ε}} k_{∗}F̂_{b}(R_{ε}, Q_{δ})

where Q_{δ} and R_{ε} run over the set of selfcentralizing pointed groups of b , k_{∗}F̂_{b}(R_{ε}, Q_{δ}) denotes the evident twisted k-vector space and, as in [15, §11], the multiplication is distributive and defined over the k^{∗}-fusions by the composition in the category when available, and by zero otherwwise.

54. This k-algebra has an obvious nilpotent two-sided ideal

54.1  ℑ̂_{G}(b) = ⨁_{Q_{δ},R_{ε}} k_{∗}F̂_{b}(R_{ε}, Q_{δ}) ,

where Q_{δ} and R_{ε} run over the set of selfcentralizing pointed groups of b such that |R| ≠ |Q| , and clearly we have

54.2  𝔉̂_{G}(b)/ℑ̂_{G}(b) ≅ ∏_{Q_{δ}} 𝔐_{n_{δ}}(k_{∗}F̂_{b}(Q_{δ})) ,

where Q_{δ} runs over a set of representatives for the G-conjugacy classes of selfcentralizing pointed groups of b and, for any Q_{δ} , n_{δ} is the cardinal of the G-conjugacy class of Q_{δ} and 𝔐_{n_{δ}}(k_{∗}F̂_{b}(Q_{δ})) denotes the matrix algebra of degree n_{δ} over k_{∗}F̂_{b}(Q_{δ}) .

55. Since ℑ̂_{G}(b) is nilpotent — no matter either its precise nature or the structure of 𝔉̂_{G}(b) — any idempotent in the direct product ∏_{Q_{δ}} 𝔐_{n_{δ}}(k_{∗}F̂_{b}(Q_{δ})) in the right member of isomorphism 54.2 can be lifted to 𝔉̂_{G}(b) ; in particular, for any selfcentralizing local pointed group Q_{δ} , denoting by ê_{Q_{δ}} the lifting of e_{Q_{δ}} (cf. 9) to Z(k_{∗}F̂_{b}(Q_{δ})) and identifying it with the corresponding central idempotent in 𝔐_{n_{δ}}(k_{∗}F̂_{b}(Q_{δ})) , the sum ê_{b} of them over a set of representatives for the G-conjugacy classes of selfcentralizing local pointed groups Q_{δ} of b can be lifted to an idempotent ě_{b} of 𝔉̂_{G}(b) , and then isomorphism 54.2 induces an isomorphism

55.1  ě_{b}𝔉̂_{G}(b)ě_{b}/ě_{b}ℑ̂_{G}(b)ě_{b} ≅ ê_{b}( ∏_{Q_{δ}} 𝔐_{n_{δ}}(k_{∗}F̂_{b}(Q_{δ})))ê_{b} ,

where Q_{δ} runs over a set of representatives for the G-conjugacy classes of selfcentralizing local pointed groups of b , and therefore the number of isomorphism classes of simple ě_{b}𝔉̂_{G}(b)ě_{b}-modules coincides with the right member of equality 9.1 and, conjecturally, with |Irr_{k}(G, b)| .

56. In conclusion, although we are convinced that the k^{∗}-fusion category of b , introduced here, and its corresponding k-algebra are important invariants of b , Alperin’s Conjecture does not support this conviction farther than the well-known equality 9.1. A more promising invariant, coming from the existence of the functor from F_{b}^{sch} (cf. 12) to the category of k^{∗}-groups, is the local Grothendieck group over k

56.1  R_{k}(F̂_{b}^{sc}) = lim_{⟵} {R_{k}(F̂_{b}(𝔮))}_{𝔮} ,

for the ℤ-rank of R_{k}(F̂_{b}^{sc}) coincides with the righ member of equality 9.1, allowing the conviction on the existence of an isomorphism

56.2  ℚ ⊗_{ℤ} R_{k}(G, b) ≅ ℚ ⊗_{ℤ} R_{k}(F̂_{b}^{sc})

compatible with the action in both members of this isomorphism of the stabilizer of b in the group of outer automorphisms of G .

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