FUSION CATEGORY ALGEBRAS

Markus Linckelmann

November 2002

Abstract. The fusion system F on a defect group P of a block b of a finite group G over a suitable p-adic ring O does not in general determine the number l(b) of isomorphism classes of simple modules of the block. We show that conjecturally the missing information should be encoded in a single second cohomology class α of the constant functor with value k^{×} on the orbit category F̄^{c} of F-centric subgroups Q of P of b which “glues together” the second cohomology classes α(Q) of AutF̄(Q) with values in k^{×} in Külshammer-Puig [11, 1.8]. We show that if α exists, there is a canonical quasi-hereditary k-algebra F̄(b) such that Alperin’s weight conjecture becomes equivalent to the equality l(b) = l(F̄(b)). By work of Broto, Levi, Oliver [3], the existence of a classifying space of the block b is equivalent to the existence of a certain second cohomology class ζ of the center functor Z on F^{c}. If both invariants α, ζ exist we show that there is an O-algebra L(b) associated with b having F̄(b) as quotient such that Alperin’s weight conjecture becomes again equivalent to the equality l(b) = l(L(b)); furthermore, if b has an abelian defect group, L(b) is isomorphic to a source algebra of the Brauer correspondent of b.

1 Twisted category algebras

We introduce in this section twisted category algebras as a straightforward generalistation of twisted group algebras. In order to settle some notation, we briefly describe the “untwisted” case first.

1.1. Let C be a finite category; that is, the object class Ob C of C is a finite set and for any two objects Q, R in C, the morphism set Hom_{C}(Q, R) is finite. Let O be a commutative ring.

Denote by F(C, O) the category of contravariant functors from C to the category of left O-modules Mod(O); the morphisms in F(C, O) are the natural transformations of functors. This is an abelian category with enough projectives. We denote by \underline{O} the constant contravariant functor mapping each object in C to O and each morphism in C to the identity map on O.

For any non negative integer n, the degree n cohomology of C with coefficients in a functor F ∈ F(C, O) is the O-module defined by means of the usual general abstract nonsense

H^{n}(C; F) = Ext_{F(C,O)}^{n}(\underline{O}, F) = H^{n}(Hom_{F(C,O)}(P, F)) ,

Typeset by AMS-TEX

where P is a projective resolution of the constant functor \underline{O} in F(C, O), and where Hom_{F(C,O)}(P, F) is the cochain complex of O-modules obtained from applying the contravariant functor Hom_{F(C,O)}(−, F) to the chain complex P. See [5], [6] for more background information on functor cohomology.

It is well-known that there is a natural isomorphism lim_{⟵} F ≅ Hom_{F(C,O)}(\underline{O}, F); that is, the inverse limit functor lim_{⟵} from F(C, O) to Mod(O) is isomorphic to the covariant functor Hom_{F(C,O)}(\underline{O}, −), and hence we have isomorphisms between their higher derived functors H^{n}(C; F) ≅ lim_{⟵}^{n}F.

It is equally well-known that F(C, O) is equivalent to the module category of the category algebra OC of C over O (terminology by P. Webb); that is, as O-module, OC is free having as basis the set of all morphisms in C, with the unitary associative O-bilinear multiplication induced by composition of morphisms in C. More precisely, for any two morphisms φ, ψ in C we define the multiplication by

φψ = ψ ∘ φ

provided that the composition Q ⟶^{φ} R ⟶^{ψ} S is defined, and φψ = 0 otherwise. The unit element of OC is

1_{OC} = ∑_{Q} Id_{Q} ,

where in the sum Q runs over the set of objects in C. This is in fact a decomposition of 1_{OC} as a sum of pairwise orthogonal (not necessarily primitive) idempotents in OC. With this notation, we have an obvious equivalence of categories

\{ \begin{matrix}F(C, O) & ≅ Mod(OC) \\ F & ↦ ⊕_{Q∈Ob C} F(Q) , \end{matrix}

where the module structure on the right side is given, for any morphism φ : Q → R in C by φ.m = F(φ)(m) if m ∈ F(R), and φ.m = 0 if m ∈ F(R′) for some object R′ ≠ R in C. The inverse of this equivalence maps a left OC-module M to the functor sending any object Q of C to Id_{Q}.M and any morphism φ : Q → R to the map Id_{R}.M → Id_{Q}.M induced by left multiplication with φ on M; this makes sense as φId_{R} = φ = Id_{Q}φ in OC.

Since C is finite, this equivalence maps the abelian subcategory f(C, O) of F(C, O) of contravariant functors from C to the category of finitely generated O-modules mod(O) onto mod(O).

Example 1.2. Let G be a finite group. Denote by \underline{G} the category having one object ∗ such that the set of endomorphisms of ∗ in \underline{G} corresponds bijectively to the set of elements of G, with composition induced by multiplication in G. Then the category algebra O\underline{G} is isomorphic to the group algebra OG.

1.3. Denote by \underline{O}^{×} the constant contravariant functor from C to the category of abelian groups Mod(ℤ), mapping each object in C to the multiplicative group of invertible elements O^{×} of O and each morphism in C to the identity on O^{×}. Let α ∈ H^{2}(C; \underline{O}^{×}). Similarly to what happens in group theory, α can be represented by a 2-cocycle, abusively still denoted by α; that is, α is a function mapping any sequence of two composable morphisms Q ⟶^{φ} R ⟶^{ψ} S in C to an element α(ψ, φ) ∈ O^{×} with the property that for any sequence of three composable morphisms

Q ⟶^{φ} R ⟶^{ψ} S ⟶^{τ} T

in C we have the equality

α(τ, ψ ∘ φ)α(ψ, φ) = α(τ ∘ ψ, φ)α(τ, ψ) .

We define the twisted category algebra O_{α}C of C by α as follows: as O-module, O_{α}C is equal to OC - that is, O-free with the morphism set of C as basis - , and the multiplication in O_{α}C is defined by

φψ = α(ψ, φ) · (ψ ∘ φ)

for any sequence of two composable morphisms Q ⟶^{φ} R ⟶^{ψ} S in C and φψ = 0 otherwise. As before, this is a unitary associative O-bilinear product; the associativity is in fact equivalent to the above condition which defines α as 2-cocycle.

1.4. If α, α′ are two cocycles on C with values in O^{×}, then α and α′ represent the same class in H^{2}(C, \underline{O}^{×}) if and only if there is a 1-cochain β (that is, a map sending any morphism in C to an element in O^{×}) satisfying α′(ψ, φ) = α(ψ, φ)β(ψ ∘φ)β(ψ)^{−1}β(φ)^{−1} for any two composable morphisms φ, ψ in C. The map sending a morphism φ in C to β(φ)φ induces then an algebra isomorphism O_{α}C ≅ O_{α′}C. In particular, if we set β(φ) = α(φ, Id_{Q}) then α′(Id_{Q}, Id_{Q}) = 1 for any object Q in C. Thus the isomorphism class of O_{α}C depends only on the class of α in H^{2}(C; \underline{O}^{×}), which justifies our notational abuse of denoting a 2-cocycle and its cohomology class by the same letter. Moreover, one may (and we do) choose α to be normalised; that is, α(Id_{Q}, Id_{Q}) = 1 for every object Q in C. Then Id_{Q} is an idempotent in O_{α}C and the unit element of O_{α}C is

1_{O_{α}C} = ∑_{Q∈Ob C} Id_{Q}.

Example 1.5. The previous construction applied to the example C = \underline{G} yields the usual definition of twisted group algebras.

Remark 1.6. If φ : Q ≅ Q′ is an isomorphism in C then the idempotents Id_{Q}, Id_{Q′} are conjugate. To see this we may assume that Q ≠ Q′. Set t = φ + φ^{−1} + ∑_{T} Id_{T}, with

T running over Ob C − {Q, Q′}. Then t is invertible and tId_{Q′} = Id_{Q}t = φ in O_{α}C.

2 Algebras over ordered categories

Let C be a finite ordered category; that is, every endomorphism of an object in C is an automorphism. The set of isomorphism classes of objects in C is then partially ordered by [Q] ≤ [R] whenever Hom_{C}(Q, R) ≠ ∅, where [Q], [R] are the isomorphism classes of objects Q, R in C. Let O be a commutative ring and let α be a normalised 2-cocycle on C with values in O^{×}. The aim of this section is to describe some basic properties of the twisted category algebra of C by α.

Set A = O_{α}C. Denote by B the free O-submodule of A having as basis the set of all isomorphisms in C. Let N be the free O-submodule of A having as basis the set of all non-isomorphisms in C. We keep this notation throughout this section.

Proposition 2.1. (i) B is a unitary subalgebra of A.

(ii) N is a nilpotent 2-sided ideal in A.

(iii) A = B ⊕ N and J(A) = J(B) ⊕ N.

Proof. The composition of any two isomorphisms is obviously again an isomorphism in C, whence (i). The composition of a non-isomorphism with any morphism is a nonisomorphism in C thanks to the fact that C is ordered. Thus N is a 2-sided ideal. Moreover, since C is finite, there is an upper bound for the length of a sequence of composable non-isomorphisms in C. Thus N is nilpotent, from which (ii) and (iii) follow. □

We can be more precise about the structure of B.

Proposition 2.2. The algebra B is Morita equivalent to the direct product of twisted group algebras ∏_{Q} O_{α(Q)}Aut_{C}(Q)^{op} , where Q runs over a set of representatives of the iso-

morphism classes of objects in C, and where α(Q) ∈ H^{2}(Aut_{C}(Q), O^{×}) is the restriction of α to Aut_{C}(Q). In particular, the algebra B is symmetric.

Proof. For any object Q in C denote by i_{Q} be the idempotent in A which is the sum of all idempotents Id_{Q′}, with Q′ running over the set of all objects in C which are isomorphic to Q in C. We have B = ∏_{Q} i_{Q}·A·i_{Q}, where Q runs over a set of representatives of the set

of isomorphism classes in C. For any two isomorphic objects Q, Q′ in C, the idempotents Id_{Q}, Id_{Q′} are conjugate in A by 1.6, and hence the algebra i_{Q} ·A·i_{Q} is Morita equivalent to the twisted group algebra Id_{Q}AId_{Q} = O_{α(Q)}Aut_{C}(Q)^{op}. The proposition follows. □

Since N is a 2-sided ideal in A, the projection of A onto B with kernel N is a canonically split surjective algebra homomorphism. In general, A need not be projective as left or right B-module. If every morphism in C is in addition a monomorphism, then Aut_{C}(Q) acts freely on the set Hom_{C}(Q, R), whenever the latter set is non-empty. Similarly, if every morphism in C is an epimorphism, then Aut_{C}(R) acts freely on Hom_{C}(Q, R) whenever the latter set is non-empty. Together with the observation that Hom_{C}(Q, R) is an O-basis of Id_{Q} · A · Id_{R}, this translates to our algebra theoretic language as follows:

Proposition 2.3. With the notation above, let Q be an object in C.

(i) If every morphism in C is a monomorphism then Id_{Q} ·A is free as left Id_{Q} ·A·Id_{Q}- module; in particular, A is projective as left B-module.

(ii) If every morphism in C is an epimorphism then A·Id_{Q} is free as right Id_{Q}·A·Id_{Q}- module; in particular, A is projective as right B-module.

The following theorem is the main result of this section. We refer to [4, Appendix] for an account and further references on quasi-hereditary algebras.

Theorem 2.4. Suppose that O = k is a field, and let e be an idempotent in B such that J(eBe) = {0}. Then the k-algebra eAe is quasi-hereditary.

We break up the proof of 2.4 in a series of easy lemmas, for which we introduce the following notation. For any positive integer n let C(n) be the full subcategory of C consisting of all objects Q in C for which there exists a sequence of n composable non-isomorphisms of the form

Q_{0} →^{ψ_{0}} Q_{1} →^{ψ_{1}} ⋯ →^{ψ_{n−1}} Q_{n} = Q .

We set C(0) = C. Clearly C(n+1) ⊆ C(n) for any n ≥ 0 and C(n) = ∅ for n large enough, as C is ordered. For any n ≥ 0 we define an idempotent e_{n} in A by

e_{n} = ∑_{Q∈Ob(C(n))} Id_{Q} .

Then e_{0} = 1_{A} and e_{n}e_{n+1} = e_{n+1} = e_{n+1}e_{n} for any n ≥ 0.

Lemma 2.5. For every integer n ≥ 0, the projective A-module Ae_{n} is a 2-sided ideal in A and we have Ae_{n+1} ⊆ Ae_{n}.

Proof. Let φ : Q → R be a morphism in C. If e_{n}φ = 0 (the product taken in A) there is nothing to prove. If e_{n}φ ≠ 0 then Q ∈ Ob(C(n)). But then clearly also R ∈ Ob(C(n)), hence e_{n}φ = φe_{n}, which shows that Ae_{n} is a 2-sided ideal in A. The inclusion Ae_{n+1} ⊆ Ae_{n} is a trivial consequence of the equality e_{n+1} = e_{n+1}e_{n}. □

Lemma 2.6. For every ring E and any idempotent f ∈ E such that Ef is a 2-sided ideal in E we have Hom_{E}(Ef, E/Ef) = {0}.

Proof. Any E-homomorphism from Ef to E/Ef lifts to a homomorphism from Ef to E, hence is induced by right multiplication with an element in E. Since Ef is a 2-sided ideal, right multiplication with an element in E maps Ef to itself, hence induces the zero map from Ef to E/Ef. □

Lemma 2.7. For every integer n ≥ 0 we have e_{n}N ⊆ Ae_{n+1}.

Proof. If Q ∈ Ob(C(n)) and φ : Q → R is a non-isomorphism in C then obviously R ∈ Ob(C(n + 1)). □

Proof of Theorem 2.4. Up to replacing e by a suitable conjugate, we may assume that e commutes with all idempotents Id_{Q}, where Q ∈ Ob(C). Then e commutes with all idempotents e_{n}, n ≥ 0. Hence, by 2.5, the k-subspace H_{n} = eAe_{n}e of eAe is a projective left eAe-module and a 2-sided ideal in eAe. In order to prove 2.4, we need to observe the following three facts:

(1) H_{n}/H_{n+1} is a projective eAe/H_{n+1}-module;

(2) Hom_{eAe/H_{n+1}}(H_{n}/H_{n+1}, eAe/H_{n}) = {0};

(3) H_{n}/H_{n+1} · J(eAe/H_{n+1}) · H_{n}/H_{n+1} = {0}. As for (1), there is nothing to prove since in fact H_{n} is projective as left eAe-module. Statement (2) is an immediate consequence of 2.6 applied to E = eAe and f = e_{n}e. Finally, (3) is equivalent to e_{n}J(eAe) ⊆ Ae_{n+1}. Now, by 2.7, we have e_{n}N ⊆ Ae_{n+1}, and since J(eAe) = J(eBe) ⊕ eNe by 2.1(iii), statement (3) follows from the hypothesis J(eBe) = {0}. □

3 Local structure of blocks and Alperin’s weight conjecture

In this section we provide some background material and notation that we are going to use in the next section for the definition of fusion category algebras.

3.1. Let O be a complete local commutative Noetherian ring whose residue field k = O/J(O) is perfect of prime characteristic p > 0. Let G be a finite group and let b be a block of OG; that is, b is a primitive idempotent in Z(OG). Denote by b̄ the canonical image of b in Z(kG). A Brauer pair is defined to be a pair (Q, e) consisting of a p-subgroup Q of G and a block e of kC_{G}(Q). Note that in particular (1, b̄) is a Brauer pair. The set of Brauer pairs is obviously a G-set with respect to the action of G by conjugation. It has been shown by Alperin and Broué [1] that this set can be endowed in a canonical way with a partial order “⊆” which is compatible with the G-action; in other words, the set of Brauer pairs is a G-poset. Note that in particular (1, b̄) is a Brauer pair. A Brauer pair (Q, e) is called a b-Brauer pair if (1, b̄) ⊆ (Q, e). The following two fundamental properties of this partial order have been proved in [1]:

3.1.1. for any Brauer pair (Q, e) and any subgroup R of Q there is a unique block f of kC_{G}(R) such that (R, f) ⊆ (Q, e);

3.1.2. any two maximal b-Brauer pairs are G-conjugate.

In other words, the set of b-Brauer pairs is a G-subposet of the set of all Brauer pairs in which the maximal pairs form a single G-conjugacy class. Furthermore, if (P, e) is a maximal b-Brauer pair, then P is called a defect group of b, notion due to R. Brauer, and the group E = N_{G}(P, e)/P C_{G}(P) is a p′-group, called inertial quotient of b.

3.2. Fix a maximal b-Brauer pair (P, e_{P}). Thus P is a defect group of the block b, for every subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e_{P}) and (Q, e_{Q}) is still a b-Brauer pair. By standard properties (cf. [1]), the idempotent e_{Q} remains a block of kN_{G}(Q, e_{Q}).

The fusion system of the block b is the category F defined as follows.

• the objects of F are the subgroups of P;

• for any two subgroups Q, R of P, a morphism from Q to R in F is an injective group homomorphism φ : Q → R for which there exists an element x ∈ G satisfying φ(u) = xux^{−1} for all u ∈ Q and ^{x}(Q, e_{Q}) ⊆ (R, e_{R}) (or equivalently, xe_{Q}x^{−1} = e_{xQx^{−1}}).

Loosely speaking, a morphism in F is a group homomorphism induced by conjugation with an element in G which is compatible with Brauer pairs. We have a canonical isomorphism Aut_{F}(Q) ≅ N_{G}(Q, e_{Q})/C_{G}(Q). If b is the principal block of OG then e_{Q} is the principal block of kC_{G}(Q), and we have N_{G}(Q, e_{Q}) = N_{G}(Q) in that case. In general, since any two maximal b-Brauer pairs are G-conjugate, the equivalence class of F does not depend on the choice of (P, e_{P}).

The orbit category of F is the quotient category F̄ defined as follows:

• the objects of F̄ are again the subgroups of P;

• the morphism set HomF̄(Q, R) is the set of orbits Inn(R)\Hom_{F}(Q, R) of the group of inner automorphisms Inn(R) of R, acting on the left of Hom_{F}(Q, R) by composition of homomorphisms, for any two subgroups Q, R of P.

We have an obvious canonical functor F ⟶ F̄ which is the identity on objects and surjective on morphisms. For any subgroup Q of P we have a canonical isomorphism AutF̄(Q) ≅ N_{G}(Q, e_{Q})/QC_{G}(Q).

A subgroup Q of P is called F-centric if Z(Q) is a defect group of kC_{G}(Q)e_{Q}, and Q is called F-radical if AutF̄(Q) has no non trivial normal p-subgroup. By Alperin’s fusion theorem (adapted to blocks in [1]) the fusion category F of the block b is completely determined by P and the automorphism groups Aut_{F}(Q) of the F-centric radical subgroups Q of P.

3.3. Alperin’s weight conjecture states that there should be an equality

l(b) = ∑_{Q} z(Q, e_{Q}) ,

where the notation is as follows: as before, l(b) is the number of isomorphism classes of simple kGb̄-modules, z(Q, e_{Q}) is the number blocks of defect 0 of k(N_{G}(Q, e_{Q})/Q)ē_{Q} (where ē_{Q} is the canonical image of e_{Q} in kN_{G}(Q, e_{Q})/Q), and where the sum is taken over a set of representatives of the F-isomorphism classes of subgroups of P.

A key observation - which is implicitly in [9] and more explicitly in [17] - is that if z(Q, e_{Q}) ≠ 0 then Q is F-centric, and one easily sees that Q is also F-radical (since normal p-subgroups are contained in all defect groups). In other words, only the Fcentric radical subgroups of P are relevant for the right hand expression of Alperin’s weight conjecture.

We denote by F^{c} the full subcategory of F consisting of F-centric subgroups of P; similarly, we denote by F̄^{c} the full subcategory of F̄ consisting of all F-centric subgroups of P. It is easy to see that a subgroup Q of P is F-centric if and only if C_{P} (Q′) = Z(Q′) for any subgroup Q′ of P which is isomorphic to Q in the category F. In particular, if Q ⊆ R ⊆ P and Q is F-centric then so is R, and we have Z(R) ⊆ Z(Q). From this follws that the map sending an F-centric subgroup Q of P to its center Z(Q) extends canonically to a contravariant functor

Z : F^{c} ⟶ mod(ℤ_{(p)}) .

Since inner automorphisms of Q act trivially on Z(Q), this functor factors uniquely through F̄^{c}.

4 Fusion category algebras

We define in this section certain suitably twisted algebras over fusion systems and related categories of blocks. At this point this is still somewhat speculative as the existence of what one would like to call a “suitable twist” is not settled in general. Nonetheless, we are certain that such twists exist - see Conjecture 4.2 below - and that the resulting algebras are invariants of blocks of finite groups carrying a large amount of information which makes them worthwhile to be considered.

4.1. We keep the notation of the preceding section. The starting point of what follows is a theorem of Külshammer and Puig in [11], which associates with every Fcentric subgroup Q of P two pieces of information which are quite different in nature in that the first is an invariant of the fusion system F, while the second is not:

4.1.1. there is a canonical class ζ(Q) ∈ H^{2}(Aut_{F}(Q), Z(Q)), or equivalently, a canonical group extension

1 ⟶ Z(Q) ⟶ L_{Q} ⟶ Aut_{F}(Q) ⟶ 1

with the property that if N_{P} (Q) is a defect group of kN_{G}(Q, e_{Q})e_{Q} then N_{P} (Q) is a Sylow-p-subgroup of L_{Q} and the above exact sequence restricts to the obvious exact sequence 1 → Z(Q) → N_{P} (Q) → N_{P} (Q)/Z(Q) → 1;

4.1.2. there is a canonical class α(Q) ∈ H^{2}(AutF̄(Q), k^{×}) such that the twisted group algebra k_{α(Q)}AutF̄(Q) is Morita equivalent to k(N_{G}(Q, e_{Q})/Q)ē_{Q}.

The work of Broto, Levi and Oliver [3] shows what it means for a block b to have a classifying space; in particular, the existence of such a classifying space is shown to be equivalent to the existence of a certain class ζ ∈ H^{2}(F^{c}, Z) whose restriction to Aut_{F}(Q) is the class ζ(Q) in 4.1.1 above, for every F-centric subgroup subgroup Q of P. In other words, ζ “glues together” the classes ζ(Q).

As in group theory, the class ζ determines an extension category L of F^{c} by the center functor Z, called centric linking system in [3]. More precisely, the objects of L are again the F-centric subgroups in P, for every F-centric subgroup Q in P we have Aut_{L}(Q) = L_{Q} and there is a functor L → F^{c} which is the identity on objects, surjective on morphisms and which induces for each F-centric Q the surjective map L_{Q} → Aut_{F}(Q) from the Külshammer-Puig exact sequence in 4.1.1. The classifying space of b is then obtained as the p-completion |L|_{p}^{∧} of the nerve of L.

By the time of this writing, neither the existence nor the uniqueness of ζ seem to be known in general.

The idea of gluing together the cohomology classes ζ(Q) may well be applied to the classes α(Q) in 4.1.2:

Conjecture 4.2. There is a second cohomology class α ∈ H^{2}(F̄^{c}, \underline{k}^{×}) whose restriction to AutF̄(Q) is the class α(Q) from 4.1.2 for any F-centric subgroup Q of P.

Neither the existence nor the uniqueness of α is established at this point in general; see the next section for a (non exhaustive) list of examples. See also Remark 5.4 for a brief explanation regarding the difficulties that arise in attempting to construct α directly from block theoretic information. Since k is perfect we have a canonical group isomorphism O^{×} ≅ k^{×} × (1 + J(O)). Since any constant functor on F̄^{c} with value in Mod(ℤ_{(p)}) is acyclic (that is, its cohomology vanishes except in degree 0), applying this to the constant functor with value 1 + J(O) shows that the canonical surjection O^{×} → k^{×} and its canonical section k^{×} → O^{×} induce in particular isomorphisms

H^{2}(F̄^{c}, \underline{O}^{×}) ≅ H^{2}(F̄^{c}, \underline{k}^{×})

which are inverse to each other. If α exists, Alperin’s weight conjecture admits the following reformulation:

Theorem 4.3. Assume that α exists as in 4.2. Then Alperin’s weight conjecture is equivalent to the following equality: the number l(b) is equal to the number of isomorphism classes of simple k_{α}F̄^{c}-modules whose diagonal entry in the Cartan matrix of k_{α}F̄^{c} is 1.

Proof. Let S be a simple k_{α}F̄^{c}-module. Let j be a primitive idempotent in k_{α}F̄^{c} such that k_{α}F̄^{c}j is a projective cover of S. Since the unit element of k_{α}F̄^{c} is the sum of the idempotents Id_{Q} we may actually choose j such that j = Id_{Q}j = jId_{Q} for some F-centric subgroup Q of P. The diagonal entry of S in the Cartan matrix is the k-dimension of jk_{α}F̄^{c}j. Since α glues together the classes α(Q) we have

Id_{Q} · k_{α}F̄^{c} · Id_{Q} = k_{α(Q)}AutF̄(Q) ,

and this algebra is Morita equivalent to k(N_{G}(Q, e_{Q})/Q)ē_{Q}. Thus the dimension of jk_{α}F̄^{c}j is also the diagonal entry of the Cartan matrix of k_{α(Q)}AutF̄(Q) corresponding to the simple k_{α(Q)}AutF̄(Q)-module Id_{Q}S. Since the twisted group algebra k_{α(Q)}AutF̄(Q) is symmetric, this Cartan number is 1 if and only if Id_{Q}S belongs to a block of defect 0 of k_{α(Q)}AutF̄(Q). The Theorem follows. □

The argument proving the above Theorem shows a little more: the projective cover k_{α}F̄^{c}j of S looks like a highest weight module: its top composition factor is S, and all other composition factors are associated with F-centric subgroups R of P such that |R| < |Q|. The slogan to restate the above Theorem is “Alperin’s weight conjecture is a highest weight conjecture”.

4.4. Assume that α exists as in 4.2; that is, such that the restriction of α to AutF̄(Q) is α(Q) for every F-centric subgroup Q of P. Denote by ē an idempotent in k_{α}F̄^{c} such that ēS = S for every simple k_{α}F̄^{c}-module S whose diagonal entry in the Cartan matrix is 1 and such that ēS′ = {0} for any other simple k_{α}F̄^{c}-module S′. This determines ē up to conjugation in k_{α}F̄^{c}, but there is in fact a canonical choice for ē: it follows from the proof of 4.3 that we can take for ē the sum of all defect zero blocks of all twisted group algebras k_{α(Q)}AutF̄(Q), with Q running over the set of F-centric subgroups of P. Set

F̄(b) = ēk_{α}F̄^{c}ē .

The following combines 4.3 and 2.4:

Theorem 4.5. Assume that α exists as in 4.2. The k-algebra F̄(b) is quasi-hereditary, and Alperin’s weight conjecture is equivalent to the equality l(b) = l(F̄(b)).

Proof. The idempotent ē collects precisely the defect zero blocks of the twisted group algebras k_{α(Q)}AutF̄(Q), hence 2.4 applies, showing that F̄(b) is quasi-hereditary. The statement on Alperin’s weight conjecture is obvious from 4.3. □

4.6. Suppose now that both α and ζ exist as above. Then, as explained in 4.1, the extension category L of F^{c} by Z determined by ζ comes along with a canonical functor L ⟶ F^{c}. Composing this with the canonical functor F → F̄ yields a functor

L ⟶ F̄^{c}

through which we can restrict α to a class H^{2}(L, \underline{O}^{×}), still called α (we can replace k^{×} by O^{×} by the remark preceding 4.3). The above functor induces thus a surjective algebra homomorphism

O_{α}L ⟶ k_{α}F̄^{c} ,

and the kernel of this algebra homomorphism is contained in the radical J(O_{α}L), because at each object we are just dividing by a normal p-subgroup.

Thus there is an idempotent e in O_{α}L which lifts ē with respect to the canonical algebra homomorphism O_{α}L → k_{α}F̄^{c}; then e is unique up to conjugation. We define the O-algebra L(b) by setting

L(b) = eO_{α}Le .

The surjective algebra homomorphism above induces a surjective algebra homomorphism L(b) ⟶ F̄(b) whose kernel is contained in the radical of L(b). In particular, l(L(b)) = l(F̄(b)). As an immediate consequence of Theorem 4.3 and the construction of L(b), Alperin’s weight conjecture reads then as follows:

Theorem 4.7. If α, ζ exist, then, with the above notation, Alperin’s weight conjecture is equivalent to the equality l(b) = l(L(b)). Moreover, if P is abelian then L(b) is a source algebra of the Brauer correspondent of b.

Proof. The first statement is a trivial consequence of the remarks in 4.6 above, and the second statement follows from the fact that if P is abelian, then P is the only Fcentric subgroup of P and Aut_{F}(P) = N_{G}(P, e_{P})/C_{G}(P) = E is a p′-group, namely the inertial quotient of b. Thus L(b) = O_{α}(P ⋊ E), which is a source algebra of the Brauer correspondent of b by results in [10] or [15]. □

As pointed out by G. R. Robinson, there should be a refinement of 4.7 which formulates Dade’s conjectures in terms of a suitable variation of L(b). In fact, Robinson observed that Dade’s projective conjecture admits a formulation in terms of chains of F-centric subgroups of P and that in order to proceed further, one needs to find a way to “amalgamate” the local subgroups N_{G}(Q, e_{Q}) (cf. [16]). The definition of the 2-cocycle α is an attempt in that direction.

One might want to speculate to what extent the algebra L(b) could be used for a generalisation of Broué’s abelian defect conjecture. One certainly cannot expect a derived or even stable equivalence between the algebras OGb and L(b), since in general, L(b) need not be symmetric and K ⊗_{O} L(b) need not be semi-simple (where in the last

statement we assume that O has characteristic zero and K is the quotient field of O).

5 Examples

5.1 If b is the principal block, both ζ and α exist and are canonical. The category L is constructed explicitly in [2] and for α we can take the constant 2-cocycle mapping every pair of composable morphisms to 1 (that is, α represents the zero cohomology class).

5.2. As pointed out by G. R. Robinson, if b is tame (that is, p = 2 and P is either generalised quaternion or dihedral or semidihedral) both ζ and α exist and are canonical. For ζ this is due to the fact that the fusion system of a tame block is that of a suitable principal block, and for α we can again take the zero class.

5.3 Whenever there is a single F-centric subgroup Q of P such that N_{G}(Q, e_{Q}) controls F-fusion - or equivalently, such that F = N_{F}(Q) - then both ζ and α exist, since they are canonically determined by ζ(Q) and α(Q), respectively. This is the case if P is abelian (since here N_{G}(P, e_{P}) controls fusion) or if G is p-solvable (by results of Puig [13]) or if b is SL(2, p)-free (by results of Kessar, Linckelmann, Robinson [7]).

Any element of H^{2}(AutF̄(Q), k^{×}) shows up as an α(Q) in this way for some block of a finite group: if we start with a given α(Q) ∈ H^{2}(AutF̄(Q), k^{×}), we can view α(Q) as an element of H^{2}(L_{Q}, k^{×}), where L_{Q} is as in 4.1.1. This gives rise to a central group extension

1 ⟶ Z ⟶ H ⟶ L_{Q} ⟶ 0

of L_{Q} by a finite cyclic p′-group Z, which represents an element of H^{2}(L_{Q}, Z) whose image in H^{2}(L_{Q}, k^{×}) via a suitable group homomorphism ϵ : Z → k^{×} is α(Q). Then e = ∑_{z∈Z} ϵ(z^{−1})z is a block of kH such that the block algebra kHe is in fact isomorphic

to the twisted group algebra k_{α(Q)}L_{Q}, the fusion system of e is precisely F and the 2-cocycle α on F for this block is the one determined by α(Q).

Remark 5.4. Let Q, R be F-centric subgroups of P. Let i ∈ (kGb)^{Q} and j ∈ (kGb)^{R} be primitive idempotents such that Br_{Q}(i)e_{Q} = Br_{Q}(i) ≠ 0 and Br_{R}(j)e_{R} = Br_{R}(j) ≠ 0. In other words, i and j belong to the unique local points of Q and R on kGb associated with e_{Q} and e_{R}, respectively.

Let φ : Q → R be a group homomorphism such that there exists an invertible element a ∈ (kGb)^{×} satisfying ^{a}(uj) = φ(u)(^{a}j) for all u ∈ Q and (^{a}i)j = ^{a}i = j(^{a}i). Then, by the main result in [14], the group homomorphism φ belongs to Hom_{F}(Q, R), and moreover every element of Hom_{F}(Q, R) arises this way. In other words, if we denote by T_{jkGi}(Q, R) the set of all elements of jkGi of the form jai for some a ∈ (kGb)^{×} for which there is a group homomorphism φ : Q → R satisfying ^{a}(uj) = φ(u)(^{a}j) for all u ∈ Q and (^{a}i)j = ^{a}i = j(^{a}i), then the correspondence sending such an element a to φ induces a surjective map

T_{jkGi}(Q, R) ⟶ Hom_{F}(Q, R) .

The group ((ikGi)^{Q})^{×} acts by right multiplication on the set T_{jkGi}(Q, R), and thus the previous map induces a surjective map

T_{jkGi}(Q, R)/((ikGi)^{Q})^{×} ⟶ Hom_{F}(Q, R) .

For Q = R and i = j this is obviously a group isomorphism

T_{ikGi}(Q, Q)/((ikGi)^{Q})^{×} ≅ Aut_{F}(Q) .

The left side in this isomorphism has a canonical central k^{×}-extension, namely the group

T_{ikGi}(Q, Q)/(i + J((ikGi)^{Q})) ;

indeed, since i is primitive in (kGb)^{Q}, the algebra (ikGi)^{Q} is local, and hence ((ikGi)^{Q})^{×} ≅ k^{×} × (i + J((ikGi)^{Q})). It is shown in [15] that this central k^{×}-extension determines precisely α(Q). It is tempting to think that this might lead to a construction of the 2-cocycle α on F - and this idea had been our starting point to conjecture the existence of α as in 4.2. As it stands, this construction doesn’t work, though.

The problem here is that in general the surjective map T_{jkGi}(Q, R)/((ikGi)^{Q})^{×} ⟶ Hom_{F}(Q, R) considered above is not a bijection. Counterexamples occur systematically when the relative multiplicity of the involved local points of Q and R is greater than 1. To see this, consider the case where Q ⊂ R and where i ∈ (jkGj)^{Q} such that there exists another primitive idempotent i′ in (jkGj)^{Q} which is both orthogonal and conjugate to i by an element a ∈ ((kGb)^{Q})^{×}. Then both ji = i and jai = i′a belong to T_{jkGi}(Q, R); their images in Hom_{F}(Q, R) are both equal to the inclusion homomorphism Q ⊂ R.

However, i and i′a cannot be in the same ((ikGi)^{Q})^{×}-orbit because left multiplication by i annihilates the entire orbit i′a((ikGi)^{Q})^{×} as i and i′ are orthogonal.

This problem suggests what could be a remedy to it: if in addition Q is normal in R, then R acts on the simple algebra kC_{G}(Q)/Z(Q)ē_{Q}-algebra, and hence this algebra is isomorphic to End_{k}(W) for some endo-permutation kR/Q-module W. Then W determines the relative multiplicities of the local points of Q and R. Thus one would need to “glue together” the endo-permutation modules W obtained in this way to an endo-permutation kP -module V in such a way that “modifying” a source algebra of b by End_{k}(V) yields an algebra which is still Morita equivalent to the source algebra and which still contains the information on F but which now has the property that all relative multiplicities between local points of centric subgroups are 1. If that could be done, the above idea might still stand a chance to lead to an explicit description of α.

References

1. J. L. Alperin, M. Broué, Local methods in block theory, Ann. of Math. 110 (1979), 143–157.

2. C. Broto, R. Levi, B. Oliver, Homotopy equivalences of p-completed classifying spaces of finite groups, preprint (2000).

3. C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems, preprint (2001).

4. S. Donkin, The q-Schur Algebra, LMS Lecture Notes Series (Cambridge University Press, Cambridge) 253 (1998).

5. P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory, Erg. der Math. 35, Springer Verlag Berlin, Heidelberg, New York (1967).

6. P. Hilton, U. Stammbach, A Course in Homological Algebra, Graduate Texts in Math. 4, Springer Verlag, Berlin, Heidelberg, New York (1971).

7. R. Kessar, M. Linckelmann, G. R. Robinson, Local control in fusion systems of p-blocks of finite groups, J. Algebra 257 (2002), 393-413.

8. R. Knörr, On the vertices of irreducible modules, Ann. Math. 110 (1979), 487–499.

9. R. Knörr, G. R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39 (1989), 48–60.

10. B. Külshammer, Crossed products and blocks with normal defect groups, Comm. Algebra 13 (1985), 147–168.

11. B. Külshammer, L. Puig, Extensions of nilpotent blocks, Invent. Math. 102 (1990), 17–71.

12. L. Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), 265–292.

13. L. Puig, Local block theory in p-solvable groups, Proc. Symp. Pure Math. 37 (1980), 385–388.

14. L. Puig, Local fusion in block source algebras, J. Algebra 104 (1986), 358–369.

15. L. Puig, Pointed groups and constructions of modules, J. Algebra 116 (1988), 7–129.

16. G. R. Robinson, Weights and characters (amalgams for blocks?), Talk given in Cambridge (2002).

17. G. R. Robinson, R. Staszewski, More on Alperin’s conjecture, Astérisque 181-182 (1990), 237–255.

Markus Linckelmann Ohio State University Math Tower 231 West 18th Avenue Columbus, OH 43210

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