A SURVEY ON BROUÉ’S

ABELIAN DEFECT GROUP CONJECTURE

Andrei Marcus “Babeş–Bolyai” University Faculty of Mathematics and Computer Science Str. Mihail Kogălniceanu nr. 1 RO-400084 Cluj–Napoca, Romania

Abstract. This article is an extended version of the talk given at the 47th Algebra Symposium, held at Nagoya University, August 4–7, 2003. We give an introduction to Broué’s abelian defect group conjecture, and we present some of the methods and the achievements on this subject.

1. Preliminaries on modular representation theory

1.1. Let G be a finite group and p a prime number. Modular representation theory studies the relations between representations of G in characteristic zero and in characteristic p via the study of the group algebra of G over a complete discrete valuation ring O having algebraically closed residue field k of characteristic p and a “big enough” quotient field K of characteristic zero, that is, containing enough roots of unity. This implies that K is a splitting field for the group algebra KH, for any subgroup H of G.

The link between representations of G over K and over k is then established by the functors “reduction modulo p” k ⊗_{O} − and “extension of scalars” K ⊗_{O} −.

All modules are assumed to be finitely generated left unitary modules, and all O-algebras are finitely generated as O-modules.

1.2. A block of the finite group G is a primitive idempotent a in the center of the group algebra OG. The algebra A = OGa is called the block algebra of the block a. In fact, most of the time I shall denote by the same a the block algebra OGa. The group algebra OG decomposes into a direct product OG = A_{0} × A_{1} × ⋯ × A_{n} of blocks, where A_{i} = a_{i}OG.

Given an indecomposable OG-modules M, there is a unique block a_{i} such that a_{i}M ≠ 0, and we say that M “belongs” to a_{i}. The trivial OG-module O belongs to the principal block A_{0} = a_{0}OG =: B_{0}(OG).

1.3. A defect group of the block A is a minimal subgroup D of G such that the homomorphism A ⊗_{OP} A → A induced by multiplication splits as homomorphism of (A, A)-bimodules. A source idempotent of A is a primitive idempotent i ∈ A^{D} such that the induced map Ai ⊗_{OD} iA → A is still split surjective as a homomorphism of (A, A)-bimodules. The algebra iAi, considered together with the group homomorphism D → U(iAi), u ↦ ui, is called a source algebra of A. Here we denoted A^{D} = {a ∈ A | uau^{−1} = a for all u ∈ D}.

1991 Mathematics Subject Classification. 20C05, 18E30 20C20, 20C25, 16W50, 16S35.

Key words and phrases. group algebras, blocks, abelian defect groups, Morita, derived, and stable equivalences, group-graded algebras, crossed product, symmetric algebras, tilting complex, Rickard equivalence.

The author’s visit to the Chiba University was supported by a J.S.P.S. Invitational fellowship.

The defect groups of A form a unique conjugation class of p-subgroups of G. The defect groups of the principal block A_{0} are the Sylow p-subgroups of G. If p^{d} is the order of a defect group of A, then d is called the defect of b, and we denote d = def(A).

Reduction modulo p gives a bijection between the blocks of OG and the blocks of kG, preserving defect groups.

The source algebra of A is Morita equivalent to A and encodes all the essential information about the local structure of the block. From Puig’s point of view, the main problem of block theory is the determination of the source algebra of a block as an interior D-algebra.

The defect groups of A control the complexity of A. Defect “zero” just means that A is isomorphic to a matrix algebra M_{r}(O), and A is of finite representation type if and only if D is cyclic. In the rest of the cases, A has infinite representation type, so the problem of comparing blocks of various subgroups of G appears to be more feasible.

1.4. By Maschke’s theorem, KA := K⊗_{O} A is a semisimple K-algebra, hence a direct product of full matrix K-algebras. Each of these matrix algebras have a unique simple module, which is said to belong to A.

Let V be a simple KA-module. It is known that dim_{K} V divides [G : Z(G)]. Dade introduced the defect def(V) of V by the equality

p^{def(V)} = (\frac{|G|}{dim_{K} V} )_{p} ,

and denote by k_{δ}(B) the number of (isomorphism classes of) simple modules of defect δ in A.

2. The Brauer construction and local subgroups

The Brauer construction is a fundamental tool establishing a link between “global” and “local” structure of blocks. It and has been generalized to modules by Feit and G-algebras by Broué and Puig.

2.1. Let P a p-subgroup of G. The Brauer homomorphism associated with P is the surjective O-algebra homomorphism

Br_{P}^{G}: (OG)^{P} → kC_{G}(P),  ∑_{x∈G} α_{x}x ↦ ∑_{x∈C_{G}(P)} \overline{α}_{x}x,

where α_{x} ∈ O and \overline{α}_{x} is the image of α_{x} in k, for any x ∈ G.

Note that D is a defect group of A if and only if D is a maximal subgroup of G such that Br_{D}^{G}(a) ≠ 0.

This construction allows to define the local structure of a block in terms of its Brauer pairs. A Brauer-pair of A = OGa is a pair (P, e) consisting of a p-subgroup P of G and a block e of kC_{G}(P) such that Br_{P}^{G}(a)e = e. With a partial order relation and morphisms suitably defined, the a-Brauer pairs form a partially ordered set, and also category called the Brauer category of the block A.

2.2. For an arbitrary G-algebra A, the Brauer map is the canonical N_{G}(P)-algebra homomorphism

Br_{P}^{A}: A^{P} → A(P) := k ⊗_{O} (A^{P} / ∑_{Q<P} Tr_{Q}^{P} A^{Q}),

while for an OG-module V , the Brauer map is the canonical homomorphism of ON_{G}(P)- modules

Br_{P}^{V} : V^{P} → V (P) := k ⊗_{O} (V^{P} / ∑_{Q<P} Tr_{Q}^{P} V^{Q}),

where Tr_{Q}^{P}: V^{Q} → V^{P} is the trace map.

2.3. The correspondence in the following theorem is called the Brauer correspondence:

Let D be a p-subgroup of G and H a subgroup of G containing N_{G}(D). For any block a of OG having D as defect group there is a unique block b of OH having D as defect group such that Br_{D}(a) = Br_{D}(b). Moreover, the correspondence a ↦ b is a bijection between the sets of blocks of OG and of OH having D as defect group.

Note that by Brauer’s 3rd Main Theorem, the Brauer correspondent of the principal block A_{0} is the principal block B_{0} of OH.

Subgroups of the form N_{G}(P), where P is a p-subgroups of G are called “p-local subgroups”.

2.4. More generally, the Brauer map gives a correspondence between blocks which is not a bijection. One can formulate this also in terms of bimodules. Let H be a subgroup of G, B = bOH a block of OH and A = aOG a block of OG. We say that b is a Brauer correspondent of a and write a = b^{G}, if A is the only block of OG with the property that B is a direct summand of A regarded as O(H × H)-modules. Notice that if D is a defect group of B and H contains C_{G}(D), then b^{G} is defined.

3. The conjectures of Alperin, Dade, Isaacs-Navarro. Abelian defect groups

Block theory was initiated by Brauer with the aim to provide a tool for the classification of finite simple groups. His famous list of problems is still stimulating the research in this field. Subsequent conjectures are clarification and refinements of his ideas. Brauer’s problems are expressing the belief that block theoretic invariants of blocks of positive defect of OG should be determined from information coming from p-local subgroups. According to Alperin, the “main problem of block theory is the determination of values of the irreducible characters of G on p-singular elements in terms of p-local information.

3.1. Alperin’s weight conjecture. Let A = aOG be a block of OG, ℓ(A) be the number of simple A-modules and f_{0}(B) the number of projective simple A-modules. Then

ℓ(A) = ∑_{Q,b} f_{0}(B),

where Q runs over the p-subgroups of G up to conjugacy and B over the Brauer correspondents of B in N_{G}(Q). This formula implies that if B has positive defect, then ℓ(A) is indeed p-locally determined.

3.2. Dade’s conjecture. There is an equivalent formulation of Alperin’s conjecture in terms of p-subgroup complexes and alternating sums, due to Knörr and Robinson. Inspired by this, Dade stated the following conjecture.

Let S_{p}(G) be the set of of stricly increasing chains C = (Q_{0} < Q_{1} < ⋯ < Q_{n}) of nontrivial p-subgroups of G and let |C| = n be the length of C. Then G acts by conjugation on S_{p}(G) and let G_{C} = N_{G}(Q_{0})∩⋯∩N_{G}(Q_{n}) be the stabilizer of C. If the maximal normal p-subgroup O_{p}(G) = 1 and A is a block of positive defect, then for all δ > 0,

∑_{C,B}(−1)^{|C|} k_{δ}(B) = 0,

where C runs over S_{p}(G) up to conjugacy and B over the Brauer correspondents of A in OG_{C}.

This implies Alperin’s conjecture, and for δ = d, it gives an older conjecture of Alperin and McKay.

3.3. The Isaacs-Navarro conjecture. This is another generalization of the Alperin-McKay conjecture. Let A = aOG be a block with defect group D, |D| = p^{d}, and let B = bOH_{G}(D) be the Brauer correspondent of A. Let k be an integer such that p ∤ k and denote by M_{k}(A) be the number of simple KA modules V with defect d such that (dim_{K} V)_{p′} ≡ ±k (mod p). Denote c = |G : N_{G}(D)|_{p′}.

Then the conjecture predicts that M_{ck}(A) = M_{k}(B). There are stronger forms of this conjecture, involving the action of the Galois group of K on characters. Uno also formulated alternating sum versions of these conjectures.

3.4. Assume that A has abelian defect group D, and let B = bON_{G}(D) be the Brauer correspondent of A. In this case, Alperin’s conjecture reduces to the equalities

• ℓ(A) = ℓ(B) (A and B have the same number of simple modules);

• k(A) = k(B) (KA and KB have the same number of simple modules). Dade’s conjecture also reduces to the equalities

• k_{δ}(A) = k_{δ}(B) (KA and KB have the same number of simple modules of defect δ),

δ ≥ 0. These suggest that there should be a deeper structural connection between A and B.

4. Equivalences between symmetric algebras

Let A and B be symmetric O-algebras, free over O (note that blocks of OG satisfy these conditions). The (A, B)-bimodule is called exact, if _{A}M and M_{B} are projective, finitely generated. Denote by M^{∨} the O-dual of M.

In this context, Broue discusses three levels of equivalence: Morita equivalence, derived equivalence and stable equivalence of Morita type. An important progress was achieved by the fundamental work of Rickard describing the equivalences the derived categories of two algebras (regarded as triangulated categories). The characterization of derived equivalences is given either in terms or one-sided tilting complexes or two-sided tilting complexes (complexes of bimodules). Equivalences between the stable module categories are much more general; the stable Morita equivalences are those induced by bimodules which are projective as left and right modules.

4.1. Morita equivalence. Recall that the abelian categories A-Mod and B -Mod are equivalent if and only if the following equivalent conditions hold.

(1) There is an exact (A, B)-bimodule M such that M ⊗_{B} M^{∨} ≃ A in A ⊗_{O} A^{op} -Mod and M^{∨} ⊗_{A} M ≃ B in B ⊗_{O} B^{op} -Mod.

(2) There is a progenerator _{A}P such that B ≃ End_{A}(P)^{op} .

4.2. Rickard equivalence. The homotopy categories H^{b}(A-mod) and H^{b}(B-mod) are equivalent as triangulated categories if and only if the following equivalent conditions hold.

(1) There is a bounded complex C of exact (A, B) bimodules (called two-sided tilting complex) such that C ⊗_{B} C^{∨} ≃ A in H(A ⊗_{O} A^{op}-mod) and C^{∨} ⊗_{A} C ≃ B in H(B ⊗_{O} B^{op}-mod).

(2) There exists an object T ∈ H^{b}(A-proj) (called one-sided tilting complex) such that End_{H(A-mod)}(T)^{op} ≃ A, Hom_{H(A-mod)}(T, T [n]) = 0 for n ≠ 0, and the direct summands of T generate H^{b}(A-proj) as a triangulated category.

The connection between the tilting complexes C and T is much more subtle, since End_{H(A-mod)}(T)^{op} acts on T only up to homotopy.

4.3. Stable Morita equivalence. By definition, the bounded complex C of exact (A, B)- bimodules induces a stable equivalence between A and B, if C ⊗_{B} C^{∨} ≃ A ⊕ P in H(A ⊗_{O} A^{op}-mod) and C^{∨} ⊗_{A} C ≃ B ⊕ Q in H(B ⊗_{O} B^{op}-mod), where P and Q are bounded complexes of projective bimodules.

Note that if V ∈ B-stmod, then the image of V in A-stmod is obtained by taking the image of C ⊗_{B} V under the composition

H^{b}(A-mod) → D^{b}(A-mod) → D^{b}(A-mod)/H^{b}(A-proj) ≃ A-stmod

of exact functors. Moreover, by using an argument of Rickard, one may replace C with an exact (A, B)-bimodule M, so that the stable equivalence is induced by the exact functor M⊗_{B}: B-mod → A-mod.

4.4. Assume that A and B are derived equivalent. Then:

a) The Grothendieck groups K_{0}(A) and K_{0}(B) are isomorphic (Rickard).

b) The centers Z(A) and Z(B) are isomorphic, and more generally the Hochschild cohomology, rings HH^{∗}(A) and HH^{∗}(B) are isomorphic (Rickard).

c) The cyclic homology HC_{∗}(A) ≅ HC_{∗}(B) (Keller).

d) ℓ(A) = ℓ(B) (Happel).

e) k(A) = k(B). Moreover, if A and B are blocks of group algebras, and if the simple KA-module X corresponds to the simple KB -module Y , then def(X) = def(Y), that is, k_{δ}(A) = k_{δ}(B) for δ ≥ 0 (Broué).

f) A and B have the same representation type.

5. Some examples

5.1. p-nilpotent groups. Assume that G = D ⋊ H, where D is a Sylow p-subgroup of G. Then B_{0}(OG) ≃ OP.

5.2. Isomorphic blocks. Let H ⊴ G, D a Sylow p-subgroup of G, and assume that p ∤ [G : H] and HC_{G}(D) = G. Then B_{0}(OG) ≃ B_{0}(OH) via restriction.

5.3. T.I. situation. Assume that G has trivial intersection Sylow p-subgroups (i.e. D ∩ gD = 1 for g ∈ G \ N_{G}(D), where D is a Sylow p-subgroup of G). Let A = B_{0}(OG) and B = B_{0}(ON_{G}(D). Then it is not very difficult to see that the bimodule _{A}A_{B} induces a Morita stable equivalence between A and B.

An interesting example for the T.I. situation is the Suzuki group G = Sz(8) with p = 2. Then A and B are stably equivalent, but not derived equivalent. In fact, Z(kA) ≃ Z(kB), but Z(A) ≄ Z(B) (Cliff).

A conjecture of Auslander predicts that if two self-injective algebras over a field are stably equivalent, then they have the same number of non-projective simple modules. This would imply Alperin’s conjecture in the case of abelian defect groups.

By using the classification of finite simple groups, Blau and Michler proved the following theorem.

5.4. Theorem. If G has T.I. Sylow p-subgroups, then ℓ(A) = ℓ(B).

6. The abelian defect group conjecture

Let A = aOG be a block of OG with defect group D, let H = N_{G}(D), and let B = bOH be the block of OH corresponding to A.

6.1. Broué’s Conjecture. If D is abelian, then the algebras A and B are Rickard equivalent.

6.2. Actually, it is conjectured that the following refinements of 6.1 are also true.

a) The complex _{A}C_{B} inducing the equivalence is splendid, that is, its components are direct summands of O(G × H)-modules of the form Ind_{δ(Q)}^{G×H} O, where Q ≤ D and δ(Q) is the diagonal subgroup of Q × Q.

These come from equivalences between the source algebras of the blocks, and their main feature is that certain summands of Br_{δ(Q)}^{G×H} induce equivalence between certain blocks of OC_{G}(Q) and OC_{H}(Q) involved in Br_{Q}(a) and Br_{Q}(b), for any subgroup Q of D. This refinement relies on the fact that in the situation of 6.1, the Brauer categories of A and B are equivalent.

By 4.4, the conjecture 6.1 implies the conjecture of Alperin and Dade in the abelian defect group case. As Broué has pointed out, the “splendid” form of 6.1 also implies the validity of the Isaacs-Navarro conjecture for blocks with abelian defect groups.

The concept of splendid equivalence introduced by Rickard in the case of principal blocks, and it was generalized to arbitrary blocks by Linckelmann and Harris. A splendid Morita equivalence is also called a Puig equivalence. Moreover, Puig studied in a very general setting the local structure of equivalences between blocks, and he observes a more general class of equivalences which are compatible with the Brauer construction, the so called basic equivalences. We refer the reader to Rouquier [47] for a detailed discussion of this topic.

b) The equivalence in 6.1 is compatible with p′ -extensions of the group G (see also 7.3 below).

c) The equivalence in 6.1 is compatible with p-central extensions of the group G (see also 7.4 below).

d) The equivalence in 6.1 takes the trivial module to the trivial module. More precisely, if A and B are the principal blocks, then C ⊗_{B} O ≃ O.

7. Methods

Here A and B are the same as in the preceding section.

7.1. Gluing local equivalences. With the same notations, let C be a splendid complex of (A, B)-bimodules and assume that A is the principal block. By a theorem of Rouquier, the complex C induces a stable equivalence between A and B if and only if Br_{δP}^{A⊗B^{op}}C induces a Rickard equivalence between kC_{H}(P)a_{P} and kC_{H}(P)b_{P} for every nontrivial subgroup P of D, where a_{P} = Br_{P} (a) and b_{P} = Br_{P} (b) are the principal blocks of kC_{H}(P)e_{P} and kC_{H}(P)f_{P} respectively.

A much more general discussion of gluing splendid tilting complexes was done by Rouquier [24]. Together with the next subsection, this raises the possibility of proving Broué’s conjecture by induction.

7.2. Lifting stable equivalences to Rickard equivalences. The following strategy devised by Okuyama is widely used. Assume that we have a stable Morita equivalence between A and B, and that we can construct a tilting complex for B with endomorphism ring C. Then B and C are also stably equivalent of Morita type, hence so are A and C. Assume that the stable equivalence between A and C send simple A-modules to simple C -modules. Then, by a theorem of Linckelmann, A and C are in fact Morita equivalent.

Various useful constructions of tilting complexes have been proposed by Okuyama and Rickard.

7.3. Clifford theory. One strategy to prove the above conjectures is to reduce them to the case of simple groups and to use the classification and the available information on their representations. To this goal, it is important to investigate the relationship between representations of a group and the representations of its normal subgroups and its factor groups. This is the object of “Clifford theory”. A reduction theory for Broué’s conjecture for principal blocks has been given by Marcus, and it is based on the properties of graded bimodules and equivalences induced by them.

a) (Going up) In our situation, assume that G is a normal subgroup of a group G̃ and let F = G̃/G be a p′ -group. Let also H̃ = NG̃(D), and assume that a (and hence b) is F -invariant. The compatibility with p′ -extensions 6.2 b) requires that the complex C inducing the equivalence between A and B extends to a complex of Δ-modules, where Δ = ON_{G̃×H̃}(δP). Then one may lift an equivalence between A and B to an equivalence between the F -graded algebras aOG̃ and bOH̃ induced by the complex (aOG̃ ⊗_{O} (bOH̃)^{op}) ⊗_{Δ} C.

b) (Going down) Assume that G has a normal subgroup G^{+} , and let H^{+} = H ∩ G^{+} . Assume also that a ∈ OG^{+} and b ∈ OH^{+} , so A and B are F -graded algebras, where F = G/G^{+} . Let C be a complex inducing an equivalence between A and B. If C is a complex of F -graded (A, B)-bimodules, then then the 1-component (where 1 ∈ F) of C induces an equivalence between aOG^{+} and bOH^{+} . One useful case is when F is a cyclic p′ -group. If there is an action of the group F̂ = Hom(F, K^{×}) on C, then C is F -graded.

7.4. Lifting equivalences to p-central extensions. Assume that G = Ĝ/P is, where P is a central p-subgroup of Ĝ, and let Ĥ be the inverse imags of H in Ĝ. The blocks a and b lift to blocks of OĜ and OĤ. By the compatibility with p-central extensions 6.2 c), one may lift an equivalence between A and B to an equivalence between ãOG̃ and b̃OH̃.

8. The status of Broué’s conjecture

8.1. Broué’s conjecture appears to be very hard to check even on particular examples. The following list contains cases where Broué’s conjecture has been verified.

(1) D cyclic (Rickard, Linckelmann, Rouquier).

(2) D ≃ C_{2} × C_{2} (Erdmann, Linckelmann, Rickard, Rouquier).

(3) A principal block, p = 2 (Rouquier, Okuyama, Okuyama-Golan, and the reduction theorem by Marcus).

(4) A principal block with defect group D ≃ C_{3} × C_{3} (Okuyama, Koshitani, Kunugi, Waki, and the reduction theorem by Marcus).

(5) G a symmetric group (Rickard, Chuang-Kessar, Chuang-Rouquier) or an alternating group (Marcus).

(6) G p-solvable (there is a Morita equivalence) (Dade, Puig, Harris-Linckelmann).

(7) non-principal blocks of G = HS, G = O′N and G = He, p = 3, D ≃ C_{3} × C_{3} (Koshitani, Kunugi, Waki, Holm).

(8) non-principal blocks of G = 2.J_{2} and G = SL_{2}(p^{2}), p = 5, D ≃ C_{5} × C_{5} (Holloway).

(9) principal blocks of G = J_{2} and G = Sp_{4}(4), p = 5, D ≃ C_{5} × C_{5} (Holloway).

(10) principal blocks of G = G_{2}(2^{n}), D ≃ C_{5} ×C_{5} , 5 | 2^{n} +1, 25 ∤ 2^{n} +1 (Usami-Yoshida).

(11) G a connected reductive algebraic group over 𝔽_{q}, where p divides q − 1 but p does not divide the order of the Weil group (Puig).

(12) Unipotent blocks of weight 2 of G = GL_{n}(q), p ∤ q, D ≃ C_{p^{α}} × C_{p^{α}} (Turner, Hida-Miyachi).

(13) principal block of G = SL_{2}(p^{n}) (Okuyama).

(14) principal block of G = SU_{3}(q^{2}), p > 3 and p | q + 1, D ≃ C_{p^{α}} × C_{p^{α}}, where p^{α} = (q + 1)_{p} (Kunugi-Waki).

(15) principal block of G = Sp_{4}(q), p > 2 and p | q+1, D ≃ C_{p^{α}} ×C_{p^{α}}, where p^{α} = (q+1)_{p} (Kunugi-Okuyama-Waki).

(16) G = GL_{2}(q), p ≠ 2 prime dividing q − 1 or q + 1 (Puig, Rouquier, Gonard).

8.2. Interesting equivalences still exist in some cases of non-abelian defect groups.

(1) Consider the Chevalley group G = G_{2}(q), where q is a power of 2 and q ≡ 2, 5 (mod 9). Here p = 3 and the Sylow 3-subgroup D of G_{2}(q) is the extraspecial group M(3) of order 27 and exponent 3. Then there is a Morita stable equivalence between the principal block of OG_{2}(q) and the principal block of ON_{G}(Z(D)). This can be used to show that the principal blocks of OG_{2}(q) and OG_{2}(2) are Puig equivalent (Usami-Nakabayashi).

(2) Let G = PGL_{3}(4), p = 3, D ≃ M(3), H = N_{G}(P), where P = D∩PSL_{3}(4) ≃ C_{3}×C_{3} is a Sylow 3-subgroup of PSL_{3}(4). Then the principal blocks of OG and OH are splendidly Rickard equivalent (Kunugi-Usami)

(3) Let G = GL_{2}(q), q ≡ ±3 (mod 8), p = 2, and let P be a Sylow 2-subgroup of SL_{2}(q) (so P is abelian). Then there is a splendid Rickard equivalence between the principal blocks of OG and ON_{G}(P) (Gonard).

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