INTRODUCTION TO BLOCK THEORY

RADHA KESSAR

Abstract. This article is intended to introduce the non-specialist to some aspects of the modular representation theory of finite groups. We will show how fusion systems arise in the context of block theory and discuss the connections known and conjectural, between local and global invariants of blocks.

1. Introduction

This article is an expanded version of two talks given in April 2005 at the Centre Bernoulli during the workshop “ Topology and Representation Theory”. In some sense block theory has been the catalyst for the new channels of communication between finite groups and homotopy theory which the theory of p-local finite groups has opened up: fusion systems were introduced by Puig in order to distill the common essence of fusion in groups and in their block algebras. This article is far from being a survey. The aim here is really to acquaint people outside the area with the language of the subject, and to give some idea of how fusion ties up with other aspects of the theory. Some proofs are included, again for the sake of imparting flavour rather than for completeness. There are also some explicitly worked out examples which would be trivial for the specialist but may be of some use to others.

In the rest of this section we lay out the basic groundwork. Sections 2 and 3 are the core sections-in these we introduce the Brauer homomorphism, Brauer pairs, defect groups and fusion systems of blocks. These two sections are quite detailed, and we give proofs or ideas thereof all statements. Section 4 gives the structure of blocks with central defect groups, as well as some results on blocks of normal subgroups. The selection of statements here has been made with a view to giving a “fusion system” statement of Alperin’s weight conjecture, which is the subject of section 5. Section 6 deals with some character theoretic ramifications of fusion coming from Brauer’s second main theorem, and we also present a version of Robinson’s ordinary weight conjecture. This section contains no proofs at all. In the final section, we describe the fusion systems of principal blocks (Brauer’s third main theorem) and blocks of symmetric groups.

Throughout the article G will denote a finite group and p a prime. Recall that for R a commutative ring with identity, the group ring RG is the free R-module with a basis consisting of elements of G, and with multiplication the unique Rlinear extension to RG of the group multiplication in G. Even though the identity 1_{RG} of RG is the same as the identity 1_{G} of the group G, we will sometimes use 1_{G} and at others 1_{RG}. All RG-modules will be left modules, unless explicitly stated otherwise.

1.1. Noetherian Group Rings. Let R be a noetherian integral domain. Then RG is also noetherian. Further, note that the set of class sums of G (i.e. the elements ∑ g where g runs over a conjugacy class of elements of G) is an R-basis of the center Z(RG) of G, so in particular Z(RG) is noetherian and we get the following.

• The identity 1_{RG} has a primitive central idempotent decomposition, i.e. there is a decomposition

1_{RG} = β_{1} + ⋯ + β_{t}

such that each β_{i} is a primitive idempotent of Z(RG), 1 ≤ i ≤ t and such that β_{i}β_{j} = 0 for any j different from i.

• The above decomposition is unique. So, any central idempotent of RG is the sum of a unique subset of certain of the idempotents appearing in the above decomposition.

• For 1 ≤ i ≤ t, set A_{i} = RGβ_{i}. Then A_{i} is indecomposable as ring and

RG = A_{1} × ⋯ × A_{t},

is the unique decomposition of RG into a product of indecomposable factors.

• RG is an RG ⊗_{R} (RG)^{op}-module via left and right multiplication by group

elements and

RG = A_{1} ⊕ ⋯ ⊕ A_{t},

is the unique decomposition of RG into RG ⊗_{R} (RG)^{op}-indecomposable summands.

The following trivial observation is handy when dealing with idempotents.

• Let γ be a central idempotent of RG and let β be a primitive central idempotent of RG. Then γβ ≠ 0 is equivalent to γβ = β, and this in turn is equivalent to β appearing in the decomposition of γ into a sum of primitive idempotents of Z(RG).

For two central idempotents β and γ of RG such that γβ = β, we will say that γ contains β. We will also use the following standard notation. Suppose that U is an RG-module which is R-free of finite R-rank n. We will denote by χ_{U} the character of U, i.e.

χ_{U} : RG → R

is the function sending an element a of RG to tr(ρ(a)), where ρ : RG → Mat_{n}(R) is a representation of RG corresponding to U (with respect to some R-basis of U). Note that χ_{U} is completely determined by its values on group elements.

1.2. p-modular systems. Let p be a prime number. A p-modular system is a triple (K, O, k) where O is a local principal ideal domain, K is the field of quotients of O and k is the residue field of O, i.e. k is the quotient of O by the unique maximal ideal J(O) of O such that the following hold:

(i) O is complete with respect to the natural topology induced by its unique maximal ideal J(O).

(ii) K has characteristic 0 and

(iii) k has characteristic p.

Example 1.1. Let L be a number field, A the ring of algebraic integers in L, 𝔭 a prime ideal of A lying over p. Let A_{𝔭} be the localization of A with respect to 𝔭 and let O = \underleftarrow{lim} A_{𝔭}/𝔭^{n}, the completion of A with respect to the 𝔭-adic topology.

Then O is a local principal ideal domain, and the corresponding triple (K, O, k) is a p-modular system.

Let G be a finite group. In order to avoid complications arising from rationality considerations, it is convenient to ask that K contains a primitive |G|-th root of unity; the above example shows that this condition may easily be met. In fact, it will be expedient to work under a yet stronger assumption. For the remainder of the article, we adopt the following notation.

Assumption 1.2. (K, O, k) is a p-modular system such that k is algebraically closed and K contains a primitive |G|-th root of unity.

For a proof of the existence of p-modular systems satisfying the above assumption, see [Se, Chapter 2, Section 3, Theorem 1 and Chapter 2, Section 5, Theorem 3].

1.3. The group algebra KG. By Maschke’s theorem [NT, Chapter 3, Theorem 1.22] KG is a semi-simple algebra. Since K contains all primitive |G|-th roots of unity by Assumption 1.2, the splitting field theorem of Brauer [NT, Chapter 3, Theorem 4.11] gives that KG is split-semi-simple. By the Wedderburn structure theorem for split semi-simple algebras, it follows that the primitive central idempotents of KG are in bijection with the isomorphism classes of simple KG-modules and that for each primitive idempotent e of Z(KG), the corresponding algebra factor KGe is a matrix algebra over K of K-dimension the square of the K-dimension of the underlying simple KG-module. Furthermore, the isomorphism class of a simple KG-module V is completely determined by its character χ_{V}. For this reason, we will abusively use the notation Irr_{K}(G) both for a set of representatives of simple KG-modules and for the set of their characters. Denoting, for each χ ∈ Irr_{K}(G), by e_{χ} the central idempotent of KG corresponding to χ, we have

KG ≅ ∏_{χ∈Irr_{K}(G)} KGe_{χ},

and for each χ ∈ Irr_{K}(G), KGe_{χ} ≅ Mat_{χ(1)}(K).

Also, note that by the fact that all simple-KG modules are realizable over the cyclotomic extension of ℚ by a primitive |G|-th root of unity, the representation theory of KG is essentially that of ℂG. In particular, there is an identification of Irr_{K}(G) with Irr_{ℂ}(G), and we will use this fact without comment.

1.4. Blocks. For an element α of O denote by ᾱ the element α + J(O). Extending the natural surjection α → ᾱ, of O onto k, and the inclusion of O in K to the corresponding group algebras, yields the following diagram of ring homomorphisms.

\xymatrix{OG \ar@{^{(}->}[r] \ar@{->>}[d] & KG \\ kG}

Here the horizontal map is inclusion and the vertical map sending an element a := ∑_{g∈G} α_{g}g of OG to the element ā := ∑_{g∈G} ᾱ_{g}g of kG is a surjective map.

Since class sums form a basis of the center of any group ring (of a finite group) restricting the above homomorphisms to Z(OG) yields the following diagram of ring homomorphisms.

\xymatrix{Z(OG) \ar@{^{(}->}[r] \ar@{->>}[d] & Z(KG) \\ Z(kG)}

This allows us to compare central idempotents in OG, KG and kG.

If β is a central idempotent of OG, then clearly β̄ ∈ Z(kG) and β̄β̄ = β̄. In fact, since the kernel of the homomorphism Z(OG) → Z(kG) is contained in the radical of Z(OG), idempotent lifting results yield the following.

• If β is a primitive idempotent of Z(OG), then β̄ is a primitive idempotent of Z(kG).

Definition 1.3. Let R = O or k. A block of RG is a primitive idempotent β of the center, Z(RG), of RG; the algebra RGβ is the corresponding block algebra.

Thus

• β → β̄ is a bijection between the blocks of OG and the blocks of kG. Now let β be a block of OG. Then β is a central idempotent of KG but β is not

in general primitive in Z(KG). So, OGβ and kGb are indecomposable algebras but KGβ is not. Let Irr_{K}(G, β) consist of those elements χ of Irr_{K}(G) which satisfy be_{χ} = e_{χ} or equivalently be_{χ} ≠ 0. Then

β = ∑_{χ∈Irr_{K}(G,β)} e_{χ};

KGβ = ∏_{χ∈Irr_{K}(G,β)} Mat_{χ(1)}(K).

If χ ∈ Irr_{K}(G, β) we say that χ belongs to β or that β contains χ. Note that since distinct blocks are orthogonal, each element of Irr_{K}(G) belongs to a unique block of OG.

Definition 1.4. The principal block of OG is the block containing the 1-dimensional trivial character of KG ; the corresponding block of kG is called the principal block of kG.

Let us record a few facts in this context.

• Let χ ∈ Irr_{K}(G) belong to the block β. If e_{χ} is an element of OG, then β = e_{χ}; otherwise Irr_{K}(G, β) contains at least two elements.

• If e_{χ} is an element of OG, then OGβ = Mat_{χ(1)}(O); otherwise neither OGβ nor kGβ̄ is a product of matrix algebras.

• If p does not divide the order of G, then for any χ ∈ Irr_{K}(G), e_{χ} is in fact an element of OG, hence the blocks of OG are the same as the primitive idempotents of KG, and

OG = ∏_{χ∈Irr_{K}(G)} Mat_{χ(1)}(O).

• If p does divide the order of |G|, then denoting by χ_{0} the 1-dimensional trivial character of KG, the corresponding central idempotent e_{χ_{0}} = \frac{1}{|G|} ∑_{g∈G} g is not an element of OG, hence the principal block of G contains at least two characters. In particular, kG is not a semi-simple algebra.

Example 1.5. If G is a p-group, then 1_{OG} is a block of OG and OG is a local algebra.

Example 1.6. Let G = S_{3}, the symmetric group on 3 letters. Then KG has up to isomorphism, three simple modules- two of dimension 1 (corresponding to the trivial and sign representations) and one of dimension 2(the quotient of the natural permuation module for G by the submodule of G-fixpoints). Let us denote the corresponding characters by χ_{0}, χ_{1} and χ_{2} respectively. So,

KG = K × K × Mat_{2}(K).

We can write the corresponding primitive central idempotents using “Fourier Inversion” [NT, Chapter 3, Lemma 2.21]:

e_{χ_{0}} = \frac{1}{6} ∑_{g∈G} g, e_{χ_{1}} = \frac{1}{6} ∑_{g∈G} sgn(g)g, e_{χ_{2}} = \frac{2}{3}1_{G} − \frac{1}{3} ∑_{g∈A_{3}−1_{G}} g.

(i) If p = 2, then OG has two blocks,

β := e_{χ_{0}} + e_{χ_{1}} = \frac{1}{3} ∑_{g∈A_{3}} g

and

γ := e_{χ_{2}} = \frac{2}{3}1_{G} − \frac{1}{3} ∑_{g∈A_{3}−1_{G}} g.

OGγ ≅ Mat_{2}(O) and OGβ is isomorphic to the group algebra of a group of order 2.

(ii) If p = 3, then OG has a unique block, i.e. 1_{OG} is primitive in Z(OG), and OG is indecomposable.

(iii) If p ≥ 5, then

OG = O × O × Mat_{2}(O).

1.5. The cde triangle. The two way traffic between characteristic 0 and characteristic p representation theory established by p-modular systems can be encoded in the following commutative diagram of group homomorphisms,

\xymatrix{R_{K}(G) \ar[dr]_{d} &  & P_{k}(G) \ar[ll]_{e} \ar[dl]^{c} \\  & R_{k}(G)}

where R_{K}(G), R_{k}(G) and P_{k}(G) denote respectively the Grothendieck groups of the categories of finitely generated KG, kG and projective kG-modules. We briefly indicate how the homomorphisms d, e and c arise and mention some of their key properties.

• Let V be a finitely generated KG-module. Since O is a principal ideal domain, V has an O-form, i.e. there exists an OG-module M which is O-free and such that V = K ⊗_{O} M. Then, M̄ = M/J(O)M is a kGmodule of the same k-dimension as the K-dimension of V. Thus M̄ is a “reduction modulo p” of V. Neither the isomorphism class of M nor that of M̄ is determined by the isomorphism class of V in general. However, the multiplicities of simple kG-modules as composition factors of M̄ are determined by the isomorphism class of V. The map d : R_{K}(G) → R_{k}(G) is the unique group homomorphism sending [V] → ∑_{d_{V S}} d_{V S}[S] , where S runs over a set of representatives of isomorphism classes of simple kGmodules and where d_{V S} denotes the multiplicity of S as a composition factor of M̄.

• The process described above cannot in general be reversed, i.e. it is not the case that for any finitely generated kG-module W, there is an O-free OG-module M with W ≅ M/J(O)M. However, it can be reversed at the level of projective kG-modules. More precisely,

– The map M → M/J(O)M induces a bijection between isomorphism classes of projective OG-modules of finite O-rank and isomorphism classes of finite dimensional projective kG-modules.

(The above fact is the module theoretic manifestation of idempotent lifting and relies on the completeness of O.) The map e : P_{k}(G) → R_{K}(G) is the unique group homomorphism which sends the image in P_{k}(G) of a projective kG-module P to the image in R_{K}(G) of K ⊗ P̃, where P̃ is a lift of P to a projective OG-module.

• The map c : P_{k}(G) → R_{k}(G) is the obvious one, i.e. c is the unique group homomorphism which sends the image in P_{k}(G) of a projective kG-module P to the image in R_{k}(G) of P.

It is easy to see that the cde triangle is commutative. Furthermore,

• The homomorphism e is transpose to d in the following sense. First, we note that P → P/J(P) induces a bijection between isomorphism classes of indecomposable projective kG-modules and isomorphism classes of simple kG-modules.

Let S be a simple kG-module and let P_{S} be a projective indecomposable kG-module with P_{S}/J(P_{S}) ≅ S. Then abusively letting Irr_{K}(G) denote a set of representatives of simple KG-modules, we have

e([P_{S}]) = ∑_{V ∈Irr_{K}(G)} d_{V S}[V]

• The map d is surjective (this last fact is a consequence of Brauer’s characterization of characters). The image of P_{k}(G) under c is a subgroup of full rank of R_{k}(G), and the map e is a split injection.

Last, but not least

• The maps c, d an e all commute with the passage from the group algebra to block algebras.

2. Brauer Pairs

This section as well as the next presents the results of Alperin and Broué [AB]. Notation. (i) Let a finite group H act on a set X. For a ∈ X and h ∈ H,

we will denote by ^{h}a the image of a under the action of h, by Orb_{H}(X) the set of orbits of H on X and by X^{H} the fixed point set of the action.

If X is a k-vector space and the action of H on X is k-linear, then for a subgroup L of H, and an element a of X^{L}, Tr_{L}^{H}(a) denotes the relative trace ∑_{h∈H/L} ^{h}a of a, and Tr_{L}^{H}(X^{L}) denotes the subspace of X^{H} consisting of such relative traces.

(ii) For any finite subset S of kG we will denote by Ŝ the sum of the elements of S.

2.1. The Brauer homomorphism.

Definition 2.1. Let Q be a subgroup of G. For an element a = ∑_{g∈G} α_{g}g of kG, set Br_{Q}(a) := ∑_{g∈C_{G}(Q)} α_{g}g ∈ kG. The map

Br_{Q} : kG → kC_{G}(Q), a → Br_{Q}(a)

is clearly a surjective k-linear map.

Let Q be a subgroup of G. The conjugation action of Q on G extends naturally to an action of Q on kG and the set of Q-class sums {X̂ : X ∈ Orb_{Q}(G)} is a k-basis of (kG)^{Q}. Note that if X, Y ∈ Orb_{Q}(G), and z ∈ C_{G}(Q) then either the coefficient of z in X̂Ŷ is 0 or |X̂| = |Ŷ| and the coefficient of z in X̂Ŷ is |X̂|. Thus, if Q is a p-group and either |X̂| > 1 or |Ŷ| > 1, then the coefficient of any element of C_{G}(Q) in X̂Ŷ is 0. This yields the following crucial property of the Brauer homomorphism.

Proposition 2.2. Let Q be a p-subgroup of G. Then for any a, a′ ∈ (kG)^{Q}, Br_{Q}(aa′) = Br_{Q}(a)Br_{Q}(a′). Consequently, the map

Br_{Q} : (kG)^{Q} → kC_{G}(Q), a → Br_{Q}(a)

is a surjective homomorphism of k-algebras.

We record another important property of the Brauer homomorphism.

Proposition 2.3. Suppose that Q is a normal p-subgroup of G. Let I be the kernel of the surjective algebra homomorphism

kG → k(G/Q)

induced by the canonical surjective group homomorphism G → G/Q. Then

Ker(Br_{Q}) ∩ (kG)^{Q} ⊆ I ⊆ J(kG).

Proof. As seen above {X̂ : X ∈ Orb_{Q}(G), |X| > 1} is a k-basis of Ker(Br_{Q}) ∩ (kG)^{Q}. So, let X ∈ Orb_{Q}(G) with |X| > 1 and let x ∈ X. Then, X̂ = Tr_{C_{Q}(x)}^{Q}(x) and C_{Q}(x) is a proper subgroup of Q. Thus

X̂ = X̂ − [Q : C_{Q}(x)]x = ∑_{y∈Q/C_{Q}(x)}(^{y}x − x).

Now note that for any x ∈ G and any y ∈ Q we have ^{y}x − x ∈ I, hence the first inclusion follows. The second inclusion is equivalent to saying that for any y ∈ Q and any simple kG-module V, (y − 1)V = 0, or that V^{Q} = V. Since Q is normal in G, V^{Q} is a kG-submodule of V, hence the simplicity of V as kG module implies that either V^{Q} = 0 or V^{Q} = V. But V^{Q} ≠ 0 since the trivial module is the only simple kQ-module.

□

Remark 2.4. Since Ker(Br_{Q}) ∩ (kG)^{Q} is the sum of relative traces Tr_{L}^{Q}((kG)^{L}), where L runs over the set of proper subgroups of Q, the map Br_{Q} on (kG)^{Q} may be identified with the natural surjection of (kG)^{Q} onto (kG)^{Q}/ ∑_{L} Tr_{L}^{Q}((kG)^{L}). This point of view gives a natural generalisation of the Brauer homomomorphism (on fixed point sets) to arbitrary Q-algebras.

2.2. Inclusion of Brauer pairs.

Definition 2.5. A Brauer pair (for G) is a pair (Q, f) where Q is a p-subgroup of G and f is a block of kC_{G}(Q). Note that G acts by conjugation on the set of Brauer pairs. For a Brauer pair (Q, f) we will denote by N_{G}(Q, f) the stabilizer in G of (Q, f) under this conjugation action. For a p-subgroup Q of G, denote by B(Q) the set of blocks of kC_{G}(Q).

Consider p-subgroups Q and R of G such that Q⊴R. Then R acts by conjugation on B(Q). For any a ∈ kG, and any x in R, Br_{R}(^{x}a) = Br_{R}(a). Since k is of characteristic p, one sees that if X is any R-orbit of B(Q) of length greater than 1, then Br_{R}(X̂) = 0. Thus, noting that Br_{Q} is k-linear on all kG,

1_{kC_{G}(R)} = Br_{R}(1_{kC_{G}(Q)}) = Br_{R}( ∑_{f∈B(Q)} f) = ∑_{f∈B(Q)^{R}} Br_{R}(f)

Now let f ∈ B(Q)^{R}, i.e. f ∈ (kG)^{R}. By the multiplicative property of the Brauer homomorphism, Br_{R}(f) = Br_{R}(ff) = Br(f)Br(f). Thus, either Br_{R}(f) is zero or Br_{R}(f) is an idempotent. Furthermore, since C_{G}(R) ≤ C_{G}(Q), Br_{R}(Z(kC_{G}(Q)) ⊆ Z(kC_{G}(R)). Hence, if Br_{R}(f) ≠ 0, then Br_{R}(f) is a central idempotent of kC_{G}(R). Finally, for f′ in B(Q)^{R}, different from f, Br_{R}(f)Br_{R}(f′) = Br_{R}(ff′) = 0. From the equation displayed above, we get the following.

Proposition 2.6. Let R be a p-subgroup of G and let e ∈ B(R). Let Q be a normal subgroup R. Then there exists a unique element f of B(Q)^{R} such that Br_{R}(f)e = e. Furthermore, if f′ ∈ B(Q)^{R} is different from f, Br_{R}(f′)e = 0.

The next proposition provides a crucial property of inclusion.

Proposition 2.7. Let R be a p-subgroup of G and let e ∈ B(R). Let Q and T be normal subgroups of R with Q ≤ T. Let f be the unique element of B(Q)^{R} such that Br_{R}(f)e = e, h the unique element of B(T)^{R} such that Br_{R}(h)e = e and f′ the unique element of B(Q)^{T} with Br_{T} (f′)h = h. Then f′ = f.

Proof. By Proposition 2.6, it suffices to show that f′ is R-stable and Br_{R}(f′)e = e. So, let x ∈ R. Then ^{x}f′ ∈ B(Q). Since f′ is T -stable and T ⊴ R, ^{x}f′ is T -stable and Br_{T} (^{x}f′) = ^{x}Br_{T} (f′). Since h is R-stable, we have

Br_{T} (^{x}f′)h = ^{x}Br_{T} (f′)h = ^{x}(Br_{T} (f′)h) = ^{x}h = h.

By Proposition 2.6, it follows that ^{x}f′ = f′, i.e. f′ ∈ (kG)^{R}. Thus, Br_{T} (f′) ∈ (kG)^{R}, and

Br_{R}(f′)e = Br_{R}(f′)Br_{R}(h)e = Br_{R}(Br_{T} (f′))Br_{R}(h)e = Br_{R}(Br_{T} (f′)h) = Br_{R}(h)e = e.

□

We are now ready to define the inclusion of Brauer pairs.

Definition 2.8. Let (Q, f) and (R, e) be Brauer pairs. Then (i) (Q, f) ⊴ (R, e) if Q ⊴ R, f is R-stable and Br_{R}(f)e = e. (ii) (Q, f) ≤ (R, e) if Q ≤ R and there exist Brauer pairs (S_{i}, d_{i}), 1 ≤ i ≤ n such

that

(Q, f) ⊴ (S_{1}, d_{1}) ⊴ (S_{2}, d_{2}) ⊴ ⋯ ⊴ (S_{n}, d_{n}) ⊴ (R, e).

Clearly, ≤ is a transitive relation on the set of Brauer pairs. Furthermore, since any inclusion Q ≤ R of p-groups can be refined to a chain of normal inclusions, it follows by Proposition 2.6 that given a Brauer pair (R, e) and a subgroup Q of R, there exists f ∈ B(Q) such that (Q, f) ≤ (R, e). In fact, the uniqueness assertion of Proposition 2.6 and Proposition 2.7, along with a inductive argument give that f unique, i.e. we have the following.

Theorem 2.9. Let (R, e) be a Brauer pair. For any subgroup Q of R, there exists a unique f ∈ B(Q) such that (Q, f) ≤ (R, e). Furthermore, if Q ⊴ R, then f is the unique f ∈ B(Q)^{R} with Br_{R}(f)e = e. The conjugation action of G on the set of Brauer pairs preserves ≤.

Remark 2.10. There is another characterisation of Brauer pair inclusion due to Broué and Puig [BP]: (Q, f) ≤ (R, e) if and only if Q ≤ R and there is a primitive idempotent i of (kG)^{R}, a primitive idempotent j of (kG)^{Q} with Br_{R}(i)e = i, Br_{Q}(j)f = j and such that ij = j (note that if Q ≤ R, then (kG)^{R} ⊆ (kG)^{Q}).

2.3. Descent to local subgroups. Let Q be p-subgroup of G and let H be a subgroup of G such that QC_{G}(Q) ≤ H. Let R be a p-subgroup of H containing Q. Then since C_{G}(R) ≤ C_{G}(Q) ≤ H, we have C_{G}(R) = C_{H}(R), hence a Brauer pair for G with first coordinate R is a Brauer pair for H and vice versa. Furthermore, it is easy to see from Proposition 2.6 and Proposition 2.7 that for a Brauer pair (R, e), (Q, f) ≤ (R, e) as Brauer pairs for G if and only if (Q, f) ≤ (R, e) as Brauer pairs for H.

3. b-Brauer pairs

Noting that B(1) is just the set of blocks of kG, by Theorem 2.9 for any Brauer pair (R, e), there is a unique block b of kG such that (1, b) ≤ (R, e). Since all blocks of G are R-stable, b may be characterized as the unique block of kG such that Br_{R}(b)e = e or indeed by Proposition 2.6 as the unique block of kG such that Br_{R}(b)e ≠ 0. Put yet another way, (1, b) ≤ (R, e) if and only if e appears in the decomposition of Br_{R}(b) into primitive idempotents of Z(kC_{G}(R)).

Definition 3.1. Let b be a block of kG.

(i) A b-Brauer pair is a Brauer pair (R, e) such that (1, b) ≤ (R, e). Equivalently, a b-Brauer pair is a Brauer pair (R, e) such that Br_{R}(b)e = e.

(ii) We denote by B(R, b) the set of blocks e of kC_{G}(R) such that (1, b) ≤ (R, e). (iii) A defect group of b is a p-subgroup of G, maximal subject to Br_{P} (b) ≠ 0.

Note that since (1, b) is stabilized by every element of G, G acts by conjugation on the set of b-Brauer pairs. Also, note from the above definition that P is a defect group of b if and only if there is a maximal Brauer pair (P, e) such that (1, b) ≤ (P, e); such a Brauer pair is called a maximal b-Brauer pair.

Example 3.2. Let p = 2. Let U be an elementary abelian group of order 9 on generators u, v. Let T = ⟨x, y | x^{4} = y^{4} = 1, xyx^{−1} = y^{3}⟩ be the quaternion group of order 8. The group T acts on U via:

xu = u^{2}, ^{x}v = v, ^{y}u = u, ^{y}v = v^{2}.

Let G = UT. Then kG has 4-blocks:

b_{0} = \frac{1}{9}(1 + û + v̂ + ûv), b_{1} = \frac{1}{9}(v̂ + ûv), b_{2} = \frac{1}{9}(û + ûv), b_{3} = \frac{1}{9}(ûv).

Below are the lattices of subgroups of T and their centralizers in G:

[figure: Q ≤ T; 1; Z(T) = ⟨x^{2}⟩ = ⟨y^{2}⟩; ⟨x⟩; ⟨y⟩; ⟨xy⟩; T; C_{G}(Q); G; G; ⟨v⟩ × ⟨x⟩; ⟨u⟩ × ⟨y⟩; ⟨xy⟩; Z(T)]

The images of three of the blocks under the Brauer homomorphisms are the following (we omit b_{2}):

[figure: Br_{Q}(b_{0}); b_{0}; b_{0}; 1 + v + v^{2}; 1 + u + u^{2}; 1; 1; Br_{Q}(b_{1}); b_{1}; b_{1}; v + v^{2}; 0; 0; 0; Br_{Q}(b_{3}); b_{3}; b_{3}; 0; 0; 0; 0]

Now we can describe the set of b-Brauer pairs of the form (Q, f) where Q ≤ T for each block b of kG (again omitting b_{2}). For the principal block b_{0} this is {(Q, e_{Q}) | Q ≤ T }, where e_{Q} is the principal block of C_{G}(Q). This is true in general, as is explained in the last section. For b_{1}, we get {(1, b_{1}), (Z(T), b_{1}), (⟨x⟩, 1 + ωv + ω^{2}v^{2}), (⟨x⟩, 1 + ω^{2}v + ωv^{2})} where ω is a primitive third root of unity in k and for b_{3} we get {(1, b_{3}), (Z(T), b_{3})}.

3.1. Defect groups and relative traces. A crucial tool for understanding the nature of defect groups is the interplay between Brauer homomorphisms and relative trace maps. The main point of connection is the fact that the center of a block algebra is a local ring.

Theorem 3.3. Let b be a block of kG.

(i) If P is a defect group of G, then b ∈ Tr_{P}^{G}(kGb)^{P} and P is minimal with respect to this property.

(ii) G acts transitively on the set of defect groups of b.

We will give a brief proof of the above theorem for which we need the following notation. Let Cl(G) denote the set of conjugacy classes of G. For a p-subgroup L of G, let Cl(G)_{≥L} denote the set of conjugacy classes of G containing an element x centralized by L, and let Cl(G)_{≤L} denote the set of conjugacy classes of G containing an element x such that L contains a Sylow p-subgroup of C_{G}(x). Clearly, Cl(G)_{≥L} and Cl(G)_{≤L} depend only on the conjugacy class of L.

Proof. Note first that since b is in Z(kG), for a p-subgroup Q of G, Br_{Q}(b) ≠ 0 if and only if Br_{^{x}Q}(b) ≠ 0 for all x ∈ G. Also, note that if S is a Sylow p-subgroup of G, then

b = \frac{1}{[G : S]} ∑_{g∈G/S} ^{g}b ∈ Tr_{S}^{G}(kG)^{S}.

In particular, there are p-subgroups R of G such that b ∈ Tr_{R}^{G}(kG)^{R}.

Now, we claim that if R and Q are p-subgroups of G, such that b ∈ Tr_{R}^{G}(kG)^{R} and such that Br_{Q}(b) ≠ 0, then ^{x}Q ≤ R for some x ∈ G. Indeed, it is easy to see that the set {X̂ : X in Cl(G)_{≤R}} is a k-basis for Tr_{R}^{G}(kG)^{R}. Thus, we may write

(1)  b = ∑_{X∈Cl(G)_{≤R}} α_{X} X̂, α_{X} ∈ k

So, Br_{Q}(b) ≠ 0 means in particular, that there exists some X ∈ Cl(G)_{≤R} such that Br_{Q}(X̂) ≠ 0, i.e. such that X ∩ C_{G}(Q) ≠ ∅. The claim is now immediate.

Next we claim that if R is minimal amongst p-subgroups of G such that b ∈ Tr_{R}^{G}(kG)^{R}, then Br_{R}(b) ≠ 0. Indeed, suppose if possible that Br_{R}(b) = 0. Considering again Equation 1, one sees that this means that for each X in Cl(G)_{≤R} such that α_{X} ≠ 0, X ∩ C_{G}(R) is empty, which in turn implies that X ∈ Cl(G)_{≤R′} for some R′ strictly contained in R. In other words,

b ∈ ∑_{R′<R} Tr_{R′}^{G}(kG)^{R′} ∩ Z(kGb).

Thus, there is a proper subgroup R′ of R such that the ideal Tr_{R′}^{G}(kG)^{R′} ∩Z(kGb) is not in the radical of Z(kGb). But this means that Tr_{R′}^{G}(kG)^{R′} ∩ Z(kGb) = Z(kGb) as Z(kGb) is a local ring. In particular, b ∈ Tr_{R′}^{G}(kG)^{R′} which contradicts the minimality of R. The theorem now follows easily. □

Let P be a p-subgroup of G. There is a natural action of N_{G}(P)/P on C_{G}(P) coming from the conjugation action of N_{G}(P) on C_{G}(P). Furthermore, for any X ∈ Cl(G), X ∩ C_{G}(P) is a union of N_{G}(P)/P -orbits, hence Br_{P} (Z(kG)) ⊆ (kC_{G}(P))^{N_{G}(P)/P}. In other words, the restriction of Br_{P} to Z(kG) gives an algebra homomorphism

Br_{P} : Z(kG) → (kC_{G}(P))^{N_{G}(P)/P} .

This is not in general a surjective map. It is however surjective on relative traces:

Lemma 3.4. (i) Let P be a p-subgroup of G. Then

Br_{P} (Tr_{P}^{G}((kG)^{P})) = Tr_{1}^{N_{G}(P)/P} (kC_{G}(P))

(ii) Let b be a block of G, P a defect group of b. Then the map

Br_{P} : Z(kGb) → (kC_{G}(P)Br_{P} (b))^{N_{G}(P)/P}

is onto.

Proof.(i) This is a consequence of the Mackey formula. It may also be seen as follows: The class sums X̂ as X runs through Cl(G)_{≤P} are a basis of Tr_{P}^{G}((kG)^{P}), and the sums Ŷ, where Y runs over the N_{G}(P)/P -orbits of elements y of C_{G}(P) whose stabilizer in N_{G}(P)/P is a p′-group form a basis of Tr_{1}^{N_{G}(P)/P} (kC_{G}(P)). Now, let X ∈ Cl(G)_{≤P}. Then either X ∩ C_{G}(P) is ∅ or P is a Sylow p-subgroup of C_{G}(x) for all x ∈ X ∩ C_{G}(P). Furthermore, if P is a Sylow p-subgroup of C_{G}(x) for all x ∈ X ∩ C_{G}(P), then C_{G}(P) ∩ X is a single N_{G}(P)-orbit (and hence a single N_{G}(P)/P orbit). The assertion in (i) follows easily.

(ii) Since P is a defect group of b, by Theorem 3.3 b ∈ Tr_{P}^{G}((kG)^{P}) whence by part (i), it follows that Br_{P} (b) ∈ Tr_{1}^{N_{G}(P)/P} (kC_{G}(P)Br_{P} (b)). It also follows from part (i) that the image of Z(kGb) under Br_{P} contains Tr_{P}^{G}((kGb)^{P}). So, the image of (Z(kGb)) under Br_{P} contains an ideal of (kC_{G}(P)Br_{P} (b))^{N_{G}(P)} which in turn contains the identity of (kC_{G}(P)Br_{P} (b))^{N_{G}(P)}. The result is immediate. □

The above lemma is used to prove the following.

Theorem 3.5. Let b be a block of G.

(i) G acts transitively on the set of maximal b-Brauer pairs.

(ii) Let (P, e) be a maximal b-Brauer pair. Then N_{G}(P, e)/P C_{G}(P) is a p′-group.

Proof. (i) Let P be a defect group of b. Since we have already shown that all defect groups of G are conjugate, it suffices to show that B(P, b) is a single orbit under the action of N_{G}(P) or equivalently under the action of N_{G}(P)/P.

Since Z(kGb) is a local ring, it follows from part (ii) of the lemma and by idempotent lifting that (kC_{G}(P)Br_{P} (b))^{N_{G}(P)/P} is a local ring, i.e. the identity Br_{P} (b) is the only idempotent of (kC_{G}(P)Br_{P} (b))^{N_{G}(P)/P}. But clearly, the elements of any N_{G}(P)-orbit of B(P, b) sum to an idempotent of (kC_{G}(P)Br_{P} (b))^{N_{G}(P)/P}. This proves (i).

(ii) Let (P, e) be a maximal b-Brauer pair and suppose, if possible, that p divides [N_{G}(P, e) : P C_{G}(P)]. By transitivity of the action of N_{G}(P) on B(P, b), p divides [N_{G}(P, f) : P C_{G}(P)] for all f ∈ B(P, b). Thus, Tr_{P C_{G}(P)}^{N_{G}(P,f)}(f) = 0 for all f ∈ B(P, b).

By transitivity of relative trace maps, Tr_{P C_{G}(P)/P}^{N_{G}(P)/P} (f) = 0 for all f ∈ B(P, b). Since B(P, b) projects to a k-basis of Z(kC_{G}(P)Br_{P} (b))/J(Z(kC_{G}(P)Br_{P} (b))), we get that

Tr_{P C_{G}(P)/P}^{N_{G}(P)/P} (Z(kC_{G}(P)Br_{P} (b))) ⊆ Tr_{P C_{G}(P)/P}^{N_{G}(P)/P} (J(Z(kC_{G}(P)Br_{P} (b)))).

But J(Z(kC_{G}(P)Br_{P} (b)) is invariant under algebra automorphisms of kC_{G}(P)Br_{P} (b), hence the above implies that

Tr_{P C_{G}(P)/P}^{N_{G}(P)/P} (Z(kC_{G}(P)Br_{P} (b))) ⊂ J(Z(kC_{G}(P)Br_{P} (b)).

The transitivity of relative trace maps and the fact that Tr_{1}^{P C_{G}(P)/P} (kC_{G}(P)) ⊂ Z(kC_{G}(P)) now imply that

Tr_{1}^{N_{G}(P)/P} (kC_{G}(P)Br_{P} (b)) ⊆ J(Z(kC_{G}(P)Br_{P} (b)).

But by Theorem 3.3 and Lemma 3.4, Br_{P} (b) ∈ Tr_{1}^{N_{G}(P)/P} (kC_{G}(P)Br_{P} (b)), a contradiction. □

3.2. Brauer’s first main theorem. By Theorem 3.5, defect groups of blocks can be locally detected. This local-global connection can be made more precise.

Theorem 3.6. [Brauer’s First Main Theorem] Let P be a p-subgroup of G. The map Br_{P} induces a bijection between the set of blocks of G with defect group P and the set of blocks of N_{G}(P) with defect group P.

Note that if b is a block of G and P is a p-subgroup of G, then Br_{P} (b) is a central idempotent of kN_{G}(P). The subtlety in the proof of the above theorem, the details of which we omit, lies in showing that if b a block of G and P is a defect group of b, then Br_{P} (b) is in fact a block of kN_{G}(P) and that P is a defect group of Br_{P} (b) as block of kN_{G}(P). This fact is a consequence of Theorem 3.5 and the following lemma.

Lemma 3.7. Suppose that Q is a normal p-subgroup of a finite group G. Then any central idempotent of kG is an element of kC_{G}(Q). Thus, the blocks of kG are precisely the sums of G-orbits of blocks of kC_{G}(Q) under the conjugation action of G. Furthermore, Q is contained in all defect groups of all blocks of kG.

Proof. Let f be an idempotent of Z(kG). Write f = f_{1} + f_{2}, where f_{1} ∈ Z(kG) ∩ kC_{G}(Q) and f_{2} ∈ Z(kG) ∩ Ker(Br_{Q}). By Proposition 2.3, f_{2} ∈ J(kG), hence f_{2}^{p^{n}} = 0 for some natural number n. So, f^{p^{n}} = f_{1}^{p^{n}} + f_{2}^{p^{n}} = f_{1}^{p^{n}}. Since f is idempotent, this means that f = f_{1}^{p^{n}} ∈ kC_{G}(Q). The second statement is a consequence of the first. The last statement is also a consequence of Proposition 2.3 since J(kG) and hence Ker(Br_{Q}) contains no central idempotent of kG. □

3.3. Fusion systems of blocks. In the sequel we freely use the notation and terminology for fusion systems as given in [Li4].

Definition 3.8. Let b be a block of G and let (P, e_{P}) be a maximal b-Brauer pair. For a subgroup Q of P, we let e_{Q} be the unique block of B(Q) such that (Q, e_{Q}) ≤ (P, e_{P}). Denote by F_{(P,e_{P})}(G, b) the category whose objects are the subgroups of P and whose morphism sets Hom_{F_{(P,e_{P} )}(G,b)}(Q, R) for Q, R ≤ P consist of all injective group homomorphisms φ : Q → R such that there is an element g ∈ G with ^{g}(Q, e_{Q}) ≤ (R, e_{R}) and such that φ(x) = ^{g}x for all x ∈ Q.

Theorem 3.9. Let b be a block of kG and let (P, e_{P}) be a maximal b-Brauer pair. (i) The category F_{(P,e_{P})}(G, b) is a fusion system on P. (ii) If (P′, e_{P′}) is another maximal b-Brauer pair, then F_{(P′,e_{P′})}(G, b) is isomor-

phic to F_{(P,e_{P})}(G, b).

Theorems 3.3 and 3.5 are the main ingredients of the proof of the above result.

Lemma 3.10. Suppose that R ≤ P is such that |N_{P} (R)| ≥ |N_{P} (^{x}R)| for any x ∈ G for which ^{x}(R, e_{R}) ≤ (P, e_{P}). Then (N_{P} (R), e_{N_{P} (R)}) is a maximal Brauer pair for the group N_{P} (R)C_{G}(R).

Proof. The backdrop for this proof is the discussion in subsection 2.3. Set H = N_{P} (R)C_{G}(R) and suppose that (S, d) is a Brauer pair for H such that (N_{P} (R), e_{N_{P} (R)}) ≤ (S, d) in H. Then, (N_{P} (R), e_{N_{P} (R)}) ≤ (S, d) in G. Since (N_{P} (R), e_{N_{P} (R)}) is a b-Brauer pair, this means that (S, d) is a b-Brauer pair. But (P, e_{P}) is a maximal b-Brauer pair and by theorem 3.5, all maximal b-Brauer pairs are conjugate in G. So, there is an x ∈ G such that (^{x}S, ^{x}d) ≤ (P, e_{P}). Thus, (^{x}R, ^{x}e_{R}) ≤ (^{x}S, ^{x}d) ≤ (P, e_{P}). Since S normalizes R, ^{x}S normalizes ^{x}R, i.e. xS ≤ N_{P} (^{x}R). But by hypothesis, |N_{P} (^{x}R)| ≤ |N_{P} (R)|. So |^{x}S| ≤ |N_{P} (R)|. On the other hand, N_{P} (R) ≤ S. Thus, N_{P} (R) = S, proving the lemma. □

Proof of theorem. Set F := F_{(P,e_{P})}(G, b). It is immediate from the definition that F is a category on P. Let u ∈ P and let (Q, e_{Q}) ≤ (P, e_{P}). Then, (^{u}Q, ^{u}e_{Q}) ≤ (^{u}P, ^{u}e_{P}) = (P, e_{P}) (as e_{P} ∈ kC_{G}(P)). By Theorem 2.9, ^{u}e_{Q} = e_{^{u}Q}. Thus conjugation by u is an element of Hom_{F}(Q, P).

It remains to show that properties I-S and II-S of [Li4, Definition 2.4] hold. Note that for Q ≤ P, Aut_{F}(Q) ≅ N_{G}(Q, e_{Q})/C_{G}(Q). In particular, I-S is immediate from Theorem 3.5 .

Let Q ≤ P and let φ : Q → P be a morphism in F such that φ(Q) is fully F-normalized. By definition, there is a g ∈ G such that ^{g}(Q, e_{Q}) = (φ(Q), e_{φ(Q)}) and such that φ is conjugation by g. Let N_{φ} be the inverse image in N_{P} (Q) of the group Aut_{P} (Q) ∩ (φ^{−1} ∘ Aut_{P} (φ(Q)) ∘ φ) [Li4, Definition 2.3]. Then,

N_{φ} = N_{P} (Q) ∩ N_{^{g^{−1}} P} (Q)C_{G}(Q), whence
gN_{φ} = N_{^{g}P} (φ(Q)) ∩ N_{P} (φ(Q))C_{G}(φ(Q)).

Set R = φ(Q) and set H = N_{P} (R)C_{G}(R).

Now (Q, e_{Q}) ≤ (N_{φ}, e_{N_{φ}}) in G implies that (R, e_{R}) ≤ (^{g}N_{φ}, ^{g}e_{N_{φ}}) in G, hence (R, e_{R}) ≤ (^{g}N_{φ}, ^{g}e_{N_{φ}}) in H. But we also have (R, e_{R}) ≤ (N_{P} (R), e_{N_{P} (R)}) in H. In particular, this means that (^{g}N_{φ}, ^{g}e_{N_{φ}}) and (N_{P} (R), e_{N_{P} (R)}) are f-Brauer pairs for the same block f of H (in fact, f = e_{R}).

On the other hand, since R is fully F-normalized, by Lemma 3.10, (N_{P} (R), e_{N_{P} (R)}) is a maximal Brauer pair for the group H. So, by Theorem 3.5 (applied to the block f of H), there is an h ∈ H such that (^{hg}N_{φ}, ^{hg}e_{N_{φ}}) ≤ (N_{P} (R), e_{N_{P} (R)}) in H and hence in G. Replacing h if necessary by a multiple of h with an element of N_{P} (R), we may assume that h ∈ C_{G}(R). Then

ψ : N_{φ} → N_{P} (R), x → ^{hg}x

is a morphism in F and ψ|Q = φ. This proves that II-S holds. Part (ii) of the theorem is an immediate consequence of the fact that all maximal b-Brauer pairs are G-conjugate. □

Local subsystems of a fusion system of a block of G correspond to fusion systems of blocks of local subgroups of G in a natural way. Note that by Lemma 3.7, for any Brauer pair (Q, e) of G, e is a block of N_{G}(Q, e). The next theorem is not hard to deduce from the preceding analysis (details can be found in [Li3]). For the notation refer to [Li4].

Theorem 3.11. Let b be a block of kG and let (P, e_{P}) be a maximal b-Brauer pair. For each Q ≤ P, let (Q, e_{Q}) ≤ (P, e_{P}) and set F := F_{(P,e_{P})}(G, b).

(i) Suppose Q ≤ P is fully F-centralized. Then C_{P} (Q) is a defect group of the block e_{Q} of kC_{G}(Q). Furthermore, (C_{P} (Q), e_{C_{P} (Q)}) is a maximal e_{Q}-Brauer pair, and F_{(C_{P} (Q),e_{C_{P} (Q)})}(C_{G}(Q), e_{Q}) = C_{F}(Q). In particular, R ≤ P is F-centric if and only if Z(R) is a defect group of the block e_{R} of kC_{G}(R).

(ii) Suppose Q ≤ P is fully F-normalized. Then N_{P} (Q) is a defect group of the block e_{Q} of kN_{G}(Q, e_{Q}), (N_{P} (Q), e_{N_{P} (Q)}) is a maximal e_{Q}-Brauer pair, and F_{(N_{P} (Q),e_{N_{P} (Q)})}(N_{G}(Q), e_{Q}) = N_{F}(Q).

4. Some structure theory.

Notation. In the rest of the article, we fix a block b of G and a maximal b-Brauer pair (P, e_{P}). We set F := F_{(P,e_{P})}(G, b).

4.1. Trivial defect group. From the point of view of representation theory, Theorem 3.3 (i) provides a measure of how far away a block algebra is from being semisimple. For the ensuing discussion, we keep in mind that a finite-dimensional k-algebra A is an (A, A)-bimodule under left and right multiplication, that is to say A is a left-module for the algebra A ⊗_{k} A^{op}.

Theorem 4.1. P = 1 if and only if kGb is isomorphic to a matrix algebra over k.

Proof. Suppose first that P = 1. By Theorem 3.3, b = Tr_{1}^{G}a for some a ∈ kG. Let M and N be kGb-modules and let

φ : M → N

be a surjective map of kG-modules. Let ψ : N → M be a k-linear splitting of φ. For each n in N, define ψ̂(n) = ∑_{x∈G} xψ(ax^{−1}n). Then, the map

ψ̂: N → M

is a kG-module splitting of φ. Thus, every kGb-module M is projective. It follows that kGb is a semi-simple ring. Since kGb is indecomposable as ring, kGb is simple. By Wedderburn’s theorem, kGb is isomorphic to a matrix algebra over k. Conversely, suppose that kGb is a matrix algebra. Then kGb is in particular a separable algebra, hence kGb is projective as kGb⊗_{k} (kGb)^{op}-module, and therefore as kG ⊗_{k} (kG)^{op}-module. From this it follows that if S is a Sylow p-subgroup of G and one considers kGb as a kS-module via the conjugation action of S then kGb is projective as kS-module. The algebra kS being local, any finite dimensional projective kS-module is free, hence kGb is isomorphic to a direct sum of copies of the regular kS module. But from this it is obvious that (kGb)^{S} = Tr_{1}^{S}(kGb). In particular, b = Tr_{1}^{S}a for some a ∈ kG. Since S is a Sylow p-subgroup of G, we get

b = \frac{1}{[G : S]}Tr_{S}^{G}b = Tr_{1}^{G}(\frac{1}{[G : S]}a).

By 3.3 it follows that b is of defect 0. □

Remark 4.2. The arguments given above are the starting point for the following characterizations of defect groups.

• P is minimal amongst subgroups of G such that the kG⊗_{k}kG^{op}-module kGb is relatively kΔP projective, where ΔP is the diagonal subgroup {(x, x^{−1}) : x ∈ P }.

• P is minimal amongst subgroups of G such that every finitely generated kGb-module is relatively P -projective.

• P is minimal amongst subgroups of G such that the k-linear map

kGb ⊗_{kP} kGb → kGb, satisfying x ⊗ y → xy,

is a split surjective homomorphism of k-algebras.

4.2. Central defect groups. We first record the following facts [NT, Chapter 3, Theorem 6.22 and Chapter 5, Theorem 8.11].

• Any block of kG is a k-linear combination of p′-elements of G.

• Let Q be a central p-subgroup of G. Then the canonical k-algebra surjection

kG → k(G/Q)

induces a bijection between blocks of kG and blocks of kG/Q.

Theorem 4.3. Suppose that P ≤ Z(G) and let \underline{b} be the image of b under the surjection of kG onto kG/P. Then the block algebra kG\underline{b} is a matrix algebra over k.

Proof. In view of Theorem 4.1, we need only observe that \underline{b} has trivial defect groups. By Theorem 3.3, b ∈ Tr_{P}^{G}(kG)^{P} whence \underline{b} ∈ Tr_{1}^{G/P} (kG/P). Now apply Theorem 3.3 to \underline{b}. □

Remarks 4.4. (i) If Q is a normal p-subgroup of G, then by Lemma 2.3 the image of b in kG/Q is a central idempotent of kG/Q but not in general a block of kG/Q. Consider, for instance, Example 1.5, with p = 3, and take for Q the cyclic subgroup of order 3 of G = S_{3}. Then 1_{kG} is a block of G, but G/Q being a group of order 2, 1_{kG/Q} is a sum of two blocks.

(ii) Theorem 4.3 has a lift to G:

• If P ≤ Z(G), then kGb is a matrix algebra over kP. (iii) A far reaching generalization of Theorem 4.3 is the structure of nilpotent

blocks due to Puig [Pu1]:

• If F = F_{P} (P), then there is an equivalence of module categories mod(kGb) ∼ mod(kP).

4.3. Blocks of normal subgroups. Let N be a normal subgroup of G. Then G acts on the set of blocks of kN via conjugation. Let c be a central idempotent of kN and let d be the sum of all elements in the G-orbit of c. Then d is a central idempotent of kG, hence a sum of blocks of kG. A block f of kG appears in the block decomposition of d in kG precisely if df = f, or equivalently if cf ≠ 0.

Definition 4.5. Let N be a normal subgroup of G, and c a block of N. If bc ≠ 0, then b is said to cover c. By comparing block decompositions of the identity in kN and kG, one sees that each block of kG covers exactly one G-orbit of blocks of kN.

A convenient observation at this stage is that one may often work under the assumption that b covers a unique block of kN. We explain briefly why this is the case.

Let c be a central idempotent of kN and d the sum of all elements in the G-orbit of c. Let I be the stabiliser in G of c and set t = [G : I]. Let G/I be a set of representatives of left cosets of I in G. Index the entries of t × t-matrices by G/I and for g, h ∈ G/I, let e_{g,h} be the t × t-matrix with 1 in the (g, h)-position and zero elsewhere. Let

ψ : kIc ⊗_{k} Mat_{t}(k) → kGd

be the k-linear map sending an element e_{g,h}⊗u to guh^{−1}. Then ψ is an isomorphism of k-algebras [Th, Lemma 16.1 and Proposition 16.6]. If bc ≠ 0, then b = ψ(f ⊗ 1_{Mat_{t}(k)}) for a block f of kI with fc ≠ 0. Furthermore, the following holds.

• ψ restricts to an isomorphism between kIf ⊗_{k} Mat_{t}(k) and kGb, so that in particular,

• The categories mod(kIf) and mod(kGb) are equivalent

• If (P′, e′) is a maximal f-Brauer pair, then F is isomorphic to F_{(P′,e′)}(I, f). Thus, there is no loss in replacing the group G and the block b of kG with the group I and the block f of kI.

Proposition 4.6. Let N be a normal subgroup of G and let c be a G-stable block of kN such that b covers c. If b has trivial defect groups, then so does c .

Proof. Note that by the proof of Theorem 4.1 it suffices to show that kNc is a projective kN ⊗_{k} (kN)^{op}-module. Let g ∈ G. Since c is G-stable, the vector space kNcg is invariant by left and right multiplication by elements of N, i.e. kNcg is a kN ⊗_{k} (kN)^{op}-submodule of kGc. Furthermore, if we denote by φ_{g} : kN ⊗_{k} (kN)^{op} → kN ⊗_{k} (kN)^{op} the k-linear map sending an element (x, y) of N × N to (x, ^{g}y), then it is easy to see that kNcg is isomorphic to Res_{φ_{g}}kNc as kN ⊗_{k} (kN)^{op}-module. In particular, kNcg is an indecomposable kN ⊗_{k} (kN)^{op}- module and kNcg is projective if and only if kNc is projective. It follows that

kGc = ⊕_{g∈G/N}kNcg

is a decomposition of the kN ⊗_{k} (kN)^{op}-module into indecomposable summands and that either kNc is projective or no kN ⊗_{k} (kN)^{op}-module summand of kGc is projective.

Now the fact that c is G-stable implies that c is the unique block of kN covered by b which in turn means that bc = b. This gives another kN ⊗_{k} (kN)^{op}-module decomposition

kGc = kGb ⊕ kGc(1 − b).

But b having trivial defect groups means that kGb is projective as kG ⊗_{k} (kG)^{op}- module and hence as kN ⊗_{k} (kN)^{op}-module. In particular, kGc has a projective kN ⊗_{k} (kN)^{op}-module summand. The proof is complete. □

Remarks 4.7. (i) In general, defect groups of a covered block are contained in defect groups of covering blocks.

(ii) Covered blocks are natural descendents from the module theoretic point of view. For instance, it is easy to see from the definition, that if U is a finite dimensional kGb-module, and N is a normal subgroup of G then the restriction of U to kN is a direct sum of kNc modules, as c runs over the the set of blocks of kN covered by b.

4.4. Some Clifford theory. Suppose that N is a normal subgroup of G and c is a block of kN covered by b. Suppose that the block c of kN has trivial defect groups. Then the algebra kNc is a matrix algebra. Philosophically speaking therefore, c should not contribute very much to the representation theory of kGc. In this subsection, we explain how in this situation, we can “kick out” N from the picture. The price we have to pay is the introduction of a certain central extension of G/N.

We keep all the notation just introduced. In addition we assume that the block c is G-stable. So, G acts on kNc via conjugation. But kNc being a simple algebra, by the Noether-Skolem theorem, all k-algebra automorphisms of kNc are inner. In other words, the group of k-algebra automorphisms Aut(kNc) of kNc is isomorphic to (kNc)^{∗}/Z(kNc)^{∗} ≅ (kNc)^{∗}/k^{∗}. Thus, the conjugation action of G on kNc gives a group homomorphism

G → (kNc)^{∗}/k^{∗}.

For each g in G, choose an inverse image i_{g} in (kNc)^{∗} of the image of g under the above homomorphism. It is easy to see that the elements i_{g} may be chosen such that for g ∈ G, and n ∈ N, i_{gn} = i_{g}nc. Let α be the corresponding 2-cocycle on G, that is α(g, h) is defined via i_{gh} = α(g, h)i_{g}i_{h}. For any g, h ∈ G, and m, n ∈ N, we have α(gm, hn) = α(g, h), thus α factors to a 2-cocycle, \underline{α} on G/N. We will denote also by α and \underline{α} respectively the images of α and \underline{α} in the cohomology groups H^{2}(G, k^{∗}) and H^{2}(G/N, k^{∗}). Let k\underline{α}G/N be the twisted group algebra corresponding to \underline{α}, that is k\underline{α}G/N is the k-algebra which is equal to kG/N as k-vector space, and with multiplication satisfying: gNhN = \underline{α}(g, h)ghN for g, h ∈ G.

Theorem 4.8. Let N be a normal subgroup of G and c a G-stable block of kN. Suppose that 1 is a defect group of c.With the above notation, let

φ : kNc ⊗_{k} k\underline{α}G/N → kGc

be the k-linear function satisfying x ⊗ gN → xi_{g}^{−1}g for x ∈ kNc and g ∈ G. Then φ is an algebra isomorphism, with φ^{−1}(gc) = i_{g} ⊗ gN for g ∈ G.

Truncation by blocks of kGb gives the following corollary.

Corollary 4.9. Keep the hypothesis of the theorem and suppose that b covers c. Then there is an equivalence of categories, mod(kGb) ∼ mod(k\underline{α}G/Nf) for some primitive central idempotent f of k\underline{α}G/N.

5. Alperin’s weight conjecture

Notation. For a finite dimensional k-algebra A, let l(A) denote the number of isomorphism classes of simple A-modules and let z(A) denote the number of isomorphism classes of simple projective A-modules. Note that by Theorem 4.1, for a finite group H and a central idempotent c of kH, z(kHc) is the number of blocks d of kH with trivial defect group contained in c, i.e. such that dc = c and this in turn is equal to the number of matrix algebras factors of kHc.

Denote by S_{p}(G) the poset of p-subgroups of G and by S_{p}(G)/G a set of representatives of the orbits of the conjugation action of G on S_{p}(G). For each p-subgroup R of G, denote by \underline{a} the image of an element a of kN_{G}(R) under the canonical surjection

kN_{G}(R) → kN_{G}(R)/R.

As before, we fix a block b of G. Recall that for any p-subgroup R of G, Br_{R}(b) is either 0 or a central idempotent of kN_{G}(R) and hence \underline{Br_{R}(b)} is either 0 or a central idempotent of kN_{G}(R)/R. Also, recall from the remarks after Theorem 4.3, that \underline{Br}_{R}(b) may be a sum of blocks of kN_{G}(R)/R even when Br_{R}(b) is a single block of kN_{G}(R).

A pair of the form (R, w), where R is a p-subgroup of G, and w is a block of kN_{G}(R)/R such that w has trivial defect group and such that w\underline{Br_{R}(b)} = w is called a weight of kGb. Clearly, G acts on the set of kGb-weights by conjugation.

Conjecture 5.1. (Alperin’s weight Conjecture)[A1] The number of isomorphism classes of simple kGb-modules equals the number of G-orbits of kGb-weights. In other words,

l(kGb) = ∑_{R∈S_{p}(G)/G} z(k(N_{G}(R)/R)\underline{Br_{R}(b)})

The number of kGb-weights can be expressed in terms of the fusion system of b. As before, let (P, e_{P}) be a maximal b-Brauer pair. For each Q ≤ P, we let e_{Q} ∈ B(Q) be such that (Q, e_{Q}) ≤ (P, e_{P}) and we set F := F_{(P,e_{P})}(G, b).

Now, let Q be an F-centric subgroup of P. Then by Theorem 3.10 and Theorem 4.3 the block \underline{e_{Q}} of kC_{G}(Q)/Z(Q) has trivial defect group. Thus we may apply the discussion preceding Proposition 4.8 to the normal inclusion of C_{G}(Q)/Z(Q) in N_{G}(Q, e_{Q})/Q and the block \underline{e_{Q}} of kC_{G}(Q)/Z(Q). Also, note that the group N_{G}(Q, e_{Q})/QC_{G}(Q) is canonically isomorphic to the group of F-outer automorphisms, Out_{F}(Q) = Aut_{F}(Q)/Inn(Q), of Q. Henceforth, we will identify the two groups through this isomorphism.

Definition 5.2. Let Q be an F-centric subgroup of P. Denote by α_{Q} the element of H^{2}(N_{G}(Q)/Q, k^{∗}) that arises from the action of N_{G}(Q)/Q on kC_{G}(Q)/Z(Q)\underline{e}_{Q} and by \underline{α}_{Q} the corresponding element of H^{2}(Out_{F}(Q), k^{∗}).

Definition 5.3. Let Obj(F^{cr}) denote the set of F-centric, F-radical subgroups of P and let Obj(F^{cr})/F denote a set of representatives of the F-isomorphism classes of Obj(F^{cr}) (see [Li4]).

Proposition 5.4. The number of G-orbits of kGb-weights is equal to

∑_{Obj(F^{cr})/F} z(k_{\underline{α}_{Q}}Out_{F}(Q)).

Thus, conjecture 5.1 is equivalent to the statement

l(kGb) = ∑_{Obj(F^{cr})/F} z(k_{\underline{α}_{Q}}Out_{F}(Q)).

The above proposition follows easily once we observe the following three facts, which we state separately.

Proposition 5.5. The number of kGb-weights is the sum of z(k(N_{G}(Q, e_{Q})/Q)\underline{e}_{Q}) as Q runs over the F-isomorphism classes of subgroups of P.

Proof. This is a series of simple reductions:

• If R is a p-subgroup of G, then Br_{R}(b) is the sum of N_{G}(R) orbits of B(R, b) and the sum of each orbit is a central idempotent of kN_{G}(R)/R. Hence z(k(N_{G}(R)/R)\underline{Br_{R}(b)}) is the sum of the z(k(N_{G}(R)/R)\underline{f̌}) as f̌ runs over the orbit sums of the action of N_{G}(R) on B(R, b).

• Suppose (R, f) is a Brauer pair and f̌ is the sum of the N_{G}(R)-orbit of f via the conjugation action of N_{G}(R) on B(Q, b). Then by the discussion in subsection 4.3 (applied to the normal inclusion of C_{G}(R)/Z(R) in N_{G}(R)/R and the central idempotent \underline{f} of C_{G}(R)/Z(R)), it follows that k(N_{G}(R)/R)\underline{f̌} is a matrix algebra over k(N_{G}(R, f)/R)\underline{f}), so that in particular, z(k(N_{G}(R)/R)\underline{f̌}) = z(k(N_{G}(R, f)/R)\underline{f}).

• Two Brauer pairs (R, f) and (R, f′) are G-conjugate if and only if f and f′ are N_{G}(R)-conjugate. Thus the number of G-orbits of kGb-weights is equal to the sum of z(k(N_{G}(R, f)/R)\underline{f}), as (R, f) runs through a set of representatives of G-conjugacy classes of b-Brauer pairs.

• By Theorem 3.5 every b-Brauer pair (R, f) is G-conjugate to (Q, e_{Q}) for some Q ≤ P and by the definition of F, two pairs (Q, e_{Q}) and (Q′, e_{Q′}) with Q, Q′ ≤ P are G-conjugate if and only if Q and Q′ are F-conjugate. Thus

□

Next, we show that only F-centric, F-radical subgroups of P contribute kGbweights.

Proposition 5.6. If Q is not F-centric, or if Q is not F-radical, then

z(k(N_{G}(Q, e_{Q})/Q)\underline{e}_{Q}) = 0.

Proof. Let Q ≤ P and let H be the inverse image in N_{G}(Q, e_{Q})/Q of O_{p}(Out_{F}(Q)) Then C_{G}(Q)/Z(Q) is a proper normal subgroup of H of index a power of p and H is a normal subgroup of N_{G}(Q, e_{Q})/Q. As pointed out just before Theorem 4.3, \underline{e_{Q}} is a block of kC_{G}(Q)/Z(Q). Also, \underline{e}_{Q} is clearly a central idempotent of kH, but in fact, since all block idempotents of kH are k-linear combinations of p′-elements of H, \underline{e}_{Q} is a block of kH.

Now suppose that z(k(N_{G}(Q, e_{Q})/Q)\underline{e}_{Q}) ≠ 0. Then by applying Theorem 4.6 to the normal inclusions of C_{G}(Q)/Z(Q) and H in N_{G}(Q, e_{Q})/Q) we get that 1 is a defect group of \underline{e_{Q}} as a block of kC_{G}(Q)/Z(Q) and as block of kH. The trivial subgroup being a defect group of \underline{e}_{Q} as a block of H means that \underline{e}_{Q} is a relative trace from 1 to H of some element of kH. Now by arguments similar to those given for the proof of Theorem 3.5, the fact that [H : C_{G}(Q)/Z(Q)] is a power of p forces C_{G}(Q)/Z(Q) to be equal to H, i.e. Q is F-radical. The condition that 1 is a defect group of \underline{e_{Q}} as a block of kC_{G}(Q)/Z(Q) implies that Z(Q) is a defect group of e_{Q} as a block of kC_{G}(Q), which by Theorem 3.10 means that Q is F-centric. □

Finally, we note that by Theorem 4.8, we have

Proposition 5.7. For any Q ∈ Obj(F^{cr}),

z(k(N_{G}(Q, e_{Q})/Q)\underline{e}_{Q}) = z(k_{\underline{α}_{Q}}Out_{F}(Q)).

Remarks 5.8. (i) If b is the principal block, the 2-cocyles \underline{α_{Q}} appearing in Definition 5.3 are all trivial.

(ii) Given Q ≤ P, there is a p′-central extension, W_{Q} of Out_{F}(Q) and a central idempotent u_{Q} (not necessarily primitive) of kW_{Q} such that the algebra k_{\underline{α}_{Q}}Out_{F}(Q) is isomorphic to kW_{Q}u_{Q}. Thus, the calculation of the right side of the weight conjecture reduces to calculating the number of blocks with trivial defect group of kW_{Q} which are contained in u_{Q}.

(iii) Suppose P is abelian. Then clearly the only subgroup of P making a contribution to the right side of Proposition 5.4 is P. Retracing our steps to the original formulation yields that the only subgroup of P contributing to the right side of Conjecture 5.1 is P. By Theorem 3.3, b is a relative trace from P to G, so \underline{Br_{P} (b)} is a relative trace from 1 to N_{G}(P)/P. Again, by Theorem 3.3 it follows that every block d of kN_{G}(P)/P such that dBr_{P} (b) = d has trivial defect groups. So, by Theorem 4.1 k(N_{G}(P)/P)\underline{Br_{P} (b)} is a semi-simple algebra, hence every simple module k(N_{G}(P)/P)\underline{Br_{P} (b)} is projective, and z(k(N_{G}(P)/P)\underline{Br_{P} (b)}) is just l(k(N_{G}(P)/P)\underline{Br_{P} (b)}). But P being a normal p-group of N_{G}(P), P acts trivially on every simple kN_{G}(P)-module (see the last part of the proof of Proposition 2.3), so l(k(N_{G}(P)/P)\underline{Br_{P} (b)}) is equal to l(kN_{G}(P)Br_{P} (b)). Thus, Alperin’s conjecture in this case is equivalent to the assertion

l(kGb) = l(kN_{G}(P)Br_{P} (b)).

In fact, Broué’s abelian defect group conjecture predicts that when P is abelian the above equality comes from an equivalence of derived categories

D^{b}mod(kGb) ∼ D^{b}mod(kN_{G}(P)Br_{P} (b)).

(iv) The weight conjecture is known to hold in the following cases.

• F = F_{P} (P) [Pu1].

• P is cyclic [Da1] or if p = 2 and P is Klein four, dihedral, semi-dihedral, or a generalized quaternion group [Br],[Ol] (these correspond exactly to the cases that the module category mod(kP) is of finite or tame type ).

It might be instructive to point out at this stage that the conjecture is open when P is elementary abelian of order p^{2}, (p odd) even under the further assumption that the number of kGb weights is 1.

(v) The conjecture has been verified for many classes of finite groups. We mention some of these (the list below is not exhaustive):

• G is p-solvable [OW],[IN].

• G is a finite group of Lie type in characteristic p [CE2].

• G is a symmetric group or a finite general linear group in a characteristic

different from p [AF],[An3]. The conjecture has also been verified for blocks of many sporadic simple groups

[An1],[AW] and for several families of classical groups [An2].

(vi) In [KR] Knorr and Robinson showed that the weight conjecture can be reformulated in terms of alternating sums indexed by certain chains of p-subgroups of G. Various generalizations and reformulations of the conjecture have appeared since (see for instance [Da2], [Da3], [Da4] , [Ro1], [Ro2] [Bo], [Li1] , [Li2] and [We]). We will present one such version (the ordinary weight conjecture) in the next section.

6. Blocks in characteristic 0

We keep the notation of the previous sections. So b is a block of kG, (P, e_{P}) is a maximal b-Brauer pair; for Q ≤ P, e_{Q} is the unique block of kC_{G}(Q) such that (Q, e_{Q}) ≤ (P, e_{P}) and F := F_{P,e_{P}} (G, b) is the corresponding fusion system.

Let β be the block of OG lifting b in O. The following result is the content of Brauer’s Second Main theorem.

Theorem 6.1. [NT, Chapter 5, Theorem 4.1] Let V be an OG-module which is free and of finite rank as O-module. Suppose that V belongs to the block β, i.e. βV = V. Then for any p-element x of G and any p′-element y of C_{G}(x), and ϵ a block of OC_{G}(x), χ_{V} (ϵxy) = 0 unless ϵ̄ ∈ B(⟨x⟩, b). Consequently,

χ_{V} (xy) = ∑_{ϵ̄∈B(⟨x⟩,b)} χ_{V} (ϵxy).

The above theorem is a striking example of local control on global invariants of blocks. We mention below a few results that depend on it.

6.1. Block Orthogonality Relations. Recall that by the second orthogonality relation for characters, if g and h are two non-conjugate elements of G, then

∑_{χ∈Irr_{K}(G)} χ(g)χ(h^{−1}) = 0.

Theorem 6.1 supplies a block version of this result [NT, Chapter 5, Theorem 4.5]:

If g, h ∈ G belong to different p-sections of G, i.e. if g^{m} and h^{n} are not conjugate for any integers m and n relatively prime to p, then

∑_{χ∈Irr_{K}(G,β)} χ(g)χ(h^{−1}) = 0.

The set Irr_{K}(G) is an orthonormal basis on the set of K-valued class functions on G with respect to the standard inner product on class functions. In particular, if χ, χ′ are distinct elements of Irr_{K}(G), then

∑_{g∈G} χ(g)χ′(g^{−1}) = 0.

This generalises to the following [NT, Chapter 5, Theorem 4.7].

Let x ∈ G be a p-element. If χ, χ′ ∈ Irr_{K}(G) lie in different blocks of OG, then

∑_{y∈C_{G}(x)_{p′}} χ(xy)χ′(x^{−1}y^{−1}) = 0,

where C_{G}(x)_{p′} denotes the set of all p′-elements of C_{G}(x).

6.2. Counting characters.

Definition 6.2. Let x, y ∈ P. We will say that x and y are F-isomorphic if there is a morphism φ : ⟨x⟩ → ⟨y⟩ in F such that φ(x) = y. Denote by P/F a set of representatives of the F-isomorphism classes of elements of P.

The number |Irr_{K}(G)| of isomorphism classes of simple KG-modules equals the number of conjugacy classes of G. It is also the case that the number of simple kGmodules is equal to the number of conjugacy classes of p′-elements of G[NT, Chapter 3, Therem 6.5]. Using these facts, along with the non-degeneracy properties of the cde-triangle and Theorem 6.1, we get the following formula.

Theorem 6.3. [NT, Chapter 5, Theorem 4.13, Theorem 9.4] We have

|Irr_{K}(G, β)| = ∑_{x∈P/F} l(kC_{G}(x)e_{⟨x⟩}).

6.3. The ordinary weight conjecture. From the above theorem we see that Alperin’s weight conjecture also predicts the number of ordinary irreducible characters of b. The Ordinary Weight Conjecture whose statement we present below gives a flavour of both the characteristic 0 aspects of the weight conjecture and versions of it in terms of alternating sums. We will need the following definition.

Definition 6.4. Let H be a finite group of order p^{a}m, where (p, m) = 1. Let χ ∈ Irr_{K}(H) and write χ(1) = p^{u}n with (p, n) = 1. The number a − u is the defect of χ. For a non-negative integer d, denote by Irr_{K}^{d}(H) the set of irreducible characters of H of defect d.

We recall all the notation introduced for Proposition 5.4. Let Q ≤ P be F-centric and F-radical. Let N_{Q} denote the set of strictly increasing chains

σ : 1 ⊲ R_{1} ⊲ ⋯ ⊲ R_{l}

of p-subgroups of Out_{F}(Q) such that for each i, 1 ≤ i ≤ l, R_{i} is normal in R_{l}. For each σ as above, denote by |σ|, the integer l+1 (so if σ is the chain consisting solely of the trivial subgroup, then |σ| = 1).

Note that Out_{F}(Q) acts on N_{Q} by conjugation, and Out_{F}(Q) also acts naturally on Irr_{K}(Q) (since inner automorphisms of Q stabilize every character of Q). For each chain σ ∈ N_{Q}, and each character μ of Irr_{K}(Q), we define an integer ω(Q, σ, μ) as follows. Let I(σ) be the stabilizer in Out_{F}(Q) of the chain σ and let I(σ, μ) be the intersection of I(σ) with the stabilizer in Out_{F}(Q) of the character μ. Denote by \underline{α}_{Q} also the restriction of \underline{α}_{Q} to I(σ, μ) (see definition 5.2) and define

ω(Q, σ, μ) := z(k_{\underline{α}_{Q}}I(σ, μ)).

For each non-negative integer d, set

w(Q, d) := ∑_{σ∈N_{Q}/Out_{F} (Q)}(−1)^{|σ|+1} ∑_{μ∈Irr_{K}^{d}(Q)/I(σ)} ω(Q, σ, μ).

Conjecture 6.5. [Ordinary weight conjecture][Ro2] Let d be a non-negative integer and let Irr_{K}^{d}(G, β) be the set of characters in Irr_{K}(G, β) of defect d. Then,

|Irr_{K}^{d}(G, β)| = ∑_{Obj(F^{cr})/F} w(Q, d).

The formulation given here is not the same as the original formulation of the ordinary weight conjecture in [Ro1], but it is shown in [Ro2] that the two are equivalent.

6.4. Detecting blocks of trivial defect. The right hand sides of the “localglobal” conjectures presented here all count subsets of blocks with trivial defect groups (see Remarks 5.4 (ii); the situation for the ordinary weight conjecture is analogous).

Below, we present some character theoretic criteria which detect blocks with trivial defect groups. Note that by Theorem 4.1 and the discussion in section 1, P = 1 if and only if |Irr_{K}(G, β)| = 1.

Theorem 6.6. [NT, Chapter 3, Theorem 6.29] Let χ ∈ Irr_{K}(G, β). The following are equivalent.

(i) P = 1.

(ii) χ(x) = 0 if x ∈ G has order divisible by p.

(iii) χ has defect 0.

Note that the implication (i) to (iii) can be deduced from Theorem 6.1.

7. Examples of fusion systems

Keep all the notation of the previous sections. In addition, as in [Li4], for a finite group H and a Sylow p-subgroup S of H, we will denote by F_{S}(H) the fusion system of H on the subgroups of S.

7.1. Principal blocks.

Theorem 7.1 (Brauer’s Third Main Theorem). Suppose that b is the principal block of kG. Then, for any p-subgroup Q of G, Br_{Q}(b) is the principal block of kG. In particular, P is a Sylow p-subgroup of G and F = F_{P} (G).

The above theorem (in the version stated above) is proved in [AB]. We sketch a proof. Applying Theorem 6.1 to the rank 1-trivial module of OG, (or by using the fact that the principal block is the unique block of kG not contained in the augmentation ideal of kG), one sees that

• For any p-element x of G, B(⟨x⟩, b) contains the principal block of kC_{G}(x), i.e. (1, b) ≤ (⟨x⟩, a) where a is the principal block of kC_{G}(x).

Now let Q be a p-subgroup of G and let a be the principal block of kC_{G}(Q). Let z be a p-element of G such that z normalizes Q. Then a is ⟨z⟩-stable, hence is a block (necessarily the principal block) of H := ⟨z⟩C_{G}(Q). Let a′ be the principal block of kC_{H}(⟨z⟩). By the previous point, applied to the group H, Br_{⟨z⟩}(a)a′ = a′. Now noting that C_{H}(< z > Q) = C_{G}(⟨z⟩Q) and that Br_{⟨z⟩}(a) = Br_{⟨z⟩Q}(a), we have an inclusion G-Brauer pairs, (Q, a) ≤ (⟨z⟩Q, a′). Arguing inductively, we get

• Let Q ≤ Q′ be p-subgroups of G and let a and a′ respectively be the principal blocks of kC_{G}(Q) and kC_{G}(Q′) respectively. Then (Q, a) ≤ (Q′, a′).

Since all maximal b-Brauer pairs are conjugate it is immediate from the above that P is a Sylow p-subgroup of G and that e_{P} is N_{G}(P)-conjugate to the principal block of kC_{G}(P). But the principal block of kC_{G}(P) is N_{G}(P)-stable, hence e_{P} is the principal block of kC_{G}(P). The theorem follows from the uniqueness of the inclusion of Brauer pairs.

7.2. Blocks of symmetric groups.

Theorem 7.2. Let G = S_{n} be a finite symmetric group. Then there is a nonnegative integer w such that pw ≤ n and such that P is a Sylow p-subgroup of S_{pw}. Furthermore, F = F_{P} (S_{pw}).

The above is implicit in Puig’s proof of the Nakayama Conjecture [Pu2] which describes the block-wise distribution of Irr_{K}(G) by the p-cores of the corresponding partitions. The crucial fact for the proof of the theorem is the following lemma.

Lemma 7.3. Let X be a finite set and let S(X) be the symmetric group on X. Let Q be a p-subgroup of S(X) which acts fixed point freely on X. Then the principal block is the unique block of kC_{S(X)}(Q).

Proof. Let H = C_{S(X)}(Q) and let e be a block of kH. Consider the inclusion

Q → ∏_{Y} S(Y)

where Y runs over the orbits of Q on X. For each orbit U of Q on X, let Q_{U} be the image of Q under the projection of ∏_{Y} S(Y) onto S(U). Set

Q̃ = ∏_{Y} Q_{Y} .

Then Z(Q̃) is a normal subgroup of H. So by Proposition 3.7, e ∈ kC_{H}(Z(Q̃)) = kC_{S(X)}(QZ(Q̃)). So we may assume that Z(Q̃) ≤ Q. It follows that

C_{S(X)}(Q) = ∏_{Y} C_{S(Y)}(Q_{Y}),

as Y runs over the Q-orbits of X. So kC_{S(X)}(Q) is isomorphic to the tensor product, over k, of the algebras kC_{S(Y)}(Q_{Y}), and under such an isomorphism, e is identified with a tensor product of blocks of the kC_{S(Y)}(Q_{Y}) as Y runs over the orbits of Q on X. Thus we may assume that Q is transitive on X. But this means that |X| = p^{m} for some m. (If |X| is not a power of p, then a Sylow p-subgroup of S(X) is not transitive on X). Let 1 ≠ σ ∈ H. Then the transitivity of Q on X implies that all orbits of σ on X are of the same length. But this means that each orbit of σ has cardinality a power of P, i.e. σ is a p-element. In other words, H is a p-group. This proves the lemma. □

Proof of theorem. Let G = S(Y), where Y is a set of cardinality n and let b, (P, e_{P}) be as before. Let Q ≤ P. Denoting by Y^{Q} the set of fixed points of Q on Y, we have

C_{G}(Q) = S(Y^{Q}) × C_{S(Y −Y^{Q})}(Q),

thus

kC_{G}(Q) ≅ kS(Y^{Q}) ⊗_{k} C_{S(Y −Y^{Q})}(Q),

and so e_{Q} is of the form e′_{Q} ⊗ e′′_{Q} where e′_{Q} is a block of kS(Y^{Q}) and e′′_{Q} is a block of kC_{S(Y −Y^{Q})}(Q). Now by the above lemma, (applied with X = Y − Y^{Q}) e′′_{Q} is the principal block of kC_{S(Y −Y^{Q})}(Q). In particular, e′′_{Q} is N_{S(Y −Y^{Q})}(Q) stable. Furthermore, note that

N_{G}(Q) = S(Y^{Q}) × N_{S(Y −Y^{Q})}(Q).

From this it is immediate that • e_{Q} is N_{G}(Q)-stable, i.e. N_{G}(Q, e_{Q}) = N_{G}(Q) and Aut_{F}(Q) = Aut_{G}(Q).

Let R be a p-subgroup of S(Y − Y^{Q}) containing P and let f′′ be the principal block of C_{S(Y −Y^{P})}(R). Then by the discussion in the previous subsection, there is a containment of Brauer pairs (for the group S(Y − Y^{P})) (P, e′′_{P}) ≤ (R, f′′). Now,

C_{G}(R) = S(Y^{P}) × C_{S(Y −Y^{P})}(R).

From this it is not hard to see, that letting f be the block of kC_{G}(R) identified with the tensor e′_{P} ⊗ f′′, there is an inclusion of G-Brauer pairs (P, e_{P}) ≤ (R, f), The maximality of (P, e_{P}) now implies

• P is a Sylow p-subgroup of S(Y − Y^{P}). Finally, note that for Q ≤ P, Y − Y^{Q} ⊂ Y − Y^{P} and hence

• Aut_{G}(Q) = Aut_{S(Y −Y^{P})}(Q). The result theorem follows from the above three assertions and Alperin’s fusion

theorem (see [Li4, Section 5]). □

Remarks 7.4. (i) Results analogous to Theorem 7.2 hold in other types of groups. For instance, • If G is a finite general linear or unitary group and p is not the defining characteristic of G, then there is a subgroup H of G such that P is a Sylow p-subgroup of H and such that F = F_{P} (H) [Bro]. This is also known to hold for most unipotent blocks of finite groups of Lie type, provided we replace the condition F = F_{P} (H) by F ⊂ F_{P} (H) [CE1].

• If G is p-solvable, then by Fong-Puig reduction it follows there is a group H and a block c of kH such that kGb and kHc have equivalent module categories, P is a Sylow p-subgroup of H and F and any fusion system of c are isomorphic to F_{P} (H) [HL].

(ii) It is an open question as to whether in general F is isomorphic to F_{P} (H) for some finite group H with P as Sylow p-subgroup.

7.3. Descent to normal subgroups. Suppose that N is a normal subgroup of G and c is a block of kN covered by the block b. One might expect that a fusion system of the block c of kNc is isomorphic to a subsystem of F. However, this is not the case in general as the next example shows.

Example 7.5. Let p = 3 and let P be the cyclic group of order 3 on generator r. Let T be the quaternion group of order eight as in Example 3.2. Let T act on P via

xr = r^{2} ^{y}r = r.

Let G = P T and let N be the normal subgroup P ⟨x⟩. Consider

b = \frac{1}{2}(1 − x^{2}) = \frac{1}{2}(1 − y^{2}).

Then b is a block of kG as well as of kN (this follows for instance from Lemma 3.7). Now C_{G}(P) = P ×⟨y⟩ Thus, Br_{P} (b) is a sum of two blocks, \frac{1}{4}(1+iy −y^{2} −iy^{3}) and \frac{1}{4}(1 − iy − y^{2} + iy^{3}) of kC_{G}(P), where i is a primitive fourth root of unity. Setting e_{P} := \frac{1}{4}(1 + iy − y^{2} − iy^{3}), it is immediate that Aut_{F}(P) = 1.

On the other hand, C_{N}(P) = P × ⟨x^{2}⟩ whence Br_{P} (b) = b is a block of kC_{N}(P). So, setting F′ equal to the category F_{(P,Br_{P} (b))}(N, b), it is clear that Aut_{F′}(P) is cyclic of order 2.

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Radha Kessar, Department of Mathematical Sciences, University of Aberdeen, Meston Building 318, Aberdeen AB24 3UE, U.K.

   E-mail address: kessar@maths.abdn.ac.uk