ON THE BRAUER INDECOMPOSABILITY OF SCOTT MODULES

RADHA KESSAR, SHIGEO KOSHITANI, MARKUS LINCKELMANN

Abstract. Let k be an algebraically closed field of prime characteristic p, and let P be a p-subgroup of a finite group G. We give sufficient conditions for the kG-Scott module Sc(G, P) with vertex P to remain indcomposable under the Brauer construction with respect to any subgroup of P. This generalizes similar results for the case where P is abelian. The background motivation for this note is the fact that the Brauer indecomposability of a p-permutation bimodule is a key step towards showing that the module under consideration induces a stable equivalence of Morita type, which then may possibly be lifted to a derived equivalence.

1. Introduction

Throughout this paper we denote by k an algebraically closed field of prime characteristic p. The Brauer construction with respect to a p-subgroup P of a finite group G sends a p-permutation kG-module M functorially to a p-permutation kN_{G}(P)-module M(P); see e. g. [3, p.402] or [16, pp.91 and 219]. Following the terminology introduced in [9], the module M is called Brauer indecomposable if the kC_{G}(Q)-module Res_{C_{G}(Q)}^{N_{G}(Q)}(M(Q)) is indecomposable or zero for any p-subgroup Q of G. As mentioned in [9], the Brauer indecomposability of p-permutation modules is relevant for the gluing technique used for proving categorical equivalences between p-blocks of finite groups as in Broué’s abelian defect group conjecture, see [10], [9] and [17]. For any subgroup H of G there is up to isomorphism a unique indecomposable direct summand of the permutation kG-module kG/H which has a Sylow-p-subgroup of H as a vertex and the trivial kG-module as a quotient. This is called the Scott kG-module with respect to H, denoted by Sc(G, H). If P is a Sylow-p-subgroup of H, then Sc(G, H) = Sc(G, P) is the up to isomorphism unique indecomposable kG-module with P as a vertex, the trivial kP -module as a source, and the trivial kG-module as a quotient. See [14, Chap.4, §8] and [3] for more details on Scott modules. We first prove a criterion for the Brauer indecomposability of Scott modules in terms of the indecomposability of Scott modules of certain local subgroups.

Theorem 1.1. Let P be a p-subgroup of a finite group G. Let F = F_{P} (G) be the fusion system of G on P. Suppose that F is saturated, and that F = F_{P} (N_{G}(P)). Then Sc(G, P) is Brauer indecomposable if and only if

Res_{C_{G}(Q)}^{N_{P} (Q)C_{G}(Q)}(Sc(N_{P} (Q)C_{G}(Q), N_{P} (Q))

is indecomposable for any subgroup Q of P.

It is shown in [9, Theorem 1.2] that if P is an abelian p-subgroup of G, and if the fusion system F_{P} (G) is saturated, then the kG-Scott module Sc(G, P) is Brauer indecomposable. The following result extends this in some cases to non-abelian P.

Theorem 1.2. Let P be a p-subgroup of a finite group G. Let F = F_{P} (G) be the fusion system of G on P. Suppose that F is saturated, and that F = F_{P} (N_{G}(P)). Suppose that, for any subgroup Q of P, at least one of the following holds:

(a) N_{P} (Q) = QC_{P} (Q).

(b) C_{G}(Q) is p-nilpotent.

Then Sc(G, P) is Brauer indecomposable.

If P is a common subgroup of two groups G and H, we denote by ΔP the ‘diagonal’ subgroup ΔP = {(u, u) | u ∈ P } of G × H.

Corollary 1.3. Let G be a finite group and P a Sylow p-subgroup of G. Set M = Sc(G × N_{G}(P), ΔP). Suppose that, for any subgroup Q of P, at least one of the following holds:

(a) N_{P} (Q) = QC_{P} (Q).

(b) C_{G}(Q) is p-nilpotent.

Then M is Brauer indecomposable.

Remark 1.4. For P abelian, this is Corollary 1.4 of [9], which follows also from [10, Theorem]. Examples of non-abelian p-groups to which the above applies are all groups of order p^{3} and metacyclic p-groups of the form M_{n+1}(p) ≅ C_{p^{n}} ⋊ C_{p}, see [6, p.190], where C_{m} denotes a cyclic group of order m, for any positive integer m. See the Example 3.2 below for a stable equivalence of Morita type which is constructed making use of Corollary 1.3.

The above results will be proved in section 3. We adopt the following notation and conventions. All modules over finite group algebras are assumed to be finitely generated unitary left modules. We write H ≤ G if H is a subgroup of a group G, and H ⊴ G if H is normal in G. The trivial kG-module will be denoted again by k. For G a group, H a subgroup of G, M a kG-module and N a kH-module, we write as usual Res_{H}^{G}(M) for the restriction of M from kG to kH and Ind_{H}^{G}(N) for the induction of N from kH to kG. For a subset S of G and an element g ∈ G, we write ^{g}S for gSg^{−1}, and for h ∈ G, we write ^{g}h = ghg^{−1}. For H, K ≤ G we write H ≤_{G} K when ^{g}H ≤ K for an element g ∈ G. As mentioned before, given a p-subgroup P of a finite group G and a kG-module M, we write M(P) for the Brauer construction with respect to P applied to M; see [4, p.402] or [16, pp.91 and 219]. We denote by F_{P} (G) the fusion system of G on P; that is, F_{P} (G) is the category whose objects are the subgroups of P and whose morphisms from Q to R are the group homomorphisms induced by conjugation by some elements of G; see [2, Definition I.2.1] and [11, p.83]. If P is a Sylow p-subgroup of G, then F_{P} (G) is saturated, see [2, Definition I.2.2]. If F_{P} (G) = F_{P} (N_{G}(P)), then the saturation of F_{P} (G) is equivalent to requiring that N_{G}(P)/P C_{G}(P) has order prime to p. For any remaining notation and terminology, see the books of [14] and [16], and also [2] and [11] for fusion systems.

2. Lemmas

This section contains some technicalities needed for the proofs of the main results in the section. We start with a very brief review of some basic properties of Scott modules. Let G be a finite group, H a subgroup of G, and P a Sylow-p-subgroup of H. Let M be a p-permutation kG-module. In particular, M has a k-basis X which is permuted by the action of P. By [3, §1] or [16, Proposition (27.6)], the image in M(P) of the subset X^{P} of P -fixed points in X is a k-basis of M(P), and we have a direct sum decomposition of kN_{G}(P)-modules Res_{N_{G}(P)}^{G}(M) = M(P) ⊕ N, where N is the span of the P -orbit sums of X \ X^{P}. For any subgroup Q of P we have X^{P} ⊆ X^{Q}. In particular, if M(P) ≠ {0}, then M(Q) ≠ {0} for any subgroup Q of P. By [3, (1.3)], if M is an indecomposable p-permutation kGmodule, then M(P) ≠ {0} if and only if P is contained in a vertex of M. By [3, (3.2) Theorem], if P is a vertex of M, then M(P) is the Green correspondent of M. Frobenius’ reciprocity implies that Hom_{kG}(Ind_{H}^{G}(k), k) ≅ Hom_{kH}(k, k) ≅ k. Thus exactly one indecomposable direct summand of Ind_{H}^{G}(k) has a quotient isomorphic to the trivial kG-module. This summand is the Scott module S(G, H). Under the above isomorphism the identity map on k (viewed as a kP -module) corresponds to the unique kG-homomorphism η : Ind_{H}^{G}(k) → k sending 1 ⊗ 1_{k} to 1 and y ⊗ 1_{k} to 0 for any y ∈ G \ H. Thus the Scott module Sc(G, H) is, up to isomorphism, the unique indecomposable direct summand of Ind_{H}^{G}(k) which is not contained in ker(η). Applying the Brauer construction to η yields a non-zero map η(P) : (Ind_{H}^{G}(k))(P) → k, because the element 1 ⊗ 1_{k} is a P -fixed element of the P -stable basis consisting of the elements y ⊗ 1_{k}, with y running over a set of representatives of the cosets G/H in G. This shows in particular that Sc(G, H) has P as a vertex and therefore must coincide with Sc(G, P). We will use these facts without further reference. The following lemma is essentially a special case of a result of H. Kawai [8, Theorem 1.7].

Lemma 2.1. Let G be a finite group, and let P and Q be p-subgroups of G such that Q ≤ P. Suppose that for any g ∈ G satisfying Q ≤ ^{g}P we have |N_{^{g}P} (Q)| ≤ |N_{P} (Q)|. Let M be an indecomposable p-permutation kG-module with vertex P. Set H = N_{G}(Q). Then Res_{H}^{G}(M) has an indecomposable direct summand X satisfying X(N_{P} (Q)) ≠ {0}, and any such summand has N_{P} (Q) as a vertex. In particular, Sc(H, N_{P} (Q)) is isomorphic to a direct summand of Res_{H}^{G}(Sc(G, P)) and of (Sc(G, P))(Q).

Proof. We have H ∩P = N_{P} (Q), and since M(P) is non-zero, so is M(H ∩P). Thus there is an indecomposable direct summand X of Res_{H}^{G}(M) such that X(H ∩ P) ≠ {0}. Let R be a vertex of X containing H ∩ P. Since P is a vertex of M, it follows that M is isomorphic to a direct summand of Ind_{P}^{G}(k). The Mackey decomposition formula implies that X is isomorphic to a direct summand of

Res_{H}^{G}(Ind_{P}^{G}(k)) = ⊕_{y} Ind_{H∩^{y}P}^{H} (k) ,

where y runs over a set of representatives of the double cosets H\G/P in G. The indecomposability of X and the Krull-Schmidt theorem imply that there is y ∈ G such that X is isomorphic to a direct summand of Ind_{H∩^{y}P}^{H} (k). Then H ∩^{y}P contains a vertex S of X. Since the vertices of an indecomposable module are conjugate, it follows that there is h ∈ H such that S = ^{h}R. The element h normalises Q, and hence Q ≤ S ≤ H ∩ ^{y}P. This implies S = N_{^{y}P} (Q). The assumptions imply further that |S| ≤ |N_{P} (Q)| ≤ |R|. Since R and S are conjugate, they have the same order, whence R = N_{P} (Q) is a vertex of X. For the second statement, suppose that M = Sc(G, P). That is, M is the up to isomorphism unique indecomposable direct summand of kG/P which is not in the kernel of the kG-homomorphism kG/P → k sending the trivial coset P to 1 and any non-trivial coset yP to 0, where y ∈ G \ P. As mentioned at the beginning of this section, the trivial coset P is a P -fixed point of the basis of kG/P consisting of the P -cosets in G, and hence applying the Brauer construction to a non-zero kG-homomorphism M → k yields a non-zero map M(P) → k. Then also the map M(R) → k induced by a non-zero kG-homomorphism M → k is nonzero. It follows that Res_{H}^{G}(M) has an indecomposable direct summand X satisfying X(R) ≠ {0} such that there is a non-zero kH-homomorphism X → k. By the first statement, R is a defect group of X. Thus X ≅ Sc(H, R). This shows that Sc(H, R) is isomorphic to a direct summand of Res_{N_{G}(Q)}^{G}(M). Since R contains Q and Q is normal in H, it follows that Q acts trivially on Sc(H, Q), and thus Sc(H, Q) is isomorphic to a direct summand of M(Q). □

In fusion theoretic terminology, the hypothesis on the maximality of |N_{P} (Q)| in the previous lemma is equivalent to requiring that Q is fully F-normalised. If F = N_{F}(P), then every subgroup of P is fully F-normalised, which explains why this hypothesis is no longer needed in the second statemement of the next lemma. The proof of the first statement of the next lemma is essentially in [9, Theorem 1.2].

Lemma 2.2. Let G be a finite group and P a p-subgroup of G. Set F = F_{P} (G). Assume that F is saturated and that F = F_{P} (N_{G}(P)).

(i) Suppose that M is an indecomposable p-permutation kG-module with vertex P. Then for any subgroup Q of P and for any indecomposable direct summand X of Res_{N_{G}(Q)}^{G}(M) satisfying X(Q) ≠ {0}, there is a vertex R of X such that Q ≤ R ≤ P.

(ii) For any subgroup Q of P, the module Sc(N_{G}(Q), N_{P} (Q)) is a direct summand of Res_{N_{G}(Q)}^{G}(Sc(G, P)) and of (Sc(G, P))(Q).

Proof. (i) Let X be an indecomposable direct summand of Res_{N_{G}(Q)}^{G}(M) such that X(Q) ≠ {0}. There is a vertex R of X such that Q ≤ R. Then X(R) ≠ {0}, hence M(R) ≠ {0}, and so R is contained in a vertex of M. Since the vertices of M are conjugate to P, it follows that there is g ∈ G such that Q ≤ R ≤ ^{g}P. Then ^{g^{−1}}Q ≤ P, which implies that the map ^{g^{−1}}Q → Q sending u to ^{g}u is an isomorphism in the fusion system F. The assumption F = F_{P} (N_{G}(P)) implies that there is an element h ∈ N_{G}(P) such that c = hg^{−1} ∈ C_{G}(Q). It follows that Q = ^{c}Q ≤ ^{c}R ≤ ^{cg}P = ^{h}P = P. Clearly ^{c}R is also a vertex of X, whence the statement.

(ii) Let g ∈ G with Q ≤ ^{g}P. By the argument in the proof of (i), there is an element h ∈ N_{G}(P) such that c = hg^{−1} ∈ C_{G}(Q). Then cg = h normalises P. Thus conjugation by c induces an isomorphism N_{^{g}P} (Q) ≅ N_{P} (Q); in particular, both groups have the same order. Therefore Lemma 2.1 implies the assertion. □

3. Proof of the main result

Let G be a finite group and let M be an indecomposable p-permutation kGmodule with vertex P. If Q is a p-subgroup of G which is not conjugate to a subgroup of P, then M(Q) = {0}. The property of M(Q) being decomposable is invariant under conjugation of Q in G. Thus if M is not Brauer indecomposable, then there is a subgroup Q of P such that M(Q) is decomposable as a kC_{G}(Q)- module. The key step towards proving the main results is the following lemma.

Lemma 3.1. Let G be a finite group and P a p-subgroup of G. Set F = F_{P} (G). Assume that F = F_{P} (N_{G}(P)) and that F is saturated. Set M = Sc(G, P). Suppose that M is not Brauer indecomposable. Let Q be a subgroup of maximal order in P such that Res_{C_{G}(Q)}^{N_{G}(Q)}(M(Q)) is decomposable. Then Q is a proper subgroup of P and setting R = N_{P} (Q), we have

Res_{RC_{G}(Q)}^{N_{G}(Q)}(M(Q)) ≅ Sc(RC_{G}(Q), R) .

In particular, Res_{RC_{G}(Q)}^{N_{G}(Q)}(M(Q)) is indecomposable with R as a vertex.

Proof. The kC_{G}(P)-module M(P) is indecomposable by [9, Lemma 4.3(ii)] (this lemma requires the hypothesis on F being saturated). Thus Q is a proper subgroup of P, hence a proper subgroup of R = N_{P} (Q), by [6, Chap.1, Theorem 2.11(ii)]. We first show that M(Q) is indecomposable as a kN_{G}(Q)-module. By Lemma 2.2 (ii) we have Res_{N_{G}(Q)}^{G}(M) = Sc(N_{G}(Q), R) ⊕ X for some kN_{G}(Q)-module X, and Sc(N_{G}(Q), R) is isomorphic to a direct summand of M(Q). We need to show that X(Q) = {0}. Arguing by contradiction, suppose that X(Q) ≠ {0}. Then there exists an indecomposable direct summand Y of X such that Y (Q) ≠ {0}. Since F is saturated and N_{F}(P) = F, it follows from Lemma 2.2(i) that Y has a vertex S such that Q ≤ S ≤ P. Then S ≤ N_{G}(Q) ∩ P = R. Note that Q is not a vertex of M since |Q| ≠ |P |. If Q = S, then Q is a vertex of Y, and hence Q is a vertex of M by the result of Burry-Carlson-Puig [14, Chap. 4, Theorem 4.6(ii)], a contradiction. Thus Q is a proper subgroup of S. Since Y is an indecomposable p-permutation kN_{G}(Q)-module with vertex S, we have Y (S) ≠ {0}, and hence X(S) ≠ {0}. Since R is a vertex of Sc(N_{G}(Q), R) and S ≤ R, it follows that (Sc(N_{G}(Q), R))(S) ≠ {0}. We have

(Res_{N_{G}(Q)}^{G}(M))(S) = (Sc(N_{G}(Q), R))(S) ⊕ X(S),

and both of the two direct summands of the right-hand-side are non-zero. This implies that (Res_{N_{G}(Q)}^{G}(M))(S) is not indecomposable; in other words, M(S) is not indecomposable as a k(N_{G}(Q)∩N_{G}(S))-module. Since C_{G}(S) ≤ C_{G}(Q)∩N_{G}(S) ≤ N_{G}(Q) ∩ N_{G}(S), it follows that M(S) is not indecomposable as a kC_{G}(S)-module. But this contradicts the assumptions since |P : S| < |P : Q|. This shows that X(Q) = {0}, and hence that M(Q) is indecomposable as a kN_{G}(Q)-module. Using Lemma 2.2 (ii), this shows that

M(Q) = Sc(N_{G}(Q), R) .

Set L = RC_{G}(Q). Since F = F_{P} (N_{G}(P)), it follows that N_{G}(Q) = (N_{G}(Q) ∩ N_{G}(P))C_{G}(Q). The subgroup N_{G}(P) ∩ N_{G}(Q) normalises R, and hence L is a normal subgroup of N_{G}(Q) and we have N_{G}(Q) = (N_{G}(R) ∩ N_{G}(Q))L. In particular, L acts transitively on the set of N_{G}(Q)-conjugates of R. Since M(Q) has R as a vertex and R ≤ L, there is an indecomposable kL-module V with vertex R such that M(Q) is isomorphic to a direct summand of Ind_{L}^{N_{G}(Q)}(V). The Mackey formula, using that L is normal in N_{G}(Q), implies that

Res_{L}^{N_{G}(Q)}(M(Q)) = ⊕_{x} ^{x}V

with x running over a subset E of N_{G}(Q) ∩ N_{G}(R). In particular, all indecomposable direct summands of Res_{L}^{N_{G}(Q)}(M(Q)) have R as a vertex. Thus applying the Brauer construction with respect to R sends every summand to a non-zero kN_{L}(R)- module. Therefore, if the set E has more than one element, then M(Q)(R) = M(R) is decomposable as a kN_{L}(R)-module, hence also as a kC_{G}(R)-module. This contradicts the assumptions, and hence X consists of a single element, or equivalently, Res_{L}^{N_{G}(Q)}(M(Q)) is indecomposable. Then necessarily Res_{L}^{N_{G}(Q)}(M(Q)) ≅ Sc(L, R), whence the result. □

Proof of Theorem 1.1. Set M = Sc(G, P). Suppose that M is Brauer indecomposable. Then M(Q) = Sc(N_{G}(Q), N_{P} (Q)) by Lemma 2.2 (ii), and M(Q) is indecomposable as a module for any subgroup of N_{G}(Q) containing C_{G}(Q). In particular, setting M_{Q} = Sc(N_{P} (Q)C_{G}(Q), N_{P} (Q)), we have M_{Q} ≅ Res_{N_{P} (Q)C_{G}(Q)}^{N_{G}(Q)}(M(Q)). By the assumptions, the restriction to kC_{G}(Q) of this module remains indecomposable. Suppose conversely that Res_{C_{G}(Q)}^{N_{P} (Q)C_{G}(Q)}(M_{Q}) remains indecomposable for all subgroups Q of P. Arguing by contradiction, suppose that M is not Brauer indecomposable. Let Q be a subgroup of maximal order of P such that Res_{C_{G}(Q)}^{N_{G}(Q)}(M(Q)) is decomposable. Set R = N_{P} (Q). By Lemma 3.1, the kRC_{G}(Q)-

module Res_{RC_{G}(Q)}^{N_{G}(Q)}(M(Q)) is indecomposable with vertex R, hence isomorphic to

M_{Q} = Sc(RC_{G}(Q), R) by Lemma 2.2 (ii). Thus Res_{C_{G}(Q)}^{RC_{G}(Q)}(M_{Q}) is decomposable, contradicting the assumptions. □

Proof of Theorem 1.2. Set M = Sc(G, P). Arguing by contradiction, let Q be a subgroup of maximal order in P such that M(Q) is not indecomposable as a kC_{G}(Q)-module. Set R = N_{P} (Q) and L = RC_{G}(Q). It follows from Lemma 3.1 that Q is a proper subgroup of P, and that Res_{L}^{N_{G}(Q)}(M(Q)) is indecomposable, with R as a vertex, hence isomorphic to Sc(L, R) by Lemma 2.2 (ii). By the construction of M(Q), the group Q acts trivially on M(Q).

Suppose first that hypothesis (a) holds; that is, R = QC_{P} (Q). Then L = QC_{G}(Q). Thus Res_{QC_{G}(Q)}^{N_{G}(Q)}(M(Q)) is indecomposable. Since Q acts trivially, it

follows that Res_{C_{G}(Q)}^{N_{G}(Q)}(M(Q)) is indecomposable, a contradiction.

Thus hypothesis (b) holds; that is, C_{G}(Q) is p-nilpotent. The indecomposable kN_{G}(Q)-module M(Q) = Sc(N_{G}(Q), R) is in the principal block as a kN_{G}(Q)- module, and its restriction to L = RC_{G}(Q) remains indecomposable by the above. Hence we can assume that O_{p′}(N_{G}(Q)) = 1. Then also O_{p′}(C_{G}(Q)) = 1. This implies that C_{G}(Q) is a p-group by (b). Hence the groups C_{G}(Q), C_{G}(R), L = RC_{G}(Q), and QC_{G}(Q) are all finite p-groups. Using that transitive permutation modules of finite p-groups are indecomposable, it follows that

Sc(L, R) = Res_{L}^{N_{G}(Q)}(M(Q)) ≅ Ind_{R}^{L}(k) .

The Mackey formula implies that

Res_{QC_{G}(Q)}^{N_{G}(Q)}(M(Q)) = Res_{QC_{G}(Q)}^{L} ∘ Res_{L}^{N_{G}(Q)}(M(Q))
= Res_{QC_{G}(Q)}^{L} ∘ Ind_{R}^{L}(k)
= Ind_{QC_{G}(Q)∩R}^{QC_{G}(Q)}(k)

This is again a transitive permutation module of the p-group QC_{G}(Q), hence indecomposable. As before, since Q acts trivially on M(Q), this implies that Res_{C_{G}(Q)}^{N_{G}(Q)}(M(Q)) is indecomposable. This concludes the proof. □

Proof of Corollary 1.3. Set H = N_{G}(P). The fusion system of G × H on ΔP is equal to that of ΔH on ΔP, and this is saturated as P is a Sylow-p-subgroup of H. Moreover, for Q a subgroup of P, we have C_{G×H}(ΔQ) = C_{G}(Q) × C_{H}(Q). Thus if C_{G}(Q) is p-nilpotent, then so is C_{G×H}(ΔQ). The result follows from Theorem 1.2. □

Example 3.2. Suppose that p = 3. Let G be a finite group. Assume that G has a Sylow 3-subgroup P such that P ≅ M_{3}(3), the extraspecial 3-group of order 27 of exponent 9. Set H = N_{G}(P). Then the k(G × H)-Scott module M = Sc(G × H, P) induces a stable equivalence of Morita type between the principal blocks B_{0}(kG) and B_{0}(kH). This is trivial if G is 3-nilpotent because both blocks are isomorphic to kP in that case. If G is not 3-nilpotent, then |N_{G}(P)/P C_{G}(P)| = 2. Let Q be a non-trivial subgroup of P. It follows from Theorem 1.2, results of Hendren [7, Propositions 5.12 and 5.13] and the Z_{3}^{∗}-theorem that M(Q) induces a Morita equivalence between B_{0}(kC_{G}(Q)) and B_{0}(kC_{H}(Q)). Hence the gluing theorem [4, Theorem 6.3] implies that M induces a stable equivalence of Morita type between the principal blocks of kG and kH. Furthermore, by [4, Proposition 5.3], such a stable equivalence of Morita type implies the equality k(B_{0}(kG)) − ℓ(B_{0}(kG)) = k(B_{0}(kH)) − ℓ(B_{0}(kH)), where k(B_{0}(kG)) and ℓ(B_{0}(kG)) denote the number of ordinary and modular irreducible characters B_{0}(kG), respectively, with the analogous notation for H instead of G. This yields a proof of a special case of a result of Hendren [7, Theorem 5.14]: if G is not 3-nilpotent, then k(B_{0}(kG)) − ℓ(B_{0}(kG)) = 8.

Remark 3.3. With the notation of 1.3, the Scott module Sc(G × N_{G}(P), ΔP) is the Green correspondent of the Scott module Sc(G × G, ΔP), which is isomorphic to the prinicipal block of kG viewed as a k(G × G)-module. One might wonder how to generalise 1.3 to arbitrary blocks. Let b be a block of kG and let (P, e_{P}) be a maximal (G, b)-Brauer pair. Set H = N_{G}(P, e_{P}). The G × H-Green correspondent with vertex ΔP of the k(G × G)-module kGb is of the form M = kGf for some primitive idempotent f in (kGb)^{ΔH} satisfying Br_{ΔP} (f)e_{P} ≠ 0 (see e. g. [1]). Note that (P, e_{P}) is also a maximal (H, e_{P})-Brauer pair. For any subgroup Q of P denote by e_{Q} the unique block of kC_{G}(Q) satisfying (Q, e_{Q}) ≤ (P, e_{P}) and by f_{Q} the unique block of kC_{H}(Q) satisfying (Q, e_{Q}) ≤ (P, e_{P}). The ‘obvious’ generalisation of 1.3 would be the statement that the kC_{G}(Q)e_{Q}-kC_{H}(Q)f_{Q}-bimodule e_{Q}M(ΔQ)f_{Q} is indecomposable. This is, however, not the case in general. In order to construct an example for which this is not the case, we first translate this indecomposability to the source algebra level.

Let j ∈ (kHe_{P})^{ΔP} be a source idempotent e_{P} as a block of kHe_{P}. Then i = jf is a source idempotent of kGb (see e. g. [5, 4.10]). Thus multiplication by f induces an interior P -algebra homomorphism from B = jkHj to A = ikGi. In particular, A can be viewed as an A-B-bimodule. Multiplication by a source idempotent, or more generally, by an almost source idempotent, is a Morita equivalence (cf. [15, 3.5] and [12, 4.1]). Moreover, the Brauer construction with respect to a fully F-centralised subgroup Q of P sends the source idempotent i to the almost source idempotent Br_{ΔQ}(i) in kC_{G}(Q)e_{Q} (cf. [12, 4.5]). Through the appropriate Morita equivalences, the kGb-kHe_{P} -bimodule M = kGf corresponds to the A-B-bimodule iMj = ikGjf = A, and the kC_{G}(Q)e_{Q}-kC_{H}(Q)f_{Q}-bimodule e_{Q}M(ΔQ)f_{Q} corresponds to the A(ΔQ)-B(ΔQ)-bimodule A(ΔQ). It follows that for Q a fully F-centralised subgroup of P, the indecomposability of e_{Q}M(ΔQ)f_{Q} is equivalent to the indecomposability of A(ΔQ) as an A(ΔQ)-B(ΔQ)-module.

We construct an example for which this fails. Suppose that p is odd. Let P be an extraspecial p-group of order p^{3} of exponent p. Let Q be a subgroup of order p^{2} in P; we have C_{P} (Q) = Q and in particular, Q is fully centralised (even centric) with respect to any fusion system on P. Set V = Inf_{P/Q}^{P}(Ω_{P/Q}(k)). Thus dim_{k}(V) = p − 1, and Q acts trivially on V. Setting S = End_{k}(V), it follows that S = S^{ΔQ} ≅ S(ΔQ). By the main result of Mazza in [13], there exists a nilpotent block of some finite group having a source algebra isomorphic to A = S ⊗_{k} kP. The Brauer correspondent of such a block has source algebra B = kP. We have A(ΔQ) (S ⊗_{k} kP)(ΔQ) ≅ S ⊗_{k} kQ and B(ΔQ) = kQ. Thus any primitive idempotent e in S = S^{ΔQ} determines a nontrivial direct bimodule summand Se ⊗_{k} kQ of A(ΔQ), and hence A(ΔQ) is not indecomposable as an A(ΔQ)-B(ΔQ)-module.

Acknowledgements. A part of this work was done while the second author was visiting the first and the third authors in the University of Aberdeen in February 2012 and in the City University London in October 2013. He would like to thank the Mathematics Institutes for their wonderful hospitality. The second author was supported by the Japan Society for Promotion of Science (JSPS) Grant-in-Aid for Scientific Research(C) 23540007, 2011–2014.

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City University London, Department of Mathematics, Northampton Square, London, EC1V 0HB, United Kingdom

E-mail address: radha.kessar.1@city.ac.uk

Department of Mathematics, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

E-mail address: koshitan@math.s.chiba-u.ac.jp

City University London, Department of Mathematics, Northampton Square, London, EC1V 0HB, United Kingdom

E-mail address: markus.linckelmann.1@city.ac.uk