November 19, 2010

Broué’s abelian defect group conjecture holds for the sporadic simple Conway group Co_{3}

Shigeo Koshitani^{a,∗}, Jürgen Müller^{b}, Felix Noeske^{c}

aDepartment of Mathematics, Graduate School of Science, Chiba University, Chiba, 263-8522, Japan b, cLehrstuhl D für Mathematik, RWTH Aachen University, 52062, Aachen, Germany

Abstract In the representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block A_{N} of the normaliser N_{G}(P) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Strong Version of Broué’s Abelian Defect Group Conjecture. In this paper, we prove that the strong version of Broué’s abelian defect group conjecture is true for the non-principal 2-block A with an elementary abelian defect group P of order 8 of the sporadic simple Conway group Co_{3}. This result completes the verification of the strong version of Broué’s abelian defect group conjecture for all primes p and for all p-blocks of Co_{3}. Keywords: Broué’s conjecture; abelian defect group; sporadic simple Conway group

1. Introduction and notation

In the representation theory of finite groups, one of the most important and interesting problems is to give an affirmative answer to a conjecture which was introduced by Broué around 1988 [5], and is nowadays called Broué’s Abelian Defect Group Conjecture. He actually conjectures the following:

Conjecture 1.1 (Strong version of Broué’s Abelian Defect Group Conjecture [5], [17]). Let p be a prime, and let (K, O, k) be a splitting p-modular system for all subgroups of a finite group G. Assume that A is a block algebra of OG with a defect group P and that A_{N} is a block algebra of ON_{G}(P) such that A_{N} is the Brauer correspondent of A, where N_{G}(P) is the normaliser of P in G. Then A and A_{N} should be derived equivalent (Rickard equivalent) provided P is abelian.

In fact, a stronger conclusion than 1.1 is expected, namely that A and A_{N} are splendidly Rickard equivalent in the sense of Linckelmann ([32], [33]), which he calls splendidly derived equivalent, see 1.12. Note that for principal block algebras, this notion coincides with the splendid equivalence given by Rickard in [46].

Conjecture 1.2 (Rickard [46], [47]). Keeping the notation, we suppose that P is abelian as in 1.1. Then there should be a splendid Rickard equivalence between the block algebras A of OG and A_{N} of ON_{G}(P).

There are several cases where the conjectures 1.1 and 1.2 of Broué and Rickard, respectively, have been verified. For example, in [19, (0.2)Theorem] it is shown that 1.1 and 1.2 are true for the principal block algebra A of an arbitrary finite group G when the defect group P of A is elementary abelian of order 9 (and hence p = 3). As extensions of this, there are results for non-principal 3-blocks with the same defect group C_{3} × C_{3}, see [20], [21], [39], [22] and [24].

On the other hand, let us look at the case where a block A has an elementary abelian defect group P of order 8, namely, P = C_{2} × C_{2} × C_{2}. The numbers of irreducible ordinary characters k(A) and of irreducible Brauer characters ℓ(A), respectively, are important in block theory. For the principal 2-blocks they have been known for some time, see [18] and [27], for instance. However, only recently, the numbers of irreducible ordinary characters k(A) and of irreducible Brauer characters ℓ(A) for non-principal 2-blocks have been determined in general, see [15]. In [15] it is proved with the help of the classification of finite simple groups, that Alperin’s weight conjecture and also the weak version (character theoretic version) of Broué’s abelian defect group conjecture for arbitrary 2-blocks with defect group C_{2} × C_{2} × C_{2} are both true. The strong version of Broué’s abelian defect group conjecture, namely, the existence of Rickard splendid equivalences between blocks corresponding via the Brauer correspondence for arbitrary 2-blocks with defect group C_{2} × C_{2} × C_{2}, is still open. There are four cases for the inertial index e of A with the defect group P = C_{2} × C_{2} × C_{2}. Namely, e = 1, 3, 7 or 21, since Aut(P) ≅ GL_{3}(2) has a unique maximal 2′-subgroup, up to conjugacy, which is isomorphic to the Frobenius group F_{21} = C_{7} ⋊ C_{3} of order 21. For the cases where e = 1 everything is known because the blocks are nilpotent, see Broué-Puig [9]. For the case e = 3, there are results of Landrock [27] and Watanabe [55].

∗ Corresponding author.

E-mail addresses: koshitan@math.s.chiba-u.ac.jp (S.Koshitani), juergen.mueller@math.rwth-aachen.de (J.Müller), Felix.Noeske@math.rwth-aachen.de (F.Noeske).

Our objective in this paper is to investigate a non-principal 2-block with elementary abelian defect group P of order 8, which has inertial index 21. An interesting candidate for this endeavour is the non-principal 2-block of Conway’s third group Co_{3}, for which we investigate whether the strong version of Broué’s abelian defect group conjecture holds. For previous results on Co_{3}, its defect groups, and 2-modular characters confer [12, p.193 Table 6], [26, §7 p.1879], [27, Theorems 3.10 and 3.11], and [53], for example.

Our main theorem of this paper is the following:

Theorem 1.3. Let G be the sporadic simple Conway group Co_{3}, and let (K, O, k) be a splitting 2-modular system for all subgroups of G, see 1.11. Suppose that A is a non-principal block algebra of OG with a defect group P which is an elementary abelian group of order 8, and that A_{N} is a block algebra of ON_{G}(P) such that A_{N} is the Brauer correspondent of A. Then A and A_{N} are splendidly Rickard equivalent, and hence the conjectures 1.1 and 1.2 of Broué and Rickard both hold.

Actually, 1.3 is the last tile in the mosaic proving both Broué’s abelian defect group conjecture and Rickard’s conjecture for Co_{3} in arbitrary characteristic. By [30], [45], and [51] the conjecture is proved for blocks of cyclic defect groups. Hence, since |G| = 2^{10}·3^{7}·5^{3}·7·11·23, see [10, p.134], it is sufficient to consider the primes p ∈ {2, 3, 5}. For odd p the only block with defect at least 2 is the principal block, whose defect groups are not abelian. For p = 2 there is precisely a unique block with a non-cyclic abelian defect group. Its defect group is isomorphic to C_{2} × C_{2} × C_{2} (see [59, Co_{3}], [26, p.1879] and [53, p.494 §2]). Therefore we may state the following immediate consequence of 1.3:

Corollary 1.4. The strong version of Broué’s abelian defect group conjecture 1.1 and even Rickard’s splendid equivalence conjecture 1.2 are true for all primes p and for all block algebras of OG if G = Co_{3}.

As a matter of fact, the main result 1.3 is obtained by proving the following:

Theorem 1.5. We keep the notation and the assumption as in 1.3. Let H be a maximal subgroup of G with H = R(3) × 𝔖_{3} ⩾ N_{G}(P), where R(3) = ^{2}G_{2}(3) ≅ SL_{2}(8) ⋊ C_{3} is the smallest Ree group, 𝔖_{3} is the symmetric group on 3 letters, and C_{3} is the cyclic group of order 3. Let B be a block algebra of OH such that B is the Brauer correspondent of A, see [40, Chap.5 Theorem 3.8]. In addition, let 𝔣 denote the Green correspondence with respect to (G × G, ΔP, G × H), and let M = 𝔣(A). Then M induces a Morita equivalence between A and B, and hence it is a Puig equivalence.

The following result is used to get 1.7 from our main result 1.5.

Theorem 1.6 (Landrock-Michler [29] and Okuyama [42]). Let p = 2, and let R(q) = ^{2}G_{2}(q) be a Ree group, where q = 3^{2n+1} for some n = 0, 1, 2, · · ·. Let (K, O, k) be a splitting 2-modular system for all subgroups of R(q), for all q at the same time, see [56, Theorem 3.6], and let B_{0}(OR(q)) be the principal block algebra of the group algebra OR(q). Then the block algebras B_{0}(OR(3)) and B_{0}(OR(q)) are Puig equivalent. In particular, Broué’s abelian defect group conjecture 1.1 and Rickard’s conjecture 1.2 hold for the principal block algebras of R(q) for any q.

Proof. This follows from [29, Theorem 5.3] and [42, Example 3.3 and Remark 3.4]. □

Corollary 1.7. We keep the notation and the assumption as in 1.3. Let R(q) = ^{2}G_{2}(q) be a Ree group, where q = 3^{2n+1} for some n = 0, 1, 2, · · ·. We may assume that (K, O, k) also is a splitting 2-modular system for all subgroups of R(q), for all q at the same time. Let B_{0}(OR(q)) be the principal block algebra of the group algebra OR(q). Then A and B_{0}(OR(q)) are Puig equivalent.

Strategy 1.8. Our starting point for this work is the observation that the 2-decomposition matrix for the non-principal block A of Co_{3} with an elementary abelian defect group of order 8, see [53], is exactly the same as that for the principal 2-block B of R(3) ≅ SL_{2}(8) ⋊ C_{3}, see [29]. Therefore it is natural to ask whether these two 2-block algebras are Morita equivalent not only over an algebraically closed field k of characteristic 2 but also over a complete discrete valuation ring O whose residue field is k. Furthermore, one might even expect that they are Puig equivalent, see 1.12. If this is the case, since the two conjectures of Broué and Rickard 1.1 and 1.2 respectively have been shown to hold for the principal 2-block of R(3) in a paper of Okuyama [42], it follows that these conjectures also hold for the non-principal 2-block of Co_{3} with the same defect group P = C_{2} × C_{2} × C_{2}.

The verification that A and B are indeed Morita equivalent relies on theorems by Linckelmann, Broué, Rickard and Rouquier. Linckelmann has shown in [31] that a stable equivalence of Morita type between A and B which maps simple modules to simple modules is in fact a Morita equivalence, see 2.1. To obtain an appropriate stable equivalence, we employ a variant of a ”glueing” theorem, which is due to (originally Broué [6, 6.3.Theorem]), Rickard [46, Theorem 4.1], Rouquier [52, Theorems 5.6 and 6.3, Remark 6.4], and Linckelmann, see [32],

[34] and 2.3: A stable equivalence between two blocks A and B may be derived from Morita equivalences between unique blocks of the centralizers of non-trivial subgroups of P in Co_{3} and R(3). Once we have obtained a stable equivalence of Morita type between A and B, it remains to show that it preserves simplicity of modules as stated above. Usually this may be a very hard task.

Contents 1.9. The paper is structured as follows: In Section 2, we give the fundamental lemmas which are used to prove our main results. Furthermore, we establish some properties of the stable equivalences we consider, and collect some further results on Morita equivalences and Green correspondence for ease of reference. In Section 3 we investigate non-principal 2blocks of the symmetric group 𝔖_{5} and the Mathieu group M_{12} whose structure will be used later on in order to get our main theorems. In Section 4 the main objective is to construct the stable equivalence of Morita type between the blocks A and B as outlined above. In order to apply glueing theorems of Rouquier and Linckelmann 2.3, we begin by analysing the 2-local structure of Co_{3} to identify the groups. Then, we combine this knowledge and what we get already in Section 3 to give a stable equivalence F as saught. Section 5 prepares the proof that F maps simple A-modules to simple B-modules. In order to prove this fact, we collect information on simple and indecomposable modules in the three blocks A, B, and A_{N}. In Section 6 we determine the F -images of the simple A-modules, thus showing that they are indeed all simple. Finally, in Section 7 we combine the previous results to give complete proofs of our main theorems 1.3, 1.4, 1.5 and 1.7. At the end of the paper, we have collected several useful properties of the stable equivalences obtained through 2.3.

Computations 1.10. A few words on computer calculations are in order. To find our results, next to theoretical reasoning we have to rely fairly heavily on computations. Of course, many of the data contained in explicit libraries and databases are of computational nature, and quite a few traces of further computer calculations are still left in the present exposition. But we would like to point out that we have found many of our intermediate results by explicit computations first, which have subsequently been replaced by more theoretical arguments.

As tools, we use the computer algebra system GAP [13], to calculate with permutation groups and tables of marks, as well as with ordinary and Brauer characters. We also make use of the data library [4], in particular allowing for easy access to the data compiled in [10], [14] and [59], and of the interface [58] to the data library [60]. Moreover, we use the computer algebra system MeatAxe [49] to handle matrix representations over finite fields, as well as its extensions to compute submodule lattices [35], radical and socle series [38], homomorphism spaces and endomorphism rings [37], and direct sum decompositions [36]. We give more comments later on where necessary.

Notation 1.11. Throughout this paper, we use the standard notation and terminology as is used in [40], [54] and [10].

Let k be a field and assume that A and B are finite dimensional k-algebras. We denote by mod-A, A-mod and A-mod-B the categories of finitely generated right A-modules, left Amodules and (A, B)-bimodules, respectively. We write M_{A}, _{A}M and _{A}M_{B} when M is a right A-module, a left A-module and an (A, B)-bimodule. In this note, a module always refers to a finitely generated right module, unless stated otherwise. We let M^{∨} = Hom_{A}(M_{A}, A_{A}) be the A-dual of M, so that M^{∨} becomes a left A-module via (aϕ)(m) = a·ϕ(m) for a ∈ A, ϕ ∈ M^{∨} and m ∈ M, and we let M^{⊛} = Hom_{k}(M, k) be the k-dual of M, so that M^{⊛} becomes a left A-module as well via (aϕ)(m) = ϕ(ma) for a ∈ A, ϕ ∈ M^{⊛} and m ∈ M. For A-modules M and N we write [M, N]^{A} for dim_{k}[Hom_{A}(M, N)]. For an A-module M and the projective cover P (S) of a simple A-module S, we write [P (S) | M]^{A} for the multiplicity of direct summands of M which are isomorphic to P (S). If A is self-injective, the stable module category \underline{mod}-A, is the quotient category of mod-A with respect to the projective A-homomorphisms, that is those factoring through a projective module.

In this paper, G is always a finite group and we fix a prime number p. Assume that (K, O, k) is a splitting p-modular system for all subgroups of G, that is to say, O is a complete discrete valuation ring of rank one such that its quotient field is K which is of characteristic zero, and its residue field O/rad(O) is k, which is of characteristic p, and that K and k are splitting fields for all subgroups of G. By an OG-lattice we mean a finitely generated right OG-module which is a free O-module. We denote by k_{G} the trivial kG-module, and similarly by O_{G} the trivial OG-lattice. If X is a kG-module, then we write X^{∗} = Hom_{k}(X, k) for the contragredient of X, namely, X^{∗} = Hom_{k}(X, k) which is again a right kG-module via (φg)(x) = φ(xg^{−1}) for x ∈ X, φ ∈ X^{∗} and g ∈ G; if no confusion may arise we also call this the dual of X. Let H be a subgroup of G, and let M and N be an OG-lattice and an OH-lattice, respectively. Then let M↓_{H}^{G} = M↓_{H} be the restriction of M to H, and let N↑_{H}^{G} = N↑^{G} = (N ⊗_{OH} OG)_{OG} be the induction (induced module) of N to G. A similar definition holds for kG- and kH-modules. For a subgroup Q of G we write Scott(G, Q) for the (Alperin-)Scott module with respect to Q in G, see [40, Chap.4 p.297].

We denote by Irr(G) and IBr(G) the sets of all irreducible ordinary and Brauer characters of G, respectively. We write 1_{G} for the trivial character of G, and we write χ^{∗} for the complex conjugate of χ ∈ Irr(G). For χ, ψ ∈ Irr(G) we denote by (χ, ψ)^{G} the usual inner product. If A is a block algebra (p-block) of OG, then we write Irr(A) and IBr(A) for the sets of all characters in Irr(G) and IBr(G) which belong to A, respectively. We denote by B_{0}(kG) the principal block algebra of kG.

Let G′ be another finite group, and let V be an (OG, OG′)-bimodule. Then we can regard V as a right O[G × G′]-module. A similar definition holds for (kG, kG′)-bimodules. We denote by ΔG = {(g, g) ∈ G × G | g ∈ G} the diagonal copy of G in G × G. For an (OG, OG′)-bimodule V and a common subgroup Q of G and G′, we set V^{ΔQ} = {v ∈ V | qv = vq for all q ∈ Q}. If Q is a p-group, the Brauer construction is defined to be the quotient V (ΔQ) = V^{ΔQ}/[∑_{R≨Q} Tr↑_{R}^{Q}(V^{ΔR}) + radO·V^{ΔQ}], where Tr↑_{R}^{Q} is the usual trace map. The Brauer homomorphism Br_{ΔQ} : (OG)^{ΔQ} → kC_{G}(Q) is obtained from composing the canonical epimorphism (OG)^{ΔQ} ↠ (OG)(ΔQ) and the canonical isomorphism (OG)(ΔQ) →^{≈} kC_{G}(Q).

For a positive integer n, 𝔄_{n} and 𝔖_{n} denote the alternating and the symmetric groups on n letters, M_{n} denotes the Mathieu group, and C_{n} and D_{n} denote the cyclic group and the dihedral group of order n, respectively. We denote by Z(G) the center of G, and by S^{g} a set g^{−1}Sg for g ∈ G and a subset S of G.

Equivalences 1.12. Let A and A′ be block algebras of OG and OG′, respectively. Then we say that A and A′ are Puig equivalent if A and A′ have a common defect group P, and if there is a Morita equivalence between A and A′ which is induced by an (A, A′)-bimodule 𝔐 such that, as a right O[G × G′]-module, 𝔐 is a trivial source module and ΔP -projective. A similar definition holds for blocks of kG and kG′. Due to a result of Puig (and independently of Scott), see [44, Remark 7.5], this is equivalent to a condition that A and A′ have source algebras which are isomorphic as interior P -algebras, see [33, Theorem 4.1].

We say that A and A′ are stably equivalent of Morita type if there exists an (A, A′)- bimodule 𝔐 such that _{A}𝔐 is projective as a left A-module, 𝔐_{A′} is projective as a right A′-module, _{A}(𝔐 ⊗_{A′} 𝔐^{∨})_{A} ≅ _{A}A_{A} ⊕ (proj (A, A)-bimod) and _{A′}(𝔐^{∨} ⊗_{A} 𝔐)_{A′} ≅ _{A′}A′_{A′} ⊕ (proj (A′, A′)-bimod).

We say that A and A′ are splendidly stably equivalent of Morita type if A and A′ have a common defect group P and the stable equivalence of Morita type is induced by an (A, A′)-bimodule 𝔐 which is a trivial source O[G × G′]-module and is ΔP -projective, see [33, Theorem 3.1].

We say that A and A′ are derived equivalent (or Rickard equivalent) if D^{b}(mod-A) and D^{b}(mod-A′) are equivalent as triangulated categories, where D^{b}(mod-A) is the bounded derived category of mod-A. In that case, there even is a Rickard complex M^{•} ∈ C^{b}(A-mod-A′), where the latter is the category of bounded complexes of finitely generated (A, A′)-bimodules, all of whose terms are projective both as left A-modules and as right A′-modules, such that M^{•} ⊗_{A′} (M^{•})^{∨} ≅ A in K^{b}(A-mod-A) and (M^{•})^{∨} ⊗_{A} M^{•} ≅ A′ in K^{b}(A′-mod-A′), where K^{b}(A-mod-A) is the homotopy category associated with C^{b}(A-mod-A). In other words, in that case we even have K^{b}(mod-A) ≅ K^{b}(mod-A′).

We say that A and A′ are splendidly Rickard equivalent if K^{b}(mod-A) and K^{b}(mod-A′) are equivalent via a Rickard complex M^{•} ∈ C^{b}(A-mod-A′) as above, such that additionally each of its terms is a direct sum of ΔP -projective trivial source modules as an O[G × G′]-module.

2. Preliminaries

In this section we give several theorems crucial to the later sections of this paper. We state these results in a more general context; in particular, G is an arbitrary finite group and (K, O, k) is a p-modular splitting system for G. As we draw upon these lemmas frequently in the sequel, we state these explicitly for the convenience of the reader and ease of reference.

As stated in the introduction, our approach centers around 2.1 which allows us to verify that a stable equivalence of Morita type is in fact a Morita equivalence. The stable equivalences investigated are obtained with the help of 2.3, and are realised by tensoring with a bimodule given through Green correspondence. We proceed to study several properties of these stable equivalences, and give some further results needed in the upcoming parts of this paper. We refer the reader also to the appendix for a more detailed discussion of further properties of stable equivalences obtained through 2.3.

Lemma 2.1 (Linckelmann [31]). Let A and B be finite-dimensional k-algebras such that A and B are both self-injective and indecomposable as algebras, but not simple. Suppose that there is an (A, B)-bimodule M such that M induces a stable equivalence between the algebras A and B.

(i) If M is indecomposable then for any simple A-module S, the B-module (S ⊗_{A} M)_{B} is non-projective and indecomposable.

(ii) If for all simple A-module S the B-module S ⊗_{A} M is simple then M induces a Morita

equivalence between A and B. (iii) If (M, M^{∨}) induces a stable equivalence of Morita type between A and B then there is

a unique (up to isomorphism) non-projective indecomposable (A, B)-bimodule M′ such that M′ | M, and (M′, M′^{∨}) induces a stable equivalence of Morita type between the algebras A and B.

Proof. (i) and (ii) respectively are given in [31, Theorem 2.1(ii) and (iii)]. Part (iii) follows by [31, Theorem 2.1(i) and Remark 2.7]. □

We obtain a suitable stable equivalence to apply 2.1 through a “glueing theorem” as given in 2.3.

Lemma 2.2 (Koshitani-Linckelmann [23]). Let A be a block algebra of kG with defect group P, and let (P, e) be a maximal A-Brauer pair such that H = N_{G}(P, e) = N_{G}(P). Let B be a block algebra of kH such that B is the Brauer correspondent of A. Let 𝔣 be the Green correspondence with respect to (G × G, ΔP, G × H), and set M = 𝔣(A), in particular M is an indecomposable (A, B)-bimodule with vertex ΔP.

Take any subgroup Q of Z(P), and set G_{Q} = C_{G}(Q) and H_{Q} = C_{H}(Q). Let e_{Q} and f_{Q} be block idempotents of kG_{Q} and kH_{Q} satisfying (Q, e_{Q}) ⊆ (P, e) and (Q, f_{Q}) ⊆ (P, e), respectively, see [54, (40.9) Corollary]. Let 𝔣_{Q} be the Green correspondence with respect to (G_{Q} × G_{Q}, ΔP, G_{Q} × H_{Q}). Then we have

e_{Q}M(ΔQ)f_{Q} = 𝔣_{Q}(e_{Q}kG_{Q})

and this is a unique (up to isomorphism) indecomposable direct summand of (e_{Q}kG_{Q})↓_{G_{Q}×H_{Q}} with vertex ΔP.

Proof. We know M = 𝔣(A) | A↓_{G×H}^{G×G} | _{kG}kG_{kH}. Hence, M(ΔQ) | (kG)(ΔQ) = kC_{G}(Q) = kG_{Q}. Thus,

e_{Q}M(ΔQ)f_{Q} | e_{Q}kG_{Q}f_{Q} | (e_{Q}kG_{Q})↓_{G_{Q}×H_{Q}}^{G_{Q}×G_{Q}}.

By [23, Theorem], e_{Q}M(ΔQ)f_{Q} is an indecomposable k[G_{Q} × H_{Q}]-module with vertex ΔP. Thus Green correspondence yields e_{Q}M(ΔQ)f_{Q} = 𝔣_{Q}(e_{Q}kG_{Q}). □

Lemma 2.3 (Linckelmann [33], [34]). Let A be a block algebra of OG with a defect group P, and let (P, e) be a maximal A-Brauer pair in G. Set H = N_{G}(P, e), Assume that

(1) P is abelian,

(2) for each Q with 1 ≠ Q ⩽ P, kC_{G}(Q) has a unique block algebra A_{Q} with the defect group P,

(3) for each Q with 1 ≠ Q ⩽ P, kC_{H}(Q) has a unique block algebra B_{Q} with the defect

group P. Let B a block algebra of OH which is the Brauer correspondent of A. For each subgroup Q of P, let e_{Q} and f_{Q} be the block idempotents of A_{Q} and B_{Q}, respectively, and hence A_{Q} = kC_{G}(Q)e_{Q} and B_{Q} = kC_{H}(Q)f_{Q}. Note that e_{P} = e = f_{P} and A_{P} = B_{P}. Let 𝔣 be the Green correspondence with respect to (G × G, ΔP, G × H), and set _{A}M_{B} = 𝔣(A), see 2.4. Moreover, let 𝔣_{Q} be the Green correspondence with respect to (C_{G}(Q) × C_{G}(Q), ΔP, C_{G}(Q) × C_{H}(Q)). Now, assume further that

(4) for each non-trivial proper subgroup Q of P, the (A_{Q}, B_{Q})-bimodule 𝔣_{Q}(A_{Q}) induces a

Morita equivalence between A_{Q} and B_{Q}. Then the (A, B)-bimodule M induces a stable equivalence of Morita type between A and B.

Proof. First, note H = N_{G}(P). Secondly, it follows from 2.2 that e_{Q}·M(ΔQ)·f_{Q} = 𝔣_{Q}(A_{Q}) for each Q ⩽ P since P is abelian by (1). Then since A_{P} = B_{P} and since A_{P} = 𝔣_{P} (A_{P}) = e·M(ΔP)·e, the (A_{P} , B_{P})-bimodule e_{P} ·M(ΔP)·e_{P} induces a Morita equivalence between A_{P} and B_{P}.

Now, for each Q ⩽ P, it follows from the uniqueness of e_{Q} and f_{Q} that

(Q, e_{Q}) ⊆ (P, e)  and  (Q, f_{Q}) ⊆ (P, e).

Next, we want to claim

E_{G}((Q, e_{Q}), (R, e_{R})) = E_{H}((Q, f_{Q}), (R, f_{R})) for Q, R ⩽ P,

where E_{G}((Q, e_{Q}), (R, e_{R})) is the set {φ : Q → R | there is g ∈ G with φ(u) = u^{g}, for all u ∈ Q, and (Q, e_{Q})^{g} ⊆ (R, e_{R})}, see [33, p.821]. This is known by using [2, Proposition 4.21 and Theorem 3.4] and [8, Theorem 1.8(1)] since P is abelian, see [21, The proof of 1.15. Lemma] for details. Therefore we can apply Linckelmann’s result [33, Theorem 3.1]. □

We remark that in [33, Theorem 3.1] and [34, Theorem A.1], Linckelmann proves more general theorems than 2.3. However, we formulate with 2.3 a version which is specifically tailored to our practical purposes, and use this ad hoc version in the sequel.

In the notation of 2.3, we have that the bimodule M realising a stable equivalence between A and B is a Green correspondent of A. In fact it is a direct summand of 1_{A} · kG · 1_{B} as the next lemma shows.

Lemma 2.4. Let A be a block algebra of kG with defect group P. Assume that (P, e) is a maximal A-Brauer pair such that H = N_{G}(P, e) = N_{G}(P). Let B be a block algebra of kH such that B is the Brauer correspondent of A. Let 𝔣 be the Green correspondence with respect to (G × G, ΔP, G × H). Then we have 𝔣(A) | 1_{A}·kG·1_{B}.

Proof. It follows from [3, Theorem 5(i)] that (A↓_{G×H}^{G×G})·1_{B} = 1_{A}·kG·1_{B} has a unique (up to isomorphism) indecomposable direct summand with vertex ΔP. Clearly, 1_{A}·kG·1_{B} | (A↓_{G×H}^{G×G}), hence by Green correspondence we have 𝔣(A) | 1_{A}·kG·1_{B}. □

We remark that a stable equivalence of Morita type induced by the Green correspondent 𝔣(A) in the context of 2.4 preserves vertices and sources, see A.3.

Lemma 2.5. Let G, H, and L be finite groups, all of which have a common non-trivial psubgroup P, and assume that H ⩽ G. Let A, B, and C be block algebras of kG, kH, and kL, respectively, all of which have P as their defect group. In addition, suppose that a pair (_{A}𝔐_{B},_{B}𝔐′_{A}) induces a stable equivalence between A and B such that _{A}𝔐_{B} | k_{ΔP} ↑^{G×H}, B𝔐′_{A} | k_{ΔP} ↑^{H×G}, and that 𝔐 and 𝔐′ preserve vertices and sources. Similarly, suppose that a pair (_{B}𝔑_{C},_{C}𝔑′_{B}) induces a stable equivalence between B and C such that _{B}𝔑_{C} | k_{ΔP} ↑^{H×L}, C𝔑′_{B} | k_{ΔP} ↑^{L×H}, and that 𝔑 and 𝔑′ preserve vertices and sources. Then we have (A, C)- and (C, A)-bimodules M and M′, respectively, which satisfy the following:

(1) _{A}(𝔐 ⊗_{B} 𝔑)_{C} = _{A}M_{C} ⊕ (proj (A, C)-bimodule) and C(𝔑′ ⊗_{B} 𝔐′)_{A} = _{C}M′_{A} ⊕ (proj (C, A)-bimodule).

(2) _{A}M_{C} and _{C}M′_{A} are both non-projective indecomposable.

(3) The pair (M, M′) induces a stable equivalence between A and C.

(4) The functors

− ⊗_{A} M : mod-A ⟶ mod-C

and

− ⊗_{C} M′ : mod-C ⟶ mod-A

preserve vertices and sources of indecomposable modules. That is, for non-projective indecomposable A- and C-modules X and Y corresponding via X ⊗_{A} M = Y ⊕ (proj) and Y ⊗_{C} M′ = X ⊕ (proj), respectively, there is a non-trivial p-subgroup Q and an indecomposable kQ-module S such that Q is a common vertex of X and Y and that S is a common source of X and Y.

(5) _{A}M_{C} | k_{ΔP} ↑^{G×L} and _{C}M′_{A} | k_{ΔP} ↑^{L×G}.

(6) In particular, if a pair (𝔐, 𝔐^{∨}) induces a stable equivalence of Morita type between A and B, and if a pair (𝔑, 𝔑^{∨}) induces a stable equivalence of Morita type between B and C, then we can replace M′ above by M^{∨} and we have that the pair (M, M^{∨}) induces a stable equivalence of Morita type between A and C.

Proof. Obviously, the pair (_{A}(𝔐 ⊗_{B} 𝔑)_{C},_{C}(𝔑′ ⊗_{B} 𝔐′)_{A}) induces a stable equivalence between A and C. Clearly, _{A}(𝔐 ⊗_{B} 𝔑), (𝔐 ⊗_{B} 𝔑)_{C}, _{C}(𝔑′ ⊗_{B} 𝔐′), and (𝔑′ ⊗_{B} 𝔐′)_{A} are all projective. Since A and C are symmetric algebras, it follows from 2.1(iii) that there are (A, C)- and (C, A)-bimodules M and M′ which satisfy the conditions (1)–(4).

Next we want to show (5). It follows from [40, Chap.5 Lemma 10.9(iii)] that

M | 𝔐 ⊗_{B} 𝔑 | (k_{ΔP} ↑^{G×H}) ⊗_{kH} (k_{ΔP} ↑^{H×L})

≅ (kG ⊗_{kP} kH) ⊗_{kH} (kH ⊗_{kP} kL) ≅ kG ⊗_{kP} [(kH)↓_{P ×P}^{H×H}] ⊗_{kP} kL

≅ kG ⊗_{kP} ( ⨁_{h∈[P \H/P]} k[P hP]) ⊗_{kP} kL ≅ ⨁_{h∈[P \H/P]} k[P hP]↑_{P ×P}^{G×L}.

Since _{A}M_{C} is indecomposable, there is an element h ∈ H such that M | k[P hP]↑_{P ×P}^{G×L}. Set (P × P)_{h} = {(u, h^{−1}uh) ∈ P × P | u ∈ P ∩ hP h^{−1}}. Then

(P × P)_{h} = {(huh^{−1}, u) ∈ P × P | u ∈ P ∩ h^{−1}P h} = (h, 1)·Δ[P ∩ P^{h}]·(h^{−1}, 1).

We get by [40, Chap.5 Lemma 10.9(iii)] that k[P hP] ≅ k_{(h,1)Δ[P ∩P^{h}](h^{−1},1)}↑^{P ×P}, and hence M | k_{(h,1)Δ[P ∩P^{h}](h^{−1},1)}↑^{G×L}. Now, since (h^{−1}, 1) ∈ H × L ⩽ G × L, we have that

M | k_{Δ[P ∩P^{h}]}↑^{G×L} ≅ kG ⊗_{kQ} kL

where Q = P ∩ P^{h}. Then for any X in mod-A the module X ⊗_{A} M has a vertex contained in Q. If Q is a proper subgroup of P then, since (M, M′) induces a stable equivalence between A and C, any module in mod-C has a vertex properly contained in P, a contradiction since P is a defect group of C. Hence Q = P, so that h ∈ N_{H}(P) ⊆ N_{G}(P). Therefore M | k_{ΔP} ↑^{G×L}. An analogous argument gives the claim for M′.

(6) Follows from (1)–(5) and 2.1(iii). □

Next, we give some results on Morita equivalences and tensor products, which will be useful in Section 4.

Lemma 2.6. The following hold:

(i) Let A, B, C and D be finite dimensional k-algebras. Assume that an (A, B)-bimodule M realizes a Morita equivalence between A and B, and so does a (C, D)-bimodule N between C and D. Then the (A⊗C, B⊗D)-bimodule M⊗N induces a Morita equivalence between A ⊗ C and B ⊗ D.

(ii) Keep the notation as in (i). Assume that P is a common p-subgroup of finite groups G and H, and that Q is a subgroup of P. Suppose moreover that A and B respectively are block algebras of kG and kH, C = D = kQ and N = _{kQ}kQ_{kQ}. If a (kG, kH)-bimodule M satisfies that M | k_{ΔP} ↑^{G×H}, then (M ⊗ N) | k_{Δ[P ×Q]}↑^{(G×Q)×(H×Q)}.

Proof. The proof of (i) is straightforward. For (ii) observe that k_{ΔP} ↑^{(G×Q)×(H×Q)} is isomorphic to k[G×Q]⊗_{k[P ×Q]} k[H ×Q], and hence to (kG⊗_{kP} kH)⊗kQ as k[G×Q]⊗k[H ×Q]-bimodules. The latter is isomorphic to k_{ΔP} ↑^{G×H} ⊗_{kQ}kQ_{kQ}. □

Note that we cannot replace the Morita equivalence in 2.6 by a stable equivalence in general, see [48, Question 3.8].

Lemma 2.7. Let G and H be finite groups, let A and B, respectively, be block algebras of kG and kH. Let X be an indecomposable kG-module in A, and let Y be an indecomposable kH-module in B. Then the following hold:

(i) If B is of defect zero, then a block algebra A ⊗ B of k[G × H] is Puig equivalent to A. (ii) Set Z = X ⊗ Y. Then Z is an indecomposable k[G × H]-module in A ⊗ B. If X and Y

are are trivial source modules, then Z is a trivial source module as well. (iii) If Y is projective, and Q is a vertex of X, then Q × ⟨1⟩ is a vertex of Z, and Z is a

trivial source module if and only if X is.

Proof. (i) By [54, p.341 line −9], k is a source algebra of B. Hence the assertion follows from Lemma 2.6(i).

(ii)–(iii) These follow from [25, Proposition 1.2]. □

Finally, we collect a few facts about Green correspondence, its compatability with Brauer correspondence, and its transitivity (see [40, Chap.4, §4], for example).

Lemma 2.8. Let P be a p-subgroup of a finite group G, and let N and H be subgroups of G with N_{G}(P) ⩽ N ⩽ H ⩽ G. Furthermore, assume that f, f_{1} and f_{2} are the Green correspondences with respect to (G, P, H), (H, P, N) and (G, P, N), respectively. Then from the definition and properties of Green correspondence and the Krull-Schmidt Theorem we get the following:

(i) We have 𝔄(G, P, N) ⊆ 𝔄(G, P, H) ∩ 𝔄(H, P, N), where 𝔄(G, P, N) and the others are defined as in [40, Chap.4, §4].

(ii) For any indecomposable kG-module X with vertex in 𝔄(G, P, N), the isomorphism

f_{1}(f(X)) ≅ f_{2}(X) holds. (iii) Let N = N_{G}(P), and let A, B, and A_{N} be block algebras of kG, kH, and kN, respec-

tively, such that they are Brauer correspondents with respect to P. Then any indecomposable kG-module X belonging to A such that a vertex of X is in 𝔄(G, P, N) has its Green correspondent f(X) belong to B.

Proof. (i) and (ii) are clear.

(iii) It follows from Green’s result [40, Chap.5 Corollary 3.11] and Brauer’s first main theorem that f_{2}(X) belongs to A_{N}. The Green correspondent f(X) has a vertex in 𝔄(G, P, N), and hence in 𝔄(H, P, N). By (ii), f_{2} = f_{1} ∘ f. Hence f_{2}(X) = f_{1} ∘ f(X) lies in the Brauer correspondent of A which is A_{N}. Therefore, by the above, the block of f(X) corresponds to A_{N}, namely, it is B. □

3. Non-principal 2-blocks of 𝔖_{5} and M_{12}

By the ”glueing” theorem given in 2.3, we want to obtain a stable equivalence of Morita type between the non-principal 2-block of Co_{3} with a defect group P = C_{2} × C_{2} × C_{2} and its Brauer correspondent in the normalizer N_{Co_{3}}(P). In order to do it, we need to consider nontrivial subgroups of P and establish Morita equivalences between unique blocks of the associated centralizers in Co_{3} and N_{Co_{3}}(P). The objective of this section is to show the existence of various Morita equivalences which will be required to apply 2.3. The relevance of the groups related to 𝔖_{5} and M_{12}, respectively, will be revealed in in 4.2 in the next section.

For the remainder of this paper, we let the characteristic p of k be 2.

Lemma 3.1. Set G = 𝔖_{5}.

(i) There exists a unique block algebra A of kG with defect one. In fact, a defect group T of A is generated by a transposition.

(ii) Set H = N_{G}(T). Then H = C_{G}(T) ≅ T × 𝔖_{3} ≅ D_{12}.

(iii) A is a nilpotent block algebra, k(A) = 2, ℓ(A) = 1, and we can write Irr(A) = {χ_{4}, χ′_{4}}

and IBr(A) = {4_{kG}}, where the number 4 denotes the degree (dimension). (iv) The unique simple kG-module 4_{kG} is a trivial source module.

(v) Let B be a block algebra of kH such that B is the Brauer correspondent of A. Then k(B) = 2, ℓ(B) = 1, and we can write Irr(B) = {θ_{2}, θ′_{2}} and IBr(B) = {2_{kH}}, where the number 2 again gives the degree (dimension).

(vi) Let 𝔣 be the Green correspondence with respect to (G×G, ΔT, G×H), and set M = 𝔣(A). Then _{A}M_{B} = 1_{A}·kG·1_{B} and M induces a Puig equivalence between A and B.

Proof. (i)–(iii) and (v) are immediate by [10, p.2], and [14, A_{5}.2 (mod 2)] or [59, A_{5}.2 (mod 2)].

(iv) It follows from [10, p.2] that 1_{H}↑^{G} = 1_{G} + χ_{4} + χ_{5}, where χ_{i} ∈ Irr(G) and χ_{i}(1) = i for i = 4, 5. Thus, by (ii), 1_{H}↑^{G}·1_{A} = χ_{4}, and hence k_{H}↑^{G}·1_{A} = 4_{kG}.

(vi) We first show that 1_{A}·kG·1_{B} induces a Morita equivalence between A and B. To this end let 1Â·OG·1B̂ be its lift to O, which is projective both as a left OG-module and as a right OH-module. Moreover, it follows from (iii), (v), and [10, p.2] that

χ_{4}↓_{H}·1_{B} = θ_{2}, χ′_{4}↓_{H}·1_{B} = θ′_{2}

by interchanging θ_{2} and θ′_{2} if necessary. Therefore

χ_{4} ⊗_{KA} (1Â·KG·1B̂) = θ_{2}, χ′_{4} ⊗_{KA} (1Â·KG·1B̂) = θ′_{2}.

Hence by [5, 0.2 Théorèm] , we get that 1Â·OG·1B̂ induces a Morita equivalence between Â and B̂, and so does 1_{A}·kG·1_{B} between A and B. As 1_{A}·kG·1_{B} is a trivial source k[G × H]-module with vertex ΔP, we infer that this even is a Puig equivalence.

Finally, let (T, e) be a maximal A-Brauer pair. Then we know N_{G}(T, e) = H by (ii). Hence 2.4 implies that M|1_{A}·kG·1_{B}. But it follows from Morita’s Theorem, see [11, Sect. 3D Theorem (3.54)] that 1_{A}·kG·1_{B} already is indecomposable as an (A, B)-bimodule, implying that M = 1_{A}·kG·1_{B}. □

Lemma 3.2. Set R = C_{2} and consider a group G = R×𝔖_{5}, and let T be as in 3.1, Q = R×T (and hence Q ≅ C_{2} × C_{2}), and set H = N_{G}(Q). Let A be a unique non-principal block algebra of kG with defect group Q, and let B be a block algebra of kH such that B is the Brauer correspondent of A. Then we get the following:

(i) H = C_{G}(Q) = Q × 𝔖_{3}.

(ii) Let 𝔣 be the Green correspondence with respect to (G×G, ΔQ, G×H), and set M = 𝔣(A). Then M ≅ 1_{A}·kG·1_{B}, and M induces a Puig equivalence between A and B.

Proof. This follows from 3.1(vi) and 2.6. □

Lemma 3.3. Let Q = C_{2} × C_{2}, and let G = Q × 𝔖_{5} and P = Q × T, where T is as in 3.1. Set H = N_{G}(P). Let A be a unique non-principal block algebra of kG with defect group P, and let B be a block algebra of kH such that B is the Brauer correspondent of A. Then we get the following:

(i) H = C_{G}(P) = Q × (T × 𝔖_{3}) = P × 𝔖_{3}.

(ii) Let f be the Green correspondence with respect to (G×G, ΔP, G×H), and set M = 𝔣(A). Then M = 1_{A}·kG·1_{B} and M induces a Puig equivalence between A and B.

Proof. This follows from 3.1(vi) and 2.6. □

We next turn to the Mathieu group M_{12}.

Lemma 3.4. Let G = M_{12}.

(i) There exists a unique block algebra A of kG with defect group Q = C_{2} × C_{2}.

(ii) We can write IBr(A) = {16, 16^{∗}, 144}, where the numbers 16 and 144 denote dimensions (degrees). Moreover, all the simple kG-modules in A are trivial source modules.

(iii) Let H = N_{G}(Q). Then H ≅ 𝔄_{4} × 𝔖_{3} ≅ (Q ⋊ C_{3}) × 𝔖_{3}.

(iv) Let B be a block algebra of kH such that B is the Brauer correspondent of A. Let 𝔣 be the Green correspondence with respect to (G×G, ΔQ, G×H), and set M = 𝔣(A). Then M induces a Puig equivalence between A and B.

Proof. (i)–(iii) except the last part of (ii) are easy by [10, p.33], and [14, M_{12} (mod 2)] or [59, M_{12} (mod 2)]. Actually, using the character table of G, it turns out that the conjugacy class 3B of G is a defect class of A. Hence Q is a Sylow 2-subgroup of the centralizer C_{G}(3B) = 𝔄_{4} ×C_{3}, while the normalizer N_{G}(3B) = 𝔄_{4} × 𝔖_{3} is a maximal subgroup of G, containing Q as normal subgroup.

It remains to show the last statement in (ii). By [10, p.33], G has a maximal subgroup L ≅ PSL_{2}(11). Then again [10, p.33] yields that 1_{L}↑^{G}·1_{A} = χ_{16} + χ_{16}^{∗}, where χ_{16}(1) = χ_{16}^{∗}(1) = 16. Set X_{kG} = k_{L}↑^{G}·1_{A}. Then X = 16 + 16^{∗} as composition factors. Since χ_{16} ≠ χ_{16}^{∗}, we get by [40, Chap.4 Theorem 8.9(i)] that [X, X]^{G} = 2. Therefore X = 16 ⊕ 16^{∗}. Hence 16 and 16^{∗} are both trivial source kG-modules. Finally, we know that k_{W} ↑^{G}·1_{A} = 144, where W is a maximal subgroup of G with W = 2_{+}^{1+4}.𝔖_{3}. This shows that 144 is also a trivial source kG-module.

(iv) All elements of Q − {1} are conjugate in H, hence the character table of G [10, p.33] shows that they all belong to the conjugacy class 2A of G. Take any element t ∈ Q − {1}, and set R = ⟨t⟩. Thus we have

C_{G}(R) ≅ R × 𝔖_{5}  and  C_{H}(R) ≅ Q × 𝔖_{3} ≅ R × (C_{2} × 𝔖_{3}).

The algebra kC_{G}(R) has a unique block algebra A_{R} with the defect group Q since k𝔖_{5} has a unique block algebra with defect group C_{2}, and similarly kC_{H}(R) has a unique block algebra B_{R} with the defect group Q since k𝔖_{3} has a unique block algebra of defect zero. Moreover, we know by 3.2 that 𝔣_{R}(A_{R}) induces a Morita equivalence between A_{R} and B_{R}, where 𝔣_{R} is the Green correspondence with respect to (C_{G}(R) × C_{G}(R), ΔQ, C_{G}(R) × C_{H}(R)). Thus it follows from 2.3 that M induces a stable equivalence of Morita type between A and B.

Now, let f be the Green correspondence with respect to (G, Q, H). Take any simple kGmodule S in A. It follows from (ii), [16, 3.7.Corollary], and [41, Lemma 2.2] that f(S) is a simple kH-module. Hence from A.3(v) and 2.1(i) we obtain that S ⊗_{A} M is a simple kHmodule in B. We then finally know that M realizes a Morita equivalence between A and B by 2.1(ii). □

Lemma 3.5. Set R = C_{2}, and let G = R × M_{12}.

(i) There exists a unique block algebra A of kG with defect group P = R × C_{2} × C_{2}. (ii) We can write IBr(A) = {16, 16^{∗}, 144}, where the numbers 16 and 144 give the dimensions (degrees). Moreover, all the simple kG-modules 16, 16^{∗}, 144 in A are trivial source modules.

(iii) Let H = N_{G}(P). Then H = R × 𝔄_{4} × 𝔖_{3} ≅ (P ⋊ C_{3}) × 𝔖_{3}. Note that P ⋊ C_{3} ≅

R × (Q ⋊ C_{3}) and Q ⋊ C_{3} ≅ 𝔄_{4}, where Q = C_{2} × C_{2}. (iv) Let B be a block algebra of kH such that B is the Brauer correspondent of A. Let 𝔣

be the Green correspondence with respect to (G × G, ΔP, G × H). Then 𝔣(A) induces a Puig equivalence between A and B.

Proof. This follows from 3.4(iv) and 2.6. □

4. Obtaining stable equivalences

This enables us to determine a stable equivalence of Morita type between the principal 2block of the smallest Ree group R(3) and the non-principal 2-block of Co_{3} with defect group C_{2} × C_{2} × C_{2} under consideration. The following hypothesis determines our standard setting which we fix here for future reference.

Hypothesis 4.1. Let G be the sporadic group Co_{3}, and let A be the block algebra of kG with defect group P = C_{2} × C_{2} × C_{2}, see [59, Co_{3}], [26, p.1879] and [53, p.494 §2]. Set N = N_{G}(P), and let A_{N} be the Brauer correspondent of A in kN. Furthermore, let (P, e) be a maximal A-Brauer pair in G.

Let Q be a subgroup of P isomorphic to C_{2} ×C_{2}, and R one which is cyclic of order 2. Let e_{Q} and f_{Q} be block idempotents of the block algebras of kC_{G}(Q) and kC_{H}(Q), respectively, such that (Q, e_{Q}) ⊆ (P, e) and (Q, f_{Q}) ⊆ (P, e), see [54, §10 p.346]. Similarly define e_{R} and f_{R} by replacing Q with R. We denote by F_{21} the Frobenius group of order 21, namely, F_{21} ≅ C_{7} ⋊C_{3}, which is a maximal subgroup of GL_{3}(2). Also, let R(3) ≅ SL_{2}(8) ⋊ C_{3} be the smallest Ree group, see [10, p.6].

We first collect information on the subgroups of Co_{3} to consider.

Lemma 4.2. Assume 4.1. Then the following hold:

(i) N ≅ (P ⋊ F_{21}) × 𝔖_{3} ≅ ((P ⋊ C_{7}) ⋊ C_{3}) × 𝔖_{3}.

(ii) There is a maximal subgroup H of G such that N ⩽ H ≅ (SL_{2}(8) ⋊ C_{3}) × 𝔖_{3}, and

P ⋊ C_{7} isomorphic to a Borel subgroup of SL_{2}(8). (iii) C_{G}(P) = C_{H}(P) = C_{N}(P) ≅ P × 𝔖_{3}. (iv) There exists a unique block algebra β of k𝔖_{3} such that β has defect zero, β ≅ Mat_{2}(k)

as k-algebras, and ekC_{G}(P) ≅ kP ⊗ β.

(v) N_{G}(P, e) = N.

(vi) The inertial quotient N_{G}(P, e)/C_{G}(P) is isomorphic to F_{21}.

(vii) All elements of P − {1} are conjugate in N. That is, any subgroup of P of order 2 is

conjugate to R in N. (viii) C_{G}(R) ≅ R × M_{12} and C_{H}(R) = C_{N}(R) ≅ R × 𝔄_{4} × 𝔖_{3} ≅ (P ⋊ C_{3}) × 𝔖_{3}. (ix) All subgroups of P of order 4 are conjugate in N. That is, any subgroup of P of order

4 is conjugate to Q in N.

(x) C_{G}(Q) ≅ Q × 𝔖_{5} and C_{H}(Q) = C_{N}(Q) = C_{H}(P) ≅ P × 𝔖_{3}.

(xi) Let B = B_{0}(kR(3)) ⊗ β, see (iv) for β. Then B is a block algebra of kH with the defect group P, the block B is the Brauer correspondent of A and of A_{N} in H, and we furthermore know that B and B_{0}(kR(3)) are Puig equivalent.

Proof. This is verified easily using GAP [13], with the help of the smallest faithful permutation representation of G on 276 points, available in [58] in terms of so-called standard generators [57]. Since in [58] also representatives of the conjugacy classes of elements, as well as of the maximal subgroups of G are provided, all above-mentioned subgroups of G can be constructed explicitly.

To begin with, using the character table of G [10, p.135], it turns out that the conjugacy class 3C of G is a defect class of A. Hence P is a Sylow 2-subgroup of the centralizer C_{G}(3C), where by [10, p.135] again we have C_{G}(3C) ≅ (SL_{2}(8) ⋊ C_{3}) × C_{3}, while the normalizer H = N_{G}(3C) ≅ (SL_{2}(8) ⋊ C_{3}) × 𝔖_{3} is a maximal subgroup of G.

Using the data on subgroup fusions available in [4], it follows that the elements of P − {1} belong to the 2B conjugacy class of G, hence [10, p.134] shows that C_{G}(R) ≅ R × M_{12}, which is another maximal subgroup of G. Moreover, it follows that C_{G}(Q) ≅ C_{2} × C_{M_{12}}(2A) ≅ C_{2}×(C_{2}×𝔖_{5}), where by [10, p.33] C_{2}×𝔖_{5} is a maximal subgroup of M_{12}. Finally, the structure of C_{H}(P), C_{H}(R), and C_{H}(Q) follows from a consideration of the action of F_{21} ⩽ GL_{3}(2) on the defect group P.

(xi) This follows by 2.7. □

Notation 4.3. We use the notation H, β and B as in 4.2(ii), (iv) and (xi), respectively. We denote the unique simple k𝔖_{3}-module in β by 2_{𝔖_{3}}.

It is now time to harvest what we have sown in our analysis of the 2-local structure of G. In 4.5, we use our previous results to obtain a stable equivalence of Morita type between the blocks A and A_{N} via 2.3. Similarly in 4.4, we derive a stable equivalence between the blocks B and A_{N}, which together with the first yields the stable equivalence sought between A and B in 4.6.

Lemma 4.4. Let 𝔣_{1} be the Green correspondence with respect to (H × H, ΔP, H × N), and set 𝔑 = 𝔣_{1}(B). Then 𝔑 induces a stable equivalence of Morita type between B and A_{N}.

Proof. By 2.4, 𝔑|1_{B}·kH·1_{A_{N}}. We know by 4.2(viii) and 4.2(x) that

C_{H}(Q) = C_{N}(Q) = P × 𝔖_{3} and C_{H}(R) = C_{N}(R) = (P ⋊ C_{3}) × 𝔖_{3}.

Let 𝔸_{Q}, 𝔸_{R}, 𝔹_{Q} and 𝔹_{R} be the block algebras of kC_{H}(Q), kC_{H}(R), kC_{N}(Q) and kC_{N}(R), respectively, such that they have P as a defect group. Then

𝔸_{Q} = 𝔹_{Q} = kP ⊗ k𝔖_{3}·β ≅ Mat_{2}(kP) and 𝔸_{R} = 𝔹_{R} = k[P ⋊ C_{3}] ⊗ k𝔖_{3}·β,

where the isomorphism is of k-algebras. Thus we obviously know that

𝔣_{Q}(𝔸_{Q}) = 𝔸_{Q} and 𝔣_{R}(𝔸_{R}) = 𝔸_{R},

where 𝔣_{Q} and 𝔣_{R} are the Green correspondences with respect to

(C_{H}(Q) × C_{H}(Q), ΔP, C_{H}(Q) × C_{N}(Q)) and (C_{H}(R) × C_{H}(R), ΔP, C_{H}(R) × C_{N}(R)),

respectively. Thus 𝔣_{Q}(𝔸_{Q}) induces a Morita equivalence between 𝔸_{Q} and 𝔹_{Q}, and 𝔣_{R}(𝔸_{R}) induces a Morita equivalence between 𝔸_{R} and 𝔹_{R}. Therefore we get the assertion by 2.3. □

Lemma 4.5. Let 𝔣_{2} be the Green correspondence with respect to (G × G, ΔP, G × N), and set 𝔐 = 𝔣_{2}(A). Then we get

(i) 𝔐 | 1_{A}·kG·1_{A_{N}}.

(ii) The bimodule e_{R}𝔐(ΔR)f_{R} induces a Morita equivalence between the block algebras

kC_{G}(R)e_{R} and kC_{N}(R)f_{R}. (iii) The bimodule e_{Q}𝔐(ΔQ)f_{Q} induces a Morita equivalence between the block algebras

kC_{G}(Q)e_{Q} and kC_{N}(Q)f_{Q}. (iv) 𝔐 induces a stable equivalence of Morita type between A and A_{N}.

Proof. (i) This follows from 4.2(v) and 2.4.

(ii) Let 𝔣_{R} be the Green correspondence with respect to (C_{G}(R) × C_{G}(R), ΔP, C_{G}(R) × C_{N}(R)). We get from (i) and 2.2 that 𝔣_{R}(e_{R}kC_{G}(R)) = e_{R}𝔐(ΔR)f_{R}. Hence we obtain the assertion by 3.5.

(iii) Analogous to the proof of (ii) if we use 3.3 instead of 3.5.

(iv) This follows by 3.5 and 3.3, (i)–(iii) and 2.3. □

Lemma 4.6. There is an (A, B)-bimodule M which satisfies the following:

(1) _{A}M_{B} is indecomposable,

(2) (_{A}M_{B},_{B}M^{∨}_{A}) induces a stable equivalence of Morita type between A and B,

(3) _{A}M_{B} | k_{ΔP} ↑^{G×H} and _{B}M^{∨}_{A} | k_{ΔP} ↑^{H×G},

(4) the stable equivalence of Morita type induced by _{A}M_{B} preserves vertices and sources,

(5) for any indecomposable X ∈ mod-A with vertex in 𝔄(G, P, N), it holds (X ⊗_{A} M)_{B} = f(X) ⊕ (proj), where f is the Green correspondence with respect to (G, P, H) (recall that 𝔄(G, P, N) ⊆ 𝔄(G, P, H) ∩ 𝔄(H, P, N) by 2.8(i)).

Proof. Let 𝔣_{2} be the Green correspondence with respect to (G × G, ΔP, G × N), and set 𝔐 = 𝔣_{2}(A). Let f_{2} be the Green correspondence with respect to (G, P, N). Moreover, let 𝔣_{1} be the Green correspondence with respect to (H × H, ΔP, H × N), and set 𝔑 = 𝔣_{1}(B). Let f_{1} be the Green correspondence with respect to (H, P, N). Then by 4.4 and 4.5 the bimodules 𝔑 and 𝔐 induce stable equivalences, so by A.3, and 2.5 there is a bimodule _{A}M_{B} such that

(∗)  _{A}(𝔐 ⊗_{A_{N}} 𝔑^{∨})_{B} = _{A}M_{B} ⊕ (proj (A, B)-bimodule)

and (1)–(4) hold.

It remains to show (5). Take any indecomposable X ∈ mod-A with a vertex which is in 𝔄(G, P, N). Then it follows from (∗) that

X ⊗_{A} (𝔐 ⊗_{A_{N}} 𝔑^{∨}) = X ⊗_{A} (M ⊕ (proj (A, B)-bimodule)).

On the other hand, by 2.8(ii) we get

(X ⊗_{A} 𝔐) ⊗_{A_{N}} 𝔑^{∨} = [f_{2}(X) ⊕ (proj A_{N}-module)] ⊗_{A_{N}} 𝔑^{∨}

= (f_{2}(X) ⊗_{A_{N}} 𝔑^{∨})_{B} ⊕ ((proj A_{N}-module) ⊗_{A_{N}} 𝔑^{∨})_{B}

= (f_{2}(X) ⊗_{A_{N}} 𝔑^{∨})_{B} ⊕ (proj B-module)

= (f_{1}^{−1}(f_{2}(X)))_{B} ⊕ (proj B-module)

= f(X) ⊕ (proj B-module)

□

5. Modules in A, B and A_{N}

In the previous section, we have shown that there is a stable equivalence of Morita type between the blocks A and B. As outlined in the introduction, our aim now is verify that this equivalence is in fact a Morita equivalence with the help of 2.1. In other words, we need to show that the associated tensor functor takes simple modules to simple modules. Therefore in this intermediate section we collect all the necessary information on the simple modules and some indecomposable modules lying in the three blocks we consider.

In addition to the notation of our standard hypothesis 4.1, we fix the following:

Lemma 5.1 (Suleiman-Wilson [53]). The 2-decomposition matrix of A is given in Table 1, where S_{1}, ⋯, S_{5} are non-isomorphic simple kG-modules in A whose degrees are 73600, 896, 896, 19712, 131584, respectively. The two simple modules S_{2} and S_{3} are dual to each other, while the remaining are self-dual. There are two pairs (χ_{6}, χ_{7}) and (χ_{18}, χ_{19}) of complex conjugate characters. All other χ’s are real-valued.

Proof. See [53, §6]. □

\begin{matrix}degree & [10, p.135] & S_{1} & S_{2} & S_{3} = S_{2}^{∗} & S_{4} & S_{5} \\ 73 600 & χ_{29} & 1 & . & . & . & . \\ 896 & χ_{6} & . & 1 & . & . & . \\ 896 & χ_{7} = χ_{6}^{∗} & . & . & 1 & . & . \\ 93 312 & χ_{32} & 1 & . & . & 1 & . \\ 20 608 & χ_{18} & . & 1 & . & 1 & . \\ 20 608 & χ_{19} = χ_{18}^{∗} & . & . & 1 & 1 & . \\ 226 688 & χ_{38} & 1 & 1 & 1 & 1 & 1 \\ 246 400 & χ_{39} & 1 & 1 & 1 & 2 & 1 \end{matrix}

Table 1. The 2-modular decomposition matrix of Co_{3}.

Remark 5.2. The 2-blocks of Co_{3} have been studied before by several other people, see [12, p.193 Table 6], [26, §7 p.1879] and [27, Theorems 3.10 and 3.11].

Notation 5.3. We use the notation χ_{29}, χ_{6}, χ_{7}, χ_{32}, χ_{18}, χ_{19}, χ_{38}, χ_{39}, and S_{1}, ⋯, S_{5} as in 5.1.

Lemma 5.4. All simple kG-modules S_{1}, ⋯, S_{5} in A have P as a vertex.

Proof. See [16, 3.7.Corollary]. □

Lemma 5.5. We get the following:

(i) A_{N} = k[P ⋊ F_{21}] ⊗ β ≅ Mat_{2}(k[P ⋊ F_{21}]), as k-algebras.

(ii) We can write Irr(F_{21}) = {k, 1, 1^{∗}, 3, 3^{∗}}.

(iii) We can write

\begin{aligned}IBr(A_{N}) = \{˜2_{0} = k_{P ⋊F_{21}} & ⊗ 2_{𝔖_{3}}, ˜2 = 1 ⊗ 2_{𝔖_{3}}, \\ & ˜2^{∗} = 1^{∗} ⊗ 2_{𝔖_{3}}, ˜6 = 3 ⊗ 2_{𝔖_{3}}, ˜6^{∗} = 3^{∗} ⊗ 2_{𝔖_{3}}\}. \end{aligned}

Note that there exists a unique simple ˜2_{0} which is self-dual. (iv) The trivial source A_{N}-modules with vertex P are precisely the simple A_{N}-modules.

Proof. (i)–(iii) are easy by 4.2 and the definition of A_{N}.

(iv) This follows from (iii) and the Green correspondence [40, Chap.4 Problem 10]. □

Lemma 5.6. Set ℜ = R(3) ≅ SL_{2}(8) ⋊ C_{3}. We get the following:

(i) For the principal block of kℜ we have

Irr(B_{0}(kℜ)) = {1_{ℜ}, χ_{1}, χ_{1}^{∗}, χ_{7a}, χ_{7b}, χ_{7c}, χ_{21}, χ_{27}},

and

IBr(B_{0}(kℜ)) = {k_{ℜ}, 1, 1^{∗}, 6, 12},

where the indices give the degrees (dimensions). The simples k_{ℜ}, 6, 12 are self-dual, and the simples k_{ℜ}, 1, 1^{∗} are trivial source kℜ-modules.

(ii) For the block B we have

Irr(B) = {χ_{2a}, χ_{2}, χ_{2}^{∗}, χ_{14a}, χ_{14b}, χ_{14c}, χ_{42}, χ_{54}},

and

\begin{aligned}IBr(B) = \{2_{0} = k_{ℜ} ⊗ 2_{𝔖_{3}}, & 2 = 1 ⊗ 2_{𝔖_{3}}, \\ & 2^{∗} = 1^{∗} ⊗ 2_{𝔖_{3}}, 12 = 6 ⊗ 2_{𝔖_{3}}, 24 = 12 ⊗ 2_{𝔖_{3}}\}, \end{aligned}

where the indices give the degrees (dimensions). The simple kH-modules 2_{0}, 2, 2^{∗} in B are trivial source modules, the simple kH-modules 2_{0}, 12, 24 are self-dual, and all the simples in B have P as their vertices.

Proof. (i) It follows from [10, p.6], and [14, L_{2}(8).3 (mod 2)] or [59, L_{2}(8).3 (mod 2)], see 4.2(xi). Clearly, k_{ℜ}, 1, 1^{∗} are trivial source kℜ-modules.

(ii) 2_{𝔖_{3}} is a trivial source k𝔖_{3}-module. Therefore the simples 2_{0}, 2, 2^{∗} are trivial source kH-modules, by (i) and 4.2(xi). Finally, use [16, 3.7.Corollary]. □

Notation 5.7. We use the notation ℜ, χ_{2a}, χ_{2}, χ_{2}^{∗}, χ_{14a}, χ_{14b}, χ_{14c}, χ_{42}, χ_{54}, ˜2_{0}, ˜2, ˜2^{∗}, ˜6, ˜6^{∗} and 2_{0}, 2, 2^{∗}, 12, 24 as in 5.5 and 5.6.

Lemma 5.8 (Landrock-Michler [29]). The radical and socle series of projective indecomposable kH-modules in B are the following:

\begin{matrix}2_{0} \\ 12 \\ 2_{0} 2 2^{∗} 24 \\ 12 12 \\ 2_{0} 2 2^{∗} 24 \\ 12 \\ 2_{0} \end{matrix} , \begin{matrix}2 \\ 12 \\ 2_{0} 2 2^{∗} 24 \\ 12 12 \\ 2_{0} 2 2^{∗} 24 \\ 12 \\ 2 \end{matrix} \begin{matrix}2^{∗} \\ 12 \\ 2_{0} 2 2^{∗} 24 \\ 12 12 \\ 2_{0} 2 2^{∗} 24 \\ 12 \\ 2^{∗} \end{matrix}  \begin{matrix}12 \\ 2_{0} 2 2^{∗} 24 \\ 12 12 12 \\ 2_{0} 2_{0} 2 2 2^{∗} 2^{∗} 24 24 \\ 12 12 12 \\ 2_{0} 2 2^{∗} 24 \\ 12 \end{matrix} \begin{matrix}24 \\ 12 \\ 2_{0} 2 2^{∗} \\ 12 \\ 2_{0} 2 2^{∗} \\ 12 \\ 24 \end{matrix}

Proof. This follows from [29, Theorem 3.9, Theorem 4.1] and 5.6. □

Lemma 5.9. Recall that R is a subgroup of P with R ≅ C_{2}, see 4.1.

(i) The Scott module Scott(ℜ, R) has the radical and socle series

\begin{matrix}k \\ 6 \\ 1 1^{∗} 12 \\ 6 \\ k \end{matrix} ↔ 1_{ℜ} + χ_{27}.

(ii) A kH-module Scott(ℜ, R) ⊗ 2_{𝔖_{3}} has the radical and socle series

\begin{matrix}2_{0} \\ 12 \\ 2 2^{∗} 24 \\ 12 \\ 2_{0} \end{matrix} ↔ χ_{2a} + χ_{54}.

Proof. By 5.6(ii), it suffices to prove (i). [10, p.6] says that ℜ has a maximal subgroup M such that M = C_{9} ⋊ C_{6}, |ℜ : M| = 28 and 1_{M}↑^{ℜ} = 1_{ℜ} + χ_{27}. Set X = k_{M}↑^{ℜ}. Then X = 2 × k + 1 + 1^{∗} + 2 × 6 + 12 as composition factors by [14, L_{3}(8).3 (mod 2)] and [59, L_{3}(8).3 (mod 2)]. It holds by [40, 4 Thm.8.9(i)] that [X, X]^{ℜ} = 2, [X, k]^{ℜ} = [k, X]^{ℜ} = 1 Thus, X/rad(X) ≅ soc(X) ≅ k_{ℜ}. Now, it follows from [29, Theorem 4.1] that P (k_{ℜ}) has the following radical and socle series:

P (k_{ℜ}) = \begin{matrix}k \\ 6 \\ k 1 1^{∗} 12 \\ 6 6 \\ k 1 1^{∗} 12 \\ 6 \\ k \end{matrix} .

Since there is an epimorphism P (k_{ℜ}) ↠ X, we infer soc(X) < soc^{2}(X) < rad^{2}(X) < rad(X) and rad(X)/rad^{2}(X) ≅ soc^{2}(X)/soc(X) ≅ 6. Thus X has the radical and socle series as asserted. By the definition of X, it holds that X = Scott(ℜ, C_{2}), see [40, Chap.4 Theorem 8.4 and Corollary 8.5]. □

Lemma 5.10. Recall that Q is a subgroup of P with Q ≅ C_{2}×C_{2}, see 4.1. Set U = Scott(ℜ, Q).

(i) We have U ↔ 1_{ℜ} + χ_{7a} + 2 × χ_{27}, and U = 4 × k_{ℜ} + 2 × 1 + 2 × 1^{∗} + 5 × 6 + 2 × 12 as composition factors.

(ii) Set V = U ⊗ 2_{𝔖_{3}}. Then V is a trivial source kH-module in B with vertex Q, V ↔ χ_{2a} + χ_{14a} + 2 × χ_{54}, and V = 4 × 2_{0} + 2 × 2 + 2 × 2^{∗} + 5 × 12 + 2 × 24, as composition factors.

Proof. (i) We know that ℜ has a subgroup 𝔄_{4}, see [10, p.6]. Clearly, Irr(𝔄_{4}) = {1_{𝔄_{4}}, ψ_{1}, ψ_{2} = ψ_{1}^{∗}, ψ_{3}} where ψ_{3} has degree 3. It follows from computations with GAP [13] that

(1)  1_{𝔄_{4}}↑^{ℜ} · 1_{B_{0}(kℜ)} = 1_{ℜ} + χ_{7a} + χ_{21} + 3 × χ_{27},
(2)  ψ_{1}↑^{ℜ} · 1_{B_{0}(kℜ)} = χ_{1} + χ_{7b} + χ_{21} + 3 × χ_{27},
(3)  ψ_{1^{∗}}↑^{ℜ} · 1_{B_{0}(kℜ)} = χ_{1^{∗}} + χ_{7c} + χ_{21} + 3 × χ_{27}.

Let X = k_{𝔄_{4}}↑^{ℜ} · 1_{B_{0}(kℜ)}. First, we want to claim that P (12) | X, where P (12) is the projective cover 12.

Set S = SL_{2}(8). By Clifford theory, we have 12 ↓_{S} = 4_{1} ⊕ 4_{2} ⊕ 4_{3}, where 4_{1}, 4_{2}, 4_{3} are non-isomorphic simple kS-modules in B_{0}(kS) of dimension 4, see [14, L_{2}(8) (mod 2)] and [59, L_{2}(8) (mod 2)]. Let V_{1} be the tautological kS-module, which is simple of dimension 2, and let V_{2} and V_{3} be its images under the action of the Frobenius automorphism of 𝔽_{8}. Then the V_{i} are pairwise non-isomorphic, and by [1, p.220] we may assume that

4_{1} = V_{1} ⊗ V_{2}, 4_{2} = V_{2} ⊗ V_{3}, 4_{3} = V_{3} ⊗ V_{1}.

Set g_{a} = (\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) ∈ S for all a ∈ 𝔽_{8}. We may assume that P = \{g_{a} | a ∈ 𝔽_{8}\} ≤ S, namely,

P is a Sylow 2-subgroup of S with P ≅ C_{2} × C_{2} × C_{2}, and that Q = {g_{0}, g_{1}, g_{α}, g_{1+α}}, where α ∈ 𝔽_{8}^{∗} is a fixed primitive root, hence Q ≅ C_{2} × C_{2}. Now the action of g_{0} + g_{1} + g_{α} + g_{1+α} = (1 + g_{1})(1 + g_{α}) ∈ kQ is easily described in terms of Kronecker products of matrices, and it turns out that this element does not annihilate any of the kQ-modules 4_{i}. Therefore 4_{i}↓_{Q} has a projective indecomposable summand, and thus we infer that 4_{i}↓_{Q} = P (k_{Q}).

We conclude 12↓_{Q} = 12↓_{S}↓_{Q} = (4_{1} ⊕ 4_{2} ⊕ 4_{3})↓_{Q} ≅ 3 × P (k_{Q}), and it follows from [50, Theorem 3] that

3 = [P (k_{Q}) | 12↓_{Q}]^{Q} = [P (12) | k_{Q}↑^{ℜ}]^{ℜ} = [P (12) | k_{Q}↑^{𝔄_{4}}↑^{ℜ}]^{ℜ}
= [P (12) | (k_{𝔄_{4}} ⊕ 1_{𝔄_{4}} ⊕ 1_{𝔄_{4}}^{∗})↑^{ℜ}]^{ℜ} = [P (12) | (k_{𝔄_{4}}↑^{ℜ} ⊕ 1_{𝔄_{4}}↑^{ℜ} ⊕ 1_{𝔄_{4}}^{∗}↑^{ℜ})]^{ℜ}.

Suppose that P (12) ∤ k_{𝔄_{4}}↑^{ℜ}. Then 3×P (12) | (1_{𝔄_{4}}↑^{ℜ} ⊕1_{𝔄_{4}}^{∗}↑^{ℜ}). Since P (12) ↔ χ_{21} +χ_{27} by [14, L_{2}(8).3 (mod 2)] and [59, L_{2}(8).3 (mod 2)], we know by (2) and (3) that 3 × χ_{21} + 3 × χ_{27} is contained in (χ_{1} + χ_{7b} + χ_{21} + 3 × χ_{27}) + (χ_{1}^{∗} + χ_{7c} + χ_{21} + 3 × χ_{27}), which contradicts the multiplicity of χ_{21}.

Therefore P (12) | k_{𝔄_{4}}↑^{ℜ}. Since P (12) ↔ χ_{21} + χ_{27} as seen above, it follows from (1) that

k_{𝔄_{4}}↑^{ℜ}·1_{B_{0}(kℜ)} = X ⊕ P (12)

for a kℜ-module X such that

X ↔ 1_{ℜ} + χ_{7a} + 2 × χ_{27}.

Now, let U = Scott(ℜ, Q), and hence U|X since Q is a Sylow 2-subgroup of 𝔄_{4}, see [40, Chap.4 Corollary 8.5]. By the definition of Scott modules and [40, 4 Thm.8.9(i)], we know (χÛ, 1_{ℜ})^{ℜ} = 1. Clearly, χÛ ≠ 1_{ℜ} since Q ≨ P. Since P is a Sylow 2-subgroup of ℜ, it follows from [40, Chap.4, Theorem 7.5] that dim_{k}(U) is even. This means that χÛ ≠ 1_{ℜ} + 2 × χ_{27} and that χÛ ≠ 1_{ℜ} + χ_{7a} + χ_{27}. If χÛ = 1_{ℜ} + χ_{7a} then χÛ(2A) = 1 + (−1) = 0 by [10, p.6], contradicting [28, II Lemma 12.6] since 2A ∈ Q. Suppose that χÛ = 1_{ℜ} + χ_{27}. Then since U is a trivial source kℜ-module, we get that U has the same radical and socle series of Scott(ℜ, R) just by the same method as in 5.9. Since [U, Scott(ℜ, R)]^{ℜ} = 2 by [40, 4 Thm.8.9(i)], we have U ≅ Scott(ℜ, R), and hence Q ≅ R by [40, Chap.4, Corollary 8.5], again a contradiction.

Therefore we know that χÛ = 1_{ℜ} + χ_{7a} + 2 × χ_{27} and U = X, so that U = 4 × k_{ℜ} + 2 × 1 + 2 × 1^{∗} + 5 × 6 + 2 × 12 as composition factors.

(ii) This follows from (i) and 4.2(xi). □

Remark 5.11. We will not need the precise structure of U = Scott(ℜ, Q). Still we would like to remark that using the table of marks library of GAP [13], and the facilities available in the MeatAxe [49] and its extensions, U can actually be constructed and analysed explicitly.

In particular, it turns out that U has Loewy length 5, but its radical and socle series do not coincide; they are

\begin{matrix}k 6 \\ k 1 1^{∗} 12 \\ 6 6 6 \\ k k 1 1^{∗} 12 \\ 6 \end{matrix}  and  \begin{matrix}6 \\ k k 1 1^{∗} 12 \\ 6 6 6 \\ k 1 1^{∗} 12 \\ k 6 \end{matrix} ,

respectively.

6. Images of simples in A via Green correspondence

In this section we prove that the crucial hypothesis of 2.1 is fulfilled for the stable equivalence of Morita type we have established in 4.6. Namely, we show that simple modules in A are taken to simple modules in B. For the first four simples this is almost immediate, as this amounts to determining the Green correspondents with respect to (G, P, H), and these are easily determined theoretically and computationally. The image of the last simple A-module however, is more difficult to determine, and we make use of our knowledge on the modules of the blocks A and B we have gained in Section 5.

Notation 6.1. We use the notation _{A}M_{B}, f, f_{1} and f_{2} as in 4.6. Let F : mod-A → mod-B denote the functor giving the stable equivalence of Morita type of 4.6, namely, in the notation of 4.6 we have F (X) = X ⊗_{A} M for each X ∈ mod-A.

Lemma 6.2. The following hold:

(i) S_{4} = 22 ⊗ S_{2}, where 22 is a simple kG-module in B_{0}(kG).

(ii) We have

22↓_{H} = (6 ⊗ k_{𝔖_{3}}) ⊕ (proj), S_{2}↓_{H} = 2 ⊕ 110 ⊕ (proj) and (6 ⊗ k_{𝔖_{3}}) ⊗ 2 = 12,

where 6 ⊗ k_{𝔖_{3}} is a simple kH-module in B_{0}(kH) = B_{0}(kR(3)) ⊗ B_{0}(k𝔖_{3}), and 110 is an indecomposable kH-module in B_{0}(kH), hence S_{2}↓_{H}·1_{B} = 2 and S_{2}^{∗}↓_{H}·1_{B} = 2^{∗}.

(iii) 12 | S_{4}↓_{H}.

Proof. (i) This is obtained by [53, p.502], see [59, Co_{3} (mod 2)], and a direct computation with Brauer characters in GAP [13].

(ii) By [14, L_{3}(8).3 (mod 2)] or [59, L_{3}(8).3 (mod 2)], except for the principal 2-block B_{0}(k[R(3)]) of kR(3) = k[SL_{2}(8) ⋊ C_{3}] there are only three 2-blocks of defect zero, consisting of the extensions of the Steinberg character of SL_{2}(8) to R(3). Hence it is easy to write down the block idempotents of kR(3), and similarly those of k𝔖_{3}. Thus, H being a small group of order 9 072, using GAP [13] the block idempotents of kH can be explicitly evaluated in a given representation. This yields the block components, which are then further analysed using the MeatAxe [49] and its extensions.

(iii) It follows from (i) and (ii) that

S_{4}↓_{H} = (22 ⊗ S_{2})↓_{H} = 22↓_{H} ⊗ S_{2}↓_{H}
= ((6 ⊗ k_{𝔖_{3}}) ⊕ (proj)) ⊗ (2 ⊕ 110 ⊕ (proj))
= ((6 ⊗ k_{𝔖_{3}}) ⊗ 2) ⊕ (other) = 12 ⊕ (other).
□

Lemma 6.3. We have f(S_{2}) = 2, f(S_{2}^{∗}) = 2^{∗}, f(S_{4}) = 12, and hence that F (S_{2}) = 2, F (S_{2}^{∗}) = 2^{∗} and F (S_{4}) = 12.

Proof. By 6.2(ii) the Green correspondents of S_{2} and S_{2}^{∗} are immediate. By 5.4 all simple A-modules have vertex P ∈ 𝔄(G, P, H), and by 6.2(ii) the direct summands of (6 ⊗ k_{𝔖_{3}}) ⊗ 110 lie in the principal block. Therefore by 6.2(iii) and 5.6(ii) the simple module 12 is the unique summand of S_{4}↓_{H} in B with vertex P. Hence f(S_{4}) = 12. By 4.6(5) and 2.1(i) the functor F maps any simple A-module to its Green correspondent in B, and so the claim follows. □

Lemma 6.4. The simples S_{2} and S_{2}^{∗} are trivial source kG-modules with S_{2} ↔ χ_{6} and S_{2}^{∗} ↔ χ_{6}^{∗}.

Proof. We know by 5.6(ii) that 2 and 2^{∗} are trivial source kH-modules. Hence, by the definition of Green correspondence, 6.3 and 5.1, we get the assertion. □

Lemma 6.5. The simple kG-module S_{1} in A is a trivial source module with S_{1} ↔ χ_{29}.

Proof. It follows from [10, p.143] that G has a maximal subgroup L with L = 2^{.}S_{6}(2). Then using GAP [13], we know that 1_{L}↑^{G}·1_{A} = χ_{29}. Hence the assertion follows by 5.1. □

Lemma 6.6. It is f(S_{1}) = 2_{0}, and hence F (S_{1}) = 2_{0}.

Proof. First, let f′_{1} be the Green correspondence with respect to (R(3), P, P ⋊ F_{21}). Clearly, f′_{1}(k_{R(3)}) = k_{P ⋊F_{21}}. Since f_{1} is the Green correspondence with respect to (H, P, N) = (R(3) × 𝔖_{3}, P, (P ⋊ F_{21}) × 𝔖_{3}), we know that f_{1}(k_{R(3)} ⊗ 2_{𝔖_{3}}) = k_{P ⋊F_{21}} ⊗ 2_{𝔖_{3}}, namely, f_{1}(2_{0}) = ˜2_{0}.

By 2.8(ii), f_{1} ∘ f = f_{2}. Thus it follows from 5.4, 6.5 and 2.8(iii) that f_{1} ∘ f(S_{1}) is a trivial source kN-module in A_{N} with vertex P. Hence 5.5(iv) implies that

f_{1} ∘ f(S_{1}) ∈ {˜2_{0}, ˜2, ˜2^{∗}, ˜6, ˜6^{∗}}.

Then since S_{1} is self-dual by 5.1, we know that f_{1}∘f(S_{1}) is also self-dual. Therefore f_{1}∘f(S_{1}) = ˜2_{0}, giving f_{1} ∘ f(S_{1}) = f_{1}(2_{0}). This implies that f(S_{1}) = 2_{0}. Hence we get the assertion from 4.6(5) and 2.1(i). □

Lemma 6.7. The following hold:

(i) Ext_{A}^{1}(S_{1}, S_{2}) = Ext_{A}^{1}(S_{1}, S_{2}^{∗}) = Ext_{A}^{1}(S_{2}, S_{1}) = Ext_{A}^{1}(S_{2}^{∗}, S_{1}) = 0.

(ii) Ext_{A}^{1}(S_{2}, S_{2}^{∗}) = Ext_{A}^{1}(S_{2}^{∗}, S_{2}) = 0.

(iii) dim_{k}[Ext_{A}^{1}(S_{1}, S_{4})] = dim_{k}[Ext_{A}^{1}(S_{4}, S_{1})] = 1.

Proof. By 6.6 and 6.3 we know the simple images of the simple modules given under the stable equivalence F of 6.1. Hence the results are immediate by looking at the B-PIMs in 5.8 and using A.1. □

Lemma 6.8. All composition factors of F (S_{5})/rad(F (S_{5})) and soc(F (S_{5})) are isomorphic to the simple module 24.

Proof. Take any simple kH-module T in B such that T ≅ 2̸4. Then we know by 5.6, 6.3 and 6.6 that T = F (S_{i}) for i ∈ {1, 2, 3, 4}, where S_{3} = S_{2}^{∗}. Then it follows from [28, II Lemma 2.7 and Corollary 2.8] and 6.1 that Hom_{B}(F (S_{5}), T) = \underline{Hom}_{B}(F (S_{5}), T) = \underline{Hom}_{B}(F (S_{5}), F (S_{i})) ≅ \underline{Hom}_{A}(S_{5}, S_{i}) = Hom_{A}(S_{5}, S_{i}) = 0. Thus we get the assertion for the head of F (S_{5}). The assertion for the socle follows by the same argument and considering Hom_{B}(T, F (S_{5})) instead.

□

We can now finally prove that also the image of the last remaining simple A–module S_{5} under F is a simple B-module.

Lemma 6.9. It is F (S_{5}) = 24.

Proof. By [10, p.134], G has a maximal subgroup 𝔘 = U_{3}(5) ⋊ 𝔖_{3}. Set X = k_{𝔘}↑^{G}·1_{A}. By calculations in GAP [13] we know that 1_{𝔘}↑^{G}·1_{A} = χ_{29} + χ_{39}, so that

(4)  X ↔ χ_{29} + χ_{39}.

Hence, by 5.1

(5)  X = 2 × S_{1} + S_{2} + S_{2}^{∗} + 2 × S_{4} + S_{5} as composition factors.

Since S_{1}, S_{2} and S_{2}^{∗} are trivial source kG-modules by 6.5 and 6.4, it follows from (4), 5.1 and [40, 4 Thm.8.9(i)] that

[S_{1}, X]^{G} = [X, S_{1}]^{G} = 1, [S_{2}, X]^{G} = [X, S_{2}]^{G} = [S_{2}^{∗}, X]^{G} = [X, S_{2}^{∗}]^{G} = 0.

If [S_{5}, X]^{G} ≠ 0 or [X, S_{5}]^{G} ≠ 0, then the self-duality of X and S_{5} implies that S_{5} | X, and hence S_{5} is a trivial source kG-module, so that S_{5} is liftable to O by [40, 4 Thm.8.9(iii)], which contradicts to 5.1. Hence

[S_{5}, X]^{G} = [X, S_{5}]^{G} = 0.

Assume [S_{4}, X]^{G} ≠ 0 or [X, S_{4}]^{G} ≠ 0. Then again the self-dualities of X and S_{4} in 5.1 say that both are non-zero. Thus we have endomorphisms ψ_{1}, ψ_{2} and ψ_{3} of X such that ψ_{1} = id_{X}, Im(ψ_{2}) ≅ S_{1} and Im(ψ_{3}) ≅ S_{4}. This means [X, X]^{G} ⩾ 3. But [40, 4 Thm.8.9(i)] and (4) yield that [X, X]^{G} = 2, a contradiction. Thus [S_{4}, X]^{G} = [X, S_{4}]^{G} = 0. These imply that

(6)  X/rad(X) ≅ soc(X) ≅ S_{1}.

Hence X is indecomposable. Set X_{0} = rad(X)/soc(X), the heart of X. Thus (5) implies

(7)  X_{0} = S_{2} + S_{2}^{∗} + 2 × S_{4} + S_{5}, as composition factors.

By 6.7(i), it holds

[X_{0}, S_{2}]^{G} = [X_{0}, S_{2}^{∗}]^{G} = [S_{2}, X_{0}]^{G} = [S_{2}^{∗}, X_{0}]^{G} = 0.

Moreover, 6.7(iii) yields that X_{0}/rad(X_{0}) | (S_{4} ⊕ S_{5}). These imply that the radical and socle series of X is one of the following:

(8)  X = \begin{matrix}S_{1} \\ S_{4} \\ S_{2} S_{2}^{∗} S_{5} \\ S_{4} \\ S_{1} \end{matrix} ,  \begin{aligned}& S_{1} \\ \begin{matrix}S_{4} \\ S_{2} S_{2}^{∗} \\ S_{4} \end{matrix} & ⊕ & S_{5} \\ & S_{1} \end{aligned} ,  \begin{matrix}S_{1} \\ S_{4} \\ S_{2} \\ S_{5} \\ S_{2}^{∗} \\ S_{4} \\ S_{1} \end{matrix}  or  \begin{matrix}S_{1} \\ S_{4} \\ S_{2}^{∗} \\ S_{5} \\ S_{2} \\ S_{4} \\ S_{1} \end{matrix} .

Now, it follows from 6.1, [28, II Lemma 2.7 and Corollary 2.8], 6.3 and (6) that

Hom_{B}(F (X), 2) = \underline{Hom}_{B}(F (X), 2) = \underline{Hom}_{B}(F (X), F (S_{2}))
≅ \underline{Hom}_{A}(X, S_{2}) = Hom_{A}(X, S_{2}) = 0.

Hence [F (X), 2)]^{B} = 0. Similarly we obtain [F (X), 2^{∗}]^{B} = 0 and [F (X), 12]^{B} = 0 and [F (X), 2_{0}]^{B} = 1. Similar for soc(F (X)), too. Thus, by 5.6, we know that

(9)  F (X)/rad(F (X)) ≅ 2_{0} ⊕ (r × 24) and soc(F (X)) ≅ 2_{0} ⊕ (r′ × 24)

for some r, r′ ⩾ 0. By 6.1, we have

(10)  F (X) = Y ⊕ (proj B-module)

for a non-projective indecomposable kH-module Y in B. Thus, by 6.6 and A.1 we have

(11)  2_{0}|Y/rad(Y) and 2_{0}|soc(Y).

Recall that 2_{0} = k_{ℜ} ⊗ 2_{𝔖_{3}} in 5.6(ii). Since B and B_{0}(kℜ) are Puig equivalent by 4.2(xi), and Y is a trivial source module by 4.6, it follows that Y ≅ Scott(ℜ, S) ⊗ 2_{𝔖_{3}} for a subgroup S of P. Clearly S ≠ 1 since Y is non-projective indecomposable. If S = P then (11) yields Y = 2_{0}, so that F (X) = 2_{0} ⊕ (proj) and F (S_{1}) = 2_{0} by 6.6. This is a contradiction since X is non-projective indecomposable and non-simple. Thus S ≅ Q or S ≅ R.

Suppose that S ≅ Q, namely Y ≅ Scott(ℜ, Q) ⊗ 2_{𝔖_{3}}. Then it follows by 5.10(ii) that

Y ↔ χ_{2a} + χ_{14a} + 2 × χ_{54},

and we have

(12)  Y = 4 × 2_{0} + 2 × 2 + 2 × 2^{∗} + 5 × 12 + 2 × 24, as composition factors.

We know by 6.6 and 6.3 that

F (S_{1}) = 2_{0}, F (S_{4}) = 12, F (S_{2}) = 2, F (S_{2}^{∗}) = 2^{∗}.

Thus it follows by (6), (8) and A.1 that we can strip off 2 × S_{1}, 2 × S_{4}, S_{2}, and S_{2}^{∗} from the top of X and from the bottom of X, and also 2 × 2_{0}, 2 × 12, 2, and 2^{∗} from the top of Y and from the bottom of Y sequentially, by looking at (8) and (12). Consequently by 2.1(i), we have F (S_{5}) = Z for an indecomposable kH-module Z in B such that Z = 2 × 2_{0} + 2 + 2^{∗} + 3 × 12 + 2 × 24 as composition factors. Then 6.8 yields Z/rad(Z) ≅ soc(Z) ≅ 24 and rad(Z)/soc(Z) = 2_{0} + 2 + 2^{∗} + 3 × 12 as composition factors, which contradicts 5.8.

Therefore S ≅ R and Y ≅ Scott(ℜ, R) ⊗ 2_{𝔖_{3}}. Hence we get by 5.9(ii) that

(13)  F (X) = Y ⊕ (proj),  Y = \begin{matrix}2_{0} \\ 12 \\ 2 2^{∗} 24 \\ 12 \\ 2_{0} \end{matrix} .

Thus by the same stripping off method taken above, we can subsequently strip off 2 × S_{1}, 2 × S_{4}, S_{2}, and S_{2}^{∗} from the top of X and the bottom of X, and also 2 × 2_{0}, 2 × 12, 2, and 2^{∗} from the top of Y and the bottom of Y, by looking at (8) and (13). Hence we arrive at F (S_{5}) = 24 ⊕ (proj), so that 2.1 yields F (S_{5}) = 24. □

Remark 6.10. We know by 1.7 that the block A of G and the principal 2-block B_{0}(kR(3)) of R(3) are Puig equivalent. Let X be the same as in the proof of 6.9. Thus it follows from 5.9(i)-(ii) and the proof of 6.9 that the radical and socle series of X is actually the first one in (8) in the proof of 6.9, and that X is a trivial source kG-module in A with vertex C_{2}.

7. Proof of the main results

Proof of 1.5. First of all, consider the blocks A and B over k, namely, A and B are block algebras of kG and kH, respectively. Hence M is a (kG, kH)-bimodule. We know by 4.6(ii) and 6.1 that the functor F defined by M realises a stable equivalence of Morita type between A and B. It follows from 5.1, 6.3, 6.6 and 6.9 that, for any simple kG-module S in A, F (S) is a simple kH-module in B. Hence, 2.1(ii) yields that _{A}M_{B} realizes a Morita equivalence between A and B. Since M is a ΔP -projective trivial source k[G×H]-module, the Morita equivalence is a Puig equivalence by [44, Remark 7.5] or [33, Theorem 4.1] (note that this was independently observed by L. Scott). Moreover, by [40, 4 Thm.8.9(i)], the Morita equivalence lifts from k to O; see also [54, (38.8)Proposition] or [43, 7.8.Lemma]. □

Proof of Corollary 1.7. This follows by 1.5, 1.6 and 2.7. □

Proof of Theorem 1.3. This follows from 1.7, 2.7 and 4.2 (i). □

Appendix A. Properties of the stable equivalences considered

In this appendix we collect some fundamental properties of the stable equivalences which are found throughout this paper, and in particular of the stable equivalence F of 6.1. For the large part, these properties are used at several steps in this paper, but they are also of independent interest, as a reference providing collection with proofs is desirable. Also, in this section, we aim to supply more general hypotheses for clarity.

The first fundamental property to be shown, is the fact that the Heller operator commutes with applying a stable equivalence in the following sense. Moreover, we give the following “stripping off”-method, which enables us to reduce the problem of determining the image of a module under a stable equivalence to determining the images of its head and socle components; the proof of 6.9 bears testimony of the utility of this lemma.

Lemma A.1. Let A and B be finite dimensional k-algebras for a field k such that A and B are both self-injective. Let F be a covariant functor such that

(1) F is exact.

(2) If X is a projective A-module, then F (X) is a projective B-module,

(3) F induces a stable equivalence from mod-A to mod-B. Then the following holds:

(i) For any positive integer n and for any A-modules X, Y ∈ mod-A, we have that

Ext_{A}^{n}(X, Y) ≅ Ext_{B}^{n}(F (X), F (Y))

as k-spaces. (ii) Let X be a projective-free A-module, and write F (X) = Y ⊕ (proj) for a projective-free B-module Y. Let S be a simple A-submodule of X, and set T = F (S). Now, if T is a simple B-module, then we may assume that Y contains T and that

F (X/S) = Y/T ⊕ (proj).

(iii) Similarly, let X be a projective-free A-module, and write F (X) = Y ⊕ (proj) for a projective-free B-module Y. Let X′ be an A-submodule of X such that X/X′ is simple, and set T = F (X/X′). Now, if T is a simple B-module, then we may assume that T is an epimorphic image of Y and that

Ker(F (X) ↠ T) = Ker(Y ↠ T) ⊕ (proj).

Proof. (i) Note first that the short exact sequence 0 → ΩX →^{i} P (X) → X → 0, where P (X) is the projective cover of X, gives rise to the short exact sequence 0 → F (ΩX) → F (P (X)) → F (X) → 0 of B-modules by (1). Hypothesis (2) implies F (P (X)) = P (F (X)) ⊕ P and F (ΩX) = Ω(F (X)) ⊕ P for a projective B-module P. Thus we have F (ΩX) ≅ Ω(F (X)) in \underline{mod}-B, and hence F (Ω^{n}X) ≅ Ω^{n}(F (X)) in \underline{mod}-B for all n ⩾ 1.

Moreover, letting i^{∗} : Hom_{A}(P (X), Y) → Hom_{A}(ΩX, Y) be the canonical k-map Then we have Ext_{A}^{1}(X, Y) ≅ Hom_{A}(ΩX, Y)/Im(i^{∗}) as k-spaces. A being self-injective, it is well-known that Im(i^{∗}) is the set of projective A-homomorphisms from ΩX to Y. We thus know that Ext_{A}^{1}(X, Y) ≅ \underline{Hom}_{A}(ΩX, Y), and similarly Ext_{A}^{n}(X, Y) ≅ \underline{Hom}_{A}(Ω^{n}X, Y) for all n ⩾ 1.

Then for all n ⩾ 1 we have

Ext_{A}^{n}(X, Y) ≅ \underline{Hom}_{A}(Ω^{n}X, Y)
≅ \underline{Hom}_{B}(F (Ω^{n}X), F (Y)) by (3)
≅ \underline{Hom}_{B}(Ω^{n}(F (X)), F (Y))
≅ Ext_{B}^{n}(F (X), F (Y)).

(ii) Using [28, II Lemma 2.7 and Corollary 2.8] we get

0 ≠ Hom_{A}(S, X) = \underline{Hom}_{A}(S, X) = \underline{Hom}_{B}(F (S), F (X)) = \underline{Hom}_{B}(T, Y) = Hom_{B}(T, Y).

Now the assertions follow from [21, 1.11.Lemma]; see also [21, 3.25.Lemma and 3.26.Lemma]. Note that statement and proof of [21, 1.11.Lemma] remain valid in our more general setting of self-injective algebras.

(iii) is similar. □

Next, we want to show that the stable equivalence of Morita type also commutes with taking the contragredient module if A and B are blocks of group algebras. This is made precise in A.2(iv), but first we place ourselves into a more general context.

Lemma A.2. Let A and B be finite dimensional k-algebras for a field k.

(i) Assume that X ∈ mod-A, and M ∈ A-mod-B, and that _{A}M is projective. Then the

correspondence

Φ : _{B}(M^{∨} ⊗_{A} X^{⊛})^{→}_{B}[(X ⊗_{A} M)^{⊛}]

defined by

[Φ(ψ ⊗_{A} θ)](x ⊗_{A} m) = θ(x·ψ(m))

for ψ ∈ M^{∨}, θ ∈ X^{⊛} and m ∈ M, is an isomorphism of left B-modules. (ii) Assume that Y ∈ A-mod, and N ∈ B-mod-A, and that N_{A} is projective. Then the

correspondence

Θ : (Y^{⊛} ⊗_{A} N^{∨})_{B} ^{→}[(N ⊗_{A} Y)^{⊛}]_{B}

defined by

[Θ(θ ⊗_{A} ψ)](n ⊗_{A} y) = θ(ψ(n) · y)

for ψ ∈ N^{∨}, θ ∈ Y^{⊛} and n ∈ N, is an isomorphism of right B-modules. (iii) If A moreover is a symmetric algebra, with symmetrising form t ∈ Hom_{k}(A, k), then as

(B, A)-bimodules we have

B(M^{∨})_{A} ≅ _{B}(M^{⊛})_{A} via the correspondence t_{∗} : f ↦ t ∘ f.

Thus we have an isomorphism of left B-modules

Ψ : _{B}(M^{⊛} ⊗_{A} X^{⊛}) ⟶^{≈}_{B}(M^{∨} ⊗_{A} X^{⊛}) ⟶^{Φ}_{B}(X ⊗_{A} M)^{⊛}

given by

t_{∗}(ψ) ⊗_{A} θ ↦ ψ ⊗_{A} θ ↦ Φ(ψ ⊗_{A} θ).

(iv) If finally A and B are block algebras of finite groups, and M is self-dual, namely, M^{∗} ≅ M as (A, B)-bimodules, then as right B-modules we have

(X^{∗} ⊗_{A} M)_{B} ≅ [(X ⊗_{A} M)^{∗}]_{B}.

Proof. We prove (i), the proof of (ii) is entirely similar. It is easy to see that Φ is well-defined and a homomorphism of left B-modules. Next, we want to claim that Φ is onto. Since _{A}M is projective, it follows from [7, Theorem 1.7(v)] that a map τ_{M} = τ_{M,M} : M^{∨} ⊗_{A} M → Hom_{A}(_{A}M,_{A}M) defined by [τ_{M}(ψ ⊗_{A} m)](m′) = ψ(m′)·m for ψ ∈ M^{∨} and m, m′ ∈ M, is onto. Since the identity map of M exists, there are a positive integer ℓ; ψ_{1}, ⋯, ψ_{ℓ} ∈ M^{∨} and m_{1}, ⋯, m_{ℓ} ∈ M such that

∑_{i=1}^{ℓ} ψ_{i}(m)·m_{i} = m  for m ∈ M.  (∗)

Now take any σ ∈ (X ⊗_{A} M)^{⊛}. Then we can define a k-linear map f_{σ} : M → X^{⊛} by [f_{σ}(m)](x) = σ(x ⊗_{A} m) for m ∈ M and x ∈ X. Take any a ∈ A. Then,

[f_{σ}(am)](x) = σ(x ⊗_{A} am) = σ(xa ⊗_{A} m).

On the other hand, [a·f_{σ}(m)](x) = [f_{σ}(m)](xa) = σ(xa ⊗_{A} m) by the definitions of f_{σ} and A(X^{⊛}). This means that f is a left A-module homomorphism. Now by (∗), we have an element α = ∑_{i=1}^{ℓ} ψ_{i} ⊗_{A} f_{σ}(m_{i}) ∈ M^{∨} ⊗_{A} X^{⊛}. Then, for x ∈ X and m ∈ M, it is straightforward to check that [Φ(α)](x ⊗_{A} m) = σ(x ⊗_{A} m). Thus, Φ(α) = σ, which means that Φ is onto. Now we have the following five epimorphisms of k-spaces:

M^{∨} ⊗_{A} X^{⊛} ↠^{Φ} (X ⊗_{A} M)^{⊛} →^{≈} X ⊗_{A} M  as k-spaces

→^{≈} (X^{⊛})^{⊛} ⊗_{A} (M^{∨})^{∨} (at least) as k-spaces

↠^{Θ} (M^{∨} ⊗_{A} X^{⊛})^{⊛} →^{≈} M^{∨} ⊗_{A} X^{⊛} as k-spaces.

Since all modules above are of finite k-dimension, all the five epimorphisms above have to be isomorphisms of k-spaces.

(iii) It is easy to see that t_{∗} is a homomorphism of (B, A)-bimodules, and that t_{∗} is injective. Hence the first assertion follows from [7, Proposition 2.7]. The second assertion now follows from this together with (i).

(iv) follows easily from (iii). □

Finally, a fundamental property of the stable equivalences obtained through 2.3 (see also 2.4) is that it preserves vertices and sources, and takes indecomposable modules to their Green correspondents.

Lemma A.3. Let H be a proper subgroup of G, and let A and B be block algebras of kG and kH, respectively. Now, let M and M′ be finitely generated (A, B)- and (B, A)-bimodules, respectively, which satisfy the following:

(1) _{A}M_{B} | 1_{A}·kG·1_{B} and _{B}M′_{A} | 1_{B}·kG·1_{A}.

(2) The pair (M, M′) induces a stable equivalence between mod-A and mod-B.

Then we get the following:

(i) Assume that X is a non-projective indecomposable kG-module in A with vertex Q. Then there exists a non-projective indecomposable kH-module Y in B, unique up to isomorphism, such that (X ⊗_{A} M)_{B} = Y ⊕ (proj), and Q^{g} is a vertex of Y for some element g ∈ G (and hence Q^{g} ⊆ H). Since Q^{g} is also a vertex of X, this means that X and Y have the same vertices.

(ii) Assume that Y is a non-projective indecomposable kH-module in B with vertex Q. Then there exists a non-projective indecomposable kG-module X in A, unique up to isomorphism, such that (Y ⊗_{B} M′)_{A} = X ⊕ (proj), and Q is a vertex of X.

(iii) Let X, Y and Q ⩽ H be the as in (i). Then there is an indecomposable kQ-module L such that L is a source of both X and Y. This means that X and Y have the same sources.

(iv) Let X, Y and Q ⩽ H be the same as in (ii). Then there is an indecomposable kQ-module L such that L is a source of both X and Y. This means that X and Y have the same sources.

(v) Let X, Y, Q and L be the same as in (iii). In addition, suppose that A and B have a common defect group P (and hence P ⊆ H) and that H ⩾ N_{G}(P). Let f be the Green correspondence with respect to (G, P, H). If Q ∈ 𝔄 = 𝔄(G, P, H), then we have (X ⊗_{A} M)_{B} = f(X) ⊕ (proj).

(vi) Let X, Y, Q and L be the same as in (ii). Furthermore, as in (v), assume that P is a common defect group of A and B, and that H ⩾ N_{G}(P), and let f and 𝔄 be the same as in (v). Now, if Q ∈ 𝔄, then we have (Y ⊗_{B} M′)_{A} = f^{−1}(Y) ⊕ (proj).

Proof. (i) Clearly, X | X↓_{Q}↑^{G}. By (2) there exists a non-projective indecomposable kH-module Y in B, unique up to isomorphism, such that (X ⊗_{A} M)_{B} = Y ⊕ (proj). Hence,

Y | X⊗_{A}M = X⊗_{kG}M | X⊗_{kG}kG_{kH} = X↓_{H} | X↓_{Q}↑^{G}↓_{H} = ⨁_{g∈[Q\G/H]}(X↓_{Q})^{g}↓_{Q^{g}∩H}↑^{H}.

The last equality follows from Mackey Decomposition. Since Y_{kH} is indecomposable, the Krull-Schmidt Theorem yields Y | (X↓_{Q})^{g}↓_{Q^{g}∩H}↑^{H} for some g ∈ G. That is, Y is (Q^{g}∩H)-projective, so that there is a vertex R of Y such that R ⩽ Q^{g} ∩ H. Since Y | Y ↓_{R}↑^{H}, it holds as above that

X | ⊗_{B}M′ = Y ⊗_{kH}M′ | Y ⊗_{kH}kG_{kG} = Y ↑^{G} | (Y ↓_{R}↑^{H})↑^{G} = Y ↓_{R}↑^{G}.

Hence, X is R-projective, so that there is a vertex S of X with S ⊆ R. Since Q is also a vertex of X, we have S = Q^{g′} for some g′ ∈ G. Namely, Q^{g′} ⊆ R. This implies that Q^{g′} = S ⊆ R ⊆ Q^{g} ∩ H ⊆ Q^{g}, and hence Q^{g′} = R = Q^{g} ∩ H = Q^{g}. This yields that Q^{g} ⊆ H.

(ii) Similar to (i).

(iii) By the assumption, Q is a common vertex of X and Y. Let L_{kQ} be a source of Y_{kH}. Then by the proof of (i), X | Y ↑^{G} | L↑^{H}↑^{G} = L↑^{G}. Hence, X | L↑^{G}. Since X has vertex Q and L is an indecomposable kQ-module, it follows that L is a source of X, too.

(iv) This follows from (iii).

(v) Let 𝔛, 𝔜 and 𝔄 be those with respect to (G, P, H) as in [40, Chap.4 §4]. Now, let X be an indecomposable kG-module in A such that a vertex of X is in 𝔄. Thus, we can assume that Q ∈ 𝔄. If X is projective then Q is trivial, so that the trivial group is not contained in 𝔛 by the definition of 𝔄, a contradiction, since H ≠ G.

Hence, X is non-projective. Thus, we get by (i) and (ii) that there is a non-projective indecomposable kH-module Y in B such that X ⊗_{A} M = Y ⊕ (proj B-mod) and that Y also has Q as its vertex. On the other hand, we know (X ⊗_{A} M) | X_{kH} = f(X) ⊕ (𝔜-proj B-mod). This implies that f(X) ⊕ (𝔜-proj B-mod) = Y ⊕ (proj B-mod) ⊕ V for a kH-module V.

Assume that Y is 𝔜-projective. Since Q is a vertex of Y, we have Q ∈_{H} 𝔜. Hence, we get by [40, Chap.4 Lemma 4.1(ii)] that Q ∈ 𝔛. Then we have Q ∉ 𝔄, a contradiction. Therefore, by the Krull-Schmidt Theorem, we have Y ≅ f(X).

(vi) We get this exactly as in (iii) just by replacing X, M, and f by Y, M′, and f^{−1}, respectively. □

Acknowledgements

We are grateful to Burkhard Külshammer for pointing out [56] to us. A part of this work was done while the first author was staying in RWTH Aachen University in 2009 and 2010. He is grateful to Gerhard Hiss for his kind hospitality. For this research the first author was partially supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C)20540008, 2008–2010. The first author still remembers that in January 1985, in Bad Honnef Bonn, Germany, there was a conference and he was informed by Peter Landrock about the non-principal 2-block of Co_{3} with defect group C_{2} × C_{2} × C_{2}, which was interesting because the block is non-principal and the inertial index is 21.

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