May 21, 2012

Broué’s abelian defect group conjecture holds for the double cover of the Higman-Sims sporadic simple group

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

Dedicated to the memory of Herbert Pahlings

Abstract

In the representation theory of finite groups, there is a well-known and important conjecture, due to Broué’, saying 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 B of the normaliser N_{G}(P) of P in G are derived equivalent. We prove in this paper, that Broué’s abelian defect group conjecture, and even Rickard’s splendid equivalence conjecture are true for the faithful 3-block A with an elementary abelian defect group P of order 9 of the double cover 2.HS of the Higman-Sims sporadic simple group. It then turns out that both conjectures hold for all primes p and for all p-blocks of 2.HS.

Keywords: Broué’s conjecture; abelian defect group; splendid derived equivalence, double cover of the Higman-Sims sporadic simple 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]. He actually conjectures the following, where the various notions of equivalences used are recalled more precisely in 1.8:

Conjecture 1.1 (Broué’s Abelian Defect Group Conjecture [5]). Let (K, O, k) be a splitting p-modular system, where p is a prime, 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 provided P is abelian.

In fact, a stronger conclusion than 1.1 is expected:

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

∗ 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).

There are several cases where the conjectures 1.1 and 1.2 have been verified, albeit the general conjecture is widely open; for an overview, containing suitable references, see [7]. As for general results concerning blocks with a fixed defect group, by [28, 45, 49, 50] conjectures 1.1 and 1.2 are proved for blocks with cyclic defect groups in arbitrary characteristic.

Moreover, in [17, (0.2)Theorem] it is shown that 1.1 and 1.2 are true for the principal block algebra of an arbitrary finite group when the defect group is elementary abelian of order 9. In view of the strategy used in [17], and of a possible future theory reducing 1.1 and 1.2 to the quasi-simple groups, it seems worth-while to proceed with this class of groups, as far as nonprincipal 3-blocks with elementary abelian defect group of order 9 are concerned. Indeed, for these cases there are partial results already known, see [13, 20, 21, 22, 24, 26, 38] for instance. The present paper is another step in that programme, our main result being the following:

Theorem 1.3. Let G be the double cover 2.HS of the Higman-Sims sporadic simple group, and let (K, O, k) be a splitting 3-modular system for all subgroups of G. Suppose that A is the faithful block algebra of OG with elementary abelian defect group P = C_{3} × C_{3} of order 9, and that B is a block algebra of ON_{G}(P) such that B is the Brauer correspondent of A. Then, A and B are splendidly derived equivalent, hence the conjectures 1.1 and 1.2 of Broué and Rickard hold.

As an immediate corollary we get:

Corollary 1.4. 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.

Our strategy to prove 1.3 is similar to the ones pursued, for example, for the Janko sporadic simple group J_{4} in [22, 1.6.Theorem] or the Harada-Norton sporadic simple group HN in [24, 1.3.Theorem]. Our starting point was actually to realise that the 3-decomposition matrix of A coincides (up to a suitable order of rows and columns) with the 3-decomposition matrix of the principal 3-block A′ of the alternating group 𝔄_{8} on 8 letters:

A A′
176  1 1 ….
176^{∗} 7 . 1 …
616 14 1 . 1 . .
616^{∗} 20 . 1 1 . .
56 28 … 1 .
1000 35 …. 1
1792 56 1 1 1 . 1
1232 64 1 . . 1 1
1232^{∗} 70 . 1 . 1 1

Here, we indicate ordinary irreducible characters just by their degrees, and complex conjugation by ^{∗}. Therefore, it is quite natural to suspect that the block algebras A and A′ are Morita equivalent, or even Puig equivalent. If this were true, then since conjectures 1.1 and 1.2 have been solved for A′ in [41, 42], this would immediately entail their validity for A as well. Indeed, we are able to prove:

Theorem 1.5. We keep the notation and the assumptions as in 1.3, and let G′ = 𝔄_{8} be the alternating group on 8 letters. Then, the block algebra A of OG and the principal block algebra A′ of OG′ are Puig equivalent.

Remark 1.6. A few remarks on 1.5 are appropriate:

(a) In order to prove 1.5 in its full strength, the detailed local analysis leads to a problem similar to the one already encountered in [22, 6.14.Question]: Viewing G and G′ as (unrelated) abstract groups would only allow to prove that the block algebras A and A′ are Morita equivalent, but not necessarily Puig equivalent. In its consequence this would mean that we were only able to verify Broué’s conjecture 1.1, but not Rickard’s conjecture 1.2 for G. To remedy this, and to circumvent [22, 6.14.Question], we use the fact that G′ can be embedded as a subgroup into G, leading to an explicit configuration of groups allowing for compatible local analysis.

(b) Note that complex conjugation induces a non-trivial permutation both on the irreducible ordinary and Brauer characters of A; in terms of columns of the decomposition matrix this amounts to interchanging the first two columns. But all ordinary and Brauer characters of A′ are real-valued. Hence the Puig equivalence asserted by 1.5 does not commute with the selfequivalences of the module categories of A and A′ induced by taking contragredient modules.

Actually, our proof of 1.5 provides two distinct Puig equivalences, one inducing the bijection between the simple A- and A′-modules as indicated in the decomposition matrix above, the other one inducing the bijection obtained by interchanging its first two columns.

(c) As far as we have experienced, it looks that most of all non-principal 3-blocks with elementary abelian defect group P of order 9 are just Morita equivalent to certain principal 3-blocks with defect group P, see [20, 21, 22, 24], for instance. One might be tempted to say that these non-principal blocks are pseudo-principal. So, the non-principal block algebra A considered here is even pseudo-principal in two ways, each leading to different ‘trivial’ character; and the principal block algebra A′ is also pseudo-principal with a different ‘trivial’ character.

However, there are non-principal 3-blocks of finite groups with defect group P which are not pseudo-principal in the above sense, that is they are not Morita equivalent to any principal 3-block: For example, it has been already noted in [16, Remark 4.4] that the non-principal 3-block with defect group P of the Higman-Sims sporadic simple group HS has this property, and the faithful 3-blocks of 4.M_{22} described in [38] have as well.

Contents 1.7. This paper is organised as follows: In §2 we recall a few of the most important ingredients of our proofs. In §3 we present the local data related to G′ = 𝔄_{8}. In §4 we present the local data related to G = 2.HS, and relate the groups G′ and G; in particular we comment on how the explicit embedding is achieved in a computational setting. In §5 we proceed to give a stable equivalence for A and its Brauer correspondent. In §6 we determine the images of the simple A-modules with respect to this stable equivalence. In §7 we finally complete the proofs of 1.3, 1.4 and 1.5, and we also give details on the phenomena in 1.6(a) and 1.6(b).

A further comment on the computational contents of the present paper is in order: As tools, we use the computer algebra system GAP [10], to calculate with permutation groups 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 [8, 12, 56], and of the interface [55] to the data library [57]. Moreover, we use the computer algebra system MeatAxe [48] to handle matrix representations over finite fields, as well as its extensions to compute submodule lattices [32, 37], radical and socle series [35], and homomorphism spaces, endomorphism rings and direct sum decompositions [33, 34].

Notation/Definition 1.8. (a) Throughout this paper, we use the standard notation and terminology as is used in [8, 39, 52]. We recall a few for convenience:

If A and B are finite dimensional k-algebras, where k is a field, we denote by mod-A, A-mod and A-mod-B the categories of finitely generated right A-modules, left A-modules 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. 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 the A-module M, so that M^{∨} becomes a left A-module via (aϕ)(m) = a·ϕ(m) for a ∈ A, ϕ ∈ M^{∨} and m ∈ M. We denote by soc(M) and rad(M) the socle and the radical of M, respectively. For simple Amodules S_{1}, ⋯, S_{n}, and positive integers a_{1}, ⋯, a_{n}, we write that ‘M = a_{1}×S_{1}+⋯+a_{n}×S_{n}, as composition factors’ when the set of all composition factors are a_{1} times S_{1}, ⋯, a_{n} times S_{n}. For another A-module L, we write M|L when M is isomorphic to a direct summand of L as an A-module. 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.

By G we always denote 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 K is of characteristic zero, and its residue field k = O/rad(O) is of characteristic p, and that K and k are splitting fields for all subgroups of G. We denote by k_{G} the trivial kG-module. If X is a kG-module, then we write X^{∗} = Hom_{k}(X, k) for the contragredient of X, namely, X^{∗} is again a 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 a kG-module and an kH-module, respectively. Then let M↓_{H}^{G} = M↓_{H} be the restriction of M to H, and let N↑_{H}^{G} = N↑^{G} = (N ⊗_{kH} kG)_{kG} be the induction (induced module) of N to G.

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. Since the character field ℚ(χ) := ℚ(χ(g) ; g ∈ G) ⊆ K of any character χ ∈ Irr(G) is contained in a cyclotomic field, we may identify ℚ(χ) with a subfield of the complex number field ℂ, hence we may think of characters having values in ℂ. In particular, we write χ^{∗} for the complex conjugate of χ, where of course χ^{∗} is the character of the KG-module contragredient to the KG-module affording χ. If A is a block algebra 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.

(b) 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 via v·(g, g′) = g^{−1}vg′ for v ∈ V and g, g′ ∈ G. We denote by ΔG = {(g, g) ∈ G × G | g ∈ G}. 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. This is equivalent to a condition that A and A′ have source algebras which are isomorphic as interior P -algebras, see [44, Remark 7.5] and [31, 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 both _{A}𝔐 and 𝔐_{A′} are projective and that _{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 [31, 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 derived 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; see [30, 31].

2. Preliminaries

Lemma 2.1 (Scott). The following holds:

(i) If M is a trivial source kG-module, then M lifts uniquely (up to isomorphism) to a trivial source OG-lattice M̂.

(ii) If M and N are both trivial source kG-modules, then [M, N]^{G} = (χM̂, χN̂)^{G}.

Proof. See [27, II Theorem 12.4 and I Proposition 14.8] and [3, Corollary 3.11.4]. □

Lemma 2.2 (Linckelmann). 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.

Proof. (i) and (ii) respectively are given in [29, Theorem 2.1(ii) and (iii)]. □

Lemma 2.3 (Fong-Reynolds). Let H be a normal subgroup of G, and let A and B be block algebras of OG and OH, respectively, such that A covers B. Let T = T_{G}(B) be the inertial subgroup (stabiliser) of B in G. Then, there is a block algebra Ã of OT such that Ã covers B, 1_{A}1Ã = 1Ã1_{A} = 1Ã, A = Ã^{G} (block induction), and the block algebras A and Ã are Morita equivalent via a pair (1_{A}·OG·1Ã, 1Ã·OG·1_{A}), that is, the Morita equivalence is a Puig equivalence and induces a bijection

Irr(Ã) → Irr(A), χ̃ ↦ χ̃↑^{G}; Irr(A) → Irr(Ã), χ ↦ χ↓_{T} ·1Ã

between Irr(Ã) and Irr(A), and a bijection

IBr(Ã) → IBr(A), ϕ̃ ↦ ϕ̃↑^{G}; IBr(A) → IBr(Ã), ϕ ↦ ϕ↓_{T} ·1Ã

between IBr(Ã) and IBr(A),

Proof. See [21, 1.5.Theorem] and [39, Chap.5, Theorem 5.10]. □

Remark 2.4. In 2.3 Ã is called a Fong-Reynolds correspondent of A and vice versa. Note that there can be more than one Fong-Reynolds correspondent in general.

Lemma 2.5. 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) (Stripping-off method, case of socle) 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).

(ii) (Stripping-off method, case of radical) 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 Asubmodule 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. See [21, 1.11.Lemma] or [25, A.1.Lemma]. □

Lemma 2.6. 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 at least one vertex in common.

(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 at least one source in common.

(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 at least one source in common.

(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(G, P, H), see [39, Chap.4, §4] 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(G, P, H), then we have (Y ⊗_{B} M′)_{A} = f^{−1}(Y) ⊕ (proj).

Proof. See [25, A.3.Lemma]. □

Lemma 2.7. Set G = 𝔄_{5} ⋊ C_{4} = (𝔄_{5} × 2).2, where the action on C_{4} of 𝔄_{5} is that C_{4}/C_{2} acts faithfully on 𝔄_{5}. Let P be a Sylow 3-subgroup of 𝔄_{5} (and hence P ≅ C_{3}).

(i) There is a faithful non-principal block algebra A of kG (that is, not having the central subgroup of order 2 in its kernel) with defect group P.

(ii) We can write Irr(A) = {χ_{1}, χ_{2}, χ_{3}} such that χ_{1}(1) = 1, χ_{2}(1) = 4, χ_{3}(1) = 5 and χ_{1}(u) = χ_{2}(u) = 1 for any element u ∈ P − {1}. Moreover, we can write IBr(A) = {φ_{1}, φ_{2}} such that the 3-decomposition matrix of A is

\begin{aligned}& φ_{1} & φ_{2} \\ χ_{1} & 1 & . \\ χ_{2} & . & 1 \\ χ_{3} & 1 & 1 \end{aligned}

(iii) Set H = N_{G}(P). Then H = 𝔖_{3} × C_{4}, and let B be a block algebra of kH which is the

Brauer correspondent of A. (iv) Set M = 𝔣(A), where 𝔣 is the Green correspondence with respect to (G × G, ΔP, G × H).

Then, M induces a Morita equivalence between A and B (and hence M induces a Puig equivalence between A and B). Furthermore, the simple kG-modules in A affording φ_{1} and φ_{2} are both trivial source kG-modules.

Proof. (i)-(iii) are easy. (iv) follows from (i)-(iii) and [19, Theorem 1.2]. □

3. Green correspondences for 𝔄_{8}

Notation 3.1. We introduce some further notation which we use through out the rest of the paper. Let G′ be the alternating group on 8 letters, namely, G′ = 𝔄_{8}. Since Sylow 3-subgroups of G′ are isomorphic to C_{3} × C_{3}, we can assume that P is a Sylow 3-subgroup of 𝔄_{8} as well, which is originally a defect group of A and also a Sylow 3-subgroup of G = 2.HS, see 4.1 and 4.3. There are exactly two conjugacy classes of G′ which contain elements of order 3, that is, P has exactly two G′-conjugacy classes of subgroups of order 3, so we call them Q and R, see 5.2(ii), and see also [8, p.22]. Let H′ = N_{G′}(P), and hence H′ = P ⋊ D_{8}; note that the subgroups of order 3 of P still fall into two H′-conjugacy classes. Let A′ and B′, respectively, be the principal block algebras of kG′ and kH′. Thus B′ = kH′.

Lemma 3.2. (i) The 3-decomposition matrix and the Cartan matrix of A′, respectively, are the following:

\begin{aligned}& k_{G′} & 7 & 13 & 28 & 35 \\ χ′_{1} & 1 & . & . & . & . \\ χ′_{7} & . & 1 & . & . & . \\ χ′_{14} & 1 & . & 1 & . & . \\ χ′_{20} & . & 1 & 1 & . & . \\ χ′_{28} & . & . & . & 1 & . \\ χ′_{35} & . & . & . & . & 1 \\ χ′_{56} & 1 & 1 & 1 & . & 1 \\ χ′_{64} & 1 & . & . & 1 & 1 \\ χ′_{70} & . & 1 & . & 1 & 1 \end{aligned}  \begin{aligned}& P (k_{G′}) & P (7) & P (13) & P (28) & P (35) \\ k_{G′} & 4 & 1 & 2 & 1 & 2 \\ 7 & 1 & 4 & 2 & 1 & 2 \\ 13 & 2 & 2 & 3 & 0 & 1 \\ 28 & 1 & 1 & 0 & 3 & 2 \\ 35 & 2 & 2 & 1 & 2 & 4 \end{aligned}

(ii) All simple kG′-modules k_{G′}, 7, 13, 28, 35 in A′ have P as their vertices.

Proof. (i) follows from [12, A_{8} (mod 3)] and [8, p.22], and for (ii) see [15, 3.7.Corollary]. □

Notation 3.3. We use the notation χ′_{1}, ⋯, χ′_{70} and k_{G′}, 7, 13, 28, 35 as in 3.2, where the numbers mean the degrees (dimensions) of characters (modules).

Lemma 3.4. The following holds:

(i) The character table of H′ = P ⋊ D_{8} ≅ (C_{3} × C_{3}) ⋊ D_{8} is given as follows:

centraliser 72 8 12 12 18 18 4 6 6
element 1A 2A 2B 2C 3A 3B 4A 6A 6B
χ_{1a}  1 1 1 1 1 1 1 1 1
χ_{1b}  1 1 −1 −1 1 1 1 −1 −1
χ_{1c}  1 1 −1 1 1 1 −1 −1 1
χ_{1d}  1 1 1 −1 1 1 −1 1 −1
χ_{2}  2 −2 0 0 2 2 0 0 0
χ_{4a}  4 0 0 2 −2 1 0 0 −1
χ_{4b}  4 0 0 −2 −2 1 0 0 1
χ_{4c}  4 0 2 0 1 −2 0 −1 0
χ_{4d}  4 0 −2 0 1 −2 0 1 0

Note that χ_{1b} is distinguished amongst the non-trivial linear characters, for example by having an element of order 4 in its kernel.

(ii) H′ = Inn(H′) ◁ Aut(H′) such that |Aut(H′)/H′| = 2, and any non-inner automorphism of H′ induces a non-inner automorphism of D_{8} = H′/P, and interchanges the two conjugacy classes of subgroups of order 3 of P. In particular, there is an induced character table automorphism of Irr(H′) interchanging

χ_{1c} ↔ χ_{1d}, χ_{4a} ↔ χ_{4c}, χ_{4b} ↔ χ_{4d}.

(iii) The 3-decomposition matrix and the Cartan matrix of B′ = kH′ = k[P ⋊ D_{8}], respectively, are the following:

\begin{aligned}& 1a & 1b & 1c & 1d & 2 \\ χ_{1a} & 1 & . & . & . & . \\ χ_{1b} & . & 1 & . & . & . \\ χ_{1c} & . & . & 1 & . & . \\ χ_{1d} & . & . & . & 1 & . \\ χ_{2} & . & . & . & . & 1 \\ χ_{4a} & 1 & . & 1 & . & 1 \\ χ_{4b} & . & 1 & . & 1 & 1 \\ χ_{4c} & 1 & . & . & 1 & 1 \\ χ_{4d} & . & 1 & 1 & . & 1 \end{aligned}  \begin{aligned}& P (1a) & P (1b) & P (1c) & P (1d) & P (2) \\ 1a & 3 & 0 & 1 & 1 & 2 \\ 1b & 0 & 3 & 1 & 1 & 2 \\ 1c & 1 & 1 & 3 & 0 & 2 \\ 1d & 1 & 1 & 0 & 3 & 2 \\ 2 & 2 & 2 & 2 & 2 & 5 \end{aligned}

(iv) All simple kH′-modules 1a, 1b, 1c, 1d, 2 in B′ have P as their vertices.

Proof. (i) and (ii) follow from an explicit computation with GAP [10], the rest is easy. □

Notation 3.5. We use the notation χ_{1a}, ⋯, χ_{4d} and 1a = k_{H′}, 1b, 1c, 1d, 2 as in 3.4, where the numbers mean the degrees (dimensions) of characters (modules).

Lemma 3.6. The block algebra B′ = kH′ = k[P ⋊ D_{8}] has exactly 18 non-isomorphic trivial source modules over k. In fact, they are given in the following list, where the diagrams are Loewy and socle series:

(i) Five PIM’s P (1a), P (1b), P (1c), P (1d), P (2):

\begin{matrix}1a \\ 2 \\ 1a 1c 1d \\ 2 \\ 1a \end{matrix} ,  \begin{matrix}1b \\ 2 \\ 1b 1c 1d \\ 2 \\ 1b \end{matrix} ,  \begin{matrix}1c \\ 2 \\ 1a 1b 1c \\ 2 \\ 1c \end{matrix} ,  \begin{matrix}1d \\ 2 \\ 1a 1b 1d \\ 2 \\ 1d \end{matrix} ,  \begin{matrix}2 \\ 1a 1b 1c 1d \\ 2 2 2 \\ 1a 1b 1c 1d \\ 2 \end{matrix} .

(ii) Five trivial source modules with vertex P: The simple modules 1a, 1b, 1c, 1d, 2.

(iii) Eight trivial source modules with cyclic vertex of order 3, where we also give the associated trivial source characters, see 2.1:

\begin{matrix}1a 1d \\ 2 \\ 1a 1d \end{matrix} ; ↕ ; χ_{1a} + χ_{1d} + χ_{4c},  \begin{matrix}1b 1c \\ 2 \\ 1b 1c \end{matrix} ; ↕ ; χ_{1b} + χ_{1c} + χ_{4d},  \begin{matrix}2 \\ 1a 1d \\ 2 \end{matrix} ; ↕ ; χ_{2} + χ_{4c},  \begin{matrix}2 \\ 1b 1c \\ 2 \end{matrix} ; ↕ ; χ_{2} + χ_{4d},

\begin{matrix}1a 1c \\ 2 \\ 1a 1c \end{matrix} ; ↕ ; χ_{1a} + χ_{1c} + χ_{4a},  \begin{matrix}1b 1d \\ 2 \\ 1b 1d \end{matrix} ; ↕ ; χ_{1b} + χ_{1d} + χ_{4b},  \begin{matrix}2 \\ 1a 1c \\ 2 \end{matrix} ; ↕ ; χ_{2} + χ_{4a},  \begin{matrix}2 \\ 1b 1d \\ 2 \end{matrix} ; ↕ ; χ_{2} + χ_{4b}.

Proof. (i) The structure of the PIM’s is immediate as soon as we know that Ext_{kH′}^{1}(1x, 1y) = 0 for all x, y ∈ {a, b, c, d}. This in turn follows from the Ext-quiver of B′, which is given as a quiver with relations in [41, Section 4, Case 2].

To find the non-projective trivial source modules, we employ [39, Chap.4, Exc.10]. From that (ii) is immediate. Moreover, this also yields the trivial source characters given in (iii), from which it is easy to see, using the vanishing of Ext_{kH′}^{1}(1x, 1y) again, that the associated modules are indecomposable. □

Lemma 3.7. An (A′, B′)-bimodule M′ defined by M′ = f_{(G′×G′,ΔP,G′×H′)}(A′) induces a splendid stable equivalence of Morita type between A′ and B′, namely by

F′ : mod-A′ → mod-B′ : X_{A′} ↦ (X ⊗_{A′} M′)_{B′}.

In particular, F′ fulfills the assumptions of 2.6, and hence its assertions as well.

Proof. This follows from [41, Example 4.3] and [42, Corollary 2]. □

Lemma 3.8. Let f′ and g′ be the mutually inverse Green correspondences with respect to (G′, P, H′). Then the Green correspondents of simple modules are the following:

\begin{aligned}g′(1a) & = & k_{G′} & ↔ & χ′_{1} & f′(k_{G′}) & = & 1a \\ g′(1b) & = & 7 & ↔ & χ′_{7} & f′(7) & = & 1b \\ g′(1c) & = & \begin{matrix}13 \\ k_{G′} 7 \\ 13 \end{matrix} & ↔ & χ′_{14} + χ′_{20} & f′(13) & = & \begin{matrix}1c \\ 2 \\ 1c \end{matrix} \\ g′(1d) & = & 28 & ↔ & χ′_{28} & f′(28) & = & 1d \\ g′(2) & = & 35 & ↔ & χ′_{35} & f′(35) & = & 2 \end{aligned}

On the left hand we also give the associated trivial source characters, see 2.1. Recall that by considering H′ just as an abstract group {1c, 1d} are indistinguishable, see 3.4, but now note that by fixing H′ ⩽ G′ and specifying f′, this defines 1c and 1d uniquely.

Proof. This follows from [53, Theorem] and [41, Example 4.3]. □

4. 3-Local structure for 2.HS

Notation 4.1. From now on, we assume that G is the covering group 2.HS of the sporadic simple Higman-Sims group HS, and hence |G| = 2^{10}·3^{2}·5^{3}·7·11, see [8, p.80].

Lemma 4.2. We obtain the following:

(i) In order to prove Broué’s abelian defect group conjecture 1.1 and Rickard’s conjecture 1.2 for G = 2.HS, it suffices to prove them for the case p = 3.

(ii) There exists a unique faithful 3-block A with non-cyclic abelian defect group P, and P is elementary abelian of order 9, and in order to prove Broué’s abelian defect group conjecture for G = 2.HS, it suffices to prove it for this 3-block A.

Proof. (i) Since the conjecture is proved when the defect group is cyclic, we know from 4.1 that it is enough to check it for the primes p ∈ {2, 3, 5}. For p = 2 it follows from [12, HS, (mod 2)] that there are a couple of blocks of G and both have non-abelian defect groups. For p = 5 it follows from [12, HS (mod 5)] that there are a couple of blocks of G which have noncyclic defect groups and the defect group is non-abelian.

(ii) Assume that p = 3. Then, again by [12, HS (mod 3)], there are three 3-blocks of G which have noncyclic defect groups. Those 3-blocks have defect groups which are elementary abelian of order 9. Two of them are non-faithful and therefore these two blocks show up in HS. For the principal 3-block of HS, the conjectures have been checked by Okuyama [41, Example 4.8]. For the non-principal 3-block of HS, they have been verified in our previous paper [20, 0.2 Theorem(ii)]. Thus, the remaining untreated case is a unique faithful 3-block A of G with noncyclic defect group. □

Notation 4.3. From now on, we assume p = 3 and we use the notation A and P as in 4.2, that is, A is a block algebra of kG with defect group P ≅ C_{3} × C_{3}. Set N = N_{G}(P), and let A_{N} be a block algebra of kN which is the Brauer correspondent of A. Let (P, e) be a maximal A-Brauer pair in G, namely, e is a block idempotent of kC_{G}(P) such that Br_{P} (1_{A})·e = e, see [1], [6] and [52, §40]. Set H = N_{G}(P, e), namely, H = {g ∈ N_{G}(P)|g^{−1}eg = e}. Let B be a block algebra of kH which is a Fong-Reynolds correspondent of A_{N}, see 2.3; note that there are exactly two distinct Fong-Reynolds correspondents of A_{N}, see 4.9(iii).

Lemma 4.4. (i) The 3-decomposition matrix of A is given as follows:

\begin{matrix}degree & [8, p.81] & S_{1} & S_{2} = S_{1}^{∗} & S_{3} & S_{4} & S_{5} \\ 176 & χ_{26} & 1 & . & . & . & . \\ 176^{∗} & χ_{27} & . & 1 & . & . & . \\ 616 & χ_{28} & 1 & . & 1 & . & . \\ 616^{∗} & χ_{29} & . & 1 & 1 & . & . \\ 56 & χ_{25} & . & . & . & 1 & . \\ 1000 & χ_{32} & . & . & . & . & 1 \\ 1792 & χ_{35} & 1 & 1 & 1 & . & 1 \\ 1232 & χ_{33} & 1 & . & . & 1 & 1 \\ 1232^{∗} & χ_{34} & . & 1 & . & 1 & 1 \end{matrix}

where S_{1}, ⋯, S_{5} are non-isomorphic simple kG-modules in A whose k-dimensions are 176, 176, 440, 56, 1000, respectively. The simples S_{1} and S_{2} are dual to each other, while the remaining are self-dual. There are three pairs (χ_{26}, χ_{27}), (χ_{28}, χ_{29}) and (χ_{33}, χ_{34}) of complex conjugate characters, and all the other χ_{i}’s are real-valued.

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

Proof. (i) follows from [12, HS (mod 3)] and [8, p.81], and for (ii) see [15, 3.7.Corollary]. □

Notation 4.5. We use the notation χ_{26}, χ_{27}, χ_{28}, χ_{29}, χ_{25}, χ_{32}, χ_{35}, χ_{33}, χ_{34}, and S_{1}, ⋯, S_{5} as in 4.4.

Lemma 4.6. The following holds:

(i) N = N_{G}(P) = 2.(2 × (P ⋊ SD_{16})) = P ⋊ L for a subgroup L of N with |L| = 2^{6} such that L ▷ Z ≅ C_{4} and L/Z ≅ SD_{16}. Moreover, L/Z acts non-trivially on Z, with kernel isomorphic to D_{8}. (Recall that SD_{16} has a unique subgroup isomorphic to D_{8}.)

(ii) C_{G}(P) = Z × P and L/Z acts faithfully on P. (Note that SD_{16} is a Sylow 2-subgroup

of GL_{2}(3).) (iii) Z = O_{3′}(C_{G}(P)) = O_{3′}(N), and we can write Irr(Z) = {ψ_{0}, ψ_{1}, ψ_{2}, ψ_{3}} such that

ψ_{i}(z) = \sqrt{−1}^{i} for i = 0, 1, 2, 3, where z is a generator of Z ≅ C_{4}, and \sqrt{−1} ∈ O is a fixed 4-th root of unity. Moreover, we have T_{N}(ψ_{i}) = G for i = 0, 2, while for j = 1, 3 we have

T_{N}(ψ_{1}) = T_{N}(ψ_{3}) ≨ G such that T_{N}(ψ_{j})/C_{G}(P) ≅ D_{8}.

(iv) kN = A_{0} ⊕ A_{2} ⊕ A_{N} as block algebras, having inertial quotients SD_{16}, SD_{16} and D_{8}, respectively. Here, A_{0} is the principal block algebra of kN, covering ψ_{0}, while A_{2} covers ψ_{2}; hence A_{N} is the faithful block algebra being the Brauer correspondent of A. Moreover, A_{0} ≅ A_{2} ≅ k[P ⋊ SD_{16}] as k-algebras, and

A_{N} ≅ Mat_{2}(k[P ⋊ D_{8}])

as k-algebras, where A_{N} has k[P ⋊ D_{8}] as its source algebra.

(v) The 3-decomposition matrix of A_{N} is given as follows:

2α 2β 2γ 2δ 4
χ_{2α}  1 ….
χ_{2β} = χ_{2α}^{∗} . 1 …
χ_{2γ}  . . 1 . .
χ_{2δ}  … 1 .
χ_{4}  …. 1
χ_{8α}  1 . 1 . 1
χ_{8β}  . 1 . 1 1
χ_{8γ} = χ_{8β}^{∗} 1 . . 1 1
χ_{8δ} = χ_{8α}^{∗} . 1 1 . 1

where the numbers mean the degrees (dimensions) of characters (modules). Note that 2β = 2α^{∗} and that 2γ, 2δ and 4 are all self-dual, but apart from this the characters of degree 2 are indistinguishable: Apart from the character table automorphism of Irr(A_{N}) induced by complex conjugation there is another one interchanging

χ_{2γ} ↔ χ_{2δ}, χ_{8α} ↔ χ_{8γ}, χ_{8β} ↔ χ_{8δ}.

Proof. (i)–(ii) follow from explicit computation with GAP [10], and (iii) is an immediate consequence.

(iv)–(v) It follows from (iii) and [36, Theorem 2] that kG = A_{0} ⊕ A_{2} ⊕ A_{N} where

A_{0} ≅ Mat_{|G:T_{G}(ψ_{0})|ψ_{0}(1)}(k^{α}[T_{G}(ψ_{0})/Z]) ≅ k^{α}[P ⋊ SD_{16}],
A_{1} ≅ Mat_{|G:T_{G}(ψ_{2})|ψ_{2}(1)}(k^{β}[T_{G}(ψ_{2})/Z]) ≅ k^{β}[P ⋊ SD_{16}],
A_{N} ≅ Mat_{|G:T_{G}(ψ_{1})|ψ_{1}(1)}(k^{γ}[T_{G}(ψ_{1})/Z]) ≅ Mat_{2}(k^{γ}[P ⋊ D_{8}]),

as k-algebras, for some α, β ∈ H^{2}(SD_{16}, k^{×}), and γ ∈ H^{2}(D_{8}, k^{×}). Since H^{2}(SD_{16}, k^{×}) = 0 by [14, Proof of Corollary (2J)], we can assume α = β = 1.

On the other hand, |H^{2}(D_{8}, k^{×})| = 2 by [11, V Satz 25.6]. But now the assertion in (v) follows by explicit computation with GAP [10], in particular we get that there are 9 irreducible ordinary characters belonging to A_{N}, and 5 irreducible Brauer characters. Hence, by [14, Page 34 Table 1] we infer γ = 1. Finally, the statement about source algebras follows from [43, Proposition 14.6], see [52, (45.12)Theorem] and [2, Theorem 13].

Note that the decomposition matrix of A_{N} given above coincides with that of B′ in 3.4. This will of course turn out to be no accident, but by the current state of knowledge we cannot avoid the explicit computation to proceed as above. □

Notation 4.7. We use the notation L and Z, as in 4.6. Moreover, let z be a generator of Z ≅ C_{4}. We also use the notation χ_{2α}, χ_{2β}, χ_{2γ}, χ_{2δ}, χ_{4}, χ_{8α}, χ_{8β}, χ_{8γ}, χ_{8δ} and 2α, 2β, 2γ, 2δ, 4 as in 4.6.

Lemma 4.8. The block algebra A_{N} has exactly 18 non-isomorphic trivial source modules over k. In fact, they are given in the following list, in which the diagrams are Loewy and socle series and we use the same notation as in 4.7.

(i) Five PIM’s: P (2α), P (2β), P (2γ), P (2δ), P (4).

\begin{matrix}2α \\ 4 \\ 2α 2γ 2δ \\ 4 \\ 2α \end{matrix} ,  \begin{matrix}2β \\ 4 \\ 2β 2γ 2δ \\ 4 \\ 2β \end{matrix} ,  \begin{matrix}2γ \\ 4 \\ 2α 2β 2γ \\ 4 \\ 2γ \end{matrix} ,  \begin{matrix}2δ \\ 4 \\ 2α 2β 2δ \\ 4 \\ 2δ \end{matrix} ,  \begin{matrix}4 \\ 2α 2β 2γ 2δ \\ 4 4 4 \\ 2α 2β 2γ 2δ \\ 4 \end{matrix} .

(ii) Five trivial source modules with a vertex P: 2α, 2β, 2γ, 2δ, 4.

(iii) Eight trivial source modules with cyclic vertex of order 3, where we also give the associated trivial source characters, see 2.1:

\begin{matrix}2α 2δ \\ 4 \\ 2α 2δ \end{matrix} ; ↕ ; χ_{2α} + χ_{2δ} + χ_{8γ},  \begin{matrix}2β 2γ \\ 4 \\ 2β 2γ \end{matrix} ; ↕ ; χ_{2β} + χ_{2γ} + χ_{8δ},  \begin{matrix}4 \\ 2α 2δ \\ 4 \end{matrix} ; ↕ ; χ_{4} + χ_{8γ},  \begin{matrix}4 \\ 2β 2γ \\ 4 \end{matrix} ; ↕ ; χ_{4} + χ_{8δ},

\begin{matrix}2α 2γ \\ 4 \\ 2α 2γ \end{matrix} ; ↕ ; χ_{2α} + χ_{2γ} + χ_{8α},  \begin{matrix}2β 2δ \\ 4 \\ 2β 2δ \end{matrix} ; ↕ ; χ_{2β} + χ_{2δ} + χ_{8β},  \begin{matrix}4 \\ 2α 2γ \\ 4 \end{matrix} ; ↕ ; χ_{4} + χ_{8α},  \begin{matrix}4 \\ 2β 2δ \\ 4 \end{matrix} ; ↕ ; χ_{4} + χ_{8β}.

Table 1. G′ = 𝔄_{8} as a subgroup of G = 2.HS.

[figure: G = 2.HS; N = N_{G}(P) = 2.(2 × (P ⋊ SD_{16})); H = N_{G}(P, e) = Z × (P ⋊ D_{8}); G′ = 𝔄_{8}; H′ = N_{G′}(P) = P ⋊ D_{8}; C_{G}(P) = Z × P; P = C_{G′}(P); 8; 4; 2; 4; 8]

Proof. This follows from 3.6 and 4.6(iv). □

Lemma 4.9. The following holds:

(i) H = Z × (P ⋊ D_{8}) = C_{G}(P) ⋊ D_{8}, hence H ◁ N such that |N/H| = 2.

(ii) We have the block decomposition

kH = B_{0} ⊕ B_{1} ⊕ B_{2} ⊕ B_{3},

where the block B_{i} covers ψ_{i} ∈ Irr(Z), for i = 0, 1, 2, 3. (iii) Hence both B_{1} and B_{3} are Fong-Reynolds correspondents of A_{N}, see 4.3. (iv) We can write IBr(B_{i}) = {1α_{i}, 1β_{i}, 1γ_{i}, 1δ_{i}, 2_{B_{i}}}, for i = 0, 1, 2, 3, so that we have

2x↓_{H}^{N} = 1x_{1} ⊕ 1x_{3}, for each x ∈ {α, β, γ, δ}, and 4↓_{H}^{N} = 2_{B_{1}} ⊕ 2_{B_{3}}.

Proof. This follows from 4.6 and 2.3. □

Notation 4.10. We use the notation 1α_{i}, 1β_{i}, 1γ_{i}, 1δ_{i}, 2_{B_{i}} as in 4.9.

Lemma 4.11. The following holds:

(i) The group G = 2.HS has a unique conjugacy class of subgroups isomorphic to G′ = 𝔄_{8}. (ii) Fixing an embedding of G′ into G, and a Sylow 3-subgroup P of G′, we have the configuration of groups as depicted in Table 1, where the numbers between two boxes are indices between the two corresponding groups.

Proof. This follows from [8, pp.80–81], 3.1, 4.6 and 4.9. □

Remark 4.12. In view of the group theoretic configuration given in 4.11, a few more detailed comments on how computations in GAP [10] are actually done are in order:

The starting point is the smallest faithful permutation representation of G on 704 points, available in terms of standard generators, see [54], in [57]; we choose this realisation of G once and for all. Moreover, there is a maximal subgroup of G isoclinic to 𝔖_{8} × 2, a generating set of which in terms of the standard generators of G is available in [57] as well. Hence going over to the derived subgroup of the latter, we find a subgroup G′ ⩽ G isomorphic to 𝔄_{8}; we keep G′ fixed all the time. Finally, we compute a Sylow 3-subgroup P of G′, and keep this fixed as well. Since the other groups appearing in the diagram in 4.11 are uniquely determined from this by group theoretic properties, we have thus achieved a concrete realisation of the above configuration of groups.

Using this setting we have all kinds of computational tools at our disposal: In particular, we are able to compute the ordinary and Brauer character tables of the groups N, H, and H′ explicitly, as well as the restriction or induction of characters between these groups. Moreover, we may fetch representations of G and G′ from [57], or compute them using the MeatAxe [48], and restrict them explicitly to N, H, and H′, respectively, in order to analyse the restrictions with the MeatAxe [48] and its extensions.

Note that the explicit results in 3.4 and 4.6 have been obtained in that setting already. Moreover, by restricting the representation 28, see 3.3, from G′ to H′ we are able to compute its Green correspondent f′(28), see 3.8, and thus to identify the representation 1d, see 3.5. In the same spirit we obtain the following, where we recall that so far, we are not able to tell 2γ and 2δ apart, see 4.6(v):

Lemma 4.13. The following holds:

(i) For i = 1, 3 the restriction functor Res↓_{H′}^{H} induced by the (B_{i}, B′)-bimodule _{B_{i}}(B_{i})_{B′} induces a Puig equivalence mod-B_{i} → mod-B′.

(ii) {1α_{1}↓_{H′}^{H}, 1α_{3}↓_{H′}^{H}} = {1a, 1b} = {1β_{1}↓_{H′}^{H}, 1β_{3}↓_{H′}^{H}}, where 1α_{i}↓_{H′}^{H} ≠ 1β_{i}↓_{H′}^{H}; hence

2α↓_{H′}^{N} = (2α^{∗})↓_{H′}^{N} = 2β↓_{H′}^{N} = 1a ⊕ 1b.

(iii) 1γ := 1γ_{1}↓_{H′}^{H} = 1γ_{3}↓_{H′}^{H} and 1δ := 1δ_{1}↓_{H′}^{H} = 1δ_{3}↓_{H′}^{H}, where {1γ, 1δ} = {1c, 1d}; hence

2γ↓_{H′}^{N} = 1γ ⊕ 1γ and 2δ↓_{H′}^{N} = 1δ ⊕ 1δ.

(iv) 2_{B_{1}}↓_{H′}^{H} = 2_{B_{3}}↓_{H′}^{H} = 2; hence 4↓_{H′}^{N} = 2 ⊕ 2.

Proof. Most of the assertions follow from 4.9, while the identification of the explicit restrictions 1x_{i}↓_{H′}^{H}, for x ∈ {α, β, γ, δ}, follows from explicit computations in GAP [10]. □

5. Stable equivalences for 2.HS

Notation 5.1. Recall, first of all, the notation G, A, P, N, H, B, e as in 4.1, 4.3 and 4.7. Let i and j respectively be source idempotents of A and B with respect to P. As remarked in [31, pp.821–822], we can take i and j such that Br_{P} (i)·e = Br_{P} (i) ≠ 0 and that Br_{P} (j)·e = Br_{P} (j) ≠ 0. Set G_{P} = C_{G}(P) = C_{H}(P) = H_{P}.

Moreover, letting Q ⩽ P be a subgroup of order 3, we set G_{Q} = C_{G}(Q) and H_{Q} = C_{H}(Q). By replacing e_{Q} and f_{Q} (if necessary), we may assume that e_{Q} and f_{Q} respectively are block idempotents of kG_{Q} and kH_{Q} such that e_{Q} and f_{Q} are determined by i and j, respectively. Namely, Br_{Q}(i)·e_{Q} = Br_{Q}(i) and Br_{Q}(j)·f_{Q} = Br_{Q}(j). Let A_{Q} = kG_{Q}·e_{Q} and B_{Q} = kH_{Q}·f_{Q}, so that e_{Q} = 1_{A_{Q}} and f_{Q} = 1_{B_{Q}}.

Lemma 5.2. The following holds:

(i) All elements in P − {1} are conjugate in N, and hence in G; actually P − {1} ⊆ 3A, where 3A is a conjugacy class of G following the notation in [8, pp.80–81]. Thus all subgroups of P of order 3 are conjugate in N, and hence in G.

(ii) The elements in P − {1} fall into two conjugacy classes of H. Thus P has exactly two H-conjugacy classes of subgroups of order 3; we call them Q and R. Note that we can use the same notation Q and R as in 3.1.

(iii) H_{Q} = C_{N}(Q) = Z ×Q×𝔖_{3} and H_{R} = C_{N}(R) = Z ×R ×𝔖_{3}, so that H_{Q}/Q ≅ C_{4} ×𝔖_{3}

and H_{R}/R ≅ C_{4} × 𝔖_{3}. (iv) G_{Q} = Q × (𝔄_{5} ⋊ Z) ≅ Q × (𝔄_{5} × 2).2, so that G_{Q}/Q ≅ (𝔄_{5} × 2).2.

Proof. (i)–(iii) follow from 4.6 and 4.9, (iv) follows from an explicit computation in GAP [10]. □

Lemma 5.3. Let M_{Q} be the unique (up to isomorphism) indecomposable direct summand of A_{Q}↓_{G_{Q}×H_{Q}}^{G_{Q}×G_{Q}}·1_{B_{Q}} with vertex ΔP. Then, a pair (M_{Q}, M_{Q}^{∨}) induces a Puig equivalence between A_{Q} and B_{Q}.

Proof. Note first that M_{Q} exists by [22, 2.4.Lemma]. Then using 5.2(iii) and (iv), as well as 2.7(iv), the assertion follows as in [22, Proof of 6.2.Lemma], by going over to the central quotients G_{Q}/Q and H_{Q}/Q and their blocks dominating A_{Q} and B_{Q}, respectively, and applying [19, 1.2.Theorem] and [18, Theorem]. □

Lemma 5.4. The (A, B)-bimodule 1_{A}·kG·1_{B} has a unique (up to isomorphism) indecomposable direct summand M with vertex ΔP. Moreover, the functor

F : mod-A → mod-B : X_{A} ↦ (X ⊗_{A} M)_{B}

induces a splendid stable equivalence of Morita type between A and B. In particular, F fulfils the assumptions of 2.6, and hence its assertions as well.

Proof. Note first that, again, _{A}M_{B} exists by [22, 2.4.Lemma]. Then the assertion follows as in [22, Proof of 6.3.Lemma], by applying [23, Theorem] in order to make use of 5.3, and using gluing through [31, 3.1.Theorem]; note that the fusion condition in the latter theorem is automatically satisfied by [21, 1.15.Lemma]. □

Notation 5.5. We use the notation M and F as in 5.4.

Lemma 5.6. The following holds:

(i) The (A, A_{N})-bimodule 1_{A}·kG·1_{A_{N}} has a unique (up to isomorphism) indecomposable direct summand M_{N} with vertex ΔP. Moreover, the functor

F_{N} : mod-A → mod-A_{N} : X_{A} ↦ (X ⊗_{A} M_{N})_{A_{N}}

induces a splendid stable equivalence of Morita type between A and A_{N}. In particular, F fulfils the assumptions of 2.6, and hence its assertions as well.

(ii) Suppose that X is an indecomposable kG-module in A such that a vertex of X belongs to 𝔄(G, P, N), and let F(X) = Y ⊕(proj) for a non-projective indecomposable kH-module Y in B, and F_{N}(X) = f(X) ⊕ (proj), where f denotes the Green correspondence with respect to (G, P, N). Then, f(X) is the correspondent of Y with respect to the Fong-Reynolds correspondence between B and A_{N}, namely f(X) ≅ Y ↑^{N}.

Proof. (i) follows by 5.4 and 2.3, and (ii) follows from 2.6. □

6. Images of simples via the functor F_{N}

Notation 6.1. For brevity, let F := F_{N} : \underline{mod}-A → \underline{mod}-A_{N} denote the functor given in 5.6. Recall that F (X) = f(X) ⊕ (proj), where f is the Green correspondence with respect to (G, P, N), whenever the Green correspondent f(X) is defined, see 5.6(ii).

Lemma 6.2. The simples S_{1} and S_{2} are trivial source kG-modules.

Proof. By [8, p.80] G has a subgroup U with U ≅ U_{3}(5). Then, by a computation with GAP [10], we know 1_{U}↑^{G}·1_{A} = χ_{26} + χ_{27}. Therefore there is a kG-module X such that X = (k_{U}↑^{G}·1_{A}) and X is liftable to an OG-lattice affording χ_{26} + χ_{27}. Thus by 4.4, it holds that X = S_{1} + S_{2}, as composition factors. ¿From 2.1(ii), we have dim_{k}[End_{kG}(X)] = 2. Hence, X = S_{1} ⊕S_{2} since S_{1} is not isomorphic to S_{2}. □

Lemma 6.3. The simple S_{4} is a trivial source kG-module.

Proof. By [8, p.80] G has a subgroup M with M ≅ M_{11}. Then, again by a computation with GAP [10], it holds that 1_{M}↑^{G}·1_{A} = χ_{25} + χ_{26} + χ_{27} + χ_{28} + χ_{29}. Set X = k_{M}↑^{G}·1_{A}. Thus it follows from [27, I Theorem 17.3] that X has a submodule S such that S ↔ χ_{25}. By 4.4, S = S_{4} and X = 2 × S_{1} + 2 × S_{2} + 2 × S_{3} + S_{4} as composition factors. Therefore the self-duality of S_{4} and X implies S|X. □

Lemma 6.4. The simple S_{5} is a trivial source kG-module.

Proof. As before it follows from [8, p.80] that G has a maximal subgroup M such that M ≅ 2.M_{22} and |G : M| = 100. By [8, p.39] and [12, M_{22} (mod 3)], we know that M has a 3-block Ã (which is called ”Block 6” in [12, M_{22} (mod 3)]) such that Ã has a defect group P̃ with P̃ ≅ C_{3} × C_{3}, where we can assume P̃ = P. Moreover, Ã has an irreducible ordinary character χ̃_{13} of degree 10, and Ã has a simple kM-module S̃ of dimension 10 corresponding to χ̃_{13}. Now, it follows from [9, Proposition 3.19] that S̃ has a trivial source. On the other hand, a computation in GAP [10] shows χ̃_{13}↑^{G} = χ_{32}. Therefore 4.4 yields that S_{5} ≅ S̃↑^{G} also has a trivial source. □

Lemma 6.5. We can assume that f(S_{1}) = 2α and f(S_{2}) = f(S_{1}^{∗}) = (2α)^{∗} = 2β.

Proof. It follows from 6.2, 4.4 and [40, Lemma 2.2] that f(S_{1}) and f(S_{2}) are simple, so {f(S_{1}), f(S_{2})} ⊆ {2α, 2β, 2γ, 2δ, 4}. Then, since 2γ, 2δ, 4 are self-dual and 2β = (2α)^{∗} by 4.6(v), and since S_{2} = S_{1}^{∗} by 4.4, it holds that {f(S_{1}), f(S_{2})} = {2α, 2β}. Thus we get the assertion. □

Lemma 6.6. We can assume that

f(S_{4}) = 2δ and f(S_{5}) = 4.

Recall that by considering N just as an abstract group {2γ, 2δ} are indistinguishable, see 4.6(v). But fixing N ⩽ G and specifying f serves to identify the latter uniquely.

Proof. It follows from 6.3, 6.4, 6.5, 4.4 and [40, Lemma 2.2] that {f(S_{4}), f(S_{5})} ⊆ {2γ, 2δ, 4}. Now, by 4.1 and Sylow’s theorem, we have |G : N| ≡ 1 (mod 3). Set T_{i} = f(S_{i}) for i = 4, 5. By the definition of Green correspondence, T_{5}↑^{G} = S_{5} ⊕ X for a kG-module X such that X is Q-projective. Hence, by 4.1 and [39, Chap.4, Theorem 7.5], it holds that

dim(S_{5}) ≡ dim(T_{5}↑^{G}) = dim(T_{5}) · |G : N| ≡ dim(T_{5}) mod 3,

and dim(S_{5}) = 1000 ≡ 1 mod 3. Thus, T_{5} = 4, so that T_{4} ∈ {2γ, 2δ}. □

Lemma 6.7. Using the assumption of 6.6, it holds that

f(S_{3}) = \begin{matrix}2γ \\ 4 \\ 2γ \end{matrix} .

Proof. As noted in the proof of 6.4, G has a maximal subgroup M such that M ≅ 2.M_{22} and |G : M| = 100. By [8, p.39] and [12, M_{22} (mod 3)], we know that M has a 3-block B̃ (which is called ”Block 7” in [12, M_{22} (mod 3)]) such that B̃ has a defect group Q̃ with Q̃ ≅ C_{3}, where we can assume Q̃ = Q. Moreover, B̃ has an irreducible ordinary character χ̃_{16} of degree 120, and B̃ has a simple kM-module T̃ of dimension 120 corresponding to χ̃_{16}. Now, it follows from [9, Proposition 3.19] that T̃ is a trivial source module with vertex Q. Hence the indecomposable summands of X := T̃↑^{G} · 1_{A} have a trivial source as well, and are Q-projective. On the other hand, a computation in GAP [10] says that

X ↔ χ̃_{16}↑^{G} · 1_{A} = χ_{26} + χ_{27} + χ_{28} + χ_{29}.

Therefore, 4.4 yields that X = 2 × S_{1} + 2 × S_{1}^{∗} + 2 × S_{3} as composition factors; note that this shows that X is projective-free. Recall that χ_{26} ↔ S_{1} and χ_{28} ↔ S_{1} + S_{3}. Hence, it holds by 2.1, 6.2 and 4.4 that, as k-spaces,

Hom_{kG}(S_{1}, X) ≅ Hom_{kG}(S_{1}^{∗}, X) ≅ Hom_{kG}(X, S_{1}) ≅ Hom_{kG}(X, S_{1}^{∗}) ≅ k,
Hom_{kG}(S_{4}, X) = Hom_{kG}(S_{5}, X) = Hom_{kG}(X, S_{4}) = Hom_{kG}(X, S_{5}) = 0.

Moreover, it follows from [27, II Lemma 2.7 and Corollary 2.8], 6.5 and 6.1 that, as k-spaces,

Hom_{kN}(F (X), 2α) ≅ \underline{Hom}_{kN}(F (X), 2α) ≅ \underline{Hom}_{kG}(X, S_{1}) ≅ Hom_{kG}(X, S_{1}) ≅ k.

Similarly, we get Hom_{kN}(F (X), (2α)^{∗}) ≅ k. Using 6.6 in the above proof, we obtain

Hom_{kN}(F (X), 2δ) = Hom_{kN}(F (X), 4) = Hom_{kN}(2δ, F (X)) = Hom_{kN}(4, F (X)) = 0.

Then, let Y be an indecomposable direct summand of X with S_{1}|(Y/rad(Y)), hence Y is non-projective. This means that we can write F (Y) = U ⊕ (proj) for a non-projective indecomposable kN-module U in A_{N}, where by 2.6 we infer that U is a trivial source kNmodule with vertex Q, since T̃ has vertex Q and Y is non-projective. Moreover, from

F (X) = F (Y) ⊕ (module) = U ⊕ (proj) ⊕ (module),

6.5 and 6.1, we get Hom_{kN}(F (Y), 2δ) = Hom_{kN}(F (Y), 4) = 0 and

Hom_{kN}(U, 2α) ≅ \underline{Hom}_{kN}(U, 2α) ≅ \underline{Hom}_{kN}(F (Y), 2α)
≅ \underline{Hom}_{kG}(Y, S_{1}) ≅ Hom_{kG}(Y, S_{1}) ≅ k.

Thus, from 4.8(iii) we conclude that

U = \begin{matrix}2α 2γ \\ 4 \\ 2α 2γ \end{matrix} .

Since U|F (X) and X is self-dual, by [25, A.2.Lemma] we have U^{∗}|F (X)^{∗} = F (X^{∗}) = F (X) as well. Since U is not self-dual, this yields that

F (X) = U ⊕ U^{∗} ⊕ (module) ⊕ (proj)

= \begin{matrix}2α 2γ \\ 4 \\ 2α 2γ \end{matrix} ⨁ \begin{matrix}(2α)^{∗} 2γ \\ 4 \\ (2α)^{∗} 2γ \end{matrix} ⨁ (module) ⨁ (proj),

Note that since X is Q-projective, neither S_{1} nor S_{1}^{∗} can possibly be a direct summand of X by 4.4(ii). Hence it follows from the stripping-off method, see 2.5, that there is a subquotient module Z of X such that Z = S_{3} + S_{3} as composition factors, and such that

F (Z) = V ⊕ V^{∗} ⊕ (module) ⊕ (proj), where V = \begin{aligned}2γ & | \\ & | & 4 & | \\ & & & | & 2γ \end{aligned} .

Since F induces a stable equivalence by 6.1, we conclude that Z is decomposable, that is Z ≅ S_{3} ⊕ S_{3}, which implies that F (S_{3}) ⊕ F (S_{3}) = V ⊕ V^{∗}, since F (S_{3}) is indecomposable by 2.2(i). Hence we infer that V ≅ V^{∗} is indecomposable, having Loewy and socle series

V = \begin{matrix}2γ \\ 4 \\ 2γ \end{matrix} .

Note that (although we do not need this fact) the above analysis also shows that

X = \begin{matrix}S_{1} \\ S_{3} \\ S_{1} \end{matrix} ⨁ \begin{matrix}S_{1}^{∗} \\ S_{3} \\ S_{1}^{∗} \end{matrix} .

□

7. Proof of main results

Lemma 7.1. Keeping the setting in 4.11 fixed, we have

2δ↓_{H′}^{N} = 1δ ⊕ 1δ = 1d ⊕ 1d and 2γ↓_{H′}^{N} = 1γ ⊕ 1γ = 1c ⊕ 1c.

Proof. By 4.13 we have to show that 1δ = 1d. In order to do so, we employ the kG-module S_{4}, for which we first show that S_{4}↓_{G′} = 28 ⊕ 28: By an explicit computation with GAP [10] we know that S_{4}↓_{G′} = 2 × 28 as composition factors, see 4.4 and 3.2. Hence we are done as soon as we show that Ext_{kG′}^{1}(28, 28) = 0. This in turn is seen as follows: Since by 3.7 the functor F′ commutes with taking Heller translates in the stable module categories, we have

Ext_{kG′}^{1}(28, 28) ≅ Ext_{kH′}^{1}(F′(28), F′(28)) ≅ Ext_{kH′}^{1}(1d, 1d) = 0,

by making use of 3.8, and the vanishing result in the proof of 3.6. (As an alternative, the vanishing of Ext_{kG′}^{1}(28, 28) can also be found in [51, Appendix p.3115].)

Now, on the one hand we get by 3.8 that

S_{4}↓_{H′} =S_{4}↓_{G′}↓_{H′} = (28 ⊕ 28)↓_{H′}
=(f′(28) ⊕ f′(28) ⊕ (𝔜(G′, P, H′)-proj))
=1d ⊕ 1d ⊕ (𝔜(G′, P, H′)-proj),

see [39, Chap.4, §4]. On the other hand, we get from 6.6 that

S_{4}↓_{H′} =S_{4}↓_{N}↓_{H′} = (f(S_{4}) ⊕ (𝔜(G, P, N)-proj))↓_{H′}
=(2δ ⊕ (𝔜(G, P, N)-proj))↓_{H′}
=(1δ ⊕ 1δ) ⊕ (𝔜(G, P, N)-proj)↓_{H′}.

Therefore, by comparing the vertices of the indecomposables showing up above and by Krull-Schmidt’s theorem, we finally know that 1δ = 1d. □

Lemma 7.2. The blocks A and A′ are Morita equivalent induced by an (A, A′)-bimodule which is ΔP -projective and is a trivial source module as an O[G × G′]-lattice.

Proof. First of all, A and A′ are splendidly stable equivalent of Morita type by either of the (A, A′)-bimodules

M̃_{i} = _{A}(M ⊗_{B} B_{i} ⊗_{B′} M′^{∨})_{A′},

where M and M′ are the same as in 5.4 and 3.7, respectively, and i = 1, 3 by 4.13(i). Hence, the following holds from 3.8, 5.6(ii) as well as 6.5, 6.6, 6.7, 4.13(ii)–(iv) and 7.1:

mod-A ⟶^{F} mod-B_{i} ⟶^{Res↓_{H′}^{H}} mod-B′ ⟶^{F′^{−1}} mod-A′

S_{1}  ↦  1α_{i}  ↦ \{ 1a ; 1b  ↦  \begin{matrix}k_{G′} \\ 7 \end{matrix} \}

S_{2} = S_{1}^{∗} ↦  1β_{i}  ↦ \{ 1b ; 1a  ↦  \begin{matrix}7 \\ k_{G′} \end{matrix} \}

S_{3}  ↦  \begin{matrix}1γ_{i} \\ 2_{B_{i}} \\ 1γ_{i} \end{matrix}  ↦  \begin{matrix}1c \\ 2 \\ 1c \end{matrix}  ↦  13

S_{4}  ↦  1δ_{i}  ↦  1d  ↦  28
S_{5}  ↦  2_{B_{i}}  ↦  2  ↦  35

Note that, by 3.6(i), the B′-module F′(13) is uniquely (up to isomorphism) determined by its Loewy series; hence we indeed have

Res↓_{H′}^{H}(F(S_{3})) ≅ F′(13).

Therefore we finally get that A and A′ are Morita equivalent by 2.2(ii). More precisely, we know also that the Morita equivalence is given by either of the bimodules M̃_{i}, satisfying the properties desired. □

Remark 7.3. A remark on the strategy employed in the proofs of 7.1 and 7.2 is in order:

(a) To derive 7.1 we use the full strength of 4.11: Indeed, using an embedding H′ ⩽ G′ and the Green correspondence f′, we have defined 1d, see 3.8, and similarly, using an embedding H ⩽ N ⩽ G and the Green correspondence f, we have defined 1δ see 6.6. But from that alone we would only be able to conclude that {1γ, 1δ} = {1c, 1d}, see 4.13. Now only additionally using an embedding G′ ⩽ G, entailing a compatible embedding H′ ⩽ H, we are able to conclude as in the proof of 7.1, whose starting point is restricting S_{4} from G to G′.

(b) In order to be able to proceed as in the proof of 7.2 we have to ensure that the functor induced by M̃_{i} maps simple A-modules to simple A′-modules, which happens if and only if

Res↓_{H′}^{H}(F(S_{4})) = Res↓_{H′}^{H}(1δ_{i}) = 1δ =^{!} 1d = F′(28),

which is proved by the full strength of 7.1. Without using the explicit configuration of groups in 4.11 we only know 1δ ∈ {1c, 1d}. (Note that this phenomenon has also been observed in [22, 6.14.Question].) As an alternative we would have to proceed as follows:

By 3.4(ii) there is an outer automorphism of H′, hence inducing a Morita self-equivalence of of kH′ = k[P ⋊ D_{8}], interchanging 1c ↔ 1d. Twisting the bimodule M̃_{i} accordingly then still yields a Morita equivalence between A and A′. But the outer automorphism applied necessarily changes the structure of kH′, which is its own source algebra, as an interior P -algebra; in other words, the twisted bimodule then no longer is ΔP -projective, hence it does no longer induce a Puig equivalence between A and A′. Thus, as already indicated in 1.6(a) we would end up with the weaker statement in 7.2 only saying that A and A′ are Morita equivalent.

Proof of 1.5. By 1.8, the assertion of 7.2 is equivalent to saying that A and A′ are Puig equivalent. Moreover, the two choices of M̃_{i}, for i = 1, 3, account precisely for the two bijections between the simple A- and A′-modules as described in 1.6(b). □

Proof of 1.3. This follows from 1.5, since by [41, Example 4.3] and [42, Theorem 3] the block algebras A′ and B′ are splendidly Rickard equivalent, and by 4.13(i) and 2.3 the block algebras B′ and B are Puig equivalent. □

Proof of 1.4. This follows from 1.3 and 4.2. □

Acknowledgements

A part of this work was done while the first author was staying in RWTH Aachen University in 2010, 2011 and 2012. 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, (B)21340003, 2009–2011, and also (C)23540007, 2011–2014.

References

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

[2] J.L. Alperin, M. Linckelmann, R. Rouquier, Source algebras and source modules, J. Algebra 239 (2001), 262–271.

[3] D.J. Benson, Representations and Cohomology I: Basic representation theory of finite groups and associative algebras, Cambridge Univ. Press, Cambridge, 1998.

[4] T. Breuer, GAP4-package CTblLib — The GAP Character Table Library, Version 1.1.3, 2004, http://www.gap-system.org/Packages/ctbllib.html.

[5] M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181–182 (1990), 61–92.

[6] M. Broué, L. Puig, Characters and local structure in G-algebras, J. Algebra 63 (1980), 306–317.

[7] J. Chuang, J. Rickard, Representations of finite groups and tilting, Handbook of tilting theory, pp.359–391, London Math. Soc. Lecture Note Ser. 332, Cambridge Univ. Press, Cambridge, 2007.

[8] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.

[9] S. Danz, B. Külshammer, Vertices, sources and Green correspondents of the simple modules for the large Mathieu groups, J. Algebra 322 (2009), 3919–3949.

[10] The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.4.12, http://www.gapsystem.org, 2008.

[11] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1966.

[12] C. Jansen, K. Lux, R. Parker, R. Wilson, An Atlas of Brauer Characters, Clarendon Press, Oxford, 1995.

[13] R. Kessar, M. Linckelmann, On stable equivalences and blocks with one simple module, J. Algebra 323 (2010), 1607–1621.

[14] M. Kiyota, On 3-blocks with an elementary abelian defect group of order 9, J. Fac. Sci. Univ. Tokyo (Section IA, Math.) 31 (1984), 33–58.

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

[16] S. Koshitani, Conjectures of Donovan and Puig for principal 3-blocks with abelian defect groups, Comm. Algebra 31 (2003), 2229–2243; and its Corrigendum, Comm. Algebra 32 (2004), 391–393.

[17] S. Koshitani, N. Kunugi, Broué’s conjecture holds for principal 3-blocks with elementary abelian defect group of order 9, J. Algebra 248 (2002), 575–604.

[18] S. Koshitani, N. Kunugi, Blocks of central p-group extensions, Proc. Amer. Math. Soc. 133 (2005), 21–26.

[19] S. Koshitani, N. Kunugi, Trivial source modules in blocks with cyclic defect groups, Math. Z. 265 (2010), 161-172.

[20] S. Koshitani, N. Kunugi, K. Waki, Broué’s conjecture for non-principal 3-blocks of finite groups, J. Pure Appl. Algebra 173 (2002), 177–211.

[21] S. Koshitani, N. Kunugi, K. Waki, Broué’s abelian defect group conjecture for the Held group and the sporadic Suzuki group, J. Algebra 279 (2004), 638–666.

[22] S. Koshitani, N. Kunugi, K. Waki, Broué’s abelian defect group conjecture holds for the Janko simple group J_{4}, J. Pure Appl. Algebra 212 (2008), 1438–1456.

[23] S. Koshitani, M. Linckelmann, The indecomposability of a certain bimodule given by the Brauer construction, J. Algebra 285 (2005), 726–729.

[24] S. Koshitani, J. Müller, Broué’s abelian defect group conjecture holds for the Harada-Norton sporadic simple group HN, J. Algebra 324 (2010), 394–429.

[25] S. Koshitani, J. Müller, F. Noeske, Broué’s abelian defect group conjecture holds for the sporadic simple Conway group Co_{3}, J. Algebra 348 (2011), 354–380.

[26] N. Kunugi, Morita equivalent 3-blocks of the 3-dimensional projective special linear groups, Proc. London Math. Soc. (3) 80 (2000), 575–589.

[27] P. Landrock, Finite Group Algebras and Their Modules, London Math. Society Lecture Note Series, Vol.84, London Math. Soc., Cambridge, 1983.

[28] M. Linckelmann, Derived equivalence for cyclic blocks over a P -adic ring, Math. Z. 207 (1991), 293–304.

[29] M. Linckelmann, Stable equivalences of Morita type for self-injective algebras and p-groups, Math. Z. 223 (1996), 87–100.

[30] M. Linckelmann, On derived equivalences and local structure of blocks of finite groups, Turkish J. Math. 22 (1998), 93–107.

[31] M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, J. Algebra 242 (2001), 819–843.

[32] K. Lux, J. Müller, M. Ringe, Peakword condensation and submodule lattices: an application of the MeatAxe, J. Symb. Comput. 17 (1994), 529–544.

[33] K. Lux, M. Szőke, Computing decompositions of modules over finite-dimensional algebras, Experiment. Math. 16 (2007), 1–6.

[34] K. Lux, M. Szőke, Computing homomorphism spaces between modules over finite dimensional algebras, Experiment. Math. 12 (2003), 91–98.

[35] K. Lux, M. Wiegelmann, Determination of socle series using the condensation method, in: Computational algebra and number theory, Milwaukee, 1996, J. Symb. Comput. 31 (2001), 163–178.

[36] K. Morita, On group rings over a modular field which possess radicals expressible as principal ideals, Science Report of Tokyo Bunrika Daigaku A4 (1951), 177–194.

[37] J. Müller, On a theorem by Benson and Conway, J. Pure Appl. Algebra 208 (2007), 89–100.

[38] J. Müller, M. Schaps, The Broué conjecture for the faithful 3-blocks of 4.M_{22}, J. Algebra 319 (2008), 3588–3602.

[39] H. Nagao, Y. Tsushima, Representations of Finite Groups, Academic Press, New York, 1988.

[40] T. Okuyama, Module correspondence in finite groups, Hokkaido Math. J. 10 (1981), 299–318.

[41] T. Okuyama, Some examples of derived equivalent blocks of finite groups, preprint (1997).

[42] T. Okuyama, Remarks on splendid tilting complexes, in: Representation theory of finite groups and related topics, edited by S. Koshitani, RIMS Kokyuroku 1149, Proc. Research Institute for Mathematical Sciences, Kyoto University, 2000, 53–59.

[43] L. Puig, Pointed groups and construction of modules, J. Algebra 116 (1988), 7–129.

[44] L. Puig, On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks, Birkhäuser Verlag, Basel, 1999.

[45] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303–317.

[46] J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. (3) 72 (1996), 331–358.

[47] J. Rickard, Triangulated categories in the modular representation theory of finite groups, in: Derived Equivalences for Group Rings, edited by S. König and A. Zimmermann, Lecture Notes in Math., Vol.1685, Springer, Berlin, 1998, pp.177–198.

[48] M. Ringe, The C-MeatAxe, Version 2.4, http://www.math.rwth-aachen.de/homes/MTX, 1998.

[49] R. Rouquier, From stable equivalences to Rickard equivalences for blocks with cyclic defect, in: Groups ’93 Galway/St Andrews Vol.2, edited by C.M. Campbell et al., London Math. Society Lecture Note Series, Vol.212 (1995), pp.512–523, London Math. Soc., Cambridge.

[50] R. Rouquier, The derived category of blocks with cyclic defect groups, in: Derived Equivalences for Group Rings, edited by S. König and A. Zimmermann, Lecture Notes in Mathematics, Vol.1685, Springer, Berlin, 1998, pp.199–220.

[51] S. Siegel, Projective modules for A_{9} in characteristic three, Comm. Alg. 19 (1991), 3099–3117.

[52] J. Thévenaz, G-Algebras and Modular Representation Theory, Clarendon Press, Oxford, 1995.

[53] K. Waki, On ring theoretical structure of 3-blocks of finite groups with elementary abelian defect groups of order 9 (written in Japanese), Master Thesis, Chiba University Japan, 1989.

[54] R. Wilson, Standard generators for sporadic simple groups, J. Algebra 184 (1996), 505–515.

[55] R. Wilson, R. Parker, S. Nickerson, J. Bray, T. Breuer, GAP4-package AtlasRep — A GAP Interface to the Atlas of Group Representations, Version 1.4.0, http://www.gap-system.org/Packages/atlasrep.html, 2008.

[56] R. Wilson, J. Thackray, R. Parker, F. Noeske, J. Müller, F. Lübeck, C. Jansen, G. Hiss, T. Breuer, The Modular Atlas Project, http://www.math.rwth-aachen.de/^{∼}MOC.

[57] R. Wilson, P. Walsh, J. Tripp, I. Suleiman, R. Parker, S. Norton, S. Nickerson, S. Linton, J. Bray, R. Abbott, Atlas of Finite Group Representations, http://brauer.maths.qmul.ac.uk/Atlas/v3.