Broué’s abelian defect group conjecture holds for the Janko simple group J_{4}

Shigeo Koshitani^{a,∗}, Naoko Kunugi^{b}, Katsushi Waki^{c}

aDepartment of Mathematics, Faculty of Science, Chiba University, Chiba, 263-8522, Japan

bDepartment of Mathematics, Aichi University of Education, Kariya, 448-8542, Japan cDepartment of Mathematical System Science, Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan

February 1, 2006

Dedicated to Michel Broué on his sixtieth birthday

Abstract

In 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 B of the normalizer N_{G}(P) of P in G are derived equivalent. This conjecture is called Broué’s abelian defect group conjecture. We prove in this paper that Broué’s abelian defect group conjecture is true for a non-principal 3-block A with an elementary abelian defect group P of order 9 of the Janko simple group J_{4}. It then turns out that Broué’s abelian defect group conjecture holds for all primes p and for all p-blocks of the Janko simple group J_{4}.

1. Introduction and notation

In representation theory of finite groups, one of the most interesting and important problems and questions is to give an affirmative answer to a conjecture, which was introduced by M. Broué late eighties last century [6], and is nowadays called Broué’s Abelian Defect Group Conjecture. He actually conjectures the following:

1.1.Conjecture (Broué’s Abelian Defect Group Conjecture) ([6, 6.2.Question] and [17, Conjecture in p.132]). Let p be a prime, and let (O, K, 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 B is a block algebra of ON_{G}(P) such that B is the Brauer correspondent of A, where N_{G}(P) is the normalizer of P in G. Then, A and B should be derived equivalent (Rickard equivalent) provided P is abelian.

In fact, a stronger conclusion than (1.1) is expected. If G and H are finite groups and if A and B are block algebras of OG and OH (or kG and kH) respectively, we say that A and B are splendidly Rickard equivalent in the sense of Linckelmann ([31], [32]), where he calls it a splendid derived equivalence. Note that this is the same as that given by Rickard in [41] when A and B are the principal block algebras, which he calls a splendid equivalence.

∗ Corresponding author.

E-mail addresses: koshitan@math.s.chiba-u.ac.jp (S.Koshitani), nkunugi@auecc.aichi-edu.ac.jp (N. Kunugi), waki@cc.hirosaki-u.ac.jp (K.Waki)

1.2.Conjecture (Rickard [41], [42, Conjecture 4, in p.193]). Keep the notation, and 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 B of ON_{G}(P).

There are several cases where the conjectures of Broué (1.1) and Rickard (1.2) are checked. For example we prove 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), see [19, (0.2)Theorem]. Then, it may be natural to ask what about the case of non-principal block algebras with the same defect group P = C_{3} × C_{3}. Namely, this paper should be considered as a continuation of such a project, which has already been accomplished for several cases in our previous papers for the O’Nan simple group and the Higman-Sims simple group in [22, 0.2.Theorem] and for the Held simple group and the sporadic simple Suzuki group in [23, Theorem]. That is to say, our main theorem of this paper is the following:

1.3.Theorem. Let G be the Janko simple group J_{4}, and let (O, K, k) be a splitting 3modular system for all subgroups of G, see the definition (1.8) below. Suppose that A is a non-principal block algebra of OG with a defect group P which is an elementary abelian group 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 Rickard equivalent, and hence the conjectures (1.1) and (1.2) of Broué and Rickard hold.

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

1.4.Theorem. Keep the notation and the assumption as in (1.3). Then, the non-prinicipal block algebra A of OJ_{4} with a defect group P = C_{3} × C_{3} and the principal block algebra A′ of OA_{8} of the alternating group A_{8} on 8 letters, are splendidly Rickard equivalent, and also, moreover, the block algebras A^{∗} and A^{′∗} are Morita equivalent, where A^{∗} and A^{′∗} are the k-algebras corresponding to A and A′, respectively.

Then, it turns out that, as a corollary to the main result (1.3), we eventually can prove that

1.5.Corollary. 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 when G = J_{4}.

1.6.Starting point and strategy. In his relatively recent paper [48], the third author completely determines the 3-decomposition matrix of the Janko simple group J_{4}. Our starting point for this work was actually to realize that the 3-decomposition matrix for a non-principal block of J_{4} with an elementary abelian defect group of order 9 is exactly the same as that for the principal 3-block of the alternating group A_{8}. Furthermore, the generalized 3-decomposition matrices of these two blocks are the same. Therefore, it is natural to suspect whether these two 3-block algebras would be Morita equivalent not only over an algebraically closed field k of characteristic 3 but also over a complete discrete valuation ring O whose residue field is k, and we might expect even they are Puig equivalent (we shall give a precise definition of Puig equivalence in (1.8) below). Anyhow, since the two conjectures of Broué and Rickard in (1.1) and (1.2) respectively have been solved for the principal 3-block of A_{8} in a celebrated paper of Okuyama [35] (which is still unpublished unfortunately, see also [43, 7.3]), it turns out that Broué’s abelian defect group conjecture (1.1) and Rickard’s splendid equivalence conjecture (1.2) shall be solved also for the non-principal 3-block of J_{4} with the same defect group C_{3} × C_{3}.

1.7.Contents. In §2, we shall give several fundamental lemmas, which are useful and powerful to prove our main results. Actually, (2.5) should be new. In §§3 and 4, we shall investigate 3-modular representations for J_{4} and we shall get trivial source (p-permutation) modules which are in the non-principal 3-block A with a defect group P = C_{3} × C_{3}. In §5, we shall list data on Green correspondents of simples in the principal 3-block A′ of A_{8}, the alternating group on 8 letters, which is, of course, well-known. Finally, in §6, we shall give complete proofs of our main results (1.3), (1.4) and (1.5).

1.8.Notation. Throughout this paper, we use the following notation and terminology. Let A be a ring. We denote by 1_{A}, Z(A) and A^{×} for the unit element of A, the center of A and the set of all units in A, respectively. We denote by rad(A) the Jacobson radical of A and by rad^{i}(A) the i-th power (rad(A))^{i} for any positive integer i while we define rad^{0}(A) = A. We write Mat_{n}(A) for the matrix ring of all n × n-matrices whose entries are in A. Let B be another ring. 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. However, by a module we mean a finitely generated right module unless otherwise stated. Let M and N be A-modules. We write N|M if N is (isomorphic to) a direct summand of M as an A-module.

From now on, let k be a field and assume that A is a finite dimensional k-algebra. Suppose that M is an A-module. Then, we denote by soc(M) the socle of M. We define soc_{0}(M) = 0 and soc_{1}(M) = soc(M). Then, we define soc_{i}(M) by soc_{i}(M)/soc_{i−1}(M) = soc(M/soc_{i−1}(M)) for any integer i ⩾ 2. Similarly, we write rad^{i}(M) for M·rad^{i}(A) for any integer i ⩾ 0. We write P (M) and I(M) for the projective cover and the injective hull (envelope) of M, respectively, and we write Ω for the Heller operator (functor), namely, ΩM is the kernel of the projective cover P (M) ↠ M. Dually, Ω^{−1}M is the cokernel of the injective hull M ↣ I(M). For simple A-modules S_{1}, ⋯, S_{n} (some of which are possibly isomorphic), we denote by U(S_{1}, ⋯, S_{n}) a uniserial A-module M such that rad^{i−1}(M)/rad^{i}(M) ≅ S_{i} for i = 1, ⋯, n. Note that A-modules of such a form U(S_{1}, ⋯, S_{n}) possibly are not uniquely determined up to isomorphism in general. For the same S_{1}, ⋯, S_{n}, we write that M = a_{1} × S_{1} + ⋯ + a_{n} × S_{n}, as composition factors for positive integers a_{1}, ⋯, a_{n} when the set of all composition factors are a_{1} times S_{1}, ⋯, a_{n} times S_{n}. For an A-module M and a simple A-module S, we denote by c_{M}(S) the multiplicity of all composition factors of M which are isomorphic to S. Let N be another A-module. Then, Hom_{A}(M, N) is the set of all right A-module-homomorphisms from M to N, which canonically is a k-vector space, and we denote by PHom_{A}(M, N) the set of all (relatively-) projective homomorphisms in Hom_{A}(M, N), which is a k-subspace of Hom_{A}(M, N). Hence, we can define the factor space, that is, we write \underline{Hom}_{A}(M, N) for the factor space Hom_{A}(M, N)/PHom_{A}(M, N). By making use of this, as well-known, we can construct the stable module category \underline{mod}-A, which is a quotient category of mod-A such that the set of all morphisms is given by \underline{Hom}_{A}(M, N).

In this paper, G is always a finite group and we fix a prime number p. Assume that (O, K, 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. We mean by an OG-lattice a finitely generated right OG-module which is a free O-module. We sometimes call it just an OG-module. For an OGmodule M, we denote by M^{∗} the factor module M/[M·rad(O)], so that M^{∗} is regarded as a right kG-module. Let X be a kG-module. Then, we write X^{∨} for the k-dual of X, namely, X^{∨} = Hom_{k}(X, k) which is again a right kG-module via (x)(φg) = (xg^{−1})φ for x ∈ X, φ ∈ X^{∨} and g ∈ G. 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} be the induction (induced module) of N to G, that is, N↑^{G} = (N ⊗_{OH} OG)_{OG}. Similar for kG- and kH-modules. We denote by Irr(G) and IBr(G) the sets of all irreducible ordinary and Brauer characters of G, respectively. Let A be 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 often mean by IBr(A) the set of all non-isomorphic simple kG-modules belonging to A. We sometimes denote by A^{∗} the block algebra of kG corresponding to A. But, we usually abuse A and A^{∗}, namely, we often mean the block algebra of kG by A as well when it is clear from the context. For ordinary characters χ and ψ of G, we denote by (χ, ψ)^{G} the inner product of χ and ψ in usual sense. Let X and Y be kG-modules. Then, we write [X, Y]^{G} for dim_{k}[Hom_{kG}(X, Y)]. We denote by k_{G} the trivial kG-module. Similar for O_{G}. For A-modules M and N we write [M, N]^{A} for dim_{k}[Hom_{A}(M, N)]. We say that M is a trivial source (p-permutation) kG-module if M is an indecomposable kGmodule whose source is k_{Q}, where Q is a vertex of M. 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, g ∈ G and g′ ∈ G′. Similar for (kG, kG′)-bimodules. We denote by ΔG the diagonal copy of G in G × G, namely, 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 hence P ⊆ G ∩ G′) 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 p-permutation (trivial source) module and ΔP - projective. Similar for blocks of kG and kG′. Due to a result of Puig (and independently of Scott), see [39, Remark 7.5], this is equivalent to a condition that A and A′ have source algebras which are isomorphic as interior P -algebras. For an (OG, OG′)-bimodule V and a common subgroup Q of G and G′, we set V^{ΔQ} = {v ∈ V | v(q, q) = v, ∀q ∈ Q} = {v ∈ V | qv = vq, ∀q ∈ Q}. If Q is a p-group, the Brauer construction is defined to be a 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 a canonical isomorphism (OG)(ΔQ) →^{≈} kC_{G}(Q). We say that A and A′ are stably equivalent of Morita type if there exists an (A, A′)-bimodule 𝔐 such that _{A}(𝔐 ⊗_{A′} 𝔐^{∨})_{A} ≅ _{A}A_{A} ⊕ (projective (A, A)-bimodule) and A′(𝔐^{∨} ⊗_{A} 𝔐)_{A′} ≅ _{A′}A′_{A′} ⊕ (projective (A′, A′)-bimodule). 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 ppermutation (trivial source) O[G × G′]-module and is ΔP -projective, see [32, Theorem 3.1]. We say that A and A′ are Rickard equivalent if A and A′ are derived equivalent, namely, D^{b}(mod-A) and D^{b}(mod-A′) are equivalent as triangulated categories.

For a positive integer n, A_{n} denotes the alternating group on n letters, M_{n} denotes the Mathieu group, and C_{n}, D_{n} and SD_{n} denote the cyclic group, the dihedral group and the semi-dihedral group of order n, respectively. For a subgroup E of Aut(G), G : E denotes a semi-direct product such that G is normal in G : E and E acts on G canonically. For g ∈ G and a subset S of G, we denote g^{−1}Sg by S^{g}, and simlarly, x^{g} = g^{−1}xg for x ∈ G. For other notation and terminology, see the books of Nagao-Tsushima [33] and Thévenaz [47].

2. Preliminaries

2.1.Lemma (Okuyama [34, Lemma 2.2]). Let S be a simple kG-module with vertex P, and let f : G → N_{G}(P) be the Green correspondence with respect to (G, P, N_{G}(P)). If S is a trivial source module, then its Green correspondent fS is again simple as kN_{G}(P)-module.

2.2.Lemma (Scott [27, II Theorem 12.4 and I Proposition 14.8] and [4, Corollary 3.11.4]).

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

2.3.Lemma (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 (stabilizer) 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 [23, 1.5.Theorem] and [33, Chap.5, Theorem 5.10]. □

2.4.Lemma. Let A be a block algebra of OG with a defect group P, let N = N_{G}(P), and let A_{N} be a block algebra of ON which is the Brauer correspondent of A. Moreover, let (P, e) be a maximal A-Brauer pair, H = N_{G}(P, e), the normalizer of (P, e) in N_{G}(P), and let B be a block algebra of OH which is the Fong-Reynolds correspondent of A_{N}, see (2.3). Then, A↓_{G×H}^{G×G}·1_{B} = 1_{A}·OG·1_{B}, as a right O[G × H]-module, has a unique (up to isomorphism) indecomposable direct summand with vertex ΔP.

Proof. A proof is given already in [23, 1.7.Lemma]. However, the proof there may be too simple, and hence we give it here in detail. It suffices to discuss on the blocks only over k. So, we assume that all block algebras here are over k.

First of all, as well-known (see [2, line −1 in p.263] or [47, Exercise (40.2)(b)]), it holds that 1_{B} = e, and hence B = kHe. It follows from a result of Alperin-Linckelmann-Rouquier [2, Theorem 5(i)] that there uniquely (up to isomorphism) exists an indecomposable (kG, kH)- bimodule X with vertex ΔP such that X|A↓_{G×H}^{G×G} and X|B↑_{H×H}^{G×H}.

Now, let X′ be an indecomposable direct summand of A↓_{G×H}^{G×G}·1_{B} with vertex ΔP. Hence X′|A↓_{G×H}^{G×G}·e since X′e = X′·1_{B} = X′. We can define the Green correspondence f : G×H → H × H with respect to (G × H, ΔP, H × H) since N_{G×H}(ΔP) ⊆ H × H. Let Y = fX′. Clearly,

Y |X′↓_{H×H}^{G×H} |(kG)↓_{H×H}^{G×H} = ⨁_{g∈[H\G/H]} k[HgH],

and Y has ΔP as a vertex, and hence Y |kH_{H×H}, see [33, Chap.5, Lemma 10.9(iii) and

Theorem 10.12]. Thus, Y = Y e|(kHe)_{H×H} since Y |X′ = X′e. It holds X′_{G×H} |Y ↑_{H×H}^{G×H}, and hence X′_{G×H} |(kHe)↑_{H×H}^{G×H} = B↑_{H×H}^{G×H}. Therefore, [2, Theorem 5(i)] again implies that X′ = X. □

The next lemma is actually a recent result of the first and the second authors with Rouquier [21]. It treats with blocks with cyclic defect groups, and is quite useful and efficient to get trivial source modules instead of doing huge calculations by computers. Then, it turns out that the next lemma is useful for our main result.

2.5.Lemma (Koshitani-Kunugi-Rouquier [21]). Let A be a block algebra of OG with a cyclic defect group P ≠ 1. Let N = N_{G}(P), and let A_{N} be a block algebra of ON such that A_{N} is the Brauer correspondent of A. Furthermore, let (P, e) be a maximal A-Brauer pair in G, let H = N_{G}(P, e), and let B be a block algebra of OH such that B is the Fong-Reynolds correspondent of A_{N}, see (2.3). Then, we get the following.

(i) The following (1) and (2) are equivalent:

(1) The Brauer tree of A is a star with exceptional vertex in the center, and there exists a non-exceptional irreducible ordinary character χ of G in A such that χ(u) > 0 for any element u ∈ P.

(2) The block algebras A and B are Puig equivalent. (ii) If one of the conditions (1) and (2) in (i) holds (and hence both hold), then all simple

kG-modules in A are trivial source modules. (iii) If one of the conditions (1) and (2) in (i) holds (and hence both hold), then there is

an indecomposable (A, A_{N})-bimodule M such that 1_{A}·OG·1_{A_{N}} = M ⊕ (proj) and M, as an O[G×N]-module, has ΔP as its vertex, and M realizes a Puig equivalence between A and A_{N}.

(iv) If one of the conditions (1) and (2) in (i) holds (and hence both hold), then there is an indecomposable (A, B)-bimodule 𝔐 such that 1_{A}·OG·1_{B} = 𝔐 ⊕ (proj) and 𝔐, as an O[G × H]-module, has ΔP as its vertex, and 𝔐 realizes a Puig equivalence between A and B.

2.6.Lemma. Let G = 2.M_{22}, so that |G| = 2^{8}·3^{2}·5·7·11.

(i) There are exactly two 3-blocks A_{1} and A_{2} of G with a defect group P such that P ≅ C_{3}, and the 3-decomposition matrices D_{A_{1}} and D_{A_{2}} respectively of A_{1} and A_{2} are

D_{A_{1}} = \begin{aligned}& 21 & 210a \\ χ_{21} & 1 & . \\ χ_{210a} & . & 1 \\ χ_{231} & 1 & 1 \end{aligned} D_{A_{2}} = \begin{aligned}& 120 & 210b \\ χ_{120} & 1 & . \\ χ_{210b} & . & 1 \\ χ_{330} & 1 & 1 \end{aligned}

and χ_{21}(u) = χ_{210a}(u) = χ_{120}(u) = χ_{210b}(u) = 3 for any element u ∈ P −{1}, where the indices of χ_{i}’s mean the degrees.

(ii) The simple kG-modules 21, 210a in A_{1} and 120, 210b in A_{2} are all trivial source modules.

Proof. (i) This follows from [8, pp.39–40], [12, p.94], and GAP [45].

(ii) This is obtained by (i) and (2.5)(i). □

2.7.Lemma. Assume that G ⩾ H, and let A and B respectively be block algebras of OG and OH with a common defect group P, and hence P ⩽ H. Suppose, moreover, that a pair (M, M^{∨}) induces a splendid stable equivalence of Morita type between A and B, where M is an (A, B)-bimodule such that M|1_{A}·OG·1_{B} as (A, B)-bimodules.

(i) If X is a non-projective trivial source kG-module in A, then (X⊗_{A}M)_{B} = Y ⊕(proj) for a non-projective indecomposable kH-module Y such that Y has a trivial source. (ii) If X is a non-projective indecomposable kG-module in A, then (X ⊗_{A} M)_{B} = Y ⊕ (proj) for a non-projective indecomposable kH-module Y such that there is a p-subgroup Q of H such that Q is a common vertex of X and Y.

Proof. (i) Easy by Mackey decomposition.

(ii) Let Q be a vertex of X, so that X | X↓_{Q}↑^{G}. Hence, again by Mackey decomposition,

Y | X↓_{Q}↑^{G} ⊗_{A} OG↓_{H} = X↓_{Q}↑^{G}↓_{H} = ⨁_{g∈[Q\G/H]} X^{g}↓_{Q^{g}∩H}↑^{H}.

Thus, Y | X^{g}↓_{Q^{g}∩H}↑^{H} for an element g ∈ G. Therefore, there is a vertex R of Y such that R ⩽ Q^{g} ∩ H. Conversely, Y | Y ↓_{R}↑^{H}. Hence,

X | Y ⊗_{B} M^{∨} | Y ↓_{R}↑^{H} ⊗_{B} OG = Y ↓_{R}↑^{H}↑^{G} = Y ↓_{R}↑^{G}.

This means that X is R-projective, so that Q^{g′} ⩽ R for an element of g′ ∈ G. Hence, Q^{g′} ⩽ R ⩽ Q^{g} ∩ H, which yields that Q^{g′} = R = Q^{g} ∩ H, so that Q^{g} ⩽ H and R = Q^{g}. Thus, R is a common vertex of X and Y. □

2.8.Lemma. Let A be a block algebra of kG, and let S be a one-dimensional right kGmodule in A. Set M = A ⊗_{k} S, and define actions of A on the both sides of M by

a′(a ⊗ s) = a′a ⊗ s, (a ⊗ s)a′ = aa′ ⊗ sa′

for a, a′ ∈ A and s ∈ S.

(i) M is an (A, A)-bimodule.

(ii) (M, M^{∨}) induces a self-Puig equivalence of A.

Proof. Easy by a result of Puig (and, independently, of Scott) [39, Remark 7.5] and [32, Theorem 4.1] since S ⊗ S^{∨} ≅ k_{G} and S is a trivial source kG-module. □

2.9.Lemma. Let k be a field, and let A be a finite-dimensional symmetric k-algebra. Moreover, suppose that S is a simple A-module and M is a projective-free A-module. Then, we have \underline{Hom}_{A}(S, M) ≅ Hom_{A}(S, M) and \underline{Hom}_{A}(M, S) ≅ Hom_{A}(M, S) as k-spaces.

Proof. This follows by [10, (3.2), (3.2*), (3.3)], see [27, II, Lemma 2.7, Corollary 2.8]. □

2.10.Lemma. Let k be an algebraically closed field, and let A and B be finite-dimensional symmetric k-algebras. Suppose that M is an (A, B)-bimodule such that _{A}M and M_{B} are both projective modules. Then a functor F : mod-A → mod-B defined by F (X′) = X′⊗_{A}M for X′_{A}, is additive and exact. Assume, furthermore, that F induces a stable equivalence between A and B.

Let X be a projective-free A-module such that X has a simple A-submodule S. Then, we can write F (X) = Y ⊕ R′ for a projective-free B-module Y and a projective B-module R′. Let T = F (S). Now, if T is a simple B-module, then we may assume that Y contains T and F (X/S) = Y/T ⊕ (proj).

Proof. We get the assertion from (2.9) and [23, 1.11.Lemma], just as in the proof of [23, 3.25.Lemma and 3.26.Lemma], see [26, Proposition 2.2]. □

2.11.Lemma (Linckelmann [29, Theorem 2.1(ii)]). Let A and B be finite-dimensional kalgebras for a field k such that A and B are both self-injective and indecomposable as algebras, and none of them are simple algebras. Suppose that there is an indecomposable (A, B)-bimodule M such that a pair (M, M^{∨}) induces a stable equivalence betwee A and B. If S is a simple A-module, then (S ⊗_{A} M)_{B} is a non-projective indecomposable B-module.

3. 3-Modular representation theory for J_{4}

3.1.Notation and assumption. From now on till the end of this paper, we use the following notation. We assume that G is the Janko simple group J_{4}, and hence |G| = 2^{21}·3^{3}·5·7·11^{3}·23·29·31·37·43 ≒ 8.6 × 10^{19}, see [8, p.190].

3.2.Lemma.

(i) For Broué’s abelian defect group conjecture for the finite group G = J_{4}, it is enough to prove it for the case p = 3.

(ii) There is a unique 3-block A of G with abelian defect group P, and P is elementary abelian of order 9.

Proof. We, first of all, may assume that p ∈ {2, 3, 11} since any p-block of G for the other prime p has cyclic defect groups of order p or 1 and Broué’s abelian defect group conjecture has been solved by results of Rickard [40, Theorem 4.2], Linckelmann [28, Theorem 1.1] and Rouquier [44, Theorem 10.1] in such cases. For p = 2, the principal 2-block of G is a unique 2-block with positive defect by [3, Lemma 5.1(b)] and Sylow 2-subgroups of G are nonabelian, see [8, p.190]. For p = 11, it follows from a direct calculation by GAP [45] that the principal 11-block of G is a unique 11-block with positive defect, see [5, p.464, Table 6], and Sylow 11-subgroups of G are nonabelian, see [8, p.190]. Therefore, it turns out that we can concentrate on the case p = 3. So, we assume from now on that p = 3. Then, we get by [3, Lemma 5.1(a)] that there is a unique 3-block of G with an abelian defect group, and hence let A be the unique 3-block of G with an abelian defect group P, so that P is elementary abelian of order 9. □

3.3.Notation. From now on we assume p = 3, and we use the notation A and P as in (3.2).

When we want to consider p-modular representation theory for a finite group, we, first of all, want to know an exact p-decomposition matrix of the finite group. In our case, we fortunately are able to get it because it has been calculated by the third author. Namely,

3.4.Theorem (Waki).

(i) There are exactly five non-isomorphic simple kG-modules S_{1}, S_{2}, S_{3}, S_{4}, S_{5} in A such that S_{1} = 4, 290, 927, S_{2} = 95, 288, 172, S_{3} = 300, 364, 890, S_{4} = 393, 877, 506, S_{5} = 390, 474, 357, where the numbers mean k-dimensions. In particular, dim_{k}S_{i} ≡ 6 (mod 9) for i = 1, 2, 3, 5; and dim_{k}S_{4} ≡ 3 (mod 9). Moreover, S_{1}, ⋯, S_{5} are all self-dual.

(ii) (Knörr [16, 3.7.Corollary]) All simple kG-modules S_{1}, ⋯, S_{5} in A have P as their

vertices. (iii) The 3-decomposition and the Cartan matrices of A, respectively, are of the form

\begin{matrix}degree & [8, p.188] & S_{1} & S_{2} & S_{3} & S_{4} & S_{5} \\ 4,290,927 & χ_{14} & 1 & . & . & . & . \\ 95,288,172 & χ_{21} & . & 1 & . & . & . \\ 300,364,890 & χ_{25} & . & . & 1 & . & . \\ 393,877,506 & χ_{27} & . & . & . & 1 & . \\ 394,765,284 & χ_{28} & 1 & . & . & . & 1 \\ 493,456,605 & χ_{30} & 1 & 1 & . & 1 & . \\ 690,839,247 & χ_{31} & . & . & 1 & . & 1 \\ 789,530,568 & χ_{35} & . & 1 & 1 & 1 & . \\ 1,089,007,680 & χ_{41} & 1 & . & 1 & 1 & 1 \end{matrix}

P (S_{1}) P (S_{2}) P (S_{3}) P (S_{4}) P (S_{5})
S_{1}  4  1  1  2  2
S_{2}  1  3  1  2  0
S_{3}  1  1  4  2  2
S_{4}  2  2  2  4  1
S_{5}  2  0  2  1  3

Proof. (i) and (iii) are recent results of the third author [48, Proposition 3.5]. □ 3.5.Notation. We use the notation χ_{14}, χ_{21}, χ_{25}, χ_{27}, χ_{28}, χ_{30}, χ_{31}, χ_{35}, χ_{41} and S_{1}, ⋯, S_{5} as in (3.4). 3.6.Lemma. The heart H(P (S_{i})) = rad(P (S_{i}))/soc(P (S_{i})) is indecomposable as a kGmodule for all i = 1, ⋯, 5. Proof. This is obtained by the Cartan matrix of A in (3.4)(iii) and results of Erdmann and Kawata [22, 1.9.Lemma], see [9, Theorem 1] and [13, Theorem 1.5]. □ 3.7.Lemma. rad(P (S_{2}))/rad^{2}(P (S_{2})) ≅ soc_{2}(P (S_{2}))/soc(P (S_{2})) ≅ S_{4}. Proof. This follows from the Cartan matrix in (3.4)(iii) and (3.6). □ 3.8.Lemma. The group G has maximal subgroups M = 2^{11} : M_{24} and L= 2^{1+12}.3.M_{22} : 2. Proof. See [8, p.190]. □ 3.9.Notation. We use tha notaion M and L as in (3.8) throughout this section. Moreover, let (−1)_{L} be a non-trivial ordinary linear character of L. 3.10.Lemma. There are characters θ_{2} θ_{8}, θ_{3}, θ_{13}, θ_{16} in Irr(M) which satisfy the following;

(i) θ_{2} and θ_{8} are in the principal 3-block of M with degrees 23 and 253, respectively. (ii) There is a 3-block B_{M} of M with a defect group Q ≅ C_{3} such that Irr(B_{M}) = {θ_{3}, θ_{13}, θ_{16}}, θ_{3}(1) = 45, θ_{13}(1) = 990, θ_{16}(1) = 1035, θ_{3}(u) = θ_{13}(u) = 3 for any element u in Q − {1}, and the 3-decomposition matrix of B_{M} is of the form

\begin{aligned}& T_{1} & T_{2} \\ θ_{3} & 1 & . \\ θ_{13} & . & 1 \\ θ_{16} & 1 & 1 \end{aligned}

where IBr(B_{M}) = {T_{1}, T_{2}}.

Proof. These are obtained by GAP [45], see [8, p.96]. □ 3.11.Notation. We use the notation θ_{2}, θ_{8}, θ_{3}, θ_{13}, θ_{16}, B_{M}, T_{1} and T_{2} as in (3.10).

In order to prove our main theorem one of the most important and useful ingredients is actually a trivial source module. We, in principle, can compute such modules by making use of ordinary characters because of powerful results due to Scott (2.2). So, in the rest of this section, we will be trying to get trivial source kG-modules in A as many as possible. This is our essential and fundamental strategy.

3.12.Lemma.

(i) 1_{M}↑^{G}·1_{A} = χ_{14} + χ_{21}. (ii) 1_{L}↑^{G}·1_{A} = χ_{14} + χ_{21} + χ_{30}. (iii) (−1)_{L}↑^{G}·1_{A} = χ_{25} + χ_{31}.

Proof. These follow from direct calculations by using GAP [45]. □

3.13.Lemma.

(i) The simple S_{1} is a trivial source kG-module in A such that S_{1} ↔ χ_{14}. (ii) The simple S_{2} is a trivial source kG-module in A such that S_{2} ↔ χ_{21}. (iii) There is a trivial source kG-module in A of the form U(S_{3}, S_{5}, S_{3}) ↔ χ_{25} + χ_{31}.

Proof. Set X = k_{M}↑^{G}·1_{A^{∗}}. Then, we get by (3.12)(i) and (3.4)(iii) that X = S_{1} + S_{2}, as composition factors. We know by (2.2) that [X, X]^{G} = (χ_{14} + χ_{21}, χ_{14} + χ_{21})^{G} = 2. Thus, X = S_{1} ⊕ S_{2}, so that we obtain (i) and (ii).

(iii) Set X = (−1)_{L}^{∗}↑^{G}·1_{A^{∗}}. Then, (3.12)(iii) and (3.4)(iii) say that X is self-dual. As above, we have by (3.12)(iii) and (3.4)(iii) that X = 2 × S_{3} + S_{5}, as composition factors. By (3.4)(iii), S_{5} is not liftable, and hence S_{5} is not a trivial source module by (2.2)(i). This means that S_{5} |̸ X. Thus, S_{5} |̸ [X/rad(X)] and S_{5} |̸ soc(X) from self-dualities of X and S_{5} in (3.4)(i). Therefore, X = U(S_{3}, S_{5}, S_{3}). □

3.14.Lemma.

(i) There is a trivial source kM-module corresponding to θ_{2} + θ_{8}, and (θ_{2} + θ_{8})↑^{G}·1_{A} =

χ_{27} + χ_{30}.

(ii) The module T_{1} is a trivial source kM-module corresponding to θ_{3}, and θ_{3}↑^{G}·1_{A} =

χ_{21} + χ_{25} + χ_{35}.

Proof. (i) The Mathieu group M_{24} has a maximal subgroup M′ ≅ M_{22}.2, see [8, p.96]. So, M′ has a non-trivial linear ordinary character (−1)_{M′}. Then, (−1)_{M′}↑^{M_{24}} = θ_{2} + θ_{8} by using GAP [45]. We can consider that θ_{2}, θ_{8} ∈ Irr(M). Hence, again by GAP [45], it holds that θ_{2}↑^{G}·1_{A} = χ_{30} and θ_{8}↑^{G}·1_{A} = χ_{27}.

(ii) It follows from (3.10)(ii) and (2.5)(i) that T_{1} is a trivial source kM-module corresponding to θ_{3}. Moreover, we get θ_{3}↑^{G}·1_{A} = χ_{21} + χ_{25} + χ_{35} by direct calculations of GAP [45]. □

3.15.Lemma. χ_{14}↓_{M}·1_{B_{M}} = 0 and χ_{28}↓_{M}·1_{B_{M}} = 2 × θ_{13} + θ_{16}.

Proof. This follows from calculations by GAP [45]. □

3.16.Lemma.

(i) There is a trivial source kG-module in A which has radical and socle series

\begin{matrix}S_{4} \\ S_{1} S_{2} \\ S_{4} \end{matrix} ↔ χ_{27} + χ_{30}.

(ii) There is a trivial source kG-module in A which has radical and socle series

\begin{matrix}S_{2} S_{3} \\ S_{4} \\ S_{2} S_{3} \end{matrix} ↔ χ_{21} + χ_{25} + χ_{35}.

(iii) There is a trivial source kG-module in A which has radical and socle series

\begin{matrix}S_{1} S_{2} \\ S_{4} \\ S_{1} S_{2} \end{matrix} ↔ χ_{14} + χ_{21} + χ_{30}.

(iv) The projective cover P (S_{2}) of S_{2} has radical and socle series

\begin{matrix}S_{2} \\ S_{4} \\ S_{1} S_{2} S_{3} \\ S_{4} \\ S_{2} \end{matrix} ↔ χ_{21} + χ_{30} + χ_{35}.

Proof. (i) By (3.14)(i), there is a trivial source kG-module X corresponding to χ_{27} + χ_{30}, so that (3.4)(iii) implies that X = S_{1} + S_{2} + 2 × S_{4}, as composition factors. By (3.14)(i) and (3.4)(iii), X is self-dual. Recall that S_{1}, S_{2}, S_{4} are self-dual by (3.4)(i). It follows from (2.2) and (3.13)(i)-(ii) that [X, S_{i}]^{G} = [S_{i}, X]^{G} = 0 for i = 1, 2. If X/rad(X) ≅ S_{4} ⊕ S_{4}, then soc(X) ≅ S_{4} ⊕ S_{4} by the self-dualities, a contradiction. This yields that X/rad(X) ≅ soc(X) ≅ S_{4}. Therefore, we get the assertion by the self-dualities.

(ii) We get by (3.14)(ii) that there is a trivial source kG-module X in A corresponding to χ_{21} + χ_{25} + χ_{35}. Hence, X = 2 × S_{2} + 2 × S_{3} + S_{4}, as composition factors, by (3.4)(iii). Note that X is self-dual from (3.14)(ii) and (3.4)(iii), and so are S_{2}, S_{3} and S_{4} from (3.4)(i). It follows from (2.2) and (3.13)(ii) that [X, S_{2}]^{G} = [S_{2}, X]^{G} = (χ_{21} + χ_{25} + χ_{35}, χ_{21})^{G} = 1. Since c_{X}(S_{5}) = 0, we get by (3.13)(iii) and (2.2) that [X, S_{3}]^{G} = [X, U(S_{3}, S_{5}, S_{3})]^{G} = (χ_{21} + χ_{25} + χ_{35}, χ_{25} + χ_{31})^{G} = 1, and hence [S_{3}, X]^{G} = 1 by the self-dualities. Similarly, by (i) and (2.2), we have

[X, S_{4}]^{G} ⩽ [X, \begin{matrix}S_{4} \\ S_{1} S_{2} \\ S_{4} \end{matrix} ]^{G} = (χ_{21} + χ_{25} + χ_{35}, χ_{27} + χ_{30})^{G} = 0,

and hence [X, S_{4}]^{G} = [S_{4}, X]^{G} = 0 by the self-dualities. These mean that X/rad(X) ≅ soc(X) ≅ S_{2} ⊕ S_{3}. Then, we know S_{2} |[rad(X)/rad^{2}(X)] by (3.7). Now, we get by (3.7), (2.2) and (i) that

[X, \begin{matrix}S_{2} \\ S_{4} \end{matrix} ]^{G} ⩽ [X, \begin{matrix}S_{4} \\ S_{1} S_{2} \\ S_{4} \end{matrix} ]^{G} = (χ_{21} + χ_{25} + χ_{35}, χ_{27} + χ_{30})^{G} = 0,

and hence, [X, U(S_{2}, S_{4})]^{G} = [U(S_{4}, S_{2}), X]^{G} = 0 by the self-dualities. This implies that rad(X)/rad^{2}(X) ≅ S̸_{3} ⊕ S_{4} and rad(X)/rad^{2}(X) ≅ S̸_{3}. Therefore, X has the radical series as desired, and hence the socle series as well from the self-dualities.

(iii) It follows from (3.12)(ii) that there is a trivial source kG-module X corresponding to χ_{14} + χ_{21} + χ_{30}. Hence, X = 2 × S_{1} + 2 × S_{2} + S_{4}, as composition factors, from (3.4)(iii). Note that X is self-dual by (3.12)(ii) and (3.4)(iii), and so are S_{1}, S_{2} and S_{4} by (3.4)(i). Just as in (ii), we get by (3.13)(i)-(ii) and (2.2) that [X, S_{i}]^{G} = 1 for i = 1, 2. If S_{4}|[X/rad(X)],

then S_{4}|X by the self-dualities since c_{X}(S_{4}) = 1, and hence (S_{2} ⊕ S_{2})|soc(X) from (3.7), a contradiction. These imply that X/rad(X) ≅ soc(X) ≅ S_{1} ⊕ S_{2}.

Suppose that X has a factor module U(S_{1}, S_{1}). Then, the self-dualities yield that U(S_{1}, S_{1})|X. Thus, X has a submodule Y which is a trivial source module and Y ≅ U(S_{1}, S_{1}). Then, (3.4)(iii) and (2.2) imply that Y ↔ 2 × χ_{14}. But, this is a contradiction since X ↔ χ_{14} + χ_{21} + χ_{30}.

Thus, X does not have a factor module U(S_{1}, S_{1}), and hence a submodule U(S_{1}, S_{1}), neither, because of the self-dualities. Now, let Z be the trivial source kG-module given in (i). Then, it follows from (i) and (2.2) that 1 = (χ_{14} + χ_{21} + χ_{30}, χ_{27} + χ_{30})^{G} = [X, Z]^{G} = [X, rad(Z)]^{G} since c_{X}(S_{4}) = 1. By (3.7), X does not have an epimorphic image U(S_{1}, S_{4}) since [S_{2}, X]^{G} = 1. Moreover, X does not have an epimorphic image U(S_{2}, S_{4}) since, if so, (3.7) shows that X has a submodule U(S_{1}, S_{1}), a contradiction. These imply that X has a

factor module whose radical and socle series are \begin{matrix}S_{1} S_{2} \\ S_{4} \end{matrix}. Therefore, since [X, S_{2}]^{G} = 1 and X

does not have a factor module U(S_{1}, S_{1}), it follows from (3.7) that X has radical and socle series as desired.

(iv) This easily follows from (ii), (iii), (3.4)(iii) and (3.7). □

4. Local structure for kJ_{4}

4.1.Notation and assumption. First of all, recall the notation in §3. Namely, G = J_{4}, A is the unique block algebra of kG with the defect group P such that 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, that is, e is a block (idempotent) of kC_{G}(P) such that Br_{ΔP} (1_{A})·e = e, see [1], [7] and [47, §40].

By results of Alperin-Broué [1, Theorem 3.4] and Broué-Puig [7, Theorem 1.8], for each subgroup P′ of P there uniquely exists a Brauer pair (P′, e_{P′}) such that (P′, e_{P′}) ⩽ (P, e). In particular, e_{P} = e. Set H = N_{G}(P, e), the stabilizer of the maximal A-Brauer pair (P, e), namely, H = {g ∈ N_{G}(P)|e^{g} = e}, where e^{g} = g^{−1}eg. Finally, let B be a block algebra of kH which is the Fong-Reynolds correspondent of A_{N}, see (2.3).

4.2.Lemma.

(i) All subgroups of G which are isomorphic to C_{3} × C_{3} are G-conjugate to P.

(ii) N = (P : 2 × 2^{3}).Σ_{4} = (2^{3} × P) : GL_{2}(3).

(iii) C_{G}(P) = 2^{3} × P.

(iv) H = (2^{3} × P) : D_{8} = C_{G}(P) : D_{8}.

(v) All elements in P − {1} are conjugate in N (so that in G).

(vi) There are exactly two conjugacy classes of H which contain elements of order 3, that is, P has exactly two H-conjugacy classes of subgroups of order 3. So, we call them Q and R.

(vii) C_{H}(Q) ≅ C_{H}(R) ≅ (2^{3} × P) : 2, so that C_{H}(Q)/Q ≅ C_{H}(R)/R ≅ (2^{3} × 3) : 2.

(viii) C_{N}(Q) ≅ C_{N}(R) ≅ (2^{3} × P) : Σ_{3}, so that C_{N}(Q)/Q ≅ C_{N}(R)/R ≅ (2^{3} × 3) : Σ_{3}. (ix) C_{G}(Q) ≅ C_{G}(R) ≅ 6.M_{22}, so that C_{G}(Q)/Q ≅ C_{G}(R)/R ≅ 2.M_{22}.

Proof. Note, first of all, that all elements in G of order 3 are conjugate by [8, p.188]. (i) See [15, lines −3 ∼ −2 in p.493]. (ii) This follows from (i), [15, line −2 in p.493] and calculations by GAP [45].

(iii) The same as in the proof of (ii).

(iv) Follows from calculations by GAP [45], see (2.3).

(v) Easy by (ii), (iii) and direct calculations.

(vi) Easy by (iii), (iv) and direct calculations.

(vii) The same as in the proof of (vi).

(viii) Easy by (ii), (iii) and (v).

(ix) This is obtained by [15, line −3 in p.493], [8, p.188 and p.40], [46, Theorem] and an easy observation. □

4.3.Notation. We use the notation Q and R as in (4.2).

4.4.Lemma. A_{N} ≅ Mat_{6}(k[P : D_{8}]), as k-algebras, and we can write that

Irr(A_{N}) = {χ_{6a}, χ_{6b}, χ_{6c}, χ_{6d}, χ_{12}, χ_{24a}, χ_{24b}, χ_{24c}, χ_{24d}}, IBr(A_{N}) = {6a, 6b, 6c, 6d, 12},

where the numbers mean the degrees (dimensions) of characters (modules).

Proof. It is enough to consider A_{N} as a block algebra over k, so that we assume this. By (4.2)(ii)-(iii), H/C_{G}(P) ≅ D_{8}. Hence results of Külshammer [25, A.Theorem] and Puig [38, Proposition 14.6] (cf. [2, Theorem 13]) imply that A_{N} ≅ Mat_{m}(k^{α}[P : D_{8}]) as k-algebras for a positive integer m and a 2-cocycle α ∈ H^{2}(D_{8}, k^{×}). Then, it follows from [14, Table 1 in p.34] and [18, Proposition 1.4] that α = 1 and m = 6 since we know |Irr(A_{N})| = 9 and the degrees of characters in Irr(A_{N}) from direct calculations by GAP [45]. □

4.5.Notation. We use the notation χ_{6a}, χ_{6b}, χ_{6c}, χ_{6d}, χ_{12}, χ_{24a}, χ_{24b}, χ_{24c}, χ_{24d} and 6a, 6b, 6c, 6d, 12 as in (4.4).

4.6.Lemma.

(i) B ≅ k[P : D_{8}], as interior P -algebras and hence k-algebras, and we can write that

Irr(B) = {χ_{1a}, χ_{1b}, χ_{1c}, χ_{1d}, χ_{2}, χ_{4a}, χ_{4b}, χ_{4c}, χ_{4d}}, IBr(B) = {1a, 1b, 1c, 1d, 2},

where the numbers mean the degrees (dimensions) of characters (modules). In particular, all simple modules 1a, 1b, 1c, 1d, 2 in B are self-dual.

(ii) The 3-decomposition and the Cartan matrices of B, 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 & 1 & 1 & 0 & 2 \\ 1b & 1 & 3 & 0 & 1 & 2 \\ 1c & 1 & 0 & 3 & 1 & 2 \\ 1d & 0 & 1 & 1 & 3 & 2 \\ 2 & 2 & 2 & 2 & 2 & 5 \end{aligned}

(iii) The radical and socle series of projective indecomposable modules (PIM’s) in B are the following:

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

Proof. (i) is easy by (4.4) and (2.3). (ii) and (iii) are also easy. □

4.7.Notation. We use the notation χ_{1a}, χ_{1b}, χ_{1c}, χ_{1d}, χ_{2}, χ_{4a}, χ_{4b}, χ_{4c}, χ_{4d} and 1a, 1b, 1c, 1d, 2 as in (4.6).

4.8.Lemma. The block algebra k[P : D_{8}] and B both have exactly 18 non-isomorphic trivial source modules over k. In fact, they are given in the following list, where the diagrams are radical and socle series and we use the same notation as in (4.7) since B ≅ k[P : D_{8}] as interior P -algebras from (4.6)(i):

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

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

(iii) Four trivial source modules with vertex R:

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

(iv) Four trivial source modules with vertex Q:

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

4.9.Lemma.

(i) The character table of (P : D_{8}) ≅ [(C_{3} × C_{3}) : D_{8}] is given as follows:

centralizer 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

(ii) We use the notation as in (i). There is an element w ∈ SD_{16} − D_{8} such that 2B^{w} = 2C, 2C^{w} = 2B, and hence χ_{1b}^{w} = χ_{1c}, χ_{1c}^{w} = χ_{1b}, so that 1b^{w} = 1c, 1c^{w} = 1b and w fixes all the other simples 1a, 1d and 2. It turns out that there is an automorphism σ of B^{∗} as a k-algebra such that σ(1b) = 1c, σ(1c) = 1b, and σ fixes all the other simples 1a, 1d and 2.

4.10.Notation. We use the notation σ as in (4.9)(ii).

5. Green correspondences for A_{8}

5.1.Notation and assumption. In the rest of this paper, we use the following notation, too. 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 the block algebra A of kG where G = J_{4}, see (3.1) and (3.3). 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, see [8, p.22]. Let H′ = N_{G′}(P), and hence H′ = P : D_{8}. Let A′ and B′, respectively, be the principal block algebras of kG′ and kH′. Thus B′ = kH′. Therefore, by (4.6) and (4.7), we can write

Irr(B′) = Irr(H′) = {χ_{1a} = 1_{H′}, χ_{1b}, χ_{1c}, χ_{1d}, χ_{2}, χ_{4a}, χ_{4b}, χ_{4c}, χ_{4d}},
IBr(B′) = IBr(H′) = {1a = k_{H′}, 1b, 1c, 1d, 2}.

Let f′ : G′ → H′ and g′ : H′ → G′ be the Green correspondences with respect to (G′, P, H′).

5.2.Lemma.

(i) We can write

Irr(A′) = {χ′_{1} = 1_{G′}, χ′_{28}, χ′_{7}, χ′_{35}, χ′_{14}, χ′_{64}, χ′_{20}, χ′_{70}, χ′_{56}}, IBr(A′) = {k = k_{G′}, 28, 7, 35, 13},

where the numbers mean the degrees (dimensions) of characters (modules). (ii) All simple kG′-modules k = k_{G′}, 28, 7, 35, 13 in A′ have P as their vertices.

Proof. (i) and (ii) respectively follow by [12, p.49] and [33, Chap.5, Theorem 1.9]. □

5.3.Notation. We use the notation χ′_{1}, χ′_{28}, ⋯, χ′_{56} and k, 28, 7, 35, 13 as in (5.2).

5.4.Lemma.

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

6. Proofs of the main results

In this section we shall give complete proofs of our main results (1.3), (1.4) and (1.5).

6.1.Notation and assumption. First of all, recall the notation A, G, P, N, e, H, Q and R in (3.1), (3.3), (4.1) and (4.3). Let G_{P} = C_{G}(P) = C_{H}(P) = H_{P}. As remarked in [32, pp.821–822], there is a source idempotent i of A such that i is a primitive idempotent of A^{ΔP} satisfying Br_{ΔP} (i)·e = Br_{ΔP} (i) ≠ 0. Namely, e is the block (idempotent) of kG_{P} determined by i. Similarly, there is a source idempotent j of B such that j is a primitive idempotent of B^{ΔP} satisfying Br_{ΔP} (j)·e = Br_{ΔP} (j) ≠ 0. Namely, e is the block (idempotent) of kH_{P} determined by j. Let 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} are, respectively, blocks of kG_{Q} and kH_{Q} such that e_{Q} and f_{Q} are determined by i and j, respectively. Namely, Br_{ΔQ}(i)·e_{Q} = Br_{ΔQ}(i) and Br_{ΔQ}(j)·f_{Q} = Br_{ΔQ}(j). Let A_{Q} = kC_{G}(Q)·e_{Q} and B_{Q} = kC_{H}(Q)·f_{Q}, so that e_{Q} = 1_{A_{Q}} and f_{Q} = 1_{B_{Q}}. Finally, let f : G → N and g : N → G be the Green correspondences with respect to (G, P, N).

6.2.Lemma. Let 𝔐_{Q} be a unique (up to isomorphism) indecomposable direct summand of A_{Q}↓_{G_{Q}×H_{Q}}^{G_{Q}×G_{Q}}·1_{B_{Q}} with vertex ΔP (note that such an 𝔐_{Q} always exists by (2.4)). Then, a pair (𝔐_{Q}, 𝔐_{Q}^{∨}) induces a Puig equivalence between A_{Q} and B_{Q}.

Proof. Set \overline{G}_{Q} = G_{Q}/Q, \overline{H}_{Q} = H_{Q}/Q and \overline{P} = P/Q. Then, \overline{G}_{Q} ≅ 2.M_{22} and \overline{H}_{Q} ≅ (2^{3} × 3) : C_{2} by (4.2)(ix) and (4.2)(vii). Let π_{1} : kG_{Q} ↠ k\overline{G}_{Q} and π_{2} : kH_{Q} ↠ k\overline{H}_{Q} be canonical surjective k-algebra homomorphisms. Set \overline{A}_{Q} = π_{1}(A_{Q}) and \overline{B}_{Q} = π_{2}(B_{Q}). Then, \overline{A}_{Q} and \overline{B}_{Q} respectively are block algebras of k\overline{G}_{Q} and k\overline{H}_{Q} with a common defect group \overline{P}, see [33, Chap.5, Theorems 8.10, 8.11] and [23, 1.3.Lemma]. From direct calculations by GAP [45], we know that A_{Q} has two 3-blocks 𝔄_{1} and 𝔄_{2} such that A = 𝔄_{1}^{G} = 𝔄_{2}^{G} (block induction), IBr(𝔄_{1}) = {21, 210a} and IBr(𝔄_{2}) = {120, 210b}, where the numbers mean the dimensions of simple kG_{Q}-modules. Thus, IBr(\overline{A}_{Q}) = \{21, 210a\} or \{120, 210b\}. In either case, it follows from (2.6)(ii) that both simple k\overline{G}_{Q}-modules in \overline{A}_{Q} are trivial source modules. By (2.6)(i) and (2.5)(iv), there is an indecomposable (\overline{A}_{Q}, \overline{B}_{Q})-bimodule \overline{𝔐}_{Q} with vertex Δ\overline{P} such that a pair (\overline{𝔐}_{Q}, \overline{𝔐}_{Q}^{∨}) realizes a Puig equivalence between \overline{A}_{Q} and \overline{B}_{Q}, and that

\overline{A}_{Q}↓_{\overline{G}_{Q}×\overline{H}_{Q}}^{\overline{G}_{Q}×\overline{G}_{Q}}·1_{\overline{B}_{Q}} = \overline{𝔐}_{Q} ⊕ (projective (\overline{A}_{Q}, \overline{B}_{Q})-bimodule).

Hence, a result of the first and the second authors [20, Theorem] yields that there is an indecomposable (A_{Q}, B_{Q})-bimodule 𝔐′_{Q} such that a pair (𝔐_{Q}′, 𝔐_{Q}′^{∨}) realizes a Puig equivalence between A_{Q} and B_{Q}, and that

A_{Q}↓_{G_{Q}×H_{Q}}^{G_{Q}×G_{Q}}·1_{B_{Q}} = 𝔐_{Q}′ ⊕ ( ⊕_{μ} X_{μ})

where each X_{μ} is an indecomposable (A_{Q}, B_{Q})-bimodule with vertex ΔQ. Clearly, 𝔐_{Q}′ does not have ΔQ as a vertex since A_{Q} and B_{Q} both have the defect group P. Hence, 𝔐_{Q} ≅ 𝔐_{Q}′ by the theorem of Krull-Schmidt because of the uniqueness of 𝔐_{Q}. □

6.3.Lemma.

(i) The (A, B)-bimodule 1_{A}·kG·1_{B} has a unique (up to isomorphism) indecomposable direct summand _{A}𝔐_{B} with vertext ΔP. Moreover, a functor F : mod-A → mod-B defined by X_{A} ↦ (X ⊗_{A} 𝔐)_{B} induces a splendid stable equivalence of Morita type between A and B. We use the notation F below as well.

(ii) If X is a non-projective trivial source kG-module in A, then F (X) = Y ⊕ (proj) for a non-projective indecomposable kH-module Y in B such that Y is also a trivial source module, and X and Y have a common vertex.

(iii) Suppose that X is an indecomposable kG-module in A such that a vertex of X belongs to Z(G, P, H), see [33, Chap.4, §4], and that F (X) = Y ⊕ (proj) for a non-projective indecomposable kH-module Y in B. Then, Y is the Fong-Reynolds correspondent of fX between B and A_{N}.

(iv) If X is a non-projective kG-module in A, then F (ΩX) = Ω(F (X)) ⊕ (proj).

Proof. (i) It follows from results of Alperin-Broué [1, Proposition 4.21] and Broué-Puig [7, Theorem 1.8(1)] (see [23, 1.15.Lemma]) that

E_{G}((P_{1}, e_{P_{1}}), (P_{2}, e_{P_{2}})) = E_{H}((P_{1}, f_{P_{1}}), (P_{2}, f_{P_{2}}))

for any subgroups P_{1} and P_{2} of P. On the other hand, we get by (2.4) that 1_{A}·kG·1_{B}, as a right k[G×H]-module, has a unique (up to isomorphism) indecomposable direct summand 𝔐 with vertex ΔP. We can write 𝔐 = kG𝔣 for a primitive idempotent 𝔣 of (kG·1_{A})^{ΔP} = A^{ΔP}, and we get Br_{ΔP} (𝔣)·e = e, see [24, Proof of Theorem]. Recall e = 1_{B}. Then, it follows from a result of the first author and Linckelmann [24, Theorem] that e_{Q}·𝔐(ΔQ)·f_{Q} is a unique (up to isomorphism) indecomposable direct summand of e_{Q}·kC_{G}(Q)·f_{Q} with vertex ΔP. Set 𝔐_{Q} = e_{Q}·𝔐(ΔQ)·f_{Q}. Hence, by (6.2), it holds that a pair (𝔐_{Q}, 𝔐_{Q}^{∨}) induces a Puig equivalence between kC_{G}(Q)·e_{Q} and kC_{H}(Q)·f_{Q}. Note that all subgroups of G of order 3 are conjugate in G from (4.2)(v). Thus, we get the assertion by [32, Theorem 3.1].

(ii) Easy by (i) and (2.7).

(iii) This follows from (2.3) and (i).

(iv) Easy since F is an exact functor. □

6.4.Notation. We use the notation 𝔐 and F as in (6.3).

6.5.Lemma.

(i) The block algebra B of kH itself is its source algebra with respect to P. Furthermore, B ≅ k[P : D_{8}] as k-algebras and interior P -algebras.

(ii) By (i), there is an isomorphism τ : B →^{≈} k[P : D_{8}] = kH′ = B′ of interior P - algebras. Then, B can be considered as a (B, B′)-bimodule via b·b_{1}·b′ = bb_{1}·τ^{−1}(b′) for b, b_{1} ∈ B and b′ ∈ B′. Moreover, a pair (_{B}B_{B′}, _{B′}B_{B}) induces a Puig equivalence between B and B′. Furthermore, we can assume that τ(1a) = k_{H′} = 1a, τ(1b) = 1b, τ(1c) = 1c, τ(1d) = 1d and τ(2) = 2 (we are using the same notation of simples in B and B′).

(iii) Set 𝔐̃ = _{A}(𝔐 ⊗_{B} B ⊗_{B′} 𝔐′^{∨})_{A′}, where 𝔐′ is a unique (up to isomorphism) indecomposable direct summand of 1_{A′}·kG′·1_{B′} with vertex ΔP, see (2.4). Let F̃ : mod-A → mod-A′ be a functor defined by F̃(X) = X ⊗_{A} 𝔐̃ for an A-module X. Then, F̃ induces a splendid stable equivalence of Morita type between A and A′.

Proof. (i) Easy, see [38, Proposition 14.6] and [2, Theorem 13(iv)].

(ii) The first part follows by (5.1) and a theorem of Puig (and, independently, of Scott), see [32, Theorem 4.1] and [39, Remark 7.5]. The latter part is obtained since the four simples 1a, 1b, 1c, 1d in B are symmetric by the Ext-quiver with relations of B given in [35, Case 2 in §4, p.9].

(iii) This follows from (ii) and (6.3)(i) □

6.6.Notation. We use the notation F̃, 𝔐̃ adn τ as in (6.5).

6.7.Lemma. fS_{1}, fS_{2} ∈ {6a, 6b, 6c, 6d}.

Proof. Set T_{1} = fS_{1}. By (3.13)(i), (2.1) and (4.4), we get T_{1} ∈ {6a, 6b, 6c, 6d, 12}. Then, dim_{k}(T_{1}↑^{G}) ≡ 4 × dim_{k}T_{1} (mod 9) since |G : N| ≡ 4 (mod 9) by (3.1) and (4.2)(ii). Clearly, T_{1}↑^{G} = S_{1} ⊕ X for a kG-module X such that X is C_{3}-projective. Thus, 9|dim_{k}X by (3.1) and [33, Chap.4, Theorem 7.5]. Then, since dim_{k}S_{1} ≡ 6 (mod 9) by (3.4)(i), we know dim_{k}(T_{1}↑^{G}) ≡ 6 (mod 9). This implies dim_{k}T_{1} = 6. Similar for fS_{2} since S_{2} is also a trivial source module by (3.13)(ii) and since dim_{k}S_{2} ≡ 6 (mod 9) by (3.4)(i). □

6.8.Lemma. We can assume that F (S_{1}) = 1a, (see (2.8)).

Proof. Set 𝔑 = _{A}𝔐 ⊗_{B} B_{B′}, see (6.5)(ii) for B_{B′}. Then, 𝔑 induces a splendid stable equivalence of Morita type between A and B′ from (6.3)(i) and (6.5)(ii). Set T′ = S_{1} ⊗_{A} 𝔑. Then, it follows from (6.7), (4.4), (4.6)(i) and (6.5)(ii) that T′ ∈ {1a = k_{H′}, 1b, 1c, 1d}.

Suppose that T′ = 1b, for instance. Clearly, 1b ⊗ 1b ≅ k_{H′} = 1a by (5.1). Hence, B′(B′ ⊗ 1b)_{B′} realizes a self-Puig equivalence of B′. Moreover, S_{1} ⊗_{A} 𝔑 ⊗_{B′} (B′ ⊗ 1b)_{B′} = 1b ⊗_{B′} (B′ ⊗ 1b)_{B′} = 1b ⊗ 1b = k. Thus, we get the assertion by (6.5)(ii) if we replace 𝔑 by 𝔑 ⊗_{B′} (B′ ⊗ 1b)_{B′}. □

6.9.Lemma. F (S_{2}) ≠ 1d and F (S_{2}) ≠ 2.

Proof. Suppose, first of all, that F (S_{2}) = 1d. (3.16)(iii) shows that there is a trivial source kG-module X in A such that

X = \begin{matrix}S_{1} S_{2} \\ S_{4} \\ S_{1} S_{2} \end{matrix}.

Then, (6.3)(ii) implies that F (X) = Y ⊕(proj) for a non-projective trivial source kH-module Y in B. Thus, it follows from (2.9), (6.3)(i) and (6.8) that

Hom_{B}(Y, 1a) ≅ \underline{Hom}_{B}(Y, 1a) ≅ \underline{Hom}_{B}(F (X), F (S_{1}))
≅ \underline{Hom}_{A}(X, S_{1}) ≅ Hom_{A}(X, S_{1}) ≅ k,

as k-spaces. Similarly, by the assumption that F (S_{2}) = 1d, we get also Hom_{B}(Y, 1d) ≅ k. Hence, (1a ⊕ 1d)|[Y/rad(Y)], contradicting (4.8).

By a similar way, we know also that F (S_{2}) ≠ 2 since there is not any trivial source kH-module W in B with (1a ⊕ 2)|[W/rad(W)] by (4.8). □

6.10.Lemma. It holds that F (S_{2}) = 1x, F (S_{3}) = 1d, F (S_{4}) = 2 and F (S_{5}) = \begin{matrix}1y \\ 2 \\ 1y \end{matrix} , where

{x, y} = {b, c}, and hence S_{3} and S_{4} are trivial source modules.

Proof. By (3.13)(ii) and (3.4)(ii), S_{2} is a trivial source kG-module in A with vertex P. Hence, it follows from (6.3)(i)-(ii) and (2.11) that F (S_{2}) is a trivial source kH-module in B with vertex P. Then, (4.8)(ii) yields F (S_{2}) ∈ {1a, 1b, 1c, 1d, 2}. Thus, we get by (6.8) and (6.9) that F (S_{2}) = 1x for some x ∈ {b, c}.

By (3.16)(iii), there is a trivial source kG-module X in A with

X = \begin{matrix}S_{1} S_{2} \\ S_{4} \\ S_{1} S_{2} \end{matrix}.

Then, just as in the proof of (6.9), F (X) = Y ⊕ (proj) for a non-projective trivial source kH-module Y in B such that (1a ⊕ 1x)|[Y/rad(Y)]. Hence, (4.8) yields

Y = \begin{matrix}1a 1x \\ 2 \\ 1a 1x \end{matrix}.

Next, we want to claim F (S_{4}) = 2. It follows from (2.10) and (6.8) that

F ( \begin{matrix}S_{1} S_{2} \\ S_{4} \\ S_{2} \end{matrix} ) = F (X/S_{1}) = Y/1a ⨁ (proj) ≅ \begin{matrix}1a 1x \\ 2 \\ 1x \end{matrix}.

Hence, by the dual of (2.10), it holds that

F ( \begin{matrix}S_{2} \\ S_{4} \\ S_{2} \end{matrix} ) = F (Ker[ \begin{matrix}S_{1} S_{2} \\ S_{4} \\ S_{2} \end{matrix} ↠ S_{1}]) ≅ Ker( \begin{matrix}1a 1x \\ 2 \\ 1x \end{matrix} ↠ 1a) ⨁ (proj) = \begin{matrix}1x \\ 2 \\ 1x \end{matrix} ⨁ (proj).

Since F (S_{2}) = 1x, we get from (2.10) that F (S_{4}) = 2 ⊕ (proj), so that F (S_{4}) = 2 by (6.3)(i) and (2.11).

By (3.16)(iv),

ΩS_{2} = \begin{matrix}S_{4} \\ S_{1} S_{2} S_{3} \\ S_{4} \\ S_{2}. \end{matrix}

Hence, (6.3)(iv) yields that F (ΩS_{2}) = Ω(F S_{2}) ⊕ (proj) = Ω(1x) ⊕ (proj). Thus,

F ( \begin{matrix}S_{4} \\ S_{1} S_{2} S_{3} \\ S_{4} \\ S_{2} \end{matrix} ) = \begin{matrix}2 \\ 1a 1x 1d \\ 2 \\ 1x \end{matrix} ⨁ (proj)

by (4.6)(iii). Then, since F (S_{4}) = 2 and F (S_{2}) = 1x, we obtain by (2.10) that

F (S_{1} ⊕ S_{2} ⊕ S_{3}) = 1a ⊕ 1x ⊕ 1d ⊕ (proj)

just as the above. Thus, since F (S_{2}) = 1x, it holds that F (S_{3}) = 1d⊕(proj) by (6.8). Then, again by (2.11), we get F (S_{3}) = 1d.

On the other hand, (3.13)(iii) shows that there is a trivial source kG-module U in A with U = U(S_{3}, S_{5}, S_{3}). Thus, since F (S_{3}) = 1d, it holds from (6.3)(ii) that F (U) = V ⊕(proj) for a non-projective trivial source kH-module V in B such that 1d|[V/rad(V)]. Clearly, V ≠ 1d

since F (S_{3}) = 1d. Furthermore, just by the same method as above, we have 1x |[V/rad(V)]. Hence, it follows from (4.8) that

V = \begin{matrix}1y 1d \\ 2 \\ 1y 1d \end{matrix}

for some y ∈ {b, c} with y ≠ x. Then, since F (S_{3}) = 1d, we get F (S_{5}) = U(1y, 2, 1y) as before. □

6.11.Notation. We use the notation x, y as in (6.10).

6.12.Lemma.

(i) If (x, y) = (b, c), then A and A′ are Puig equivalent.

(ii) If (x, y) = (c, b), then A and A′ are Morita equivalent though we do not know whether they are Puig equivalent.

Proof. First, define a k-algebra automorphism σ̃ of B by σ̃ = id_{B} if (x, y) = (b, c); and σ̃ = σ if (x, y) = (c, b), see (4.9)(ii). Then, we get the following diagram by (6.8), (6.10), (4.9)(ii), (6.5)(ii) and (5.4);

mod-A ⟶^{F} mod-B \underrightarrow{σ̃} mod-B ⟶^{g′∘τ} mod-A′ S_{1}  ↦  1a  ↦  1a  ↦  k_{G′} S_{2}  ↦  1x  ↦  1b  ↦  28 S_{3}  ↦  1d  ↦  1d  ↦  7 S_{4}  ↦  2  ↦  2  ↦  35 S_{5}  ↦  \begin{matrix}1y \\ 2 \\ 1y \end{matrix}  ↦  \begin{matrix}1c \\ 2 \\ 1c \end{matrix}  ↦  13 .

(i) A result of Linckelmann [29, Theorem 2.1(iii)] and (6.5)(iii) yield that A and A′ are Morita equivalent via a pair (𝔐̃, 𝔐̃^{∨}). Hence, we know that the Morita equivalence is a Puig equivalence by a result of Puig (and, independently, of Scott) [39, Remark 7.5], see [32, Theorem 4.1].

(ii) It follows from (6.3)(i) and a result of Linckelmann [29, Theorem 2.1(iii)] that A and A′ are Morita equivalent. Note that we do not know whether they are Puig equivalent, since A and A′ are twisted by the automorphism σ of B which does not take trivial source modules to trivial source modules in general, see (4.9)(ii). □

6.13.Proofs of (1.3) and (1.4). We get (1.4) by (6.12). Note that a Puig equivalence lifts from k to O by a result of Puig [37, 7.8.Lemma] (see [47, (38.8)Proposition]), and so does a splendid Rickard equivalence by a result of Rickard [41, Theorem 5.2] (see [11, p.75, lines −17∼−16]). Hence, it is enough to consider blocks A, B and A′ only over k. We know from (6.12) and [36, Corollary 2] that A and B are splendidly Rickard equivalent by taking a nice subset I_{0} = {1y} of I = {1a, 1b, 1c, 1d, 2} just as Okuyama did in [35, Example 4.3]. □

6.14.Remark and Question. We can know that the generalized decomposition matrices of A and A′ are precisely the same by using GAP [45] though we can not prove that they are Puig equivalent, see (6.12)(ii). Is it true that the block algebras A and A′ are Puig equivalent?

6.15.Proof of (1.5). This follows from (3.2)(i) and (1.3). □

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