Broué’s abelian defect group conjecture for the Held group and the sporadic Suzuki group

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

a Department of Mathematics, Faculty of Science, Chiba University, Chiba, 263-8522, Japan

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

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 p-block B of N_{G}(P) are derived equivalent. We demonstrate in this paper that Broué’s conjecture holds for non-principal 3-blocks A with elementary abelian defect group P of order 9 of the simple Held group and the sporadic simple Suzuki group.

0. Introduction and notation

In representation theory of finite groups, one of the most important problems is to give an affirmative answer, if it is true, to a conjecture introduced by M. Broué [6]. He actually conjectures the following:

(0.1) Conjecture (Broué’s abelain defect group conjecture) (Broué [6, 6.2.Question], see [12, Conjecture in p.132]). For a prime p, let A be a p-block of a finite group G with defect group P, and let B be a p-block of N_{G}(P) such that B is the Brauer correspondent of A. Then, A and B would be derived equivalent provided P is abelian.

There are several cases where Broué’s abelian defect group conjecture (0.1) is checked. For instance, we prove that (0.1) holds for the principal 3-block A of an arbitrary finite group G when the defect group P of A is elementary abelian of order 9, see [14, (0.2)Theorem]. Then, the next question may be what about the case of non-principal 3-blocks with the same defect group which is elementary abelian of order 9. Actually, in the previous paper [16] we answer to this question in special cases. Namely, we prove there that Broué’s abelian defect group conjecture (0.1) holds for non-principal 3-blocks A with elementary abelian defect group P of order 9 when G is the simple O’Nan group O′N and the simple Higman-Sims group HS. In this paper we continue such a project, namely, we want to check Broué’s abelian defect group conjecture for, at least, sporadic simple groups. That is, we demonstrate here that Broué’s abelian defect group conjecture (0.1) holds for non-principal 3-blocks A with elementary abelian defect group P of order 9 when G is the simple Held group He and the sporadic simple Suzuki group Suz. Namely, our main result is the following.

∗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).

Theorem. Let (O, K, k) be a splitting 3-modular system for all subgroups of a finite group G, see the definition below (0.2).

(i) Assume that G = He, the simple Held group. If A is a non-principal block algebra of OG with elementary abelian defect group P of order 9, and if B is its Brauer corresponding block algebra of O[N_{G}(P)], then A and B are Puig (splendidly Morita) equivalent.

(ii) Assume that G = Suz, the sporadic simple Suzuki group. If A is a non-principal block algebra of OG with elementary abelian defect group P of order 9, and if B is its Brauer corresponding block algebra of O[N_{G}(P)], then A and B are splendidly Rickard equivalent. Moreover, A is Puig equivalent to all principal block algebras of O[PSL_{3}(q)], where q is a power of a prime satisfying 3|(q − 1) and 3^{2} |̸(q − 1).

(0.2) Notation. Throughout this paper we use the following notation and terminology. Let A be a ring. Then, 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 of A, respectively. We denote also by rad(A) the Jacobson radical of A. We write mod-A for the category of finitely generated right A-modules. We write M_{A} when M is a right A-module, and _{A}M when M is a left A-module. Moreover, we write _{A}M_{B} if M is an (A, B)-bimodule, where B is another ring. For A-modules M and M′ we write M′|M when M′ is a direct summand of M as an A-module. We denote by soc(M) the socle of an A-module M if it exists.

We denote by G a finite group. We fix a prime number p, and let (O, K, k) be a splitting pmodular system for all subgroups of G, namely, O is a complete discrete valuation ring of rank one with quotient field K of characteristic zero and with residue field k of characteristic p such that K and k are both splitting fields for all subgroups of G. All modules here are finitely generated right modules unless stated otherwise. We mean by an OG-lattice a finitely generated right OG-module which are a free O-module. We sometimes call it just an OG-module. When M is an OG-module, we write M^{∗} for M/M·rad(O), so that M^{∗} is considered as a right kG-module. For an OG-module M and a kG-module X, we write M^{∨} and X^{∨} for the O-dual and k-dual of M and X, respectively, that is, M^{∨} = Hom_{O}(M, O) and X^{∨} = Hom_{k}(X, k), so that M^{∨} and X^{∨} are a right OG-module and a right kG-module again, respectively, via (m)(φg) = (mg^{−1})φ for φ ∈ M^{∨}, g ∈ G and m ∈ M, and similar for X^{∨}. For a subgroup H of G, and for an OG-module M and an OH-module N, let M↓_{H}^{G} = M↓_{H} be a restriction of M to H, and let N↑_{H}^{G} = N↑^{G} be an induction (induced module) of N to G, namely, M↓_{H} = (M ⊗_{OG} OG)_{OH} and N↑^{G} = (N ⊗_{OH} OG)_{OG}. We denote by Irr(G) and IBr(G) the sets of all irreducible ordinary and Brauer characters of G, respectively. Sometimes, we mean by IBr(G) the set of all non-isomorphic simple (irreducible) kG-modules. Now, we assume moreover that A is a block algebra (p-block) of OG. Then, Irr(A) and IBr(A) mean the sets of all characters in Irr(G) and IBr(G) which belong to A, respectively. Therefore, we mean by IBr(A) also the set of all non-isomorphic simple kG-modules which belong to A. Clearly, A^{∗} is a block algebra of kG. For a subset S of G, we write Ŝ for zero if S is an empty set and for ∑_{s∈S} s ∈ OG or ∈ kG if S is a non-empty set. We write O_{G} and k_{G}, respectively, for the trivial OG- and kG-modules. For ordinary characters χ and ψ of G, let (χ, ψ)^{G} be the inner product of χ and ψ in usual sense. For an ordinary character χ of G, we denote by \overline{χ} the complex conjugate of χ. Let X and Y be kG-modules. Then, we write [X, Y]^{G} for dim_{k}[Hom_{kG}(X, Y)] and \underline{Hom}_{kG}(X, Y) for a factor space Hom_{kG}(X, Y)/PHom_{kG}(X, Y) where PHom_{kG}(X, Y) is a set of all projective kG-homomorphisms from X to Y, and we denote by j(X) the Loewy (radical) length of X, which is equal to the socle length. We denote by P (X) the projective cover of X. We write U = U(S_{1}, ⋯, S_{n}) for a uniserial kG-module such that U·[rad(kG)]^{i−1}/U·[rad(kG)]^{i} ≅ S_{i} for i = 1, ⋯, n, where S_{1}, ⋯, S_{n} are (possibly isomorphic) simple kG-modules. Let H be another finite group (not necessarily a subgroup of G). If M is an (OG, OH)-bimodule, then we can consider M as a right O[G × H]- module via m(g, h) = g^{−1}mh for m ∈ M, g ∈ G and h ∈ H.

We denote by ΔG the diagonal copy of G in G × G, namely, ΔG = {(g, g) ∈ G × G | g ∈ G}, so that ΔG ≅ G. For a positive integer n, Σ_{n} and A_{n} respectively denote the symmetric group and the alternating group on n letters, and C_{n}, SD_{n} Q_{n} and D_{n} mean the cyclic group, the semi-dihedral group, the quaternion group and the dihedral group of order n, respectively.

Let R be a common subgroup of G and H. For an O[G × H]-module X, X^{ΔR} is defined by

X^{ΔR} = {x ∈ X|x(r, r) = x, ∀r ∈ R} = {x ∈ X|r^{−1}xr = x, ∀r ∈ R}.

We need the Brauer construction (functor) and the Brauer homomorphism for our aim. Let R be a common p-subgroup of finite groups G and H. Then, the Brauer construction (functor)

F_{ΔR}^{G×H}(−) = −(ΔR) : mod-O[G × H] ⟶ mod-k[N_{G×H}(ΔR)]

with respect to (G × H, ΔR) is defined by, for each O[G × H]-lattice X,

X(ΔR) = F_{ΔR}^{G×H}(X) = [X^{ΔR}/( ∑_{R′≨R} Tr ↑_{ΔR′}^{ΔR} (X^{ΔR′}))] ⨂_{O} k,

where Tr is a trace map, see [30, p.341], [31, 4.1.1 (p.117)], [23, pp.835–836] and [33, p.91 and p.219]. If X = OG, then (OG)(ΔR) = F_{ΔR}^{G×G}(OG) ≅ kC_{G}(R). Since C_{G×H}(ΔR) = C_{G}(R) × C_{H}(R) ⊆ N_{G×H}(ΔR), we can regard X(ΔR) as a (kC_{G}(R), kC_{H}(R))-bimodule.

On the other hand, for a p-subgroup R of G, we can define the Brauer homomorphism

Br_{ΔR}^{G} = Br_{ΔR} : (OG)^{ΔR} → kC_{G}(R) ≅ F_{ΔR}^{G×G}(OG) = (OG)(ΔR)

with respect to (G, R) by, for each R-conjugacy class C of G,

Br_{ΔR}^{G}(Ĉ) = { \atop C ∩ C_{G}(R)} ∈ kC_{G}(R),

see [33, p.91 adn p.219]. The Brauer homomorphism

Br_{ΔR} : (OG)^{ΔR} ↠ kC_{G}(R)

is a surjective O-algebra-homomorphism, see [33, (37.5)Proposition]. Remark that, in [33], Br_{ΔR} and (OG)^{ΔR} are denoted by br_{R} and (OG)^{R}, respectively. For other notation and terminology see the books of Nagao-Tsushima [24] and Thévenaz [33].

1. Preliminaries

(1.1) Lemma. Let X be a kG-module such that the multiplicity of each composition factor is one.

(i) X is indecomposable if and only if dim_{k}[End_{kG}(X)] = 1.

(ii) X is completely reducible (semisimple) if and only if dim_{k}[End_{kG}(X)] = ℓ(X), where ℓ(X) is the composition length of X.

Proof. (i) Assume that X is indecomposable. Let 0 ≠ f ∈ End_{kG}(X). Then Imf is a nonzero submodule of X, and therefore X = Kerf ⊕ Imf since Kerf and Imf have no common composition factors by the assumption. Therefore Kerf = 0, so that f is an isomorphism. This implies End_{kG}(X) ≅ k since End_{kG}(X) is a local k-algebra. The other half is trivial.

(ii) Let X = X_{1} ⊕ X_{2} ⊕ ⋯ ⊕ X_{r} be a decomposition of X into indecomposable kG-modules. Then, by the assumption, we have

End_{kG}(X) = End_{kG}(X_{1}) ⊕ End_{kG}(X_{2}) ⊕ ⋯ ⊕ End_{kG}(X_{r}).

Therefore, the assertion follows from (i). ■

(1.2) Lemma. Let A be a p-block of G with defect group P, and let H be a subgroup of G such that N_{G}(P) ⊆ N. Let A_{N} be the Brauer corresponding block of A in N, namely A = A_{N}^{G} (block induction) and A_{N} has P as its defect group. Moreover, let f : G → N be the Green correspondence with respect to (G, P, N), see [24, Chapter 4, p.276]. If M is an indecomposable OG-lattice in A with vertex P, then its Green correspondent fM belongs to A_{N}.

Proof. Let M̃ = fM. Then, M̃ is in a p-block Ã of N and M̃ has P as its vertex since f preserves vertices. It follows from a result of Nagao and Green [24, Chapter 5, Theorem 3.12] that Ã^{G} = A since C_{G}(P) ⊆ N. Let P̃ be a defect group of Ã. Thus, P̃ ⊆_{G} P by [24, Chapter 5, Lemma 3.3]. On the other hand, M̃ is relatively P̃-projective by [24, Chapter 5, Theorem 1.9(i)], which means that P ⊆_{N} P̃. Therefore, P is a defect group of Ã, and hence Ã = A_{N} by the extended Brauer’s first main theorem [24, Chapter 5, Theorem 3.8]. ■

(1.3) Lemma. Let A be a p-block of G with defect group P, let H be a subgroup of G containing P C_{G}(P), and let Z be a p-subgroup of P with Z ⊆ Z(G). Set \overline{G} = G/Z, \overline{H} = H/Z and \overline{P} = P/Z since Z ⊆ P. Let B a p-block of H with defect group P. Moreover, let π : OG ↠ O\overline{G} be an epimorphism of O-algebras induced by the canonical group-epimorphism G ↠ \overline{G}. We consider A and B respectively as block algebras of OG and OH. Set \overline{A} = π(A) and \overline{B} = π(B).

(i) It holds that \overline{A} and \overline{B} respectively are block algebras of O\overline{G} and O\overline{H}, both of which have \overline{P} as their defect groups.

(ii) If B^{G} = A, then \overline{B}\overline{G} = \overline{A}.

Proof. (i) This follows, for instance, [24, Chapter 5, Theorems 8.10 and 8.11].

(ii) Let Cl_{p′}(G) be the set of all p′-classses of G, and we use the notation s_{H}, s_{H}^{∗}, ω_{A} and ω_{A}^{∗} as in [24, Chapter 5, §3, p.319]. For a subset C of G, let \overline{C} = π(C). We, first of all, note that,

(1)  C_{1} ∩ C_{2} = ∅ ⇔ \overline{C}_{1} ∩ \overline{C}_{2}  for C_{1}, C_{2} ∈ Cl_{p′}(G),

from a uniqueness of decompositions of elements in G into p-parts and p′-parts. We next want to show that

(2)  s\overline{H} ∘ π(Ĉ) = π ∘ s_{H}(Ĉ) for C ∈ Cl_{p′}(G).

Fix any C as above. We can write C ∩ H = D_{1} ∪ ⋯ ∪ D_{m} (disjoint union) for some D_{1}, ⋯, D_{m} ∈ Cl_{p′}(H). Then,

π ∘ s_{H}(Ĉ) = π({ \atop C ∪ H})

= π(∑_{i=1}^{m} D̂_{i})

= ∑_{i=1}^{m} π(D̂_{i})

= ∑_{i=1}^{m} ˆ \overline{D}_{i} by [24, Chapter 5, Lemma 8.9(iii)].

On the other hand,

s_{H} ∘ π(Ĉ) = s\overline{H}(ˆ\overline{C}) by [24, Chapter 5, Lemma 8.9(iii)]

= { \atop \overline{C} ∩ \overline{H}}

= ; \overline{C ∩ H} since \overline{C} ∩ \overline{H} = \overline{C ∩ H}

= ; \overline{∪_{i=1}^{m}D_{i}} where the union is disjoint

= { \atop ∪_{i=1}^{m}\overline{D_{i}}} by (1), where the union is disjoint

= ∑_{i=1}^{m} ˆ \overline{D_{i}}.

Thus, we get (2). Now, we have

ω\overline{B}^{∗} ∘ s\overline{H}^{∗}(1\overline{A}^{∗}) = ω\overline{B}^{∗} ∘ s\overline{H}^{∗} ∘ π^{∗}(1_{A}^{∗})
= ω\overline{B}^{∗} ∘ π^{∗} ∘ s_{H}^{∗}(1_{A}^{∗})
by (2) and Osima’s theorem [24, Chapter 3, Theorem 6.22(ii)]
= ω_{B}^{∗} ∘ s_{H}^{∗}(1_{A}^{∗}) by [24, Chapter 5, Lemma 8.5]
= ω_{A}^{∗}(1_{A}^{∗})
since B = A^{G}, see [24, Chapter 5, §3, p.320]
= 1_{k}.

Now, since \overline{H} ⊇ C\overline{G}(\overline{P}), it follows from [24, Chapter 5, Theorem 3.6] that \overline{B}\overline{G} is defined. Set

\overline{A′} = \overline{B}\overline{G}. Hence, as before, ω\overline{B}^{∗} ∘ s\overline{H}^{∗} = ω\overline{A′}^{∗}, so that ω\overline{A′}^{∗}(1\overline{A}^{∗}) = 1 from the above, which implies

that \overline{A} = \overline{A′}, see [24, Chapter 3, §6, p.244]. Therefore, \overline{A} = \overline{B}\overline{G}. ■

The next lemma, due to Okuyama, is very useful for provinig Broué’s conjecture, since all simples in a p-block with abelian defect group have vertices which are the abelian defect group, thanks to Knörr [11].

(1.4) Lemma (Okuyama [25, 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 a simple kN_{G}(P)-module.

(1.5) Theorem (Fong-Reynolds). Let H ◁ 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 of B in G. Then, there is a block algebra Ã of T such that Ã covers B, 1_{A}1Ã = 1Ã1_{A} = 1Ã, Ã^{G} = A 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 equivalent and induces a bijection

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

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

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

between IBr(Ã) and IBr(A).

Proof. Let N = 1Ã·OG·1_{A}. By [24, Chapter 5, Theorem 5.10], we know that N induces a bijection Irr(Ã) → Irr(A) via χ̃ ↦ χ̃ ↑^{G}. Hence, Broué’s theorem [5, 0.2.Théorèm] implies that N realizes a Morita equivalence between Ã and A. Since Ã and A are symmetric O-algebras, it follows from [3, 31.Exercise 7, p.345 and 22.1.Theorem(3)] that a pair (N, N^{∨}) induces the Morita equivalence between Ã and A. Therefore, N^{∨} = 1_{A}·OG·1Ã induces a bijection Irr(A) → Irr(Ã), namely, it is given by χ ↦ χ ↓_{H} ·1Ã. Similarly for irreducible Brauer characters. ■

(1.6) Lemma. Let G = A_{7}, and let A be a 3-block of G with defect group P of order 3 such that Irr(A) = {χ_{6}, χ_{15}, χ_{21}} where χ_{i}(1) = i for each i. Let N = N_{G}(P), and let A_{N} be the Brauer corresponding block of A in N. Moreover, let H = N_{G}(P, e), where e is a block idempotent of a block of OC_{G}(P) such that (P, e) is a maximal A-Brauer pair, and let B be the Fong-Reynolds corresponding block of A_{N} in H.

(i) N = (2^{2} × P).Σ_{3} ≅ (A_{4} × 3) : 2 and IBr(A) = {6, 15} and the simple kG-modules 6 and 15 in A are both trivial source modules.

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

(iii) We have H = (2^{2} × P) : 2. Let 𝔐 be an (A, B)-bimodule defined by 𝔐 = M↓{G×N \atop G×H}·1_{B} = M·1_{B} where M is as in (ii). Then, 1_{A}·OG·1_{B} = 𝔐 ⊕ (projective (A, B)-bimodule), 𝔐 is an indecomposable (A, B)-bimodule with vertex ΔP, and 𝔐 induces a Morita (and hence Puig) equivalence between A and B.

Proof. (i) Irr(A_{N}) = {χ̃_{3a}, χ̃_{3b}, χ̃_{6}} where the suffices mean the degrees, and IBr(A_{N}) = {3a, 3b}. Since P ◁ N, we know by [20, III Lemma 10.3] that the simple kN-modules 3a and 3b are both trivial source modules. Now, by a calculation of [32], we get 3b ↑^{G} ·1_{A} = 15, which means that the simple 15 in A is a trivial source kG-module. On the other hand, by [7, p.10], k_{A_{6}} ↑^{G}= k_{G} ⊕ 6, so that the simple 6 in A is also a trivial source kG-module.

(ii) Let M̃ = 1_{A}·OG·1_{A_{N}}. Then, a pair (M̃, M̃^{∨}) induces a stable equivalence between A and A_{N} since P is a T.I. set in G. On the other hand, we know from (i), (1.4) and (1.2) that f(6) and f(15) are both simple kN-modules in A_{N}, where f : G → N is the Green correspondence with respect to (G, P, N). Therefore, it follows from a theorem of Linckelmann [21, Theorem 2.1(iii)] that M̃ has a unique indecomposable non-projective direct summand M such that a pair (M, M^{∨}) realizes a Morita equivalence between A and A_{N}.

(iii) This is easy from (ii) and (1.5). ■

(1.7) Lemma. Let A be a block algebra of OG with defect group P, let N = N_{G}(P), and let A_{N} be a block algebra of OA_{N} which is the Brauer correspondent of A. Moreover, let (P, e) be a maximal A-Brauer pair, H = N_{G}(P, e), and let B be a block algebra of OH which is the Fong-Reynolds correspondent of A_{N} in (1.5). Then, A↓_{G×H}^{G×G}·1_{B}, as a right O[G×H]-module, has a unique indecomposable direct summand with vertex ΔP.

Proof. This is obtained by a combination of the Green correspondnce f : G → N with respect to (G, P, N), (1.2) and the Fong-Reynolds correspondence N → H in (1.5). (see [1, Theorem 5(i)]). ■

(1.8) Lemma (Scott) ([20, II Theorem 12.4 and I Proposition 14.8]).

(i) If M is a trivial source kG-module, then M uniquely lifts to a trivial source OG-lattice M̂. (ii) If M and N are both trivial source kG-modules, then [M, N]^{G} = (χ_{M}, χ_{N})_{G}.

(1.9) Lemma. Let X be a kG-module and S a simple kG-module such that the multiplicity of S in the composition factors of X is one. If [X, S]^{G} ≠ 0 and [S, X]^{G} ≠ 0, then S|X.

Proof. Easy. ■

(1.10) Lemma (Brauer-Conlon-Green-Nagao) (see [24, Chapter 5 Theorems 3.8 and 3.10, Corollary 3.11 and Theorem 3.12]). Let R ∈ {k, O}. Let P be a p-subgroup of G, and let H = N_{G}(P). Assume that A and B respectively are block algebras of RG and RH with defect group P such that A and B correspond each other via the Brauer correspondence, namely, A = B^{G} (block induction). Let f : G → H and g : H → G be the Green correspondences with respect to (G, P, H). Let ℨ = {Q | Q ⩽ P, Q ⩽̸ P^{x} ∩ P for all x ∈ G − H}. Then we get the following.

(i) If X is an indecomposable RG-module in A such that a vertex of X is in ℨ, then its Green correspondent fX of X belongs to B.

(ii) If Y is an indecomposable RH-module in B such that a vertex of Y is in ℨ, then its Green correspondent gY of Y belongs to A.

(1.11) Lemma ([19, Proposition 2.2]). Let A be a finite dimensional symmetric k-algebra. Let S be a simple A-module, U an A-module, and R a projective A-module. Suppose that there is an injective A-homomorphism φ : S ↣ U ⊕ R such that π_{U} ∘ φ ≠ 0, where π_{U} : U ⊕ R ↠ U is the canonical projection. Then, there is an A-submodule V of U ⊕ R such that U ⊕ R = V ⊕ R, V ≅ U and Imφ ⊆ V.

Proof. Let π_{R} : U ⊕ R ↠ R be the canonical projection, and let φ_{U} = π_{U} ∘ φ : S → U and φ_{R} = π_{R} ∘ φ : S → R. Since φ_{U} ≠ 0 and since S is simple, φ_{U} is a monomorphism. Thus, since R is injective, there is an A-homomorphism ψ : U → R such that a diagram

(*)  \begin{aligned}0 & ⟶ & S & ⟶^{φ_{U}} & U \\ & & ↓^{φ_{R}} & ↓^{ψ} \\ & & R & R \end{aligned}

is commutative. Let V = {(u, ψ(u)) ∈ U ⊕ R | u ∈ U}. Clearly, V is a submodule of U ⊕ R such that V ≅ U and V ∩ R = 0. Hence, there is a direct sum V ⊕ R in U ⊕ R, so that V ⊕ R = U ⊕ R. Moreover,

Imφ = φ(S) = {φ(s) | s ∈ S}
= {(φ_{U}(s), φ_{R}(s)) ∈ U ⊕ R | s ∈ S}
= {(φ_{U}(s), ψ(φ_{U}(s))) ∈ U ⊕ R | s ∈ S} by (*)
⊆ V.

We are done. ■

(1.12) Lemma ([13, Theorem(19)]). Let A^{∗} be a block algebra of kG with defect group P, H = N_{G}(P), and let B^{∗} be a block algebra of kH such that B^{∗} is the Brauer correspondent of A^{∗}.

(i) All trivial source kH-modules in B^{∗} with vertex P are exactly all simple kH-modules in B^{∗}. (ii) Let T_{1}, ⋯, T_{ℓ} be all non-isomorphic simple kH-modules in B^{∗}. Then, all trivial source kG-

modules in A^{∗} with vertex P are exactly gT_{1}, ⋯, gT_{ℓ}, where g : H → G is the Green correspondence with respect to (G, P, H).

Proof. (i) This follows from a result of the first author [13, Theorem(19)].

(ii) This is obtained from (i), (1.10) and [24, Chapter 4, Problem 10]. ■

(1.13) Lemma. Let p = 3, and let G = PSL_{3}(4) = L_{3}(4).

(i) All elements of order 3 of G are conjugate in G.

(ii) A Sylow 3-subgroup P of G is a T.I. set in G.

(iii) Let Q be a subgroup of G of order 3. Then, there are exactly two non-isomorphic trivial source kG-modules with vertex Q in the principal block B_{0}(kG) of kG. Actually, they are of the forms

\begin{matrix}k \\ 19 \\ k \end{matrix} ,  \begin{matrix}19 \\ k 15_{1} 15_{2} 15_{3} \\ 19 \end{matrix}  (Loewy and socle series)

where IBr(B_{0}(kG)) = {k = k_{G}, 15_{1}, 15_{2}, 15_{3}, 19}.

Proof. (i) and (ii) Easy by [7, p.24].

(iii) By [19, Lemma 4.1(i)], IBr(B_{0}(kG)) = {k, 15_{1}, 15_{2}, 15_{3}, 19}. It follows from [19, Lemmas 5.2, 5.3, 3.2 and 5.1] that there are at least two trivial source kG-modules in B_{0}(kG) with vertex Q, whose Loewy and socle structures are as desired. Therefore, it suffices to show that the number of trivial source kG-modules with vertex Q is two. We may assume that Q ⊆ P, so that P = C_{3} × C_{3} by [19, Lemma 3.1(i)]. By [19, Lemma 3.1(iii) and (v)], it holds C_{G}(Q) = P and N_{G}(Q) = P : C_{2} where C_{2} acts fixed-point-freely on P. Thus, N_{G}(Q)/Q ≅ Σ_{3}. Then, the number of non-isomorphic simple k[N_{G}(Q)/Q]-modules is exactly two. Hence, the proof is completed by making use of [24, Chapter 4, Problem 10] since Green correspondence preserves vertices. ■

2. Broué’s conjecture for the simple Held group

(2.1) Notation and assumption. From now on till the end of this section, we use the following notation. We assume that G is the simple Held group, namely, G = He, |G| = 2^{10} · 3^{3} · 5^{2} · 7^{3} · 17, and |Out(G)| = 2 (see [7, p.104]). We will consider 3-modular representations of G, which means that we assume p = 3.

(2.2) Lemma (Jansen). The simple Held group G has seven 3-blocks, namely, the principal 3block B_{0}(G), a non-principal 3-block A with elementary abelian defect group P of order 9, three 3-blocks with defect 1, and two 3-blocks with defect zero.

Proof. See [8] or [10]. ■

(2.3) Notation. We use the notation A and P as in (2.2).

(2.4) Lemma (Jansen). The 3-decomposition matrix of A has the following form:

51 51^{∨} 1029 1029^{∨} 1275 1275^{∨} 1920
χ_{51}  1  .  .  .  .  .  .
\overline{χ}_{51}  .  1  .  .  .  .  .
χ_{1029} .  .  1  .  .  .  .
\overline{χ}_{1029} .  .  .  1  .  .  .
χ_{1275} .  .  .  .  1  .  .
\overline{χ}_{1275} .  .  .  .  .  1  .
χ_{1920} .  .  .  .  .  .  1
χ_{4080} 1  1  1  1  .  .  1
χ_{6528} .  .  1  1  1  1  1

Proof. See [8] or [10]. ■

(2.5) Notation. We use the notation 51, 51^{∨}, 1029, 1029^{∨}, 1275, 1275^{∨}, 1920 as in (2.4), by which we mean simple kG-modules in A. Let N = N_{G}(P), and let A_{N} be the Brauer corresponding block of A. Let i be a source idempotent of A in A^{ΔP}. Fix any subgroup Q of P. Then, we denote by e_{Q} a block idempotent of a block of kC_{G}(Q) determined by i, namely, Br_{ΔQ}^{G}(i)e_{Q} = Br_{ΔQ}^{G}(i). Let (P, e) be a maximal A-Brauer pair such that e is a block idempotent of a block of kC_{G}(P) determined by i, that is, Br_{ΔP}^{G} (i)e = Br_{ΔP}^{G} (i). Note that e = e_{P}. Let H = N_{G}(P, e), namely, H is a set of all elements g in N_{G}(P) such that g fixes e by conjugation (see [33, §40 p.346]). Then, e is still a block idempotent of O[P ·C_{G}(P)] and OH having P as a defect group. Let j be a source idempotent of the block OHe of OH in (OHe)^{ΔP}, and let f_{Q} be a block idempotent of a block of kC_{H}(Q) determined by j, namely, Br_{ΔQ}^{H}(j)e_{Q} = Br_{ΔQ}^{H}(j).

(2.6) Lemma. (i) The simples 51 and 51^{∨} are trivial source kG-modules.

(ii) The simples 1029 and 1029^{∨} are trivial source kG-modules.

(iii) The simple 1920 is a trivial source kG-module.

Proof. (i) By [7, p.104], G has a maximal subgroup M′ = PSp_{4}(4) : 2. Let X = k_{M′} ↑^{G} ·1_{A}^{∗}. By using [32], we know that 1_{M′} ↑^{G} ·1_{A} = χ_{51} + \overline{χ}_{51}. Hence, X = 51 + 51^{∨}, as composition factors by (2.4). By (1.8)(ii), dim_{k}X = 2. Therefore, (1.1)(ii) implies the assertion.

(ii) Let M′ be the same as in the proof of (i). Let (−1)_{M′} be the non-trivial K-character of M′ with degree one. Then, again by using [32], we have (−1)_{M′} ↑^{G} ·1_{A} = χ_{1029} + \overline{χ}_{1029}. Let Y = (−1)_{M′}^{∗} ↑^{G} ·1_{A}^{∗}. Then, Y = 1029 + 1029^{∨}, as composition factors by (2.4). Hence, again by (1.1)(i), we obtain the assertion.

(iii) It follows from [7, p.104] that G has a maximal subgroup M = 2^{2}.L_{3}(4).Σ_{3}. Then, again by [32], 1_{M} ↑^{G} ·1_{A} = χ_{51} + \overline{χ}_{51} + χ_{1920}. Let Z = k_{M} ↑^{G} ·1_{A}^{∗}. Then, (2.4) shows that Z = 51 + 51^{∨} + 1920, as composition factors. By (1.8)(ii), dim_{k}Z = 3. Thus, we get from (1.1)(ii) that 1920 is a trivial source kG-module. ■

Our strategy is to show that all seven simple kG-modules in A are trivial source (p-permutation) modules. In (2.6), we have already done it for most of the simples. If we could do it for these two simples 1275 and 1275^{∨} as well, then Broué’s conjecture for the 3-block A of G = HS would be, more or less, done. However, it is not so easy for the other two simples 1275 and 1275^{∨}. To prove this we need still several lemmas, which are going to be given in the following. The point is that we look at a particular maximal subgroup M of G above in which the defect group P of A is a T.I. (trivial intersection) set. This particular fact in M may help us.

(2.7) Lemma. Let N and H as in (2.5), and let Q be a subgroup of P of order 3.

(i) All elements in P − {1} are conjugate in H (which are in a conjugate class 3A in [7, p.105]. So that all subgroups of P of order 3 are conjugate to Q in G.

(ii) N = (2^{2} × P).GL_{2}(3).

(iii) H = (2^{2} × P).SD_{16}.

(iv) C_{G}(P) = C_{N}(P) = C_{H}(P) = 2^{2} × P.

(v) C_{G}(Q) = Q.A_{7} = 3.A_{7}, C_{N}(Q) = (2^{2} × P):Σ_{3}, C_{H}(Q) = (2^{2} × P):2 and |N_{G}(Q) : C_{G}(Q)| = |N_{H}(Q) : C_{H}(Q)| = 2.

(vi) G has a maximal subgroup M such that N ⫋ M and M = 2^{2}.L_{3}(4).Σ_{3}.

(vii) P is a T.I. set in M, where M is the same as in (vi).

Proof. (i) This follows by (2.4), the character table of G in [7, p.105] and [24, Chapter 4, Theorem 7.4].

(ii)–(v) follow by [32], see [7, p.105].

(vi) This is obtained by using [32], see [7, p.104].

(vii) By (vi), P ⊆ M. It follows by using [32] that all elements in P − {1} are in a conjugacy class, say ˜3A of M and that |˜3A| = 2^{6}·5·7 = 2240. Now, |M : N_{M}(P)| = 2^{9}·3^{3}·5·7/2^{6}·3^{3} = 280 by (vi) and (ii). This means that |{P^{m}|m ∈ M}| = 280, so that

(*)  ⋃_{i=1}^{280}(P − {1})^{m_{i}} ⊆ ˜3A

where {m_{1}, ⋯, m_{280}} is a coset representatives of N_{M}(P) in M. Now, take any element u ∈ P −{1} and m ∈ ˜3A. Then, m ∈ P^{m′} for some m′ ∈ M, and hence m ∈ P^{m_{i}} for some i with 1 ⩽ i ⩽ 280. Namely, an equality holds in (*) above. This implies that

⋃_{i=1}^{280}(P − {1})^{m_{i}} = ˜3A

and the left hand side is a disjoint union since 8 × 280 = 2240 = |˜3A|. Therefore, we get the assertion. ■

(2.8) Notation and assumption. We use the notation Q and M as in (2.7). Since N ⊆ M ⊆ G, let Ã be a 3-block of M which is the Brauer correspondent of A, so that Ã^{G} = A. Let A_{N} be a 3-block of N which is the Brauer correspondent of A and M, so that A_{N}^{G} = A and A_{N}^{M} = Ã. Let B a 3-block of H which is the Fong-Reynolds corresponding block of A_{N}, see (1.5). Furthermore, let G_{Q} = C_{G}(Q) and H_{Q} = C_{H}(Q). Since C_{G}(P) ⊆ G_{Q}, there is a 3-block A_{Q} of G_{A} with A_{Q}^{G} = A. Then, P is a defect group of A_{Q}, see [24, Chapter 5, Theorem 9.5]. Since H_{Q} = C_{G}(Q) ∩ H = G_{Q} ∩ H = N_{G_{Q}}(P, e), let B_{Q} be a 3-block of H_{Q} such that B_{Q}^{G_{Q}} = A_{Q}. Throughout this paper we use the notation Ã, B, G_{Q}, H_{Q}, A_{Q} and B_{Q} as above. As before, we consider Ã, B, A_{Q} and B_{Q} as block algebras of OM, OH, OG_{Q} and OH_{Q}, respectively, while we denote by Ã^{∗}, B^{∗}, A_{Q}^{∗} and B_{Q}^{∗} the block algebras over k corresponding to Ã, B, A_{Q} and B_{Q}, respectively. By the structure of M and [7, p.23], M has subgroups L, U and Ũ such that

M ⫌ L ⫌ U ⫌ Ũ, L = 2^{2}.L_{3}(4).3, U = 2^{2}.L_{3}(4), Ũ = 2^{2} : A_{6}.

Moreover, let V = 2.L_{3}(4), so that V is the canonical epimorphic image of U ↠ V. Clearly, U ◁ L ◁ M and |M : L| = 2, |L : U| = 3 and |U : Ũ| = 21. We use these notation, say L, U, Ũ and V in this sense throughout this section.

(2.9) Lemma. The 3-decomposition matrix of Ã has the following form:

18 18^{∨} 30a 30a^{∨} 30b 30b^{∨} 132
χ_{30a}  .  .  1  .  .  .  .
\overline{χ}_{30a}  .  .  .  1  .  .  .
χ_{30b}  .  .  .  .  1  .  .
\overline{χ}_{30b}  .  .  .  .  .  1  .
χ_{168}  1  1  .  .  .  1
χ_{192a} .  .  1  1  .  .  1
χ_{192b} .  .  .  .  1  1  1
χ_{210}  1  .  1  .  1  .  1
\overline{χ}_{210}  .  1  .  1  .  1  1

where Irr(Ã) = \{χ_{30a}, \overline{χ}_{30a}, χ_{30b}, \overline{χ}_{30b}, χ_{168}, χ_{192a}, χ_{192b}, χ_{210}, \overline{χ}_{210}\}, IBr(Ã) = \{18, 18^{∨}, 30a, 30a^{∨}, 30b, 30b^{∨}, 132}, and the numbers mean the degrees and the k-dimensions, respectively. Proof. This is calculated by [32]. ■ (2.10) Notation. We use the notation 18, 18^{∨}, 30a, 30a^{∨}, 30b, 30b^{∨}, 132 as in (2.9). (2.11) Lemma. We have the following:

(i) The block Ã of M covers a 3-block A_{L} of L such that

Irr(A_{L}) = {χ_{30a}, χ_{30b}, χ_{84a}, χ_{84b}, χ_{192}, χ_{210}},
IBr(A_{L}) = {18, 30a, 30b, 66a, 66b},
30a ↑_{L}^{M} = 30a ⊕ 30a^{∨},
30b ↑_{L}^{M} = 30b ⊕ 30b^{∨},
18 ↑_{L}^{M} = 18 ⊕ 18′,
66a ↑_{L}^{M} = 66b ↑_{L}^{M}

(ii) Under the notation in (i), the 3-decomposition matrix of A_{L} has the following form:

18 30a 30b 66a 66b
χ_{30a} .  1  .  .  .
χ_{30b} .  .  1  .  .
χ_{84a} 1  .  .  1  .
χ_{84b} 1  .  .  .  1
χ_{192} .  1  1  1  1
χ_{210} 1  1  1  1  1

Proof. Easy by (2.9) and Clifford’s theorem, or using [32]. ■ (2.12) Notation. We use the notation for simple kL-modules in the block A_{L} as in (2.11). (2.13) Lemma. We have the following:

(i) The block A_{L} of L covers a 3-block A_{U} of U such that

Irr(A_{U}) = {χ_{10a}, χ_{10b}, χ_{28a}, χ_{28b}, χ_{64}, χ_{70}}
IBr(A_{U}) = {6, 10a, 10b, 22a, 22b}
6 ↑_{U}^{L}= 18, 10a ↑_{U}^{L} = 30a, 10b ↑_{U}^{L}= 30b, 22a ↑_{U}^{L}= 66a, 22b ↑_{U}^{L}= 66b.

(ii) Under the notation in (i), the 3-decomposition matrix of A_{U} has the following form:

6 10a 10b 22a 22b
χ_{10a} .  1  .  .  .
χ_{10b} .  .  1  .  .
χ_{28a} 1 .  .  1  .
χ_{28b} 1 .  .  .  1
χ_{64}  .  1  1  1  1
χ_{70}  1 1  1  1  1

Proof. Easy by (2.11) and Clifford’s theorem, or by using [32]. ■

(2.14) Notation. We use the notation A_{U} as in (2.13).

(2.15) Lemma. The group V has a subgroup isomorphic to A_{6}.

Proof. By [7, p.23], V has a subgroup V′ such that V′ is a central extension of A_{6} by C_{2}, namely C_{2} ⊆ Z(V′) and V′/C_{2} ≅ A_{6}. Now, it follows from [7, p.5] that a unique non-split central extension 2.A_{6} of A_{6} by C_{2} has elements of order 8 since 2.A_{6} ≅ SL_{2}(9). On the other hand, V has no elements of order 8 by [7, p.24]. Therefore, V′ must be a split extension, that is, V′ = C_{2} × A_{6}. ■

(2.16) Lemma. The group V has a 3-block A_{V} with defect group P such that A_{V} = π(A_{U}), where π is the canonical O-algebra-epimorphism π : OU ↠ OV induced by the group-epimorphism U ↠ V. Thus, 3-decomposition matrix of A_{V} is the same as that of A_{U} in (2.13).

Proof. Easy, see [10, 3-decomposition matrix for L_{3}(4). ■

(2.17) Notation. We use the notation A_{V} as in (2.16).

(2.18) Lemma. k_{A_{6}} ↑^{V} ·1_{A_{V}} ^{∗} = 10a ⊕ 10b. That is, the simples 10a and 10b in A_{V} are trivial source kV -modules.

Proof. First of all, recall that V contains A_{6} from (2.15). Let X = k_{A_{6}} ↑^{V} ·1_{A_{V}} ^{∗} = 10a ⊕ 10b. It follows by [32] that

1_{A_{6}} ↑^{V} = χ_{1} + χ_{6} + χ_{35} + χ_{10a} + χ_{10b} + χ_{36},

where the characters of the right hand side are elements in Irr(V) and the numbers mean their degrees, see [10, 3-decomposition matrix for 2.L_{3}(4)]. Hence, by (2.16), 1_{A_{6}} ↑^{V} ·1_{A_{V}} = χ_{10a} + χ_{10b}. Thus, X = 10a+10b, as composition factors. Since dim_{k}[End_{kV} (X)] = (χ_{10a}+χ_{10b}, χ_{10a}+χ_{10b})^{V} = 2, we get by a theorem of Scott (1.8) that X = 10a ⊕ 10b. ■

(2.19) Lemma. k_{(2.A_{6})} ↑^{U} ·1_{A_{U}} ^{∗} = 10a ⊕ 10b. Namely, the simples 10a and 10b in A_{U} are trivial source kU-modules.

Proof. Clear by (2.18) since k_{(2.A_{6})} ↑^{U} ≅ k_{A_{6}} ↑^{V} as right kU-modules, where we can consider kV -modules as kU-modules by using the inflation induced by the canonical epimorphism U ↠ V. ■

(2.20) Lemma. The simples 30a, 30a^{∨}, 30b, 30b^{∨} in the block Ã are trivial source kM-modules.

Proof. It follows from (2.13) and (2.11) that, for i ∈ {a, b},

10i ↑_{U}^{M}= 10i ↑_{U}^{L}↑_{L}^{M}= 30i ↑_{L}^{M}= 30i ⊕ 30i^{∨}.

Hence, we get the assertion by (2.19). ■

(2.21) Lemma. (i) By exchanging the notation 30a and 30b in Ã (if necessary), we have the following:

χ_{30i} ↑_{M}^{G} ·1_{A} = χ_{1275}, \overline{χ}_{30i} ↑_{M}^{G} ·1_{A} = \overline{χ}_{1275},

for i ∈ {a, b}. So that, we have

30a ↑_{M}^{G} ·1_{A}^{∗} = 1275, 30a^{∨} ↑_{M}^{G} ·1_{A}^{∗} = 1275^{∨}.

(ii) The simples 1275 and 1275^{∨} in A are trivial source kG-modules.

Proof. (i) follows by using [32], and (ii) is obtained by (2.20) and (i). ■

At the end of this section we prove our main theorem, say Broué’s conjecture for a non-principal 3-block A of the Held simple group G = He with elementary abelian defect group of order 9. First, we need to prepare some notation which is going to be necessary for our aim.

(2.22) Notation and assumption. First of all, recall the notation A, P, N, H, Q, M and Ã given in (2.3) and (2.8). In order to prove our main result we need to investigate p-local structure of G and A. Recall also

G_{Q} = C_{G}(Q) = Q.A_{7} = 3.A_{7}, H_{Q} = C_{H}(Q) = N_{G_{Q}}(P, e) = (2^{2} × P) : 2,

A_{Q} and B_{Q}, see (2.8) and (2.7)(v).

(2.23) Lemma. (i) The non-principal 3-block A_{Q} has P as defect group, and the 3-decomposition matrix of A_{Q} is the following:

6 15
χ_{6a}  1 .
χ_{6b}  1 .
\overline{χ}_{6b}  1 .
χ_{15a} . 1
χ_{15b} . 1
\overline{χ}_{15b} . 1
χ_{21a} 1 1
χ_{21b} 1 1
\overline{χ}_{21b} 1 1

(ii) The non-principal 3-block B_{Q} has P as defect group and satisfies that B_{Q} ≅ O[P : 2] as Oalgebras, where P : 2 is the semi-direct product of P by C_{2} such that C_{2} acts non-fixed-point-freely on P, and the 3-decomposition matrix of B_{Q} is the following:

1a 1b
χ_{1a} 1  .
χ_{1c} 1  .
\overline{χ}_{1c} 1  .
χ_{1b} .  1
χ_{1d} .  1
\overline{χ}_{1d} .  1
χ_{2a} 1 1
χ_{2b} 1 1
\overline{χ}_{2b} 1 1

(iii) We get that B_{Q}^{G_{Q}} = A_{Q}, A_{Q}^{G} = A and B_{Q}^{H} = B.

Proof. This is obtained by (2.7)(iv), 3-decomposition matrix for A_{7} in [10, p.13] and a result of Külshammer [18] and Puig [29]. ■

(2.24) Lemma. Let 𝔐_{Q} be a unique indecomposable direct summand of A_{Q}↓{G_{Q}×G_{Q} \atop G_{Q}×H_{Q}}·1_{B_{Q}} with vertex ΔP, see (1.7) and [1, Theorem 5(i)]. Then, 𝔐_{Q} is an indecomposable (A_{Q}, B_{Q})-bimodule and induces a Morita equivalence (hence a Puig equivalence) between A_{Q} and B_{Q}.

Proof. By the definition, 𝔐_{Q} is an indecomposable (A_{Q}, B_{Q})-bimodule. Let \overline{G}_{Q} = G_{Q}/Q ≅ A_{7}, \overline{H}_{Q} = H_{Q}/Q ≅ (2^{2} × 3) : 2 and \overline{P} = P/Q, see (2.7)(v). Let π_{1} : OG_{Q} ↠ O\overline{G}_{Q} and π_{2} : OH_{Q} ↠ O\overline{H}_{Q} be the canonical epimorphisms of O-algebras. Furthermore, 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 3-blocks of \overline{G}_{Q} and \overline{H}_{Q} with defect group \overline{P}

from (1.3)(i). By (2.23)(iii) and (1.3)(ii), \overline{B}_{Q}^{\overline{G}^{Q}} = \overline{A}_{Q}. Therefore, we get from (1.6)(iii) that \overline{A}_{Q} and \overline{B}_{Q} are Morita equivalent via \overline{𝔐}_{Q} where \overline{𝔐}_{Q} is a unique non-projective indecomposable direct summand of 1_{\overline{A}_{Q}}·O\overline{G}_{Q}·1_{\overline{B}_{Q}}, namely, it is defined by

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

Hence, [14, Theorem] yields that A_{Q} and B_{Q} are Morita equivalent via an indecomposable (A_{Q}, B_{Q})- bimodule 𝔐′_{Q} which is defined by

1_{A_{Q}}·OG_{Q}·1_{B_{Q}} = 𝔐′_{Q} ⨁ ( ⊕ (indecomposable with vertex ΔQ))

and 𝔐_{Q}′ is a unique indecomposable direct summand of the above with vertex ≠ ΔQ. Therefore, 𝔐′_{Q} has vertex ΔP, so that 𝔐′_{Q} = 𝔐_{Q} by Krull-Schmidt theorem. ■

We next prove the final lemma, which is pretty much usable for our main theorem.

(2.25) Lemma. Let 𝔐 be an (A, B)-bimodule which is a unique indecomposable direct summand of A↓_{G×H}^{G×G}·1_{B} with vertex ΔP, see (1.7). Let i and j be source idempotents of A in A^{ΔP} and B

in B^{ΔP}, respectively, such that 𝔐|(Ai ⊗_{OP} jB) as (A, B)-bimodules, see (2.5). Moreover, for each subgroup R of P, let e_{R} and f_{R} respectively be unique block idempotents of kC_{G}(R) and kC_{H}(R) determined by i and j, namely, Br_{ΔP}^{G} (i)·e_{R} = Br_{ΔP}^{G} (i) and Br_{ΔP}^{H} (j)·f_{R} = Br_{ΔP}^{H} (j), see [23, p.821] and (2.5). Then, a pair (𝔐, 𝔐^{∨}) induces a stable equivalence of Morita type between A and B.

Proof. Our method here is applying a theorem of Linckelmann [23, Theorem 3.1] to our situation. That is, we want to check that the pair (𝔐, 𝔐^{∨}) satisfies the two conditions in (ii) in his theorem. We remark that, by (2.7)(ii), there are only two non-trivial subgroups of P up to H-conjugacy, which are P and Q (recall that Q is a subgroup of P of order 3 in (2.8)). Therefore, it is enough to show that

(1) For any R ∈ {P, Q}, e_{R}·(ΔP)·f_{R} induces a Morita equivalence between e_{R}kC_{G}(R) and f_{R}C_{H}(R).

(2) For any two subgroups R, S ∈ {P, Q, ⟨1⟩}, we have

E_{G}((R, e_{R}), (S, e_{S})) = E_{H}((R, f_{R}), (S, f_{S})).

We also note that we can write 𝔐 = OGf for a primitive idempotent f of A^{ΔH} such that f·1_{B} = 1_{B}·f = f.

Condition (1) for R = P. By (2.7)(iv), C_{G}(P) = C_{H}(P), so that e_{P} = f_{P}. Let f′ = Br_{ΔP} (f). Hence,

e_{P} ·𝔐(ΔP)·f_{P} = e_{P} ·(OGf)(ΔP)·e_{P}
= e_{P} ·(OG)(ΔP)·f′·e_{P}
= e_{P} kC_{G}(P)f′ since f′ ∈ kC_{G}(P)
|(e_{P} kC_{G}(P))_{C_{G}(P)×C_{H}(P)} = (e_{P} kC_{G}(P))_{C_{G}(P)×C_{G}(P)}

since f′ ∈ (kC_{G}(P))^{C_{H}(P)} = Z(kC_{G}(P)). This means that

e_{P} ·𝔐(ΔP)·f_{P} = e_{P} ·𝔐(ΔP)|e_{P} kC_{G}(P)_{C_{G}(P)×C_{G}(P)}.

It follows from [17, Theorem] that e_{P} ·𝔐(ΔP)·f_{P} ≠ 0. Therefore, e_{P} ·𝔐(ΔP)·f_{P} = e_{P} kC_{G}(P). Since e_{P} kC_{G}(P) = f_{P} kC_{H}(P), this itself realizes a Morita equivalence between e_{P} kC_{G}(P) and f_{P} kC_{H}(P).

Condition (1) for R = Q. Let X = e_{Q}·𝔐(ΔQ)·f_{Q}, which is considered as a (kC_{G}(Q), kC_{H}(Q))-bimodule. It follows from a result of the first author and Linckelmann [17, Theorem] that X is a unique indecomposable direct summand of e_{Q}kC_{G}(Q)f_{Q} = A_{Q}^{∗}↓_{G_{Q}×H_{Q}}^{G_{Q}×G_{Q}}·f_{Q} with vertex ΔP. Now, by (2.24), X induces a Morita equivalence between A_{Q}^{∗} = e_{Q}kG_{Q} and B_{Q}^{∗} = f_{Q}kH_{Q}.

We have checked the condition (1). Next, we want to check the condition (2). Take any two subgroups R, S ∈ {P, Q, ⟨1⟩}. Clearly, we know that E_{G}((R, e_{R}), (S, e_{S})) = 1 for any S if R = ⟨1⟩;

and that E_{G}((R, e_{R}), (S, e_{S})) = ∅ if |R| > |S|.

Therefore, it suffices to check the condition (2) for the cases (a) R = S = P,

(b) R = S = Q.
(c) R = Q, S = P,

Condition (2) for Case (a). Clearly, N_{G}(P, e_{P}) = N_{H}(P, f_{P}) = H and C_{G}(P) = C_{H}(P), so that e_{P} = f_{P} = e. If g ∈ G and it satisfies (P, e_{P})^{g} ⊆ (P, e_{P}), then we easily know that g ∈ N_{G}(P, e_{P}) = H. Thus, it follows from (2.7)(ii), (iv) and (iii) that

E_{G}((P, e_{P}), (P, e_{P})) ≅ N_{G}(P, e_{P})/C_{G}(P)
= H/C_{G}(P) = H/C_{H}(P)
= N_{H}(P, f_{P})/C_{H}(P)
≅ E_{H}((P, f_{P}), (P, f_{P})).

Condition (2) for Case (b). Since C_{G}(Q) = 3.A_{7} has a unique non-principal 3-block with defect group P (which is actually A_{Q}) by (2.7)(v). we know that N_{G}(Q) = N_{G}(Q, e_{Q}). Hence,

E_{G}((Q, e_{Q}), (Q, e_{Q})) = {g·C_{G}(Q) | g ∈ G, (Q, e_{Q})^{g} ⊆ (Q, e_{Q})}
≅ N_{G}(Q, e_{Q})/C_{G}(Q)
= N_{G}(Q)/C_{G}(Q) ≅ C_{2}

by (2.7)(v). On the other hand, by (2.7)(v), C_{H}(Q) = (2^{2} × P) : 2 and this group has a unique non-principal 3-block with defect group P such that the degrees of ordinary irreducible characters are one and two, which is actually B_{Q}. This means that N_{H}(Q, f_{Q}) = N_{H}(Q). Therefore,

E_{H}((Q, f_{Q}), (Q, f_{Q})) = {h·C_{H}(Q) | f ∈ G, (Q, f_{Q})^{h} ⊆ (Q, f_{Q})}
≅ N_{H}(Q, f_{Q})/C_{H}(Q)
= N_{H}(Q)/C_{H}(Q) ≅ C_{2}

by (2.7)(v). These imply that E_{G}((Q, e_{Q}), (Q, e_{Q})) ≅ E_{H}((Q, f_{Q}), (Q, f_{Q})).

Condition (2) for Case (c). Recall that all elements of order 3 in P are conjugate in H and hence in G. Therefore, by [23, lines 23-24 in p.838] and the proof of Condition (2) for Case (b), we get

E_{G}((Q, e_{Q}), (P, e_{P})) = {g·C_{G}(Q) | g ∈ G, (Q, e_{Q})^{g} ⊆ (P, e_{P}))}
= {g·C_{G}(Q) | g ∈ G, Q^{g} ⊆ P, g^{−1}e_{Q}g = e_{g^{−1}Qg}}
≅ C_{2}
≅ E_{H}((Q, f_{Q}), (P, f_{P})).

■

Now, we finally are ready to prove one of our main theorems.

Proof of Theorem(i). First, note that B in the statement of Theorem(i) in the introduction and B in §2 are different. By (1.7), there is an indecomposable (A, B)-bimodule 𝔐, which is a unique direct summand of A↓_{G×H}^{G×G}·1_{B} with vertex ΔP. It follows by (2.4), (2.6)(i)–(iii) and (2.21)(ii) that all simple kG-modules in A are trivial source modules. Hence, by (1.2) and a result of Okuyama (1.4), we know that, for any simple kG-module S in A, its Green correspondent fS is a simple kNmodule in A_{N}, which implies that (S ⊗_{kG} 𝔐)_{B} = (fS)↓_{H}^{N}·1_{B} is a simple kH-module in B, where f : G → N is the Green correspondence with respect to (G, P, N). Therefore, we get from (2.25) and a result of Linckelmann [21, Theorem 2.1(iii)] than 𝔐 induces a Morita equivalence between A and B. Let 𝔐′ = 𝔐 ⊗_{OH} ON = 𝔐 ⊗_{OH} ON·1_{A_{N}}, which turns out an (A, A_{N})-bimodule. Then, 𝔐′ induces a Morita equivalence between A and A_{N} by (1.5). Clearly, 𝔐′, as an O[G × N]-module, has a trivial source since 𝔐′|OG↓_{G×H}^{G×G}↑^{G×N} = O_{ΔG}↑^{G×G}↓_{G×H}↑^{G×N}. Hence, 𝔐′ has a vertex ΔP since both A and A_{N} have P as their defect groups. These yields that A and A_{N} are Puig equivalent (splendidly Morita equivalent) via 𝔐′. We are done. ■

3. Broué’s conjecture for the sporadic simple Suzuki group

(3.1) Notation and assumption. From now on till the end of this section, we use the following notation. We assume that G is the sporadic simple Suzuki group, namely, G = Suz, |G| = 2^{13} · 3^{7} · 5^{2} · 7 · 11 · 13, and |Out(G)| = 2 (see [7, p.131]). We will consider 3-modular representations of G, which means that we assume p = 3.

(3.2) Lemma. (i) (Jansen-Müller) The sporadic simple Suzuki group G has three 3-blocks, namely, the principal 3-block with defect 7, one 3-block A with elementary abelian defect group P of order 9, and one 3-block of defect one.

(ii) The group P in (i) is a T.I. set in G.

Proof. See [9] and [2, Proposition 6.1(ii)] for (i) and (ii), respectively. ■

(3.3) Notation. We use the notation A and P as in (3.2).

(3.4) Lemma (Jansen-Müller). (i) There are exactly five non-isomorphic simple kG-modules S, T, S_{1}, S_{2}, S_{3} in A such that S = 15795, T = 72657, S_{1} = 160380, S_{2} = 5103a, S_{3} = 5103b, where the numbers mean k-dimensions.

(ii) All simple kG-modules S, T, S_{1}, S_{2}, S_{3} have P as their vertices.

(iii) The 3-decomposition matrix and the Cartan matrix of A are the following:

\begin{aligned}& S & T & S_{1} & S_{2} & S_{3} \\ χ_{15795} & 1 & . & . & . & . \\ χ_{88452} & 1 & 1 & . & . & . \\ χ_{248832} & 1 & 1 & 1 & . & . \\ χ_{93555a} & 1 & 1 & . & 1 & . \\ χ_{93555b} & 1 & 1 & . & . & 1 \\ χ_{243243} & . & 1 & 1 & 1 & 1 \end{aligned}  \begin{aligned}& P (S) & P (T) & P (S_{1}) & P (S_{2}) & P (S_{3}) \\ S & 5 & 4 & 1 & 1 & 1 \\ T & 4 & 5 & 2 & 2 & 2 \\ S_{1} & 1 & 2 & 2 & 1 & 1 \\ S_{2} & 1 & 2 & 1 & 2 & 1 \\ S_{3} & 1 & 2 & 1 & 1 & 2 \end{aligned}

Proof. (i), (ii) and (iii) are obtained by [9], [11, 3.7.Corollary(i)] and [9], respectively. ■

(3.5) Notation. We use the notation S, T, S_{1}, S_{2}, S_{3} as in (3.4). We sometimes use also the notation S_{0} = S. Moreover, we use the notation χ_{15795}, χ_{88452}, χ_{248832}, χ_{93555a}, χ_{93555b}, χ_{243243} as in (3.4). Note that the subindices of χ_{i}’s mean the degrees.

(3.6) Lemma. The group G has maximal subgroups M and L such that M = (A_{4} × L_{3}(4)) : 2_{1} and L = 3_{2}.U_{3}(4) : 2_{3}.

Proof. See [7, p.131]. ■

(3.7) Notation. We use the notation M and L as in (3.6) throughout this section. Moreover, let (−1)_{M} be a non-trivial linear ordinary character of M.

(3.8) Lemma. (i) 1_{M} ↑^{G} ·1_{A} = 2 × χ_{15795} + χ_{88452}.

(ii) (−1)_{M} ↑^{G} ·1_{A} = χ_{88452} + χ_{243243}.

(iii) 1_{L} ↑^{G} ·1_{A} = χ_{15795}.

(iv) k_{L} ↑^{G} ·1_{A}^{∗} = S, so that S is a trivial source module.

Proof. (i)–(iii) are obtained by [7, p.131]. (iv) is easy by using (iii) and (3.4)(iii). ■

(3.9) Lemma. For any simple kG-module S′ in A, Ext_{kG}^{1}(S′, S′) = 0.

Proof. Easy by (3.4)(iii) and a result of Brandt [4]. ■

(3.10) Lemma. For any simple kG-module S′ in A, the heart [P (S′)rad(kG)]/S′ of P (S′) is an indecomposable kG-module.

Proof. Similar to the proof in [16, (3.3)Lemma]. ■

(3.11) Lemma. For any i, j ∈ {0, 1, 2, 3}, Ext_{kG}^{1}(S_{i}, S_{j}) = 0.

Proof. Similar to [16, (3.4)Lemma]. ■

(3.12) Lemma. (i) k_{M} ↑^{G} ·1_{A}^{∗} = S ⊕ U(S, T, S), S ↔ χ_{15795}, and U(S, T, S) ↔ χ_{15795} + χ_{88452}.

(ii) ((−1_{M})^{∗}) ↑^{G} ·1_{A}^{∗} = \begin{matrix}T \\ S S_{1} S_{2} S_{3} \\ T \end{matrix} (Loewy and socle series) ↔ χ_{88452} + χ_{243243}.

Proof. (i) Let X = k_{M} ↑^{G} ·1_{A}^{∗}. By (3.8)(i) and (3.4)(iii), X = S + S + S + T, as composition factors. Then, it follows from (3.8)(iv), a result of Scott (1.8)(ii) and (3.8)(i) that

[S, X]^{G} = [X, S]^{G} = (2 × χ_{15795} + χ_{88452}, χ_{15795})^{G} = 2.

If [X, T]^{G} ≠ 0, then the self-dualities of X and T and (1.9) imply that T |X, so that T is a trivial source kG-module, and hence T is liftable by (1.8)(i), contradicting (3.4)(iii). This yields that [X, T]^{G} = [T, X]^{G} = 0. Hence, j(X) ⩾ 3. If j(X) = 4, then X = U(S, T, S, S) or U(S, S, T, S), which is a contradiction to the self-dualities of X, S and T. Therefore, X has the radical (Loewy) series and the socle series of the forms

\begin{matrix}S S \\ T \\ S \end{matrix}  and  \begin{matrix}S \\ T \\ S S \end{matrix} ,

respectively. Let J = rad(kG). Then, X has submodules Y and Z such that XJ^{2} ⊆ Y ∩ Z, Y/XJ^{2} ≅ S ≅ Z/XJ^{2} and Y + Z = X. Thus, j(Y) = 3 or j(Z) = 3 since j(X) = 3. Hence, we may assume j(Y) = 3, so that Y = U(S, T, S), which means that soc(Y) ≅ S. By the socle series of X, X has a submodule W such that W ≅ S and soc(X) = soc(Y) ⊕ W. If Y ∩ W ≠ 0, then W ⊆ Y since W is simple, and hence W ⊆ soc(Y), a contradiction. This implies that Y ∩ W = 0 and so that X = Y ⊕ W.

(ii) Let X = (−1)_{M}^{∗} ↑^{G} ·1_{A}^{∗}. Then, (3.8)(ii) and (3.4)(iii) imply that X = S+T +T +S_{1}+S_{2}+S_{3}, as composition factors. Hence, just as in the proof of (i), we get [X, S]^{G} = [S, X]^{G} = 0, and [X, S_{i}]^{G} = [S_{i}, X]^{G} = 0 for i = 1, 2, 3. Let U be the trivial source kG-module appearing in (i), namely,

U = U(S, T, S) ↔ χ_{15795} + χ_{88452}.

Then, again by (3.8)(ii) and (1.8)(ii), we have

[X, U]^{G} = [U, X]^{G} = (χ_{15795} + χ_{88452}, χ_{88452} + χ_{243243})^{G} = 1.

This shows that

\begin{matrix}S \\ T \end{matrix} ↪ X ↠ \begin{matrix}T \\ S \end{matrix} .

Let J = rad(kG). Then, (T ⊕ T) |̸(X/XJ). Therefore, X/XJ ≅ T ≅ soc(X). Thus, we get the assertion by (3.11). ■

(3.13) Lemma. Let H = N_{G}(P), and let B a block algebra of OH such that B is the Brauer correspondent of A. Then, we get the following.

(i) H = (P : (4 × A_{6})).2_{1} = (P × A_{6})Q_{8}, (where 2_{1} is a notation in [7, p.131] and this is a cyclic group of order 2). Furthermore, H is a maximal subgroup of G.

(ii) |G : H| = 17297280.

(iii) All elements of order 3 in P are conjugate in G, and they are in the conjugacy class 3C in the notation of [7, p.128].

(iv) IBr(B) = {9a, 9b, 9c, 9d, 18}, where the numbers mean k-dimensions.

(v) B ≅ Mat_{9}(O[P : Q_{8}]) as O-algebras, and a source algebra of B is isomorphic to O[P : Q_{8}] as interior P -algebras.

(vi) All projective indecomposable kH-modules in B have the following Loewy and socle series:

\begin{matrix}9i \\ 18 \\ 9j′ 9k′ 9ℓ \\ 18 \\ 9i \end{matrix}  where \{i, j′, k′, ℓ\} = \{a, b, c, d\},  \begin{matrix}18 \\ 9a 9b 9c 9d \\ 18 18 18 \\ 9a 9b 9c 9d \\ 18 \end{matrix} .

(vii) Let f : G → H and g : H → G be the Green correspondences with respect to (G, P, H). Then, for any non-projective indecomposable kH-module Y in B, we have Y ↑^{G} = gY ⊕(projective). Furthermore, let M_{0} be a unique indecomposable direct summand of a k[G × H]-module 1_{A}^{∗}·kG·1_{B}^{∗} with vertex ΔP (actually, M_{0} is the Green correspondent of A^{∗} with respect to (G×G, ΔP, G×H). Similarly, let N_{0} be a unique indecomposable direct summand of a k[H × G]-module 1_{B}^{∗}·kG·1_{A}^{∗} with vertex ΔP. Then, a pair (M_{0}, N_{0}) induces a stable equivalence of Morita type between mod-A^{∗} and mod-B^{∗}.

Proof. (i) and (ii) are obtained by [7, p.131].

(iii) Easy by (3.4)(iii), the character table of G in [7, p.128] and [24, Chapter 5, Corollary 1.10]. (iv) This follows from direct calculation or [32].

(v) This follows from (iii) and a theorem of Puig [29, Proposition 14.6] (see [33, (45.12)Theorem] and [1, Theorem 13(iv)]), since the Schur multiplier of Q_{8} is trivial.

(vi) Easy.

(vii) This follows from (3.2)(ii), [22, Theorem 4.6] and [21, Theorem 2.1(i)]. ■

(3.14) Notation. We use the notation H, B, 9a, 9b, 9c, 9d, 18, f, g, M_{0} and N_{0} as in (3.13). Let U be the trivial source kG-module in A appearing in (3.12)(i) such that U ↔ χ_{15795} + χ_{88452}, and let V be the trivial source kG-module in A appearing in (3.12)(ii) such that V ↔ χ_{88452} + χ_{243243}. Furthermore, let Q be a subgroup of P with Q ≅ C_{3}, see (3.13)(iii).

(3.15) Lemma. fS ∈ {9a, 9b, 9c, 9d}.

Proof. We know from (3.8)(iv) and (3.4) that S is a trivial source simple kG-module in A with vertex P. Hence, by Okuyama’s result (1.4) and (1.10)(i), fS is a simple kH-module in B. Let S̃ = fS. It follows from (3.13)(vii), (3.1) and (3.4)(i) that dim(S̃ ↑^{G}) ≡ 486 (mod 3^{7}). Then, since dim S̃ = 9 or 18 by (3.13)(iv), we get the assertion from (3.13)(ii) and (3.13)(iv). ■

(3.16) Lemma. (i) There are exactly five trivial source kG-modules in A with vertex P. Actually, they are g(9a), g(9b), g(9c), g(9d) and g(18).

(ii) It holds that dim g(9i) ≡ 9 (mod 3^{7}) for i ∈ {a, b, c, d} and dim g(18) ≡ 18 (mod 3^{7}).

Proof. (i) This follows from (1.12)(ii) and (3.13)(iv).

(ii) Easy by (3.13)(vii). ■

(3.17) Lemma. Both U and V are trivial source kG-modules in A and have Q as their vertices.

Proof. By (3.12)(i), U is a trivial source kG-module in A and dim U ≡ 1458 (mod 3^{7}). Hence, U is not projective by (3.1). Moreover, (3.16)(ii) yields that P is not a vertex of U, so that U has Q as its vertex by (3.13)(iii). Similarly for V. ■

(3.18) Notation. Let G′ = PSL_{3}(4) = L_{3}(4). Since Sylow 3-subgroups of G′ are C_{3} × C_{3}, we may assume that P satisfies P ∈ Syl_{3}(G′). Let H′ = N_{G′}(P), so that H′ = P : Q_{8}, see [19, Lemma 3.1(v)]. Let A′ and B′ be the principal block algebras of OG′ and OH′, respectively, and hence B′ = OH′. Then, we can write IBr(A′) = {k = k_{G′}, 15_{1}, 15_{2}, 15_{3}, 19} and IBr(B′) = {k = k_{H′}, 1_{1}, 1_{2}, 1_{3}, 2}. Furthermore, let f′ : G′ → H′ and g′ : H′ → G′ be the Green correspondences with respect to (G′, P, H′). Let U′ and V′ be the trivial source kH′-modules in (1.13)(iii) such that

U′ = \begin{matrix}k \\ 19 \\ k \end{matrix}  and  V′ = \begin{matrix}19 \\ k 15_{1} 15_{2} 15_{3} \\ 19 \end{matrix} .

(3.19) Lemma. There is a source idempotent j of B with j ∈ B^{ΔP} and it satisfies the following: (i) jBj ≅ OH′ as interior P -algebras. (ii) _{OH}Bj_{OH′} |O_{ΔP} ↑^{H×H′}, where Bj is considered as a right OH′-module via the isomorphism

in (i).

(iii) A (B, B′)-bimodule Bj induces a Morita equivalence between B and B′.

(iv) If X is a trivial source kH-module in B with vertex R for a subgroup R of P, then (X ⊗_{kH} Bj)_{kH′} is a trivial source kH′-module in B′ with vertex R.

Proof. (i) By a result of Green [24, Chapter 5, Theorem 10.8], B|O_{ΔP} ↑^{H×H}= OH ⊗_{OP} OH, so that B|OH ⊗_{OP} 1_{B}·OH. Hence, there is a source idempotent j of B with j ∈ B^{ΔP} such that B|OH ⊗_{OP} jOH as in [23, p.822, lines 17–23]. Then, jBj ≅ OH′ = B′ as interior P -algebras by (3.13)(v), proving (i).

(ii) By (i),

Bj|OH ⊗_{OP} jOHj ≅ OH ⊗_{OP} OH′ = O_{ΔP} ↑^{H×H′},

which proves (ii).

(iii) This is easy by (i) and a well-known result due to Puig.

(iv) By (i) and (ii), we easily know that trivial source kH-modules in B are sent to trivial source kH′-modules in B′ via the Morita equivalence in (iii). Let X′ = (X ⊗_{kH} Bj)_{kH′}. Then, by the symmetry of Morita equivalence and Mackey decomposition, it is easy to know that there is an element h ∈ H such that R^{h} = h^{−1}Rh ⊆ P and R^{h} is a vertex of X′. Hence, (1.13)(i) implies that R^{h} and R are conjugate in H′. Therefore, R is a vertex of X′ as well. ■

(3.20) Notation. We use the notation j as in (3.19).

(3.21) Lemma. We get the following.

(i) There is a unique non-projective indecomposable direct summand M′_{0} of a k[G′ × H′]-module 1_{A′}^{∗}·kG′·1_{B′}^{∗} with vertex ΔP, and there is also a unique non-projective indecomposable direct summand N′_{0} of a k[H′ × G′]-module 1_{B′}^{∗}·kG′·1_{A′}^{∗} with vertex ΔP. Moreover, a pair (M′_{0}, N′_{0}) induces a stable equivalence of Morita type between A^{∗} and B^{∗}.

(ii) Let _{A^{∗}}M_{A′}^{∗} = _{kG}M_{kG′} = M_{0} ⊗_{kH} B^{∗}j^{∗} ⊗_{kH′} N′_{0}. Then, M is indecomposable as an (A^{∗}, A′^{∗})-bimodule and a pair (M, M^{∨}) realizes a stable equivalence of Morita type between A^{∗} and A′^{∗}.

Proof. (i) This follows by (1.13)(ii), [22, Theorem 4.6] and [21, Theorem 2.1(i)].

(ii) Easy by (3.13)(vii), (3.19)(iii), (i) and [21, Theorem 2.1(ii)]. ■

(3.22) Notation. We use the notation M′_{0}, N′_{0} and M as in (3.21). Moreover, we define a functor F : mod-A^{∗} → mod-A′^{∗} via F (−) = − ⊗_{A^{∗}} M, so that F induces a stable equivalence of Morita type between A^{∗} and A′^{∗} by (3.21)(ii).

(3.23) Lemma. We may assume that F (S) = k_{G′}.

Proof. By (3.15), we may assume that fS = 9a, so that S ⊗_{kG} M_{0} = 9a ⊕ (projective). Therefore, S ⊗_{kG} M_{0} = 9a by [21, Theorem 2.1(ii)]. Now, by (3.19)(iii), B^{∗}j^{∗} realizes a Morita equivalence between B^{∗} and kH′. Let W_{kH′} = 9a ⊗_{kH} B^{∗}j^{∗}. Then, by looking at Loewy and socle series of projective indecomposable modules over B and kH′ in (3.13)(vi), we know W ∈ {k_{H′}, 1_{1}, 1_{2}, 1_{3}}. Suppose, first, that W = 1_{1}. Then, since 1_{1} is self-dual, 1_{1} ⊗_{k} 1_{1} = k_{H′}. Thus, replace B^{∗}j^{∗} by kH(B^{∗}j^{∗} ⊗_{k} 1_{1})_{kH′}. Then, S ⊗_{kG} M_{0} ⊗_{kH} (B^{∗}j^{∗} ⊗_{k} 1_{1}) = k_{H′}. Moreover, since N′_{0} coincides with the Green correspondence g′ : H′ → G′, we know that

S ⊗_{kG} M_{0} ⊗_{kH} (B^{∗}j^{∗} ⊗_{k} 1_{1}) ⊗_{kH′} N′_{0} = k_{G′}

by [21, Theorem 2.1(ii)]. ■

(3.24) Remark and notation. By (3.21), (3.22) and (3.23), if necessary, we put

M = M_{0} ⊗_{kH} (B^{∗}j^{∗} ⊗_{k} 1_{ν}) ⊗_{kH′} N′_{0}

for some ν ∈ {1, 2, 3} so that F (S) = S ⊗_{A} M = k_{G′}.

(3.25) Lemma. (i) F (U) = U′ ⊕ (projective).

(ii) F (U(S, T)) = U(k, 19) ⊕ (projective) and F (U(T, S)) = U(19, k) ⊕ (projective).

(iii) F (T) = 19.

Proof. (i) By (3.17), (3.19)(iv), the definition of F in (3.22) and (3.24), we know that F (U) = X′ ⊕ R′ for a trivial source kG′-module X′ in A′^{∗} with vertex Q, and a projective kG′-module R′ in A′^{∗}. Then,

k ≅ Hom_{kG}(S, U)
= \underline{Hom}_{kG}(S, U) since S is simple, see [20, II Corollary 2.8]
≅ \underline{Hom}_{kG′}(F (S), F (U)) since F induces a stable equivalence
= \underline{Hom}_{kG′}(k, F (U)) by (3.23).

Hence, there exists a non-projective kG′-epimorphism F (U) ↠ k, and hence X′ = U′ by (1.13)(iii). (ii) By applying the functor F to a short exact sequence

0 → S →^{φ} U → U/S → 0

in mod-A^{∗}, we get a short exact sequence

0 → F (S) →^{F (φ)} F (U) → F (U/S) → 0

in mod-A′^{∗}. Put φ′ = F (φ). By (i), we can write F (U) = U′ ⊕ R′ for a projective A′^{∗}-module R′. As in the proof of (i), we have an isomorphism Hom_{A^{∗}}(S, U) ≅ \underline{Hom}_{A′}_{∗}(F (S), F (U)). This isomorphism maps φ to φ′ + PHom_{A′}^{∗}(F (S), F (U)). Since φ is non-zero, the map φ′ cannot be a projective homomorphism. This implies that π_{U′} ∘φ′ ≠ 0, where π_{U′} : U′ ⊕R′ → U′ is the canonical projection.

Hence, it follows from (1.11) that we may assume Imφ′ ⊆ U′. Therefore, F (U/S) ≅ (U′/Imφ′) ⊕ R′ = U(k, 19) ⊕ R′. Similarly for F (rad(U)).

(iii) As in (ii), we have a short exact sequence

0 → F (S) → F (rad(U)) → F (T) → 0,

which is rewritten as

0 → k →^{ψ} W ⊕ (projective) → F (T) → 0

by (3.23) and (ii), where W = U(19, k). Hence, again by (1.11), we may assume that Imψ ⊆ W. This yields F (T) ≅ W/k ⊕ (projective) ≅ 19 ⊕ (projective), which means F (T) ≅ 19 since F (T) is non-projective indecomposable by [21, Theorem 2.1(ii)]. ■

(3.26) Lemma. (i) F (V) = V′ ⊕ (projective).

(ii) F (rad(V)) = rad(V′) ⊕ (projective) and F (V/T) = V′/19 ⊕ (projective).

(iii) We can assume that F (S_{1}) = 15_{1}, F (S_{2}) = 15_{2} and F (S_{3}) = 15_{3}.

Proof. (i) and (ii) are proved as in the proofs in (3.25)(i) and (3.25)(ii), respectively. (iii) By applying F to a short exact sequence

0 → T → rad(V) → S ⊕ S_{1} ⊕ S_{2} ⊕ S_{3} → 0,

we obtain a short exact sequence

0 → F (T) → F (rad(V)) → F (S ⊕ S_{1} ⊕ S_{2} ⊕ S_{3}) → 0.

Then, it follows from (3.25)(iii), (ii) and (3.23) that there is a short exact sequence

0 → 19 →^{φ} rad(V′) ⊕ (projective) → k ⊕ F (S_{1}) ⊕ F (S_{2}) ⊕ F (S_{3}) → 0.

Hence, again by (1.11), we can assume that Imφ ⊆ rad(V′). This means that

k ⊕ F (S_{1}) ⊕ F (S_{2}) ⊕ F (S_{3}) ≅ (rad(V′)/19) ⊕ (projective)
= k ⊕ 15_{1} ⊕ 15_{2} ⊕ 15_{3} ⊕ (projective).

Therefore, [21, Theorem 2.1(ii)] implies that {F (S_{1}), F (S_{2}), F (S_{3})} = {15_{1}, 15_{2}, 15_{3}}. We are done. ■

Proof of Theorem(ii). By (3.22), the functor F : mod-A^{∗} → mod-A′^{∗} defined by F (−) = − ⊗_{A^{∗}} M induces a stable equivalence of Morita type. Furthermore, it follows from (3.4)(i), (3.23), (3.25)(iii) and (3.26)(iii) that any simple kG-module in A is sent to a simple kG′-module in A′ via F. Therefore, a result of Linckelmann [21, Theorem 2.1(iii)] implies that F, actually, induces a Morita equivalence. Thus, by a theorem of Puig and Scott in [23, Theorem 4.1], the bimodule M induces a Puig equivalence between A^{∗} and A′^{∗} (which means that source algebras of A^{∗} and A′^{∗} are isomorphic as interior P -algebras). Thus, a theorem of Puig [28, 7.8.Lemma] (or [33, (38.8)Proposition]) implies that the block algebras A and A′ over O are also Puig equivalent. Recall that A′ = B_{0}(O[PSL_{3}(4)], the principal block algebra of O[PSL_{3}(4)]. Now, Okuyama proves that A′ and B′ are splendidly (Rickard) equivalent in [26, Example 4.6] and [27]. Thus, by looking at a diagram

A ⟶^{Puig equiv.} A′
↓^{splendid equiv.}
B ⟵_{Puig equiv.} B′

we finally know that A and B are splendidly Rickard equivalent. Thus, by a result of the second author [19, Theorem 1.2], we are done. ■

Acknowledgments

This paper was in part prepared while the second author was staying at School of Mathematics, University of Bristol as a Research Fellow of the Japan Society for Promotion of Science (JSPS). She is grateful to Jeremy Rickard for his kind hospitality and also to the JSPS for the financial support.

The first author was in part supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for JSPS Fellows 01016, 2001–2003; and the JSPS, Grant-in-Aid for Scientific Research C(2) 14540009, 2002–2003. The second author was in part supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for JSPS Fellows 07600, 2001–2002.

References

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

[2] J. An and C.W. Eaton, Blocks with trivial intersection defect groups, Preprint (2002).

[3] F.W. Anderson and K.R. Fuller, Rings and categories of modules (second edition), (Springer, Berlin, 1992).

[4] J. Brandt, A lower bound for the number of irreducible characters in a block, J. Algebra 74 (1982), 509–515.

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

[6] M. Broué, Isométries de caractères et équivalences de Morita ou dérivées, Publ. Math. IHES 71 (1990), 45–63.

[7] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of finite groups (Clarendon Press, Oxford, 1985).

[8] C. Jansen, Ein Atlas 3-modularer Charaktertafeln, Dissertation (RWTH Aachen, Aachen, 1995).

[9] C. Jansen and J. Müller, The 3-modular decomposition numbers of the sporadic simple Suzuki group, Commun. Algebra 25 (1997), 2437–2458.

[10] C. Jansen, K. Lux, R. Parker and R. Wilson, An atlas of Brauer characters (Oxford Univ. Press, Oxford, 1995).

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

[12] S. König and A. Zimmermann, Derived equivalences for group rings, in: Lecture Notes in Mathematics, Vol.1685 (Springer, Berlin, 1998).

[13] S. Koshitani, On the kernels of representations of finite groups II, Glasgow Math. J. 32 (1990), 341–347.

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

[14] S. Koshitani and N. Kunugi, Blocks of central p-group extensions, to appear in Proc. Amer. Math. Soc.

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

[17] S. Koshitani and M. Linckelmann, Indecomposability of a module given by Brauer construction, Preprint (2003).

[18] B. Külshammer, Crossed products and blocks with normal defect groups, Commun. Algebra 13 (1985), 147–168.

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

[20] P. Landrock, Finite group algebras and their modules, in: London Mathematical Society Lecture Note Series, Vol.84 (London Math. Soc., Cambridge, 1983).

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

[22] M. Linckelmann, On stable and derived equivalences of blocks and algebras, An. St. University Ovidius Constantza 4 (1996), 74–97.

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

[24] H. Nagao and Y. Tsushima, Representations of Finite Groups (Academic Press, New York, 1990).

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

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

[27] 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.

[28] L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77–116.

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

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

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

[32] M. Schönert (et al.), GAP 3.4 manual (RWTH Aachen, Aachen, 1994).

[33] J. Thévenaz, G-Algebras and modular representation theory (Clarendon, Oxford, 1995).