Broué’s conjecture for non-principal 3-blocks of finite groups

Shigeo Koshitani

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

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

Naoko Kunugi

c/o S.Koshitani, Department of Mathematics, Faculty of Science, Chiba University, Chiba 263-8522, Japan

                      E-mail mkunugi@g.math.s.chiba-u.ac.jp

and

Katsushi Waki

Department of Mathematical System Science, Faculty of Science and Technology, Hirosaki University, Hirosaki, 036-8561, Japan

                        E-mail slwaki@cc.hirosaki-u.ac.jp

April, 2001

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 D, then A and its Brauer correspondent p-block B of N_{G}(D) are derived equivalent. We demonstrate in this paper that the Broué’s conjecture holds for two non-principal 3-blocks A with elementary abelian defect group D of order 9 of the O’Nan simple group and the Higman-Sims simple group. Moreover, we determine these two non-principal block algebras over a splitting field of characteristic 3 up to Morita equivalence.

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 [4]. He actually conjectures the following:

(0.1) Conjecture (M. Broué [4, 6.2.Question], see [18, Conjecture in p.132]). For a prime p, let A be a p-block of a finite group G with defect group D, and let B be a p-block of N_{G}(D) such that B is the Brauer correspondent of A. Then, A and B would be derived equivalent provided D is abelian.

There are only several special cases where the Broué’s conjecture (0.1) has been checked even in the case that A is the principal p-block, see [5, p.15] and [18, p.136]. For non-principal p-blocks, the conjecture (0.1) is thought almost only in the cases that D is cyclic in a paper by Rickard [29], that G is a p-solvable group in a paper by Harris-Linckelmann [9], and that D is the Klein four group C_{2} × C_{2} in [30]. The purpose of this article is to make observations for non-principal 3-blocks A of two sporadic simple groups, say the O’Nan and the Higman-Sims simple groups, with elementary abelian defect group D of order 9. Namely, our main result is the following:

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

(i) Assume that G = O′N, the O’Nan simple group. If OA = Â is the non-principal block algebra of OG with elementary abelian defect group D of order 9 and if OB = B̂ is the Brauer correspondent of OA in N_{G}(D), then OA and OB are splendidly equivalent.

(ii) (see Holm [11, p.60]) The same as in (i) holds for G = HS, the Higman-Sims simple group.

It is announced in [11] by Holm that he himself has proved (0.2)(ii) (probably over k instead of O). Since he has not written any proof of it, it may be useful to give a whole proof in this article. Furthermore, as a corollary of (0.2)(i), we also get the following:

(0.3) Corollary. Keep the notation in (0.2)(i). Let q be a power of a prime such that q ≡ 4 or 7 (mod 9). Then, all the principal block algebras B_{0}(O[L_{3}(q)]) of the projective special linear group L_{3}(q) are splendidly Morita (hence Puig) equivalent to the non-principal block algebra OA of O[O′N], and the block algebra A over k is completely determined up to Morita equivalence.

Moreover, by a little bit more calculations after getting (0.2)(ii), we can determine the structure of the block algebra A as k-algebra, completely up to Morita equivalence. In her thesis [10] K. Hicks almost determines A up to Morita equivalence. However, there is one parameter β_{1} there [10, p.133]. We can actually determine β_{1} such that β_{1} = −1 in this article without using [10]. That is,

(0.4) Theorem. The non-principal block algebra A of k[HS] of the Higman-Sims simple group HS is completely determined up to Morita equivalence (see (5.10) Theorem).

The strategy of, for instance, (0.2)(i) is the following. Let A be the non-principal block algebra of kG with defect group D = C_{3} × C_{3}, where G = O′N, and let B be its Brauer correspondent in k[N_{G}(D)]. Then, B is Morita equivalent to B′ = k[D:Q_{8}], which is the Brauer correspondent of the principal block algebra A′ of kG′ where G′ = L_{3}(4). Then, each Green correspondent f(S_{i}) of a simple kG-module S_{i} in A corresponds to a Green correspondent f′(S′_{i}) of a simple kG′-module S′_{i} in A′ via the above Morita equivalence. Hence, by looking at Okuyama’s method in [26, Example 4.6], we can take the same one also in A and B in a parallel way. Hence, we finally know that A and B are derived (even splendidly) equivalent and furthermore that A and A′ are Morita (even splendidly Morita) equivalent by passing the Morita equivalent block algebras B and B′.

Throughout this article we use the following notation and terminology. Let G be a finite group and p a prime. Let (K, O, k) be a splitting p-modular system for all subgroups of G, namely, O is a discrete valuation ring of rank one with unique maximal ideal J(O) = (π) for π ∈ O, K is the quotient field of O with characteristic zero and k is its residue field O/(π) with characteristic p. Here, a module always means a finitely generated right module, unless stated otherwise. We mean by an OG-lattice a right OG-module which is O-free of finite rank. When a kG-module M uniquely lifts to an OG-lattice, we denote the OG-lattice by M̂. If M̂ is an OG-lattice, we write χ_{M} for the ordinary character (namely, K-character) of G afforded by the right KG-module M̂ ⊗_{O} K. In particular, for a projective kG-module P, we sometimes write Φ_{P} for the K-character of G afforded by P̂, instead of χ_{P}. When D is a p-subgroup of G, we use a term a Green correspondence with respect to (G, D, H) following [24, IV §4].

Let A be a finite dimensional k-algebra, and let M and N be A-modules. We denote by 1_{A}, J(A) and ℓ(A), respectively, the unit element of A, the Jacobson radical of A and the number of all non-isomorphic simple A-modules. We write P (M) for the projective cover of M, ΩM for the kernel of the projective cover P (M) ↠ M, that is, Ω is the Heller operator, and Soc(M) for the socle of M. For a positive integer i, we define rad^{i}(M) by M·J(A)^{i} and Soc_{i}(M) by Soc_{1}(M) = Soc(M) and Soc_{i}(M)/Soc_{i−1}(M) = Soc(M/Soc_{i−1}(M)) if i ⩾ 2. We write j(M) for the Loewy (and hence socle) length of M, namely, j(M) is the least positive integer j such that M·J(A)^{j} = 0, equivalently, Soc_{j}(M) = M. The i-th Loewy layer M·J(A)^{i−1}/M·J(A)^{i} is denoted by L_{i}(M) for i = 1, ⋯, j(M). Let S_{1}, ⋯, S_{n} be simple A-modules, possibly S_{i} ≅ S_{j} for a pair (i, j) with i ≠ j. Then, we write U(S_{1}, ⋯, S_{n}) for a uniserial A-module M such that L_{i}(M) ≅ S_{i} for i = 1, ⋯, n. Note that U(S_{1}, ⋯, S_{n}) is not necessarily detemined up to isomorphism in general. We write N|M if N is a direct summand of M as A-module. We write M = S_{1} + ⋯ + S_{n}, as composition factors when {S_{1}, ⋯, S_{n}} is the set of all composition factors of M (possibly S_{i} ≅ S_{j} for i ≠ j). The multiplicity of S_{i} in the composition factors of M is denoted by c_{M}(S_{i}). The Cartan invariant with respect to S_{i} and S_{j} is denoted by c(S_{i}, S_{j}) or simply by c_{S_{i},S_{j}}, namely, c(S_{i}, S_{j}) = C_{P (S_{i})}(S_{j}). We say that M has a filtration

M = N_{1} | N_{2} | N_{3}

for A-modules N_{1}, N_{2} and N_{3} when M has submodules M_{1}, M_{2} and M_{3} satisfying M ⊇ M_{1} ⊇ M_{2} ⊇ M_{3}, M/M_{1} ≅ N_{1}, M_{1}/M_{2} ≅ N_{2} and M_{3} ≅ N_{3}. We write that M = N ⊕ (proj) if M is a direct sum of N and a projective A-module.

Let n be a positive integer. We denote by C_{n}, D_{n}, Q_{n} and SD_{n}, respectively, the cyclic group, the dihedral group, the generalized quaternion group and the semi-dihedral group of order n. We write Σ_{n} and A_{n} for the symmetric and the alternating group on n letters. The Mathieu group of degree n is denoted by M_{n}. For a prime p and a non-negative integer r we write ν_{p}(n) = r when p^{r}|n and p^{r+1} |̸n. For a p-block A of G, Irr(A) and IBr(A), respectively, denote the sets of all irreducible ordinary and Brauer characters of G belonging to A, and we write k(A) and ℓ(A), respectively, for the numbers of elements of the sets Irr(A) and IBr(A). We write G = H:K when G is a semi-direct product of H by K such that H ◁ G. We write Δ(G) for the diagonal subgroup {(g, g) ∈ G × G | g ∈ G} of G × G. The trivial kG-module is denoted by k_{G}. We write B_{0}(kG) for the principal block algebra of the group algebra kG.

Let M and N be kG-modules. Then, Hom_{kG}(M, N) and dim_{k}[Hom_{kG}(M, N)] are denoted by (M, N)^{G} and [M, N]^{G}, respectively. We denote by \underline{Hom}_{kG}(M, N) the k-vector space (M, N)^{G}/{all projective hom’s }. We write M^{∗} for Hom_{k}(M, k), the k-dual of M, which is also a right kG-module as usual. For a projective indecomposable kG-module P, [P |M] denotes the multiplicity of P in M as direct summands. For ordinary characters χ and ψ of G, (χ, ψ)_{G} denotes the inner product of χ and ψ in G. For other notation and terminology, see the book of Nagao and Tsushima [24].

1. Lemmas

In this section we list several lemmas which are useful to get our main result. Throughout this section we fix a finite group G.

(1.1) Lemma (Scott)([21, 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}.

(1.2) Lemma (Green-Landrock-Scott)(see [21, II Lemma 12.6]). Let M be a trivial source kG-module, so that M uniquely lifts to a trivial source OG-lattice M̂ (see (1.1)(i)).

(i) Let Q be a p-subgroup of G. Then, dim_{k}[(Soc(M↓_{Q})] = (χ_{M}↓_{Q}, 1_{Q})_{Q}.

(ii) Let x be a p-element in G. Then χ_{M}(x) equals the number of indecomposable direct summands of the k⟨x⟩-module M↓_{⟨x⟩} which are isomorphic to the trivial k⟨x⟩-module k_{⟨x⟩}. In particular, χ_{M}(x) is a non-negative integer.

(iii) Let x be a p-element in G. Then, χ_{M}(x) > 0 if and only if x belongs to some vertex of M.

(1.3) Lemma (Zassenhaus and others)(see [21, I Theorem 17.3]). Let M̂ be an OG-lattice. Assume that χ_{M} = χ_{1} + χ_{2} where each χ_{i} is an ordinary character of G. Then, M̂ contains an O-pure OG-submodule N̂_{i} such that χ_{i} = χ_{N_{i}} for each i. Let M = M̂/M̂π and N_{i} = N̂_{i}/N̂_{i}π for each i. Then, M contains kG-submodules M_{1} and M_{2} with M_{i} ≅ N_{i} for i = 1, 2.

(1.4) Lemma (Thompson)(see [21, I Corollary 17.4]). Let S be a simple kG-module, and assume that Φ_{P (S)} = χ_{1} + χ_{2} for ordinary characters χ_{i} of G. Then, there exists an OG-lattice M̂_{1} such that χ_{M_{1}} = χ_{1} and Soc(M̂_{1}/M̂_{1}π) ≅ S.

(1.5) Lemma (Landrock)(see [21, I Lemma 9.10]). Let S and T be simple kG-modules. Then, [L_{i}(P (S)), T]^{G} = [L_{i}(P (T^{∗})), S^{∗}]^{G} for all i = 1, 2, · · ·.

(1.6) Lemma (Robinson [31, Theorem 3]). Let H be a subgroup of G, and let S and T respectively be a simple kG-module and a simple kH-module. Then, [P (S) | T ↑^{G}] = [P (T) | S↓_{H}] .

(1.7) 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 D be a p-subgroup of G, and let H = N_{G}(D). Assume that A and B respectively are block algebras of RG and RH with defect group D 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, D, H). Let ℨ = {Q | Q ⩽ D, Q ⩽̸ D^{x} ∩ D 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.

(1.8) Lemma (Green) (see [24, Chapter 5 Theorem 1.9]). Let M be an indecomposable kGmodule in a block algebra A of kG with defect group D. If ν_{p}(dim_{k} M) = ν_{p}(|G|) − ν_{p}(|D|), then M has D as its vertex.

(1.9) Lemma (Erdmann-Kawata)(see [7, Theorem 1] and [16, Theorem 1.5]). Let A be a block algebra of kG which is of wild representation type. Suppose, moreover, that there does not exist any simple kG-module S in A such that the Cartan invariants of S satisfy c(S, S) = 2; and c(S, S′) = 1 or 0 for every simple kG-module S′ in A with S′ ≅ S̸. Then, for any simple kG-module T in A, the heart H(P (T)) = [P (T)·J(kG)]/T of P (T) is an indecomposable kG-module.

Proof. Suppose that there is a simple kG-module T in A such that H(P (T)) is decomposable. Let Θ be a connected component of the stable Auslander-Reiten quiver of A containing T. Since A is of wild representation type, a theorem of Erdmann [7, Theorem 1] shows that the tree class of Θ is of A_{∞}-type. Thus, as remarked in [16, ℓ.11 of Introduction], T does not lie at the end of Θ. Therefore, a theorem of Kawata [16, Theorem 1.5] implies that there exists a simple kG-module S in A such that c(S, S) = 2 and c(S, S′) = 0 or 1 for any simple kG-module S′ ≅ S̸ in A, a contradiction. □

2. Preliminaries for 3-modular representations of the O’Nan simple group

From now on except in §5 we fix notation, and we introduce several fundamental results (lemmas) on 3-modular representations of the O’Nan simple group.

(2.1) Notation. We assume that G is the O’Nan simple group, namely G = O′N, |G| = 2^{9}·3^{4}·5·7^{3}·11·19·31, and |Out(G)| = 2, so that we fix an element σ ∈ Aut(G) − G (see [28] or [6, pp.132–133]). Note that Sylow 3-subgroups of G are elementary abelian of order 3^{4} (see [28, p.422, ℓ.23]).

(2.2) Lemma. The O’Nan simple group G has eight 3-blocks, namely, the principal 3-block B_{0}(G), a non-principal 3-block A with elementary abelian defect group D of order 9, and six 3-blocks of defect zero.

Proof. See [15, §3 pp.73–75] and [4, A2.2, p.88]. □

(2.3) Notation. We use the notation A and D as in (2.2). We denote J(kG) by J.

(2.4) Lemma. (i) It holds that k(A) = 6, and we can write Irr(A) = {χ_{10944}, χ_{52668}, χ_{58311_{1}}, χ_{58311_{2}}, χ_{58311_{3}}, χ_{58653}}.

(ii) Each χ_{i} ∈ Irr(A) has a value on 3A as the following, where 3A is a unique conjugacy class of G with elements of order 3.

\begin{matrix}in [6, p.133] & χ_{2} & χ_{11} & χ_{12} & χ_{13} & χ_{14} & χ_{15} \\ here & χ_{10944} & χ_{52668} & χ_{58311_{1}} & χ_{58311_{2}} & χ_{58311_{3}} & χ_{58653} \\ 3A & 9 & 18 & −9 & −9 & −9 & 9 \end{matrix}

(iii) Moreover, σ ∈ Aut(G) − G acts on the set Irr(A) such that χ_{58311_{2}}^{σ} = χ_{58311_{3}} and χ^{σ} = χ for all χ ∈ Irr(A) − {χ_{58311_{2}}, χ_{58311_{3}}}.

Proof. This follows from [15, §3 pp.74–75] and [6, p.133] (see [4, p.88]). □

(2.5) Notation. We use the notation χ_{10944}, χ_{52668}, χ_{58311_{1}}, χ_{58311_{2}}, χ_{58311_{3}}, χ_{58653} as in (2.4). Note that the subindices of χ_{i}’s mean the degrees.

(2.6) Lemma. (i) It holds ℓ(A) = 5, and we can write IBr(A) = {S, T, S_{1}, S_{2}, S_{3}}, namely, S, T, S_{1}, S_{2} and S_{3} are all non-isomorphic simple kG-modules in A such that

in [15, p.73] φ_{13}  φ_{18}  φ_{7}  φ_{8}  φ_{9} here  S  T  S_{1}  S_{2}  S_{3} k-dim  10944 41724 5643 5643 5643  .

(ii) All simple kG-modules in A are self-dual, and have D as their vertices.

(iii) σ ∈ Aut(G) − G acts on the set IBr(A) such that S_{2}^{σ} = S_{3} and S′^{σ} = S′ for all S′ ∈ IBr(A) − {S_{2}, S_{3}}.

(iv) The 3-decomposition matrix of A has the following form.

S T S_{1} S_{2} S_{3}
χ_{10944} 1 . .  .  .
χ_{52668} 1 1 .  .  .
χ_{58311_{1}} 1 1 1 .  .
χ_{58311_{2}} 1 1 .  1 .
χ_{58311_{3}} 1 1 . . 1
χ_{58653} . 1 1 1 1

(v) The Cartan matrix of A has the following form.

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

Proof. Follows by [15, p.75, Table 6; p.73, Table 2], [17, 3.7 Corollary(i)] and [6, p.133]. □

(2.7) Lemma. (i) Let H = N_{G}(D). Then, H = D:(4 × A_{6})·2 = [(D:4) × A_{6}].2, and H is a maximal subgroup of G. Furthermore, C_{G}(D) = D × A_{6} and H/C_{G}(D) ≅ Q_{8}.

(ii) σ in (2.1) can be chosen as an element in H·2 which is a maximal subgroup of Aut(G) = G.2. Namely, σ↓_{H} ∈ Aut(H) − Inn(H).

(iii) D is a T.I. set in G.

Proof. (i) By (2.2), the inertial quotient of A is Q_{8}. Then, by [6, p.132] (or [37] and [38]), we get the first part. Hence, we may assume that D is the group R in [28, p.446]. Then, C_{G}(D) = D × A_{6} by [28, Lemmas 5.3 and 5.5] (see [4, p.88]).

(ii) We get this by [6, p.132].

(iii) This follows from the proof of (i) and [28, Lemma 5.7(ii)]. □

(2.8) Notation. We use the notation S, T, S_{1}, S_{2}, S_{3} as in (2.6) and H as in (2.7). We fix a 3-block B of H corresponding to A via the Brauer correspondence. Namely, B^{G} = A. We denote J(kH) by J̃.

(2.9) Lemma. (i) We can write Irr(B) = {χ̃_{5}, χ̃_{6}, χ̃_{13}, χ̃_{14}, χ̃_{18}, χ̃_{25}} where χ̃_{i} means an irreducible ordinary character of H such that

\begin{matrix}centr. & 25920 & 576 & 2880 & 64 & 324 & 3240 & 81 & 81 & 288 & 1440 \\ class & 1A & 2A & 2B & 2C & 3A & 3B & 3C & 3D & 4A & 4B \\ χ̃_{5} & 9 & 1 & 9 & 1 & . & 9 & . & . & 1 & 9 \\ χ̃_{6} & 9 & 1 & 9 & 1 & . & 9 & . & . & 1 & 9 \\ χ̃_{13} & 9 & 1 & 9 & 1 & . & 9 & . & . & 1 & −9 \\ χ̃_{14} & 9 & 1 & 9 & 1 & . & 9 & . & . & 1 & −9 \\ χ̃_{18} & 18 & 2 & −18 & −2 & . & 18 & . & . & 2 & . \\ χ̃_{25} & 72 & 8 & . & . & . & −9 & . & . & 8 & . \end{matrix} (cont.)

\begin{matrix}centr. & 32 & 32 & 16 & 8 & 8 & 180 & 36 & 72 & 16 & 16 \\ class & 4C & 4D & 4E & 4F & 4G & 5A & 6A & 6B & 8A & 8B \\ χ̃_{5} & 1 & 1 & 1 & 1 & 1 & −1 & . & 1 & −1 & −1 \\ χ̃_{6} & 1 & 1 & 1 & −1 & −1 & −1 & . & 1 & 1 & 1 \\ χ̃_{13} & 1 & −1 & −1 & 1 & −1 & −1 & . & 1 & −1 & −1 \\ χ̃_{14} & 1 & −1 & −1 & −1 & 1 & −1 & . & 1 & 1 & 1 \\ χ̃_{18} & −2 & . & . & . & . & −2 & . & 2 & . & . \\ χ̃_{25} & . & . & . & . & . & −8 & . & −1 & . & . \end{matrix} (cont.)

centr. 16 16 20 36 36 36 45 45 20  20
class  8C 8D 10A 12A 12B 12C 15A 15B 20A 20B
χ̃_{5}  −1 −1 −1  1  .  . −1 −1 −1  −1
χ̃_{6}  1  1 −1  1  .  . −1 −1 −1  −1
χ̃_{13}  1  1 −1  1  .  . −1 −1  1  1
χ̃_{14} −1 −1 −1  1  .  . −1 −1  1  1
χ̃_{18}  .  .  2  2  .  . −2 −2  .  .
χ̃_{25}  .  .  . −1  .  .  1  1  .  .

(ii) σ↓_{H} ∈ Aut(H) acts on the set Irr(B) such that χ̃_{13}^{σ} = χ̃_{14} and χ̃^{σ} = χ̃ for all χ̃ ∈ Irr(B) − {χ̃_{13}, χ̃_{14}}.

(iii) The relations of irreducible characters in Irr(A) and Irr(B) with respect to inductions and restrictions are the following.

χ̃_{5}  χ̃_{6}  χ̃_{13}  χ̃_{14}  χ̃_{18}  χ̃_{25}
χ_{10944}  6  6  5  5  6  33
χ_{52668} 22 20 20  20  34 148
χ_{58311_{1}} 25 21 21  21  38 165
χ_{58311_{2}} 21 22 21  22  38 163
χ_{58311_{3}} 21 22 22  21  38 163
χ_{58653} 18 20 20  20  42 161

(iv) Let 3A be a unique conjugacy class of G with elements of order 3, and let ˜3B be a conjugacy class of H in (i). Then, C_{G}(3A) = C_{H}( ˜3B) = D × A_{6}.

Proof. (i) and (iii) are obtained by GAP [34]. (ii) is easy by [6, p.133], (i), and (2.4)(iii). (iv) is easy by (i) and (2.7)(i). □

(2.10) Notation. We use the notation χ̃_{i}’s for each i as in (2.9).

(2.11) Lemma. We get the following.

(i) B̂ ≅ Mat_{9}(O[D:Q_{8}]) as O-algebras, where B̂ is the block algebra of OH corresponding to B.

(ii) We can write IBr(B) = {φ̃_{5}, φ̃_{6}, φ̃_{13}, φ̃_{14}, φ̃_{18}} such that the 3-decomposition matrix of B has the following form.

φ̃_{5} φ̃_{6} φ̃_{13} φ̃_{14} φ̃_{18}
χ̃_{5} 1 .  .  .  .
χ̃_{6} .  1 .  .  .
χ̃_{13} .  .  1  .  .
χ̃_{14} .  .  .  1  .
χ̃_{18} .  .  .  .  1
χ̃_{25} 1 1 1  1  2

so that dim_{k} φ̃_{i} = 9 for i = 5, 6, 13, 14 and dim_{k} φ̃_{18} = 18.

(iii) The Cartan matrix of B has the following form:

P (φ̃_{5}) P (φ̃_{6}) P (φ̃_{13}) P (φ̃_{14}) P (φ̃_{18})
φ̃_{5}  2  1  1  1  2
φ̃_{6}  1  2  1  1  2
φ̃_{13}  1  1  2  1  2
φ̃_{14}  1  1  1  2  2
φ̃_{18}  2  2  2  2  5

(iv) The Loewy and socle series for the projective indecomposable kH-modules in B are the following.

P (φ̃_{i}) = φ̃_{i} ; φ̃_{18} ; φ̃_{j} φ̃_{k′} φ̃_{ℓ} ; φ̃_{18} ; φ̃_{i} for {i, j, k′, ℓ} = {5, 6, 13, 14},  and  P (φ̃_{18}) = φ̃_{18} ; φ̃_{5} φ̃_{6} φ̃_{13} φ̃_{14} ; φ̃_{18} φ̃_{18} φ̃_{18} ; φ̃_{5} φ̃_{6} φ̃_{13} φ̃_{14} ; φ̃_{18}.

(v) There are exactly five trivial source kH-modules in B such that their vertices are D. They are, actually, all simple, namely, φ̃_{5}, φ̃_{6}, φ̃_{13}, φ̃_{14} and φ̃_{18}.

(vi) σ↓_{H} ∈ Aut(H) − Inn(H) acts on the set IBr(B) such that φ̃_{13}^{σ} = φ̃_{14} and φ̃^{σ} = φ̃ for all φ̃ ∈ IBr(B) − {φ̃_{13}, φ̃_{14}}.

Proof. (i) follows from (2.7)(i), a result of Külshammer [19, A.Theorem] and [13, V 25.3 Satz(b)]. (ii)–(v) are easy since the defect group D of B is normal in H (see [21, III Lemma 10.3]). □

(2.12) Notation. We use the notation φ̃_{i}’s as in (2.11).

(2.13) Lemma. By [6, p.132] (or [37] and [38]), G has maximal subgroups L and M such that L = L_{3}(7)·2 and M = 4·L_{3}(4)·2. We then get the following.

(i) 1_{L}↑^{G}·1_{A} = χ_{10944} + χ_{52668}.

(ii) 1_{M}↑^{G}·1_{A} = 2 × χ_{10944} + 2 × χ_{52668} + 2 × χ_{58311_{1}} + χ_{58311_{2}} + χ_{58311_{3}}.

(iii) (−1)_{M}↑^{G}·1_{A} = χ_{10944} + χ_{58311_{1}} + χ_{58653}, where (−1)_{M} is a non-trivial linear ordinary character of M afforded by the extension M ▷ 4·L_{3}(4).

Proof. We get the assertions by using GAP [34]. □

(2.14) Lemma. There are exactly two trivial source kH-modules in B with vertix < ˜3B >≅ C_{3}, where ˜3B is the same as in (2.9)(i) and we can take < ˜3B >⪇ D. And they are, actually,

\begin{matrix}φ̃_{5} & φ̃_{6} & φ̃_{13} & φ̃_{14} \\ φ̃_{18} & φ̃_{18} \\ φ̃_{5} & φ̃_{6} & φ̃_{13} & φ̃_{14} \end{matrix} ⟷ χ̃_{5} + χ̃_{6} + χ̃_{13} + χ̃_{14} + χ̃_{18}, \begin{matrix}φ̃_{18} & φ̃_{18} \\ φ̃_{5} & φ̃_{6} & φ̃_{13} & φ̃_{14} \\ φ̃_{18} & φ̃_{18} \end{matrix} ⟷ 2 × χ̃_{18} + χ̃_{25}

3. Projective indecomposable modules in A In this section we investigate Loewy and socle structure of projective indecomposable kG-modules

(PIM’s) in A. The information helps so much to get our main result (0.2)(i). We keep all the notation used in §2, namely in (2.1), (2.3), (2.5), (2.8), (2.10) and (2.12). The aim of this section is, actually, to prove the following proposition. (3.1) Proposition. The Loewy and socle series of projective indecomposable kG-modules in A are the following.

P (S) = \begin{matrix}S \\ T T \\ SSSS_{1}S_{2}S_{3} \\ T T \\ S \end{matrix} ,  P (T) = \begin{matrix}T \\ SSS_{1}S_{2}S_{3} \\ T T T \\ SSS_{1}S_{2}S_{3} \\ T \end{matrix} ,  P (S_{i}) = \begin{matrix}S_{i} \\ T \\ SS_{j}S_{k′} \\ T \\ S_{i} \end{matrix} for \{i, j, k′\} = \{1, 2, 3\}.

We need several lemmas to prove the above proposition. (3.2) Lemma. For any simple kG-module S′ in A, Ext_{kG}^{1}(S′, S′) = 0. Proof. Since k(A) − ℓ(A) = 1 by (2.4)(i) and (2.6)(i), the assertion follows from a result of Brandt [3, Theorem B]. □ (3.3) Lemma. There is a kG-module W such that W = U(S, T) and W^{∗} = U(T, S). Proof. Easy by (2.6)(iv) and a Thompson’s result (1.4). □ (3.4) Lemma. Let L be as in (2.13). Then, k_{L}↑^{G}·1_{A} = U(S, T, S) ↔ χ_{10944} + χ_{52668} and this is a trivial source kG-module. Proof. Let U = k_{L}↑^{G}·1_{A}. By (2.13)(i) and (1.1)(i), χ_{U} = χ_{10944} + χ_{52668}. If T |Soc(U), then the self-dualities imply T |U since c_{U}(T) = 1, so that T is a trivial source module, contradicting (1.1)(i) and (2.6)(iv). Hence, T |̸Soc(U), so that T |̸L_{1}(U) again by the self-dualities. Therefore U has Loewy and socle series as desired. □. (3.5) Lemma. For any simple kG-module S′ in A, the heart H(P (S′)) = [P (S′)·J]/S′ of P (S′) is indecomposable as kG-module. Proof. Follows from [2, Theorem 4.4.4], (2.6)(v) and (1.9). □ (3.6) Lemma. Ext_{kG}^{1}(S_{i}, S_{j}) = 0 for all i, j ∈ {0, 1, 2, 3}, possibly i = j, where S_{0} = S. Proof. Note, first, that S_{0}, S_{1}, S_{2}, S_{3} are all self-dual by (2.6)(ii). Since c(S_{i}, S_{j}) = 1 for i ≠ j by (2.6)(v), we get the assertion from (3.5), (3.2) and (2.6)(ii). □ (3.7) Lemma. There are kG-modules U(S, T, S_{i}) for i = 1, 2, 3. Proof. Easy by (1.4), (2.6)(iv) and (3.6). □ (3.8) Lemma. dim_{k}[Ext_{kG}^{1}(S_{i}, T)] = 1 for i = 1, 2, 3. Proof. Easy by (3.7), (2.6)(ii), (3.5) and (2.6)(v). □ (3.9) Lemma. P (S_{i}) has Loewy and socle series as in (3.1) for i = 1, 2, 3. Proof. By (2.6)(i), IBr(A) = {S, T, S_{1}, S_{2}, S_{3}}. Let P_{1} = P (S_{1}). It follows from (3.2), (3.6), (3.7) and (3.8) that L_{2}(P_{1}) = T and S|L_{3}(P_{1}). Thus, by (2.6)(iv) and (1.4), P_{1} has a factor module W suth that W = T + S_{1} + S_{2} + S_{3}, as composition factors. Then, (3.6) implies that j(W) = 3 and L_{i}(W) is S_{1}, T, S_{2} ⊕ S_{3} for i = 1, 2, 3, respectively. By (3.2), we get T |̸L_{3}(P_{1}). So, P_{1} has Loewy and socle series as desired by (2.6)(ii). Similar for P (S_{2}) and P (S_{3}). □ (3.10) Lemma. P (S) and P (T) have Loewy and socle series as in (3.1).

Proof. Let P = P (S) and Q = P (T). It follows from (2.6)(v), (3.9), (1.4), (3.6) and (3.4) that the Loewy series of P is

P = S ; T ⋯; S_{1}S_{2}S_{3}S ⋯; T ⋯; ⋮ ; S ,  where T, T, S, S are left.

Clearly, j(P) ⩽ 9.

Step 1: j(P) ≠ 9. Suppose j(P) = 9. Since Ext_{kG}^{1}(S, S) = Ext_{kG}^{1}(T, T) = 0 by (3.2), the Loewy series of P is

P = \begin{matrix}S \\ T \\ S_{1}S_{2}S_{3}S \\ T \\ S \\ T \\ S \\ T \\ S \end{matrix} .

Hence, (2.6)(ii), (3.9) and (1.5) imply that Q has Loewy series of the form

Q = \begin{matrix}T \\ S_{1}S_{2}S_{3}S \\ T \\ S_{1}S_{2}S_{3}S \\ T \\ S \\ T \\ S \\ T \end{matrix} .

There is a kG-epimorphism φ : Q/QJ^{6} ↠ QJ^{4}. By (3.9), Kerφ = S_{1} ⊕ S_{1} ⊕ S_{2} ⊕ S_{2} ⊕ S_{3} ⊕ S_{3}. Hence, there is a kG-submodule V of QJ^{5} such that QJ^{6} ⊆ V ⊆ QJ^{5} and V/QJ^{6} ≅ Kerφ. By (2.6)(ii), we get Q ≅ Q^{∗}, so that j(V) = 4. Hence, the Loewy and socle series of V are respectively

V = \begin{matrix}S_{1}S_{1}S_{2}S_{2}S_{3}S_{3} \\ T \\ S \\ T \end{matrix} (Loewy series)  and  V = \begin{matrix}S_{1}S_{2}S_{3} \\ T \\ SS_{1}S_{2}S_{3} \\ T \end{matrix} (socle series).

Since S_{1}|(V/Soc_{3}(V)), we get by [21, I Lemma 8.5(ii)] that V has a kG-submodule W such that j(W) = 4 and L_{1}(W) ≅ S_{1}. Let R = P (S_{1})/S_{1}. Then, by (3.9), there is an epimorphism ψ : R ↠ W. Since R has a unique simple submodule by (3.9), ψ is an isomorphism. Namely, R ≅ W ⊆ V. Then, (3.9) implies that S_{2} ⊕ S_{3} ⊆ V J^{2}/T, contradicting the Loewy series of V above. Hence j(P) ≠ 9.

Step 2: j(P) ≠ 8. Assume j(P) = 8. As in Step 1, the Loewy series of P is

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}S ⋯ \\ T ⋯ \\ ⋯ \\ ⋯ \\ ⋯ \\ S \end{matrix} , where T, T, S, S are left.

By (3.2), we get S |̸L_{2}(P) and S |̸L_{7}(P), so that T |L_{7}(P). Namely, the Loewy series of P is

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}S ⋯ \\ T ⋯ \\ ⋯ \\ ⋯ \\ T ⋯ \\ S \end{matrix} , where T, S, S are left.

If (T ⊕ T)|L_{2}(P), then P J^{4}/P J^{6} = U(S, S), contradicting (3.2). Hence, L_{2}(P) = T, so that (1.5) and (2.6)(ii) imply [L_{2}(Q), S]^{G} = 1. Thus, L_{3}(P) = S_{1} ⊕ S_{2} ⊕ S_{3} ⊕ S. Then, again by (2.6)(ii), (T ⊕ T) |̸L_{7}(P). Therefore, the Loewy series of P is

P = \begin{matrix}S \\ T \\ S_{1}S_{2}S_{3}S \\ T ⋯ \\ ⋯ \\ ⋯ \\ T \\ S \end{matrix} , where T, S, S are left.

If T |L_{6}(P), then (3.2) shows that P has a submodule X such that Soc(X) = S and X/Soc(X) = T ⊕ T. So that L_{1}(X^{∗}) = S and X^{∗}J = T ⊕ T by the self-dualities, contradicting the above. Hence, T |̸L_{6}(P), so that S|L_{6}(P). Thus, the Loewy series of P is

P = \begin{matrix}S \\ T \\ S_{1}S_{2}S_{3}S \\ T ⋯ \\ ⋯ \\ S ⋯ \\ T \\ S \end{matrix} , where T, S are left.

Clearly, S |̸L_{4}(P) by (3.9) and (3.2). If (S ⊕ S)|L_{6}(P), then Soc(Y) = T and Y/Soc(Y) = S ⊕ S by (3.2), where Y = P J^{5}/S, so that L_{1}(Y^{∗}) = T and Y^{∗}J = S ⊕ S by the self-dualities, contradicting [L_{2}(Q), S]^{G} = 1 above. Therefore L_{6}(P) ≅ S, which implies that the Loewy series of P is

P = \begin{matrix}S \\ T \\ S_{1}S_{2}S_{3}S \\ T ⋯ \\ ⋯ \\ S \\ T \\ S \end{matrix} , where T, S are left.

If T |L_{4}(P), then P J^{4}/P J^{6} = U(S, S), contradicting (3.2). Hence, T |L_{5}(P), so that the Loewy series of P is

P = \begin{matrix}S \\ T \\ S_{1}S_{2}S_{3}S \\ T ⋯ \\ T ⋯ \\ S \\ T \\ S \end{matrix} , where S is left.

If S|L_{5}(P), then L_{1}(Z) = T and ZJ = T ⊕ S where Z = P J^{3}/P J^{5}, contradicting (3.2). Hence the Loewy series of P is

P = \begin{matrix}S \\ T \\ S_{1}S_{2}S_{3}S \\ T S \\ T \\ S \\ T \\ S \end{matrix} .

Then, L_{1}(W) = S_{1} ⊕ S_{2} ⊕ S_{3} ⊕ S and W J = S where W = (P J^{4}/P J^{2})/T . This contradicts (3.9) and (3.2) as above. Therefore, j(P) ≠ 8.

Step 3: j(P) ≠ 7. Suppose j(P) = 7. As in Step 1, the Loewy series of P is

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}S ⋯ \\ T ⋯ \\ ⋯ \\ ⋯ \\ S \end{matrix} , where S, S, T, T are left.

Since S |̸L_{6}(P) by (3.2), it follows that T |L_{6}(P) and S |̸L_{2}(P). If (T ⊕ T)|L_{6}(P), then L_{2}(P) = T and there exists a module X with L_{1}(X) = S and XJ = T ⊕T by the self-dualities, a contradiction. Hence L_{6}(P) = T, which implies that the Loewy series of P is

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}S ⋯ \\ T ⋯ \\ ⋯ \\ T \\ S \end{matrix} , where S, S, T are left.

If T |L_{5}(P), then by (3.2) we get that L_{2}(P) = T and that P has a submodule Y with Soc(Y) = S and Y/Soc(Y) = T ⊕ T, so that L_{1}(Y^{∗}) = S and L_{2}(Y^{∗}) = T ⊕ T by the self-dualities, a contradiction. Thus, T |̸L_{5}(P), which implies S|L_{5}(P). Therefore, the Loewy structure of P is

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}S ⋯ \\ T ⋯ \\ S ⋯ \\ T \\ S \end{matrix} , where S, T are left.

If S|L_{4}(P), then by looking at P J^{2}/P J^{4} we know from (3.2) and (3.9) that T |L_{3}(P), so that L_{2}(P) = T, which means Ext_{kG}^{1}(T, T) ≠ 0, contradicting (3.2). Hence S |̸L_{4}(P).

Assume (T ⊕ T)|L_{4}(P). Then, L_{2}(P) = T, so that (1.5) and (2.6)(ii) show [L_{2}(Q), S]^{G} = 1. Hence, by looking at P J/P J^{4}, we get (S ⊕ S) |̸L_{3}(P). This implies (S ⊕ S)|L_{5}(P), so that the Loewy series of P is

P = \begin{matrix}S \\ T \\ S_{1}S_{2}S_{3}S \\ T T \\ SS \\ T \\ S \end{matrix} .

Hence (3.2) shows that Soc(Z) = T and Z/Soc(Z) = S⊕S where Z = P J^{4}/P J^{6}, so that L_{1}(Z^{∗}) = T and Z^{∗}J = S ⊕ S by the self-dualities, contradicting (1.5) and the above.

Therefore, (T ⊕ T) |̸L_{4}(P). If T |L_{3}(P), then L_{i}(V) = S, T, S_{1} ⊕ S_{2} ⊕ S_{3} ⊕ T ⊕ ⋯ for i = 1, 2, 3, respectively, where V = P/P J^{3}, contradicting the fact Ext_{kG}^{1}(T, T) = 0 by (3.2). Thus, T |̸L_{3}(P), so that L_{2}(P) = T ⊕ T. This shows that the Loewy series of P is

P = \begin{matrix}S \\ T T \\ S_{1}S_{2}S_{3}S ⋯ \\ T \\ S ⋯ \\ T \\ S \end{matrix} , where S is left.

If L_{5}(P) = S ⊕ S, then (3.2) implies that L_{3}(P) = S_{1} ⊕ S_{2} ⊕ S_{3} ⊕ S, and that Soc(W) = S, Soc_{2}(W)/Soc(W) = T, W/Soc_{2}(W) = S⊕S, where W = P J^{4}, so that (2.6)(ii) shows (S⊕S)|L_{3}(P), a contradiction. Hence, L_{5}(P) ≠ S ⊕ S, and hence the Loewy and socle series of P are respectively

P = \begin{matrix}S \\ T T \\ S_{1}S_{2}S_{3}SS \\ T \\ S \\ T \\ S \end{matrix} (Loewy series)  and  P = \begin{matrix}S \\ T \\ S \\ T \\ S_{1}S_{2}S_{3}SS \\ T T \\ S \end{matrix} (socle series).

Now, let H = H(P) = P J/S, the heart of P. Then H has submodules V_{1} and V_{2} such that HJ ⊆ V_{i}, V_{i}/HJ ≅ T for i = 1, 2 and V_{1} + V_{2} = H. Then, since j(H) = 5, j(V_{1}) = 5 or j(V_{2}) = 5. Thus, we may assume j(V_{1}) = 5. Then, since V_{1} ⊆ H, the socle series of V_{1} is

V_{1} = \begin{matrix}T \\ S \\ T \\ S_{1}S_{2}S_{3}SS \\ T \end{matrix} (socle series).

By looking at the socle series of H and V_{1}, H has a submodule W such that W ≅ T and Soc(H) = Soc(V_{1}) ⊕ W. Thus, V_{1} ∩ W ⊆ Soc(V_{1}), so that V_{1} ∩ W ⊆ Soc(V_{1}) ∩ W = 0. This implies that there exists a direct sum V_{1} ⊕ W ⊆ H, so that H = V_{1} ⊕ W. Therefore, H is decomposable, contradicting (3.5). Thus, j(P) ≠ 7.

Step 4: j(P) ≠ 6. Assume j(P) = 6. Then, as in Step 1, the Loewy series of P is

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}S ⋯ \\ T ⋯ \\ ⋯ \\ S \end{matrix} , where S, S, T, T are left.

By (3.2), the Loewy series of P is then

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}S ⋯ \\ T ⋯ \\ T ⋯ \\ S \end{matrix} , where S, S, T are left.

If (T ⊕ T)|L_{5}(P), then we get that L_{2}(P) = T by (3.2) and that L_{1}(X) = S and XJ = T ⊕ T by the self-dualities where X = (P J^{4})^{∗}, a contradiction. So by (3.2), L_{5}(P) = T. Namely, the Loewy series of P is

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}S ⋯ \\ T ⋯ \\ T \\ S \end{matrix} , where S, S, T are left.

Then, P J^{3} has a submodule Y such that P J^{4} ⊆ Y and Y/P J^{4} ≅ T. By (3.2), L_{1}(Y^{∗}) = S and Y^{∗}J = T ⊕ T by the self-dualities. Thus, L_{2}(P) = T ⊕ T by (3.2). That is, the Loewy series of P is

P = \begin{matrix}S \\ T T \\ S_{1}S_{2}S_{3}S ⋯ \\ T ⋯ \\ T \\ S \end{matrix} , where S, S are left.

Then, S|L_{4}(P) since Ext_{kG}^{1}(T, T) = 0 by (3.2). Hence, the Loewy series of P is

P = \begin{matrix}S \\ T T \\ S_{1}S_{2}S_{3}S ⋯ \\ T S ⋯ \\ T \\ S \end{matrix} , where S is left.

If (S ⊕ S)|L_{3}(P), then P J^{2}/P J^{4} has Loewy series \begin{matrix}S_{1}S_{2}S_{3}SS \\ T S \end{matrix} or \begin{matrix}S_{1}S_{2}S_{3}S \\ T SS \end{matrix}, contradicting (3.6).

Therefore j(P) ≠ 6.

By Steps 1–4, j(P) = 5. As in Step 1, by (3.2), the Loewy series of P is

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}SSS ⋯ \\ T ⋯ \\ S \end{matrix} , where T, T are left.

By (3.2), we know L_{3}(P) = S_{1} ⊕ S_{2} ⊕ S_{3} ⊕ 3 × S. Namely, the Loewy series of P is

P = \begin{matrix}S \\ T ⋯ \\ S_{1}S_{2}S_{3}SSS \\ T ⋯ \\ S \end{matrix} , where T, T are left.

If L_{2}(P) = T, then L_{1}(Z) = S and ZJ = T ⊕ T ⊕ T where Z = (P J^{3})^{∗} by the self-dualities, a contradiction. Hence, (T ⊕ T)|L_{2}(P). That is, the Loewy series of P is

P = \begin{matrix}S \\ T T ⋯ \\ S_{1}S_{2}S_{3}SSS \\ T ⋯ \\ S \end{matrix} , where T is left.

Next, assume that L_{2}(P) = T ⊕ T ⊕ T. Hence, the Loewy and socle series of P are, respectively,

P = \begin{matrix}S \\ T T T \\ S_{1}S_{2}S_{3}SSS \\ T \\ S \end{matrix} (Loewy series)  and  P = \begin{matrix}S \\ T \\ S_{1}S_{2}S_{3}SSS \\ T T T \\ S \end{matrix} (socle series).

Let H = H(P) = P J/S, the heart of P. Then, H has submodules V_{1} and V_{2} such that HJ ⊆ V_{i}, V_{1}/HJ ≅ T ⊕ T, V_{2}/HJ ≅ T and V_{1} + V_{2} = H. Since j(H) = 3, we know j(V_{1}) = 3 or j(V_{2}) = 3.

Assume, first, that j(V_{1}) = 3. Then, Soc(V_{1}) = T ⊕ T, Soc_{2}(V_{1})/Soc(V_{1}) = S_{1} ⊕ S_{2} ⊕ S_{3} ⊕ 3 × S, V_{1}/Soc_{2}(V_{1}) = T. Thus, H has a submodule W such that W ≅ T and Soc(H) = Soc(V_{1}) ⊕ W. Then, V_{1} ∩ W ⊆ Soc(V_{1}) ∩ W = 0, so that there is a direct sum V_{1} ⊕ W ⊆ H. Hence H = V_{1} ⊕ W. Thus, H is decomposable, contradicting (3.5).

Assume, next, that j(V_{2}) = 3. Then, Soc(V_{2}) = T, Soc_{2}(V_{2})/Soc(V_{2}) = S_{1} ⊕ S_{2} ⊕ S_{3} ⊕ 3 × S, V_{2}/Soc_{2}(V_{2}) = T. Thus, H has a submodule W′ such that W′ ≅ T ⊕T and Soc(H) = Soc(V_{2})⊕W′. Therefore, V_{2} ∩ W′ ⊆ Soc(V_{2}) ∩ W′ = 0, so that there is a direct sum V_{2} ⊕ W′ ⊆ H. Hence H = V_{2} ⊕ W′. Thus, H is decomposable, contradicting (3.5).

Hence, we finally know that P has Loewy series as stated in (3.1). Thus, (2.6)(v) and (1.5) show that the Loewy series of Q is

Q = T ; S_{1}S_{2}S_{3}SS ⋯; T ⋯; S_{1}S_{2}S_{3}SS ⋯; ⋮ ; T , where T, T are left.

By (3.2), j(Q) = 5. Thus again by (3.2), the Loewy series of Q is as stated in (3.1). This completes the proof of (3.10). □

4. Green correspondents of simples in A and B, and a proof of the main result

In this section we calculate structure of Green correspondents of simple kG-modules in A.

(4.1) Lemma. Let M be as in (2.13), and let Y = (−1)_{M}↑^{G}·1_{A} where (−1)_{M} is a non-trivial one-dimensional kM-module given by the extension M ▷ 4·L_{3}(4). Then it follows Y = S ⊕ P (S_{1}). In particular, S is a trivial source kG-module affording χ_{10944}.

Proof. By (2.13)(iii), (1.1)(i) and (2.6)(iv), it holds

(1)  χ_{Y} = χ_{10944} + χ_{58311_{1}} + χ_{58653}

and

(2)  Y = S + (S + T + S_{1}) + (T + S_{1} + S_{2} + S_{3}),  as composition factors.

Then, (1) and a theorem of Scott (1.1)(ii) show that

(3)  [Y, Y]^{G} = 3.

It follows from (2.4)(ii) and (1.2)(ii) that only the following three cases can occur. Namely,

(a) Y = Y_{1} ⊕ Y_{2} for indecomposable kG-modules Y_{1} and Y_{2} such that χ_{Y_{1}} = χ_{10944} + χ_{58311_{1}} and

χ_{Y_{2}} = χ_{58653}.

(b) Y is indecomposable.

(c) Y = Y_{1} ⊕ Y_{2} for indecomposable kG-modules Y_{1} and Y_{2} such that χ_{Y_{1}} = χ_{10944} and χ_{Y_{2}} = χ_{58311_{1}} + χ_{58653}. Case (a). Then χ_{Y_{1}}(3A) = 0 by (2.4)(ii), which means from (1.2)(iii) that Y_{1} is a projective

kG-module, contradicting (2.6)(iv).

Case (b). From (2.6)(iv) and (1.1)(i), S_{i} is not a trivial source module for any i = 1, 2, 3. Since c_{Y} (S_{2}) = c_{Y} (S_{3}) = 1, (2.6)(ii) implies that

(4)  S_{i} |̸L_{1}(Y)  and  S_{i} |̸Soc(Y)  for i = 2, 3.

By (2.6)(iv), χ_{10944} affords S, so that (1.3) implies

(5)  S|L_{1}(Y)  and  S|Soc(Y).

Again from (2.6)(iv), Y is not projective. Hence, (3.1) and (4) show that j(Y) = 3 or 4.

Suppose Soc(Y) = S. Then, Y ↪ P (S)·J. Thus, it follows from (2) and (2.6)(v) that 2 = c_{Y} (S_{1}) ⩽ c(S, S_{1}) = 1, a contradiction. Hence, Soc(Y) ≠ S.

Next, assume that there are three simple kG-modules L_{1}, L_{2} and L_{3} such that (L_{1} ⊕ L_{2} ⊕ L_{3})|Soc(Y). Then, since (L_{1} ⊕ L_{2} ⊕ L_{3}) ⊊ Y, we have [Y, Y]^{G} ⩾ 4 by (2.6)(ii), contradicting (3). Hence, Soc(Y) contains at most two direct summands of simple modules.

Therefore, by (5), we have Soc(Y) = S⊕V for a simple kG-module V. If V ≅ S, then [Y, Y]^{G} ⩾ 5 by (2.6)(ii), contradicting (3). Hence, by (2) and (4), Soc(Y) = S ⊕ S_{1} or S ⊕ T.

Assume Soc(Y) = S ⊕ S_{1}. By (2.6)(ii), L_{1}(Y) ≅ S ⊕ S_{1}. Now, Y ↪ P (S)·J ⊕ P (S_{1})·J. Thus, by (3.1) and (2), the socle series of Y is

Y = ⋮ ; S_{1}S_{2}S_{3}S ⋯; T ⋯; SS_{1} , where T is left  (socle series).

Since S_{2} |̸L_{1}(Y) by (4), the socle series of Y is

Y = \begin{matrix}T \\ S_{1}S_{2}S_{3}S \\ T \\ SS_{1} \end{matrix} , (socle series).

Hence T |L_{1}(Y), a contradiction.

Therefore, Soc(Y) = S ⊕T. Then, as above, L_{1}(Y) ≅ S ⊕T. Now, Y ↪ P (S)·J ⊕P (T)·J. Since S_{i} |̸L_{1}(Y) for i = 1, 2, 3, (3.1) implies that the socle series of Y is

Y = ⋮ ; S_{1} ⋯; S_{1} ⋯; ST , where T, S_{2}, S_{3}, S are left (socle series).

Since S_{1} |̸L_{1}(Y), we have j(Y) = 4. Namely, the socle series of Y is

Y = \begin{matrix}⋯ \\ S_{1} ⋯ \\ S_{1} ⋯ \\ ST \end{matrix} , where T, S_{2}, S_{3}, S are left (socle series).

Again since S_{i} |̸L_{1}(Y) for all i, we know T |(Y/Soc_{3}(Y)). That is, the socle series of Y is

Y = \begin{matrix}T ⋯ \\ S_{1} ⋯ \\ S_{1} ⋯ \\ ST \end{matrix} , where S_{2}, S_{3}, S are left (socle series).

Let V = Soc_{3}(Y)/Soc_{1}(Y). Then, c_{V} (T) = 0 by (2), and Soc(V) = S_{1} ⊕ ⋯ and V/Soc(V) = S_{1} ⊕ ⋯, contradicting (3.1).

Therefore, only Case (c) occurs. Hence, (2.6)(iv) implies that Y_{1} ≅ S. Now, χ_{Y_{2}}(3A) = 0 from (2.4)(ii). Hence, as in the proof of the Case (a), Y_{2} is a projective indecomposable kG-module. Thus, again by (2.6)(iv), Y_{2} = P (S_{1}). This completes the proof of (4.1). □

(4.2) Notation. In the rest of this paper except §5, let f : G → H and g : H → G be the Green correspondences with respect to (G, D, H).

(4.3) Lemma. (i) For any non-projective indecomposable kG-module U in A, U↓_{H}·1_{B} = fU ⊕ (proj).

(ii) For any non-projective indecomposable kH-module V in B, V ↑^{G}·1_{A} = gV ⊕ (proj).

Proof. Easy by (2.7)(iii) and [24, Chapter 4 Theorem 4.3(i)]. □

(4.4) Lemma. It holds the following.

(i) S↓_{H}·1_{B} = φ̃_{6} ⊕ (6 × P (φ̃_{5}) ⊕ 5 × P (φ̃_{6}) ⊕ 5 × P (φ̃_{13}) ⊕ 5 × P (φ̃_{14}) ⊕ 6 × P (φ̃_{18})). In particular, fS = φ̃_{6} and gφ̃_{6} = S.

(ii) φ̃_{6}↑^{G}·1_{A} = S ⊕ (5 × P (S) ⊕ 15 × P (T) ⊕ P (S_{1}) ⊕ 2 × P (S_{2}) ⊕ 2 × P (S_{3})).

Proof. By (1.7), (4.1) and [25, Lemma 2.2] by Okuyama, fS is a simple kH-module in B.

Assume that fS = φ̃_{18}. Then, by (4.3)(ii), φ̃↑^{G}·1_{A} = S ⊕ (proj). (2.9)(iii) implies that [(χ̃_{18}↑^{G})·1Â](1) ≡ 18 (mod 81). This is a contradiction by (2.6)(iv) and (2.11)(ii).

Therefore, fS ∈ {φ̃_{5}, φ̃_{6}, φ̃_{13}, φ̃_{14}} by (2.11)(ii). We then obtain by (2.9)(iii) and (2.11)(ii) that [P (φ̃_{18})|S↓_{H}] = 6.

Assume, first, that fS = φ̃_{5}. Then, gφ̃_{5} = S. Note that φ̃_{5}↑^{G}·1_{A} = S ⊕ P for a projective kG-module P by (4.3)(ii). Thus, by (2.9)(iii), we know

Φ_{P} = 5 × χ_{10944} + 22 × χ_{52668} + 25 × χ_{58311_{1}} + 21 × χ_{58311_{2}} + 21 × χ_{58311_{3}} + 18 × χ_{58653}.

Then, it follows by (2.6)(iv) that [P (S)|φ̃_{5}↑^{G}] = 5, so that [P (T)|φ̃_{5}↑^{G}] = 17. However,

(Φ_{P} − 5 × Φ_{P (S)} − 17 × Φ_{P (T)}, χ_{58311_{2}})_{G} = 21 − 5 − 17 < 0

from (2.6)(iv). This is a contradiction. Hence, fS ≠ φ̃_{5}.

Suppose, next, that fS = φ̃_{13}. This is a contradiction since σ fixes S but does not fix φ̃_{13} from (2.6)(iii) and (2.11)(vi). Hence, fS ≠ φ̃_{13}.

Similarly, we get fS ≠ φ̃_{14}. Therefore, fS = φ̃_{6}. The rest is obtained similarly as above. □

(4.5) Lemma. (i) There are exactly five trivial source kG-modules in A with vertex D, which are gφ̃_{5}, gφ̃_{6}, gφ̃_{13} gφ̃_{14} and gφ̃_{18}.

(ii)  dim_{k} gφ̃_{i} ≡ \{ \begin{matrix}9 (mod 81), & for i = 5, 6, 13, 14 \\ 18 (mod 81), & for i = 18. \end{matrix}

(iii)  [S, gφ̃_{i}]^{G} = [gφ̃_{i}, S]^{G} = \{ \begin{matrix}1, & for i = 6 \\ 0, & for i = 5, 13, 14, 18. \end{matrix}

Proof. (i) This follows from [24, Chapter 4 Exercise 10] and (2.11)(v) .

(ii) Easy by (2.11)(ii) and (2.7)(iii) since the order of a Sylow 3-subgroup is 81.

(iii) By (1.6), we know [S, φ̃_{i}↑^{G}]^{G} = [S↓_{H}, φ̃_{i}]^{H} and [P (S)|φ̃_{i}↑^{G}] = [P (φ̃_{i})|S↓_{H}] for i = 5, 6, 13, 14, 18. Hence, (4.4)(i)-(ii) implies that the values [S, gφ̃_{i}]^{G} are as desired. □.

(4.6) Lemma. Let M be as in (2.13), and let X = k_{M}↑^{G}·1_{A}. Then,

X = S  ⊕  \begin{matrix}T \\ SSS_{1} \\ T \end{matrix}  (Loewy and socle series) ⊕ P (S)
↕  ↕
χ_{10944}  χ_{52668} + χ_{58311_{1}}

Proof. It follows from (1.1)(i) and (2.13)(ii) that

(1)  χ_{X} = 2 × χ_{10944} + 2 × χ_{52668} + 2 × χ_{58311_{1}} + χ_{58311_{2}} + χ_{58311_{3}}.

Hence, by (1.1)(ii), (4.1) and (1), we know

(2)  [S, X]^{G} = [X, S]^{G} = 2.

Since c_{X}(S) = 8 > 2 by (1) and (2.6)(iv), (2) shows that

(3)  j(X) ⩾ 3.

(2.4)(ii) and (1) imply

(4)  χ_{X}(3A) = 18.

First, suppose that X is indecomposable as kG-module. Since ν_{3}(dim_{k} X) = 2, we get by (1.8) that D is a vertex of X. By (1), dim_{k} X ≡ 18 (mod 81). Thus, (4.5)(i)-(ii) shows X ≅ gφ̃_{18}, so that [S, X]^{G} = 0 by (4.5)(iii), contradicting (2).

Therefore, X is decomposable.

Next, assume that X is projective-free. Then, X is written as X = X_{1} ⊕ X_{2} ⊕ ⋯ ⊕ X_{n} for an integer n ⩾ 2 and indecomposable kG-modules X_{1}, ⋯, X_{n}. By (1.2)(iii), χ_{X_{i}}(3A) > 0 for all i. Hence, (2.4)(ii) and (1) imply that χ_{X_{i}}(3A) = 9m_{i} for an integer m_{i} ⩾ 1 for each i. Thus, n = 2 by (4). Namely, X = X_{1} ⊕X_{2} and χ_{X_{i}}(3A) = 9 for i = 1, 2. Now, by (1) and (2.4)(ii), there are exactly two 18’s, two 9’s and four −9’s appearing in the values χ_{X}(3A). Therefore, only the following four cases occur:

χ_{X_{1}}(3A)  χ_{X_{2}}(3A)
(a) 18 + 9 − 9 − 9 18 + 9 − 9 − 9
(b) 18 − 9  18 + 9 + 9 − 9 − 9 − 9
(c) 9 + 9 − 9  18 + 18 − 9 − 9 − 9
(d) 9  18 + 18 + 9 − 9 − 9 − 9 − 9

Case (a). By (1) and (2.4)(ii), χ_{X_{1}} = χ_{10944} + χ_{52668} + χ_{58311_{i}} + χ_{58311_{j}} for some i, j ∈ {1, 2, 3} (possibly, i = j = 1). Then, ν_{3}(dim_{k} X_{1}) = 2, so that D is a vertex of X_{1} from (1.8). Since dim_{k} X_{1} ≡ 9 (mod 81), (4.5)(i)-(ii) implies X_{1} ∈ {gφ̃_{5}, gφ̃_{6}, gφ̃_{13}, gφ̃_{14}}. On the other hand, by (1.1)(ii) and (4.1), we know [S, X_{1}]^{G} = (χ_{10944}, χ_{X_{1}})_{G} = 1. These imply X_{1} = gφ̃_{6}, so that X_{1} = S from (4.4)(i). Hence, χ_{X_{1}} = χ_{S} = χ_{10944} by (4.1) and (2.6)(iv), a contradiction.

Case (b). As in Case (a), we get χ_{X_{2}} = 2 × χ_{10944} + χ_{52668} + χ_{58311_{i}} + χ_{58311_{j}} + χ_{58311_{k′}}

for i, j, k′ ∈ {1, 2, 3}. Then, just as in Case (a), X_{2} ∈ {gφ̃_{5}, gφ̃_{6}, gφ̃_{13}, gφ̃_{14}}. Hence, [S, X_{2}]^{G} = (χ_{10944}, χ_{X_{2}})_{G} = 2 as in Case (a), contradicting (4.5)(iii).

Case (c). As in Case (a), (χ_{X_{1}}, χ_{10944})_{G} = 2. Hence, we get a contradiction by (4.5)(iii) just as in Case (b).

Case (d). As in Case (a), X_{1} ≅ X_{2} ≅ S. Hence χ_{X} = 2 × χ_{10944}, contradicting (1).

Thus, X contains a non-zero projective direct summand. So, X = P (S′) ⊕ V for a simple kGmodule S′ in A and a kG-module V ≠ 0. Since (χ_{X}, χ_{58653})_{G} = 0 by (1), (2.6)(iv) implies that S′ = S, so that χ_{V} = χ_{10944} + χ_{52668} + χ_{58311_{1}} by (1) and (2.6)(iv). Namely, X = P (S) ⊕ V.

Assume V is indecomposable. Since ν_{3}(dim_{k} V) = 2, D is a vertex of V from (1.8). Clearly, dim_{k} V = 18 (mod 81). Hence, as the above, V = gφ̃_{18} by (4.5)(i)-(ii). Hence, [S, V]^{G} = 0 by (4.5)(iii). On the other hand, as the above, [S, V]^{G} = (χ_{10944}, χ_{V})_{G} = 1, a contradiction.

Hence, V is decomposable. So, V = V_{1}⊕V_{2} for kG-modules V_{1} and V_{2} such that V_{i} ≠ 0 for i = 1, 2. It follows from (1.2)(iii) and (2.4)(ii) that we may assume χ_{V_{1}} = χ_{10944} and χ_{V_{2}} = χ_{52668} + χ_{58311_{1}}. Hence, (1.1)(i) and (2.6)(iv) say V_{1} = S. Therefore, X = P (S) ⊕ S ⊕ V_{2}. Then, again by (1.2)(iii) and (2.4)(ii), V_{2} is indecomposable. Clearly, (2.6)(iv) implies that V_{2} = (S + T) + (S + T + S_{1}), as composition factors. By (2.6)(iv) and (1.1)(i), S_{1} is not a trivial source kG-module. Hence, [S_{1}, V_{2}]^{G} = [V_{2}, S_{1}]^{G} = 0 by the self-dualities. Thus, j(V_{2}) ⩾ 3. Moreover, by (4.1), (1.1)(ii) and (2.6)(iv), and also by the self-dualities, we know that [S, V_{2}]^{G} = [V_{2}, S]^{G} = (χ_{10944}, χ_{V_{2}})_{G} = 0. By (1.3), V_{2} has a kG-submodule W such that W = S + T + S_{1}, as composition factors. Thus, Soc(W) = T. Hence, by (3.1), W/Soc(W) = S ⊕ S_{1}. By (3.1), j(V_{2}) ⩽ 4. If j(V_{2}) = 4, then the socle series of V_{2} is

V_{2} = \begin{matrix}T \\ S \\ SS_{1} \\ T \end{matrix} (socle series)

since [V_{2}, S]^{G} = 0, but the structure of Soc_{3}(V_{2})/T contradicts (3.1). Thus, j(V_{2}) = 3. Then, again since [V_{2}, S]^{G} = 0, the socle series of V_{2} is

V_{2} = \begin{matrix}T \\ SSS_{1} \\ T \end{matrix} (socle series).

By the self-dualities, the shape of the Loewy series of V_{2} is also the same. □

(4.7) Lemma. Let V be the trivial source kG-module in (4.6) such that

V = \begin{matrix}T \\ SSS_{1} \\ T \end{matrix} (Loewy and socle series)  ⟷  χ_{52668} + χ_{58311_{1}}.

(i) It holds that V has D as its vertex.

(ii) V ↓_{H}·1_{B} = φ̃_{5} ⊕ (46 × P (φ̃_{5}) ⊕ 41 × P (φ̃_{6}) ⊕ 41 × P (φ̃_{13}) ⊕ 41 × P (φ̃_{14}) ⊕ 72 × P (φ̃_{18})), so that fV = φ̃_{5} and gφ̃_{5} = V.

(iii) φ̃_{5}↑^{G}·1_{A} = V ⊕ (6 × P (S) ⊕ 15 × P (T) ⊕ 3 × P (S_{1})).

Proof. (i) Since ν_{3}(dim_{k} V) = 2 = ν_{3}(|G|) − ν_{3}(|D|), (1.8) implies (i).

(ii) and (iii): Clearly, dim_{k} V ≡ 9 (mod 81). Hence, by (i), (4.4) and (4.5)(ii), we get V ∈ {gφ̃_{i} | i = 5, 13, 14}.

Suppose that V = gφ̃_{13}. Then, (φ̃_{14})↑^{G} = (φ̃_{13}^{σ})↑^{G} = (φ̃_{13}↑^{G})^{σ} by (2.11)(vi). Therefore, V^{σ} = gφ̃_{14}. On the other hand, since the maximal subgroup M of G is unique up to G-conjugacy by [6, p.132] (or [37] and [38]), we know that (k_{M}↑^{G}·1_{A})^{σ} ≅ k_{M}↑^{G}·1_{A}, so that V^{σ} ≅ V, a contradiction.

Similarly, we have a contradiction if V = gφ̃_{14} since φ̃_{13}^{σ} = φ̃_{14} by (2.11)(vi).

Thus, V = gφ̃_{5}. Then, the rest is easy by (4.3)(ii), (2.9)(iii), (2.11)(ii) and (2.6)(iv). □

(4.8) Lemma. Let L be the same as in (2.13), and let U = k_{L}↑^{G}·1_{A}. Let Q be a vertex of U (since U is indecomposable by (3.4)).

(i) U = U(S, T, S) ↔ χ_{10944} + χ_{52668} and Q is the cyclic group of order 3.

(ii) The Green correspondent fU of U satisfies

U↓_{H}·1_{B} = fU ⊕ (27 × P (φ̃_{5}) ⊕ 25 × P (φ̃_{6}) ⊕ 24 × P (φ̃_{13}) ⊕ 24 × P (φ̃_{14}) ⊕ 40 × P (φ̃_{18})),

(fU)↑^{G}·1_{A} = U ⊕ (54 × P (S) ⊕ 175 × P (T) ⊕ 24 × P (S_{1}) ⊕ 20 × P (S_{2}) ⊕ 20 × P (S_{3})),

and fU has Loewy and socle series

fU = \begin{matrix}φ̃_{5} & φ̃_{6} & φ̃_{13} & φ̃_{14} \\ φ̃_{18} & φ̃_{18} \\ φ̃_{5} & φ̃_{6} & φ̃_{13} & φ̃_{14} \end{matrix} ⟷ χ̃_{5} + χ̃_{6} + χ̃_{13} + χ̃_{14} + χ̃_{25}.

Proof. (i) By (3.4), it is sufficient to show Q = C_{3}. Clearly, dim_{k} U ≡ 27 (mod 81), so that U is not projective by (2.1). We can choose Q as subgroup of D. If Q = D, then U is a trivial source kG-module in A with vertex D, a contradiction by (4.5)(i)–(ii).

(ii) By (4.3)(ii), U↓_{H}·1_{B} = fU ⊕ P̃ for a projective kH-module P̃ in B. Then, fU is a trivial source kH-module in B with vertex Q, see [24, Chapter 4 Theorem 4.3(i)(c)]. Then, by (2.14), fU = X or fU = Y, where X and Y are the modules appearing in (2.14) in this order. It follows from (2.7)(iii), [1, p.90, Corollary 5], (4.4)(i) and [21, II Corollary 2.8] that 0 ≠ (S, U)^{G} = \underline{Hom}_{kG}(S, U) ≅ \underline{Hom}_{kH}(fS, fU) = \underline{Hom}_{kH}(φ̃_{6}, fU) = (φ̃_{6}, fU)^{H}, so that φ̃_{6}|Soc(fU). Thus, fU = X. Hence we know the Loewy and socle series of fU, and we know also the decomposition of χ_{fU} from (2.14)(i). Then, the rest in (ii) is easily obtained by (2.9)(iii) and (2.6)(iv). □

(4.9) Lemma. We get the following.

(i) T ↓_{H}·1_{B} = fT ⊕ (15 × P (φ̃_{5}) ⊕ 15 × P (φ̃_{6}) ⊕ 14 × P (φ̃_{13}) ⊕ 14 × P (φ̃_{14}) ⊕ 28 × P (φ̃_{18})), and fT has Loewy and socle series

fT = \begin{matrix}φ̃_{13} & φ̃_{5} & φ̃_{14} \\ φ̃_{18} & φ̃_{18} \\ φ̃_{14} & φ̃_{5} & φ̃_{13} \end{matrix}

.

(ii) (fT)↑^{G}·1_{A} = T ⊕ (44 × P (S) ⊕ 145 × P (T) ⊕ 22 × P (S_{1}) ⊕ 16 × P (S_{2}) ⊕ 16 × P (S_{3})).

Proof. We get [P (φ̃_{5})|T ↓_{H}] = [P (φ̃_{6})|T ↓_{H}] = 15 by (4.7)(iii), and (4.4)(ii) and (1.6). Hence, T ↓_{H}·1_{B} = fT ⊕ (15 × P (φ̃_{5}) ⊕ 15 × P (φ̃_{6}) ⊕ m_{13} × P (φ̃_{13}) ⊕ m_{14} × P (φ̃_{14}) ⊕ m_{18} × P (φ̃_{18})) for non-negative integers m_{13}, m_{14} and m_{18} by (2.6)(ii) and (4.3)(i). Hence, (4.4)(i) implies that

\begin{matrix}S↓_{H}·1_{B} & | & T ↓_{H}·1_{B} & | & S↓_{H}·1_{B} \\ = ( φ̃_{6} | fT | φ̃_{6} ) ⨁ \binom{27 × P (φ̃_{5})⊕25 × P (φ̃_{6}) ⊕(10 + m_{13}) × P (φ̃_{13})}{⊕(10 + m_{14}) × P (φ̃_{14})⊕(12 + m_{18}) × P (φ̃_{18})} . \end{matrix}

Let U be the same as in (4.8). Then, by (4.8)(ii),

\begin{matrix}fU ⊕ (14 × P (φ̃_{13}) ⊕ 14 × P (φ̃_{14}) ⊕ 28 × P (φ̃_{18})) \\ = ( φ̃_{6} | fT | φ̃_{6} ) ⨁ (m_{13} × P (φ̃_{13}) ⊕ m_{14} × P (φ̃_{14}) ⊕ m_{18} × P (φ̃_{18})). \end{matrix}

Then, m_{13} = m_{14} = 14 and m_{18} = 28, and fU has Loewy and socle series as desired by (4.8)(ii). (ii) Since (fT)↑^{G}·1_{A} = T ⊕ (proj) by (2.6)(ii) and (4.3)(ii), we get the assertion by (2.6)(iv). □

(4.10) Lemma. (i) [fS_{1}, φ̃_{i}]^{H} = [φ̃_{i}, fS_{1}]^{H} = 0 for i = 5, 6.

(ii) [P (φ̃_{18})|S_{1}↓_{H}] = [P (S_{1})|φ̃_{18}↑^{G}] = 4.

(iii) [fS_{1}, φ̃_{18}]^{H} = [φ̃_{18}, fS_{1}]^{H}≠0.

Proof. By (2.6)(ii), let M̃ = fS_{1}. Then, by (4.3)(i), S_{1}↓_{H}·1_{B} = M̃ ⊕Q̃ for a projective kH-module Q̃ in B.

(i) It follows [M̃, φ̃_{5}]^{H} = [S_{1}↓_{H}, φ̃_{5}]^{H} − [P (φ̃_{5})|S_{1}↓_{H}] = 0 by (4.7)(iii) and (1.6). Similarly, by (4.4)(ii) and (1.6), we know [M̃, φ̃_{6}]^{H} = 0.

(ii) Let V be the same as in (4.7). Then, we get from (4.9)(i) and (4.4)(i) that

fT = \begin{matrix}φ̃_{13} & φ̃_{5} & φ̃_{14} \\ φ̃_{18} & φ̃_{18} \\ φ̃_{14} & φ̃_{5} & φ̃_{13} \end{matrix} , V ↓_{H}·1_{B} = X̃ ⊕ ( \begin{matrix}42 × P (φ̃_{5}) & ⊕40 × P (φ̃_{6}) & ⊕38 × P (φ̃_{13}) \\ 38 × P (φ̃_{14}) & ⊕68 × P (φ̃_{18}) & ⊕Q̃ \end{matrix} )

for a kH-module X̃ in B such that

X̃ = fT | φ̃_{6} ⊕ φ̃_{6} ⊕ M̃ | fT .

By (1.6), (4.7)(iii) and (4.4)(ii), we can write that Q̃ = 3 × P (φ̃_{5}) ⊕ P (φ̃_{6}) ⊕ m_{13} × P (φ̃_{13}) ⊕ m_{14} × P (φ̃_{14}) ⊕ m_{18} × P (φ̃_{18}) for non-negative integers m_{13}, m_{14} and m_{18}. So, (4.7)(ii) and the theorem of Krull-Schmidt imply that φ̃_{5} ⊕ (P (φ̃_{5}) ⊕ 3 × P (φ̃_{13}) ⊕ 3 × P (φ̃_{14}) ⊕ 4 × P (φ̃_{18})) =

X̃ ⊕ (m_{13} × P (φ̃_{13}) ⊕ m_{14} × P (φ̃_{14}) ⊕ m_{18} × P (φ̃_{18})).

Now, we want to claim that P (φ̃_{18}) |̸X̃. Let P̃ = P (φ̃_{18}).

Suppose P̃|X̃. There are kH-submodules Ỹ and Z̃ of X̃ such that X̃ ⊇ Ỹ ⊇ Z̃, X̃/Ỹ ≅ Z̃ ≅ fT and Ỹ /Z̃ ≅ φ̃_{6} ⊕ φ̃_{6} ⊕ M̃. Since φ̃_{18} |̸Soc(Z̃) by (4.9)(i), there is a direct sum P̃ ⊕ Z̃ ⊆ X̃. Thus,

P̃ ≅ (P̃ ⊕ Z̃)/Z̃ ⊆ X̃/Z̃ = (X̃/Ỹ) | (Ỹ /Z̃) .

Let Ũ = (X̃/Z̃)/(φ̃_{6} ⊕ φ̃_{6}). Since P̃ ∩ (φ̃_{6} ⊕ φ̃_{6}) = 0, we similarly know that

P̃ ≅ (P̃ ⊕ φ̃_{6} ⊕ φ̃_{6})/(φ̃_{6} ⊕ φ̃_{6}) ⊆ Ũ = fT | M̃ .

Hence, Ũ has a submodule Ṽ such that Ũ/Ṽ ≅ φ̃_{5} ⊕ φ̃_{13} ⊕ φ̃_{14} and Ṽ = \begin{matrix}φ̃_{18} & φ̃_{18} \\ φ̃_{5} & φ̃_{13} & φ̃_{14} \end{matrix} | M̃ . It follows

from (2.11)(iv) and [21, I Lemma 8.5(ii)] that P̃ has a submodule W̃ such that j(W̃) = 4 and L_{1}(W̃) = φ̃_{6}. Then, W̃ ⊆ M̃. Since j(M̃) ⩽ 4 by (2.11)(iv), we finally know that j(M̃) = 4 and φ̃_{6}|L_{1}(M̃), contradicting (i).

Hence, m_{18} = 4, which implies the rest of (ii) by (1.6).

(iii) Assume [M̃, φ̃_{18}]^{H} = [φ̃_{18}, M̃]^{H} = 0. Then, (2.11)(ii), (1.7), (i), (2.6)(iii), (2.11)(vi) and (2.6)(iv) show that L_{1}(M̃) = n × φ̃_{13} ⊕ n × φ̃_{14} for a positive integer n. By the self-dualities, Soc(M̃) ≅ L_{1}(M̃). Since M̃ ⊆ n × (P (φ̃_{13})·J̃ ⊕ P (φ̃_{14})·J̃) and since [M̃, φ̃_{18}]^{H} = 0, (2.11)(iv) implies that

M̃ ⊆ n × {( φ̃_{5}φ̃_{6}φ̃_{14} ; φ̃_{18} ; φ̃_{13} ) ⊕ ( φ̃_{5}φ̃_{6}φ̃_{13} ; φ̃_{18} ; φ̃_{14} )}, (socle series).

Then, by (i), M̃ ⊆ n×(U(φ̃_{14}, φ̃_{18}, φ̃_{13})⊕U(φ̃_{13}, φ̃_{18}, φ̃_{14})). Therefore, cM̃(φ̃_{i}) = 0 for i = 5, 6. Let

X̃ be the same as in the proof of (ii). Then, cX̃(φ̃_{6}) = 2. Now, it follows from (2.6)(iii), (2.11)(vi) and the proof of (ii) that

(*)  φ̃_{5} ⊕ (P (φ̃_{5}) ⊕ 3 × P (φ̃_{13}) ⊕ 3 × P (φ̃_{14})) = X̃ ⊕ (m × P (φ̃_{13}) ⊕ m × P (φ̃_{14}))

for a non-negative integer m. Hence, m ⩽ 3. Then, (2.11)(iii) shows that the multiplicity of φ̃_{6} at the left hand side of (*) is 7. On the other hand, at the right hand side of (*), the multiplicity is 2 + m + m, so this is an even number, a contradiction. □

(4.11) Lemma. It holds the following.

(i)

gφ̃_{18} = \begin{matrix}S_{1}S_{2}S_{3} \\ T T \\ S_{1}S_{2}S_{3} \end{matrix} (Loewy and socle series) ⟷ 2 × χ_{58653}

and φ̃_{18}↑^{G}·1_{A} = gφ̃_{18} ⊕ (6 × P (S) ⊕ 28 × P (T) ⊕ 4 × P (S_{1}) ⊕ 4 × P (S_{2}) ⊕ 4 × P (S_{3})).

(ii) (gφ̃_{18})↓_{H}·1_{B} = φ̃_{18} ⊕ \binom{36 × P (φ̃_{5})⊕40 × P (φ̃_{6}) ⊕40 × P (φ̃_{13})}{⊕40 × P (φ̃_{14})⊕83 × P (φ̃_{18})}.

(iii) [fS_{1}, φ̃_{18}]^{H} = [φ̃_{18}, fS_{1}]^{H} = 1.

Proof. Let X = gφ̃_{18}. Note that X is a trivial source kG-module in A by (4.5)(i).

(i) By (4.3)(ii), we get φ̃_{18}↑^{G}·1_{A} = X ⊕ P for a projective kG-module P in A. By (1.6), (4.4)(i), (4.9)(i), (4.10)(ii), (2.6)(iii), (2.9)(ii) and (2.11)(ii), it follows

(1)  P = 6 × P (S) ⊕ 28 × P (T) ⊕ 4 × P (S_{1}) ⊕ m × P (S_{2}) ⊕ m × P (S_{3})

for a non-negative integer m. Now, we get from (2.11)(ii), (2.9)(iii), (2.6)(iv) and (1.1)(i) that

(2)  X ↔ (4 − m) × χ_{58311_{2}} + (4 − m) × χ_{58311_{3}} + (10 − 2m) × χ_{58653}.

We know [P (S)|φ̃_{18}↑^{G}] = [P (φ̃_{18})|S↓_{H}] = 6 = [φ̃_{18}, S↓_{H}]^{H} by (1.6) and (4.4)(i), which means that [X, S]^{G} = [S, X]^{G} = 0 by the self-dualities. Similarly, [P (T)|φ̃_{18}↑^{G}] = [P (φ̃_{18})|T ↓_{H}] = 28 = [φ̃_{18}, T ↓_{H}]^{H} by (1.6) and (4.9)(i), and hence [X, T]^{G} = [T, M]^{G} = 0 by the self-dualities. That is,

(3)  S |̸L_{1}(X), T |̸L_{1}(X), S |̸Soc(X), T |̸Soc(X).

Next, we want to show (χ_{X}, χ_{58311_{2}})_{G} = 0. Assume (χ_{X}, χ_{58311_{2}})_{G} ≠ 0. Then, we get from (1.3) and (2.6)(iv) that X has a submodule X′ such that X′ = S + T + S_{2}, as composition factors. Hence (3) and (3.1) imply that X′ = U(S, T, S_{2}). Then, again by (3), j(X) ⩾ 4 so that j(X) = 4 by (3.1). Thus, X has socle series of the form Soc(X) = S_{2} ⊕ ⋯, Soc_{2}(X)/Soc(X) = T ⊕ ⋯, Soc_{3}(X)/Soc_{2}(X) = S ⊕ · · ·. By (3), T |̸(X/Soc_{3}(X)). We get from the above that X has a submodule Y such that Soc_{2}(X) ⊆ Y ⊊ Soc_{3}(X) and Soc_{3}(X)/Y ≅ S. Let X̄ = X/Y. Then, cX̄(T) = 0. Hence, by (3.1), X̄ is semi-simple, so that S|L_{1}(X). This contradicts (3).

Thus, m = 4, so that (2) implies that X ↔ 2 × χ_{58653}. Hence, by (1.1)(ii) and (2.6)(iv),

(4)  [X, X]^{G} = 4  and  X = 2 × (S_{1} + S_{2} + S_{3} + T), as composition factors.

If (S_{i} ⊕ S_{i})|L_{1}(X), then we get (S_{i} ⊕ S_{i})|Soc(X), so that [X, X] ⩾ 5, contradicting (4). Hence,

(5)  [S_{i}, L_{1}(X)]^{G} = [L_{1}(X), S_{i}]^{G} ⩽ 1 for i = 1, 2, 3.

If j(X) = 4, then we know from (3) and (4) that {2 × (S_{1} ⊕ S_{2} ⊕ S_{3})}|Soc(X), so that X/Soc_{3}(X) = T ⊕ ⋯ ⊕ T by (3.1), contradicting (3). Hence, j(X) = 3 by (3) and (3.1). Hence we know from (3),

(4) and (5) that X has Loewy and socle series as desired.

(ii) Easy by (2.6)(ii), (4.3)(i), (i), (2.9)(iii) and (2.11)(ii).

(iii) By (4.10)(ii), [P (φ̃_{18})|S_{1}↓_{H}] = 4. By (i), [S_{1}↓_{H}, φ̃_{18}]^{H} = [S_{1}, φ̃_{18}↑^{G}]^{G} = 5. Hence, [fS_{1}, φ̃_{18}]^{H} = 1 as in the proof of (4.10)(ii). Then, [φ̃_{18}, fS_{1}]^{H} = 1. □

(4.12) Lemma. We get the following.

(i) S_{1}↓_{H}·1_{B} = fS_{1} ⊕ (3 × P (φ̃_{5}) ⊕ P (φ̃_{6}) ⊕ 2 × P (φ̃_{13}) ⊕ 2 × P (φ̃_{14}) ⊕ 4 × P (φ̃_{18})) and fS_{1} has

Loewy and socle series φ̃_{18} ; φ̃_{6}φ̃_{5} ; φ̃_{18} .

(ii) (fS_{1})↑^{G}·1_{A} = S_{1} ⊕ (23 × P (S) ⊕ 86 × P (T) ⊕ 13 × P (S_{1}) ⊕ 11 × P (S_{2}) ⊕ 11 × P (S_{3})).

Proof. Let M̃ = fS_{1}.

(i) By (4.3)(i), S_{1}↓_{H}·1_{B} = M̃ ⊕ P̃ for a projective kH-module P̃ in B. As usual, we get from (1.6), (4.7)(iii), (4.4)(ii), (4.11)(ii), (2.7)(ii), (2.9)(ii), (2.11)(ii) and (2.6)(iii) that P̃ = 3 × P (φ̃_{5}) ⊕ P (φ̃_{6}) ⊕ m × P (φ̃_{13}) ⊕ m × P (φ̃_{14}) ⊕ 4 × P (φ̃_{18}) for a non-negative integer m. Then, we know from (2.6)(iv) and (2.9)(iii) that

cM̃(φ̃_{i}) = \{ \begin{matrix}5 − 2m & for i = 5, 6 \\ 6 − 3m & for i = 13, 14 \\ 10 − 4m & for i = 18 \end{matrix} .

Hence, m ∈ {0, 1, 2}, which implies cM̃(φ̃_{5}) ⩾ 1. Note by (4.10)(i) and (4.11)(iii) that

(2)  [φ̃_{i}, M̃]^{H} = [M̃, φ̃_{i}]^{H} = \{ \begin{matrix}0 & for i = 5, 6 \\ 1 & for i = 18 \end{matrix} .

It holds j(M̃) ⩽ 4 by (2.11)(iv). These imply j(M̃) = 3 or 4. Therefore, only the following six cases can occur.

\begin{matrix}(a) : & j(M̃) = 4, & m = 2, & (b) : & j(M̃) = 3, & m = 2, & (c) : & j(M̃) = 4, & m = 1 \\ (d) : & j(M̃) = 3, & m = 1, & (e) : & j(M̃) = 4, & m = 0, & (f) : & j(M̃) = 3, & m = 0 \end{matrix}

Case (a). By (1) and (2), M̃ = U(φ̃_{18}, φ̃_{i}, φ̃_{j}, φ̃_{18}) for {i, j} = {5, 6}, contradicting (2.11)(iv). Case (b). By (1) and (2), M̃ has Loewy and socle series as desired. Case (c). Clearly, by (1), M̃ = 3 × φ̃_{5} + 3 × φ̃_{6} + 3 × φ̃_{13} + 3 × φ̃_{14} + 6 × φ̃_{18}, as composition

factors. It follows from (1) and (2) that [M̃, φ̃_{18}]^{H} = 1 and c_{M̃J̃/M̃J̃^{3}}(φ̃_{5}) = 3. By looking at Soc_{2}(P (φ̃_{5})) in (2.11)(iv), we know [L_{2}(M̃), φ̃_{5}]^{H} ⩽ 1. Hence, c_{M̃J̃^{2}}(φ̃_{5}) ⩾ 2. On the other hand, [L_{4}(M̃), φ̃_{18}]^{H} ⩽ 1 by (2). These imply φ̃_{5}|Soc(M̃) by (2.11)(iv). This contradicts (2).

Case (d). By (1) and (2), [L_{1}(M̃), φ̃_{18}]^{H} = 1 and (3 × φ̃_{5})|L_{2}(M̃). Since [φ̃_{18}, L_{3}(M̃)]^{H} ⩽ 1 by (2), we get (2 × φ̃_{5})|Soc(M̃) by (2.11)(iv), contradicting (2).

Case (e). By (1), M̃ = 5 × φ̃_{5} + 5 × φ̃_{6} + 6 × φ̃_{13} + 6 × φ̃_{14} + 10 × φ̃_{18}, as composition factors. Since [L_{1}(M̃), φ̃_{18}]^{H} = 1 by (2), (2.11)(iv) implies that [L_{2}(M̃), φ̃_{5}]^{H} ⩽ 1. Hence, again by (2), (4 × φ̃_{5})|L_{3}(M̃). Thus, just as in Case (d), (3 × φ̃_{5})|Soc(M̃), contradicting (2).

Case (f). By (1) and (2), [L_{1}(M̃), φ̃_{18}]^{H} = 1 and (5 × φ̃_{5})|L_{2}(M̃), contradicting (2.11)(iv). (ii) Clearly, by (i) and (4.4)(i), [P (S)|M̃↑^{G}] = [M̃↑^{G}, S]^{G} = 23. By (i), [P (S_{1})|M̃↑^{G}] =

[M̃↑^{G}, S_{1}]^{G} − 1 = 13 by (i). Therefore, it follows from (2.11)(ii), (2.9)(iii) and (2.6)(v) that [P (T)|M̃↑^{G}] = 86 and [P (S_{i})|M̃↑^{G}] = 11 for i = 2, 3. □

(4.13) Lemma. We obtain the following.

(i) S_{i}↓_{H}·1_{B} = fS_{i} ⊕ (2 × P (φ̃_{6}) ⊕ 2 × P (φ̃_{13}) ⊕ 2 × P (φ̃_{14}) ⊕ 4 × P (φ̃_{18})) for i = 2, 3, and

fS_{2} = φ̃_{18} ; φ̃_{6}φ̃_{14} ; φ̃_{18} (Loewy and socle series)  and  fS_{3} = φ̃_{18} ; φ̃_{6}φ̃_{13} ; φ̃_{18} (Loewy and socle series).

(ii) (fS_{i})↑^{G}·1_{A} = S_{i} ⊕ (22 × P (S) ⊕ 85 × P (T) ⊕ 12 × P (S_{1}) ⊕ 13 × P (S_{2}) ⊕ 13 × P (S_{3}))

Proof. Easy by (2.9)(iii), (2.11)(ii)-(iii), (1.6), (4.7)(iii), (4.4)(i)-(ii), (4.11)(i) and (2.6)(iv). □

(4.14) Lemma. It holds the following.

(i) φ̃_{i}↑^{G}·1_{A} = gφ̃_{i} ⊕ (5 × P (S) ⊕ 14 × P (T) ⊕ 2 × P (S_{1}) ⊕ 2 × P (S_{2}) ⊕ 2 × P (S_{3})) for i = 13, 14, and

gφ̃_{13} = \begin{matrix}T \\ SSS_{3} \\ T \end{matrix} (Loewy and socle series) and gφ̃_{14} = \begin{matrix}T \\ SSS_{2} \\ T \end{matrix} (Loewy and socle series).

(ii) (gφ̃_{i})↓_{H}·1_{B} = φ̃_{i} ⊕ (43 × P (φ̃_{5}) ⊕ 42 × P (φ̃_{6}) ⊕ 41 × P (φ̃_{13}) ⊕ 41 × P (φ̃_{14}) ⊕ 72 × P (φ̃_{18})) for i = 13, 14.

Proof. These follow by (4.3)(i)-(ii), (4.5)(i), (1.1)(ii), (2.9)(iii), (4.4)(i), (1.6), (4.9)(i), (4.12)(i), (4.13)(i)-(ii), (2.6)(iv) and (3.1). □

Now, we finally get to a situation where we can prove our main result.

Proof of (0.2)(i). Let M_{0} be a unique indecomposable direct summand of a k[G × H]-module 1_{A}·kG·1_{B} with vertex ΔD (actually, M_{0} is the Green correspondent of A in k[G × H] with respect to (G × G, ΔD, 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 ΔD. Then, it follows from (2.7)(iii) and [23, Theorem 4.6] by Linckelmann that a pair (M_{0}, N_{0}) induces a stable equivalence of Morita type between A and B.

Let B′ = B_{0}(k[D:Q_{8}]) = k[D:Q_{8}]. Then, (2.11)(i) says that B is Morita equivalent to B′. Recall that IBr(B) = {φ̃_{6}, φ̃_{5}, φ̃_{14}, φ̃_{13}, φ̃_{18}} and IBr(B′) = {k = 0, 1, 2, 3, 4} by (2.11)(ii) and [26, §4 Case 3], respectively. We know by (4.4)(i), (4.9)(i), (4.12)(i) and (4.13)(i) that

fS = φ̃_{6},  fT = \begin{matrix}φ̃_{13} & φ̃_{5} & φ̃_{14} \\ φ̃_{18} & φ̃_{18} \\ φ̃_{14} & φ̃_{5} & φ̃_{13} \end{matrix} ,  fS_{1} = φ̃_{18} ; φ̃_{6}φ̃_{5} ; φ̃_{18} ,  fS_{2} = φ̃_{18} ; φ̃_{6}φ̃_{14} ; φ̃_{18} ,  fS_{3} = φ̃_{18} ; φ̃_{6}φ̃_{13} ; φ̃_{18} .

On the other hand, the calculation by Schneider [33, Theorem] (see [20, Lemma 6.6] and [26, Example 4.6]) says that

f′(k_{L_{3}(4)}) = k,  f′(19) = 2 1 3 4 4 3 1 2 ,  f′(15_{1}) = \begin{matrix}4 \\ k 1 \\ 4 \end{matrix} ,  f′(15_{2}) = \begin{matrix}4 \\ k 2 \\ 4 \end{matrix} ,  f′(15_{3}) = \begin{matrix}4 \\ k 3 \\ 4 \end{matrix} ,

where {k_{L_{3}(4)}, 19, 15_{1}, 15_{2}, 15_{3}} = IBr(B_{0}(k[L_{3}(4)])) and f′ : L_{3}(4) → N_{L_{3}(4)}(D) = D:Q_{8} is the Green correspondence with respect to (L_{3}(4), D, D:Q_{8}). Hence, we identify D with a Sylow 3subgroup of L_{3}(4) and we know N_{L_{3}(4)}(D) = D:Q_{8}. Therefore, just by tracing a method Okuyama takes in [26, Example 4.6 (1)–(10)], we finally know that there are a symmetric k-algebra B_{4} and a (B, B_{4})-bimodule M such that B and B_{4} are stably equivalent afforded by M and that (f(S′) ⊗_{B} M)_{B_{4}} = (simple) ⊕ (projective) for all S′ ∈ {S, T, S_{1}, S_{2}, S_{3}} = IBr(A). Of course, Okuyama in [26, Example 4.6 (1)–(10)] says that there are a symmetric k-algebra B′_{4} (which is denoted by A^{(4)} in [26, Example 4.6]) and a (B′, B′_{4})-bimodule M′ such that M′ realizes a stable equivalence between B′ and B′_{4} and that (f′(T′) ⊗_{B′} M′)_{B′_{4}} = (simple) ⊕ (projective) for all T′ ∈ IBr(B_{0}(L_{3}(4))) = {k_{L_{3}(4)}, 19, 15_{1}, 15_{2}, 15_{3}}. Since B_{4} and B′_{4} are Morita equivalent by the proof of [26, Example 4.6], we get by a result of Linckelmann [22, Theorem 2.1(iii)] that A and B are derived equivalent, and as byproduct, that A and B_{0}(k[L_{3}(4)]) are Morita (even splendidly Morita) equivalent. Therefore, by a result of the second author [20, Theorem 1.2], we know also that A is Morita equivalent to B_{0}(k[L_{3}(q)]) for any power q of a prime such that q ≡ 4 or 7 (mod 9). Furthermore, by reviewing the above proof and by making use of another result of Okuyama [27, Theorem 3], we conclude that A and B are, actually, splendidly equivalent, and hence Â and B̂ are splendidly equivalent as O-algebras by a result of Rickard [30, Theorem 5.2] (see a paper of M.E. Harris [8, p.75]). □

Proof of (0.3). This follows from the proof of (0.2)(i) above and a result of Okuyama [26, Example 4.6(10)]. Namely, the basic algebra denoted by A^{(4)} in [26, Example 4.6(10)] is Morita (actually, splendidly Morita) equivalent to A. □

(4.15) Remark. (0.2)(i) implies that there exists an isotypy between A and B by (2.7), (2.9)(iv), [4, A.2.2, p.88] and [8, Proposition 1.9].

5. A non-principal 3-block of the Higman-Sims simple group

In this section we prove that the Broué’s conjecture (0.1) holds for a non-principal 3-block A of the Higman-Sims simple group which has the same elementary abelian defect group D of order 9 as in §§2–4. It is announced by T. Holm [11, §5 p.60] that he has proved it. However, he has not written it yet in detail, and furthermore, just by a bit more calculations, we can determine the block algebra A up to Morita equivalence. K. Hicks almost determines the basic algebra of A in her thesis [10, Chapter 5]. But there is one parameter β_{1} which she does not determine (see [10, p.133]), while it can be taken as −1 here. Therefore, it may be meaningful to write a proof of the above fact.

(5.1) Notation and assumption. Throughout this section let G = HS, the Higman-Sims simple group of order 2^{9}·3^{2}·5^{3}·7·11. By [6, p.80] we know the following. Namely, G has a subgroup L such that L ≅ M_{22}. We fix a Sylow 3-subgroup D of G contained in L, which is elementary abelian of order 9. Let H = N_{G}(D) and H′ = N_{L}(D). Then H = (D:SD_{16}) × C_{2}, H′ = D:Q_{8}, and C_{G}(D) = D × C_{2}.

(5.2) Notation. The group algebra kG has six blocks, namely, the principal block B_{0}(kG), a non-principal block A with elementary abelian defect group D of order 9, a block of defect one and three blocks of defect zero. On the other hand, kH has two blocks, that is, the principal block B_{0}(kH) and a non-principal block B such that B is the Brauer correspondent of A, and hence D is a defect group of B (see [12, §5 pp.325–329]). We use the notation D, A and B throughout this section.

We use also the following notation which stands for simple modules. Namely, IBr(A) = {77, 49, 49^{∗}, 154_{1}, 154_{2}, 770, 770^{∗}} where the numbers mean the k-dimensions, IBr(B) = {0^{+}, 0^{−}, 1^{+}, 1^{−}, 2, 3 = 2^{∗}, 4} where 0^{+}, 0^{−}, 1^{+}, 1^{−} are of k-dimension one; and 2, 3, 4 are of k-dimension two. Furthermore, we can write IBr(B_{0}(L)) = {k_{L}, ˜55, ˜49, ˜49^{∗}, ˜231} and IBr(B_{0}(H′)) = {k_{H′}, 1_{1}, 1_{2}, 1_{3}, 2}, where the numbers mean the k-dimensions (see [12, §5] and [35, Theorem 2.3]).

(5.3) Lemma (Okuyama). The block algebra B of kH has the following Ext-quiver with relations, and hence the projective indecomposable kH-modules in B have the following Loewy and socle series.

the Ext-quiver of B = \begin{aligned}& & 0^{+} \\ & a^{+} & & c^{+} \\ & & 0^{−} \\ & a^{−} & & c^{−} \\ 2 & \begin{matrix}x \\ z \end{matrix} & 4 & \begin{matrix}y \\ w \end{matrix} & 3 \\ & d^{−} & & b^{−} \\ & & 1^{−} \\ & d^{+} & & b^{+} \\ & & 1^{+} \end{aligned}

relations: a^{±}d^{∓} = 0, b^{±}c^{∓} = 0, zx = wy, a^{±}xz = 0, xzd^{±} = 0, b^{±}yw = 0, ywc^{±} = 0, c^{+}a^{+} = c^{−}a^{−} + yz, d^{+}b^{+} = d^{−}b^{−} + xw, c^{+}a^{+}x + c^{−}a^{−}x = 0, d^{+}b^{+}y + d^{−}b^{−}y = 0, wc^{+}a^{+} + wc^{−}a^{−} = 0, zd^{+}b^{+} + zd^{−}b^{−} = 0. The projective indecomposable kH-modules in B are

0^{+} 2 1^{+} 4 3 0^{+} ,  \begin{matrix}0^{−} \\ 2 \\ 1^{−} 4 \\ 3 \\ 0^{−} \end{matrix} ,  1^{+} 3 0^{+} 4 2 1^{+} ,  \begin{matrix}1^{−} \\ 3 \\ 0^{−} 4 \\ 2 \\ 1^{−} \end{matrix} ,  \begin{matrix}2 \\ 1^{+} 4 1^{−} \\ 3 2 3 \\ 0^{+} 4 0^{−} \\ 2 \end{matrix} ,  \begin{matrix}3 \\ 0^{+} 4 0^{−} \\ 2 3 2 \\ 1^{+} 4 1^{−} \\ 3 \end{matrix} ,  \begin{matrix}4 \\ 2 3 \\ 1^{+} 1^{−} 4 0^{−} 0^{+} \\ 3 2 \\ 4 \end{matrix}

Proof. See Okuyama’s calculation [26, §4 Case 4]. □

(5.4) Lemma. Let M_{0} and N_{0} be the Green correspondents of A with respect to (G×G, ΔD, G×H) and (G × G, ΔD, H × G), respectively. Then, M_{0} and N_{0} induce a stable equivalence of Morita type between A and B.

Proof. By [6, p.81], all elements of order 3 in G are conjugate. Let Q be a subgroup of D of order 3. Then, since Σ_{8} ↪ G and since |C_{G}(Q)| = 360 by [6, pp.80–81], we easily know that C_{G}(Q) = C_{3} × Σ_{5}. Clearly, C_{H}(Q) = C_{3} × Σ_{3} × C_{2} by (5.1). Then, the non-principal 3-blocks of kΣ_{5} and k[Σ_{3} × C_{2}] with the same defect group C_{3} are Morita (even splendidly Morita) equivalent via a pair (M_{0}(ΔD), N_{0}(ΔD)) where M_{0}(ΔD) is the Brauer construction with respect to D (see [5, p.23] and [30, p.341]). Hence, it follows by a theorem of Broué [5, 6.3.Theorem] (cf. a proof of [30, Theorem 4.1 and Lemma 4.2]) that the pair (M_{0}, N_{0}) realizes a stable equivalence of Morita type between A and B. □

(5.5) Lemma. Let f_{L} : L → H′ and g_{L} : H′ → L be the Green correspondences with respect to (L, D, H′). Then,

g_{L}(k_{H′}) = k_{L},  g_{L}(1_{1}) = ˜55,  g_{L}(1_{2}) = ˜49^{∗} ; k_{L} ˜55 ; ˜49 ,  g_{L}(1_{3}) = ˜49 ; k_{L} ˜55 ; ˜49^{∗} ,  g_{L}(2) = ˜231 ; ˜49 ˜49^{∗} ; ˜231 ,

and these five indecomposable kL-modules in B_{0}(kL) are trivial source modules.

Proof. Easy. □

(5.6) Lemma. (i) A simple kG-module 22 in B_{0}(kG) of k-dimension 22 is a trivial source module with vertex D.

(ii) Simple kG-modules 77, 154_{1}, 154_{2}, 770 and 770^{∗} in A are trivial source modules with vertex D.

(iii) There are indecomposable kG-modules X and Y such that X and Y are trivial source modules with vertex D, and that X = U(49, 77, 49^{∗}) and Y = U(49^{∗}, 77, 49).

Proof. (i) and (ii). By [36, Lemma 5.1(4)], simple kG-modules 22 and 77 are both trivial source modules. Hence [36, Lemma 5.1(5)] implies that ˜21_{kL} is a trivial source module. Now, ˜21 belongs to a 3-block Ã of kL such that a defect group of Ã is C_{3} and IBr(Ã) = {˜21, ˜210} by [14, p.94]. Thus, ˜210_{kL} is also a trivial source module. Therefore, by (5.5) and [36, Lemma 5.8], 770↓_{L} is a 3-permutation module (a direct sum of trivial source kL-modules). Since 3 |̸|G:L| by (5.1), 770 itself is a trivial source kG-module, and hence so is 770^{∗}. Next, we get by [36, Lemma 5.10] and (5.5) that 154_{i}↓_{L} is a trivial source module for i = 1, 2. So, as above, 154_{i} is a trivial source kG-module for i = 1, 2.

(iii) Let X and Y be uniserial kG-modules of Loewy length 3 appearing as direct summands of 22 ⊗ 154_{2} and 22 ⊗ 154_{1} in [36, Lemma 5.12], respectively. Then, (i), (ii) and (1.8) imply that X and Y are our desired kG-modules. □

(5.7) Lemma. We get the the following:

(i) 154_{1} ⊗_{A} M_{0} = 0^{−}, 154_{2} ⊗_{A} M_{0} = 1^{−}, 770^{∗} ⊗_{A} M_{0} = 2, 770 ⊗_{A} M_{0} = 3, 70 ⊗_{A} M_{0} = 4. (ii) 49 ⊗_{A} M_{0} = U(0^{+}, 2, 1^{+}), 49^{∗} ⊗_{A} M_{0} = U(1^{+}, 3, 0^{+}).

Proof. (i) Since the functor − ⊗_{A} M_{0} coincides the Green correspondence modulo projectives, it preserves the k-dimension modulo 3 and dualities. Therefore, (i) follows from (5.6)(ii).

(ii) By (5.6)(iii), we have X ⊗_{A} M_{0} = 0^{+} ⊕ (proj), Y ⊗_{A} M_{0} = 1^{+} ⊕ (proj). By [36, Theorem], a kG-module Ω(49) has a filtration

Ω(49) = 77 | 770 | X .

Therefore, Ω(49 ⊗_{A} M_{0}) ≡ Ω(49) ⊗_{A} M_{0} ≡ U(4, 3, 0^{+}) (modulo proj). Hence, by (5.3), we have 49 ⊗_{A} M_{0} = Ω^{−1}(U(4, 3, 0^{+})) = U(0^{+}, 2, 1^{+}). Similarly, we have 49^{∗} ⊗_{A} M_{0} = U(1^{+}, 3, 0^{+}). □

Proof of (0.2)(ii). Let I = IBr(B) as in (5.2). Following Okuyama [26, §2 (III)], take a nice subset I_{0} of I such that I_{0} = {0^{+}, 1^{+}}. Then, we know from a result of Okuyama [26, Lemma 2.1(2)] that there are a symmetric k-algebra B′ which is derived equivalent to B, and a (B, B′)-bimodule M′ which induces a stable equivalence such that (S ⊗_{A} M_{0} ⊗_{B} M′)_{B′} = (simple) ⊕ (proj) for any S ∈ I. Therefore, it follows from a result of Linckelmann [22, Theorem 2.1(iii)] that A and B′ are Morita equivalent, which means that A and B are derived equivalent. Then, as before, by reviewing the above proof, we again know that this derived equivalence is actually a splendid equivalence by using Okuyama’s result [27, Theorem 3]. Thus, the derived equivalence lifts from k to O by a result of Rickard [30, Theorem 5.2] (see a paper of M.E. Harris [8, p.75]) □

(5.8) Remark. We can get (0.2)(ii) also by using a method of Rouquier [32, Theorem 7].

(5.9) Lemma. Keep the notation in the proof of (0.2)(ii) above. Then, the complex for B in the proof of (0.2)(ii) due to Okuyama is P (I_{0})^{•} = ⊕_{i∈I}P (i)^{•} such that

−2nd  −1st  0th  1st
P (0^{+})^{•} : ⋯ ⟶ 0 ⟶ P (0^{+}) ⟶  0 ⟶ 0 ⟶⋯
P (0^{−})^{•} : ⋯ ⟶ 0 ⟶ 0  ⟶ P (0^{−})⟶ 0 ⟶⋯
P (1^{+})^{•} : ⋯ ⟶ 0 ⟶ P (1^{+}) ⟶  0 ⟶ 0 ⟶⋯
P (1^{−})^{•} : ⋯ ⟶ 0 ⟶ 0  ⟶ P (1^{−})⟶ 0 ⟶⋯
P (2)^{•} : ⋯ ⟶ 0 ⟶ P (1^{+}) ⟶^{d^{+}} P (2) ⟶ 0 ⟶⋯
P (3)^{•} : ⋯ ⟶ 0 ⟶ P (0^{+}) ⟶^{c^{+}} P (3) ⟶ 0 ⟶⋯

P (4)^{•} : ⋯ ⟶ 0 ⟶ \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} ⟶^{(wc^{+}, zd^{+})} P (4) ⟶ 0 ⟶⋯

where d^{+} : P (1^{+}) → P (2) is given by d^{+}(v) = d^{+}v for v ∈ P (1^{+}) = e_{1^{+}}B, similar for c^{+} : P (0^{+}) → P (3), and

\begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} ⟶^{(wc^{+}, zd^{+})} P (4)

is given by

( \begin{matrix}u \\ v \end{matrix} ) ↦ (wc^{+}, zd^{+})( \begin{matrix}u \\ v \end{matrix} ) = wc^{+}u + zd^{+}v for ( \begin{matrix}u \\ v \end{matrix} ) ∈ \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} .

Proof. Easy by [26, Lemma 2.1 and its proof] and the structure of projective indecomposable B-modules in (5.3). □

(5.10) Theorem (see (0.4) and Hicks [10, p.133]). The basic algebra B′ of A has the following Ext-quiver and relations:

0^{+}  2
ā  c̄  z
a  c  w
4  ¯1  ¯0
b  d  x
b̄  d̄  y
1^{+}  3

aā = 0, ac̄ = 0, bb̄ = 0, bd̄= 0, cā = 0, db̄ = 0

ad̄y = 0, bc̄w = 0, xdā = 0, xdd̄= 0, zcb̄ = 0, zcc̄ = 0, cc̄w = 0, dd̄y = 0,

− āa + b̄b + c̄c − d̄d = 0,

cd̄= wx, dc̄ = yz,

ad̄d = ab̄b, cd̄d = −cb̄b, d̄dā = b̄bā, d̄dc̄ = −b̄bc̄,

bc̄c = bāa, dc̄c = −dāa, c̄cb̄ = āab̄, c̄cd̄= −āad̄,

cc̄c = cb̄b, dd̄d = dāa, c̄cc̄ = b̄bc̄, d̄dd̄= āad̄,

where the twelve arrows here are defined by using the notation in (5.9) such that

\begin{matrix}P (4)^{•} : & \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} & ⟶^{(wc^{+}, zd^{+})} & P (4) \\ a ↓ & ↓ (e_{0^{+}}, 0) & ↓ 0 \\ P (0^{+})^{•} : & P (0^{+}) & ⟶_{0} & 0 \end{matrix}  \begin{matrix}P (0^{+})^{•} : & P (0^{+}) & ⟶^{0} & 0 \\ ā ↓ & ↓( \begin{matrix}0 \\ b^{+}c^{+} \end{matrix} ) & ↓ 0 \\ P (4)^{•} : & \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} & ⟶_{(wc^{+}, zd^{+})} & P (4) \end{matrix}

\begin{matrix}P (4)^{•} : & \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} & ⟶^{(wc^{+}, zd^{+})} & P (4) \\ b ↓ & ↓ (0, e_{1^{+}}) & ↓ 0 \\ P (1^{+})^{•} : & P (1^{+}) & ⟶_{0} & 0 \end{matrix}  \begin{matrix}P (1^{+})^{•} : & P (1^{+}) & ⟶^{0} & 0 \\ b̄↓ & ↓( \begin{matrix}a^{+}d^{+} \\ 0 \end{matrix} ) & ↓ 0 \\ P (4)^{•} : & \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} & ⟶_{(wc^{+}, zd^{+})} & P (4) \end{matrix}

\begin{matrix}P (4)^{•} : & \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} & ⟶^{(wc^{+}, zd^{+})} & P (4) \\ c ↓ & ↓ (b^{+}c^{+}, 0) & ↓ x \\ P (2)^{•} : & P (1^{+}) & ⟶_{d^{+}} & P (2) \end{matrix}  \begin{matrix}P (2)^{•} : & P (1^{+}) & ⟶^{d^{+}} & 0 \\ c̄↓ & ↓( \begin{matrix}0 \\ e_{1^{+}} \end{matrix} ) & ↓ z \\ P (4)^{•} : & \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} & ⟶_{(wc^{+}, zd^{+})} & P (4) \end{matrix}

\begin{matrix}P (4)^{•} : & \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} & ⟶^{(wc^{+}, zd^{+})} & P (4) \\ d ↓ & ↓ (0, a^{+}d^{+}) & ↓ y \\ P (3)^{•} : & P (0^{+}) & ⟶_{c^{+}} & P (3) \end{matrix}  \begin{matrix}P (3)^{•} : & P (0^{+}) & ⟶^{c^{+}} & P (3) \\ d̄↓ & ↓( \begin{matrix}e_{0^{+}} \\ 0 \end{matrix} ) & ↓ w \\ P (4)^{•} : & \begin{matrix}P (0^{+}) \\ ⊕ \\ P (1^{+}) \end{matrix} & ⟶_{(wc^{+}, zd^{+})} & P (4) \end{matrix}

\begin{matrix}P (3)^{•} : & P (0^{+}) & ⟶^{c^{+}} & P (3) \\ x ↓ & ↓ 0 & ↓ −b^{−} \\ P (1^{−})^{•} : & 0 & ⟶_{0} & P (1^{−}) \end{matrix}  \begin{matrix}P (0^{−})^{•} : & 0 & ⟶^{0} & P (0^{−}) \\ y ↓ & ↓ 0 & ↓ −c^{−} \\ P (3)^{•} : & P (0^{+}) & ⟶_{c^{+}} & P (3) \end{matrix}

\begin{matrix}P (2)^{•} : & P (1^{+}) & ⟶^{d^{+}} & P (2) \\ z ↓ & ↓ 0 & ↓ a^{−} \\ P (0^{−})^{•} : & 0 & ⟶_{0} & P (0^{−}) \end{matrix}  \begin{matrix}P (1^{−})^{•} : & 0 & ⟶^{0} & P (1^{−}) \\ w ↓ & ↓ 0 & ↓ d^{−} \\ P (2)^{•} : & P (1^{+}) & ⟶_{d^{+}} & P (2) \end{matrix}

Proof. This follows by using (5.9), the Ext-quiver of B and its relations in (5.3). □

(5.11) Remark. The parameter β_{1} in [10, p.133] is taken as β_{1} = −1 by (5.10).

(5.12) Remark. By (5.1), A and B satisfy [4, 4.2.Hypothèses]. Hence, (0.2)(ii) and [8, Proposition 1.9] imply that there exists an isotypy between A and B.

Acknowledgements The authors thank S. Yoshiara for answering to a question. During the preparation of this work the first author was partially supported by the JSPS (Japan

Society for Promotion of Science), Grant-in-Aid for Scientific Research (C) 11640042, 2000.

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Appendix. In order to make the diagrams in (5.3) and (5.10) completed, one may need the following data:

(5.3) The following 12 arrows should be here:

\begin{matrix}0^{+} ⟶^{a^{+}} 2, & 0^{−} ⟶^{a^{−}} 2, & 2 ⟶^{d^{+}} 1^{+}, & 2 ⟶^{d^{−}} 1^{−}, & 1^{+} ⟶^{b^{+}} 3, & 1^{−} ⟶^{b^{−}} 3, \\ 3 ⟶^{c^{+}} 0^{+}, & 3 ⟶^{c^{−}} 0^{−}, & 2 ⟶^{x} 4, & 4 ⟶^{z} 2, & 3 ⟶^{y} 4, & 4 ⟶^{w} 3 \end{matrix}

(5.10) The following 12 arrows should be here:

\begin{matrix}0^{+} ⟶^{a} 4, & 4 ⟶^{ā} 0^{+}, & 1^{+} ⟶^{b} 4, & 4 ⟶^{b̄} 1^{+}, & 2 ⟶^{c} 4, & 4 ⟶^{c̄} 2, \\ 3 ⟶^{d} 4, & 4 ⟶^{d̄} 3, & ¯1 ⟶^{x} 3, & 3 ⟶^{y} ¯0, & ¯0 ⟶^{z} 2, & 2 ⟶^{w} ¯1 \end{matrix}