29 June, 2009

Broué’s abelian defect group conjecture holds for the Harada-Norton sporadic simple group HN

Shigeo Koshitani^{a,∗}, Jürgen Müller^{b}

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

Dedicated to Professor Tetsuro Okuyama on his sixtieth birthday

Abstract

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the normaliser N_{G}(P) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Broué’s abelian defect group conjecture. We prove in this paper that Broué’s abelian defect group conjecture is true for a nonprincipal 3-block A with an elementary abelian defect group P of order 9 of the Harada-Norton simple group HN. It then turns out that Broué’s abelian defect group conjecture holds for all primes p and for all p-blocks of the Harada-Norton simple group HN.

Keywords: Broué’s conjecture; abelian defect group; Harada-Norton simple group

1. Introduction and notation

In representation theory of finite groups, one of the most important and interesting problems is to give an affirmative answer to a conjecture, which was introduced by M. Broué around 1988 [8], and is nowadays called Broué’s Abelian Defect Group Conjecture. He actually conjectures the following:

1.1.Conjecture (Broué’s Abelian Defect Group Conjecture) ([8, 6.2.Question] and [23, Conjecture in p.132]). Let p be a prime, and let (K, O, k) be a splitting p-modular system for all subgroups of a finite group G. Assume that A is a block algebra of OG with a defect group P and that B is a block algebra of ON_{G}(P) such that B is the Brauer correspondent of A, where N_{G}(P) is the normaliser of P in G. Then, A and B should be derived equivalent (Rickard equivalent) provided P is abelian.

In fact, a stronger conclusion than 1.1 is expected. If G and H are finite groups and if A and B are block algebras of OG and OH (or kG and kH) respectively, we say that A and B are splendidly Rickard equivalent in the sense of Linckelmann ([39], [40]), where he calls it a splendid derived equivalence, see the end of 1.8. Note that this is the same as that given by Rickard in [57] when A and B are the principal block algebras, which he calls a splendid equivalence.

∗ Corresponding author.

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

1.2.Conjecture (Rickard [57], [58, Conjecture 4, in p.193]). Keep the notation, and suppose that P is abelian as in 1.1. Then, there should be a splendid Rickard equivalence between the block algebras A of OG and B of ON_{G}(P).

There are several cases where the conjectures of Broué 1.1 and Rickard 1.2 are checked. For example we prove that 1.1 and 1.2 are true for the principal block algebra A of an arbitrary finite group G when the defect group P of A is elementary abelian of order 9 (and hence p = 3), see [26, (0.2)Theorem]. Then, it may be natural to ask what about the case of nonprincipal block algebras with the same defect group P = C_{3} ×C_{3}. Namely, this paper should be considered as a continuation of such a project, which has already been accomplished for several cases in our previous papers for the O’Nan simple group and the Higman-Sims simple group in [29, 0.2.Theorem], for the Held simple group and the sporadic simple Suzuki group in [30, Theorem], and for the Janko’s simple group J_{4} [31, Theorem 1.3], see also [47] and [34]. That is to say, our main theorem of this paper is the following:

1.3.Theorem. Let G be the Harada-Norton simple group HN, and let (K, O, k) be a splitting 3-modular system for all subgroups of G, see the definition 1.8 below. Suppose that A is a non-principal block algebra of OG with a defect group P which is an elementary abelian group C_{3} × C_{3} of order 9, and that B is a block algebra of ON_{G}(P) such that B is the Brauer correspondent of A. Then, A and B are splendidly Rickard equivalent, and hence the conjectures 1.1 and 1.2 of Broué and Rickard hold.

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

1.4.Theorem. Keep the notation and the assumption as in 1.3. Then, the non-principal block algebra A of OG with a defect group P = C_{3} × C_{3} and the principal block algebra A′ of OHS of the Higman-Sims simple group are Puig equivalent, that is A and A′ are Morita equivalent which is realized by a ΔP -projective p-permutation O[G × HS]-module, in other words, A and A′ have isomorphic source algebras as interior P -algebras.

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

1.5.Corollary. Broué’s abelian defect group conjecture 1.1 and even Rickard’s splendid equivalence conjecture 1.2 are true for all primes p and for all block algebras of OG when G = HN.

1.6.Starting point and strategy. A story of the birth of this paper is actually very similar to that of the Janko’s simple group J_{4} which is given in [31, 1.6]. Namely, relatively recently G. Hiss, J. Müller, F. Noeske and J.G. Thackray [17] have determined the 3-decomposition matrix of the group HN with defect group C_{3} × C_{3}, see 4.1. Our starting point for this work was actually to realize that the 3-decomposition matrix for the non-principal block of HN with an elementary abelian defect group of order 9 is exactly the same as that for the principal 3-block of the Higman-Sims simple group HS. Furthermore, the generalised 3decomposition matrices of these two blocks are the same. Therefore, it is natural to suspect whether these two 3-block algebras would be Morita equivalent not only over an algebraically closed field k of characteristic 3 but also over a complete discrete valuation ring O whose residue field is k, and we might expect even that they are Puig equivalent (we shall give a precise definition of Puig equivalence in 1.8 below). Anyhow, since the two conjectures of Broué and Rickard in 1.1 and 1.2 respectively have been solved for the principal 3-block of HS in a paper of Okuyama [51] it turns out that Broué’s abelian defect group conjecture 1.1 and Rickard’s splendid equivalence conjecture 1.2 shall be solved also for the non-principal 3-block of HN with the same defect group C_{3} × C_{3}.

1.7.Contents. In §2, we shall give several fundamental lemmas, which are useful and powerful to prove our main results. In §§3 and 4, we shall investigate 3-modular representations for HN and we shall get trivial source (p-permutation) modules which are in the non-principal 3-block A of HN with a defect group P = C_{3} × C_{3}. In §5, we shall list data on Green correspondents of simples in the principal 3-block A′ of HS, which are known by a result of [64, Theorem], see [51, Example 4.8]. Finally, in §§6-8, we shall give complete proofs of our main results 1.3, 1.4 and 1.5.

To achieve our results, next to theoretical reasoning we have to rely on fairly heavy computations. As tools, we use the computer algebra system GAP [12], to calculate with permutation groups as well as with ordinary and Brauer characters. We also make use of the data library [7], in particular allowing for easy access to the data compiled in [10], [19] and [67], and of the interface [66] to the data library [68]. Moreover, we use the computer algebra system MeatAxe [60] to handle matrix representations over finite fields, as well as its extensions to compute submodule lattices [42], radical and socle series [45], homomorphism spaces and endomorphism rings [44], and direct sum decompositions [43]. We give more detailed comments on the relevant computations in the spots where they enter the picture.

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

From now on, let k be a field and assume that A is a finite dimensional k-algebra. Suppose that M is an A-module. Then, we denote by soc(M) the socle of M. We define soc_{0}(M) = 0 and soc_{1}(M) = soc(M). Then, we define soc_{i}(M) by soc_{i}(M)/soc_{i−1}(M) = soc(M/soc_{i−1}(M)) for any integer i ⩾ 2. Similarly, we write rad^{i}(M) for M·rad^{i}(A) for any integer i ⩾ 0. By using this, we define L_{i}(M) by rad^{i−1}(M)/rad^{i}(M) for i = 1, 2, · · ·. We call L_{i}(M) the i-th Loewy layer of M. We denote by j(M) the Loewy length of M, namely j(M) is the least positive integer j satisfying rad^{j}(M) = 0. We write P (M) and I(M) for the projective cover and the injective hull (envelope) of M, respectively, and we write Ω for the Heller operator (functor), namely, ΩM is the kernel of the projective cover P (M) ↠ M. Dually, Ω^{−1}M is the cokernel of the injective hull M ↣ I(M). For simple A-modules S_{1}, ⋯, S_{n} (some of which are possibly isomorphic) we write that M = a_{1}×S_{1}+⋯+a_{n}×S_{n}, as composition factors for positive integers a_{1}, ⋯, a_{n} when the set of all composition factors are a_{1} times S_{1}, ⋯, a_{n} times S_{n}. For an A-module M and a simple A-module S, we denote by c_{M}(S) the multiplicity of all composition factors of M which are isomorphic to S. We write c(S, T) for c_{P (S)}(T) for simple A-modules S and T, namely, this is so-called the Cartan invariant with respect to S and T.

To describe the structure of an A-module, we either indicate the radical and socle series, in cases where these series coincide and are sufficient for our analysis, or we draw an Alperin diagram [1]. An A-module need not have an Alperin diagram, but if it does then it is a compact way to give a more detailed structural description of the module under consideration; note that the Alperin diagram is closely related to the Hasse diagram of the incidence relation amongst the local submodules in the sense of [46], hence for explicit examples is easily determined using the techniques described in [42]. Note, however, that by giving any kind of diagram an A-module in general is not uniquely determined up to isomorphism.

Let N be another A-module. Then, Hom_{A}(M, N) is the set of all right A-module-homomorphisms from M to N, which canonically is a k-vector space, and we denote by PHom_{A}(M, N) the set of all (relatively) projective homomorphisms in Hom_{A}(M, N), which is a k-subspace of Hom_{A}(M, N). Hence, we can define the factor space, that is, we write \underline{Hom}_{A}(M, N) for the factor space Hom_{A}(M, N)/PHom_{A}(M, N). By making use of this, as well-known, we can construct the stable module category \underline{mod}-A, which is a quotient category of mod-A such that the set of all morphisms is given by \underline{Hom}_{A}(M, N).

In this paper, G is always a finite group and we fix a prime number p. Assume that (K, O, k) is a splitting p-modular system for all subgroups of G, that is to say, O is a complete discrete valuation ring of rank one such that its quotient field is K which is of characteristic zero and its residue field O/rad(O) is k which is of characteristic p, and that K and k are splitting fields for all subgroups of G. We mean by an OG-lattice a finitely generated right OG-module which is a free O-module. We sometimes call it just an OG-module. Let X be a kG-module. Then, we write X^{∨} for the k-dual of X, namely, X^{∨} = Hom_{k}(X, k) which is again a right kG-module via (x)(φg) = (xg^{−1})φ for x ∈ X, φ ∈ X^{∨} and g ∈ G. Similarly, we write χ^{∨} for the dual (complex conjugate) of χ for an ordinary character χ of G. Let H be a subgroup of G, and let M and N be an OG-lattice and an OH-lattice, respectively. Then, let M↓_{H}^{G} = M↓_{H} be the restriction of M to H, and let N↑_{H}^{G} = N↑^{G} be the induction (induced module) of N to G, that is, N↑^{G} = (N ⊗_{OH} OG)_{OG}. Similar for kG- and kH-modules.

We denote by Irr(G) and IBr(G) the sets of all irreducible ordinary and Brauer characters of G, respectively. Let A be a block algebra (p-block) of OG. Then, we write Irr(A) and IBr(A) for the sets of all characters in Irr(G) and IBr(G) which belong to A, respectively. We often mean by IBr(A) the set of all non-isomorphic simple kG-modules belonging to A. We sometimes denote by A^{∗} the block algebra of kG corresponding to A. But, we usually abuse A and A^{∗}, namely, we often mean the block algebra of kG by A as well when it is clear from the context. For ordinary characters χ and ψ of G, we denote by (χ, ψ)^{G} the inner product of χ and ψ in usual sense. Let X and Y be kG-modules. Then, we write [X, Y]^{G} for dim_{k}[Hom_{kG}(X, Y)]. We denote by k_{G} the trivial kG-module. Similar for O_{G}. For A-modules M and N we write [M, N]^{A} for dim_{k}[Hom_{A}(M, N)].

We say that M is a trivial source (p-permutation) kG-module if M is an indecomposable kG-module whose source is k_{Q}, where Q is a vertex of M. Let G′ be another finite group, and let V be an (OG, OG′)-bimodule. Then we can regard V as a right O[G × G′]-module via v(g, g′) = g^{−1}vg′ for v ∈ V, g ∈ G and g′ ∈ G′. Similar for (kG, kG′)-bimodules. We denote by ΔG the diagonal copy of G in G × G, namely, ΔG = {(g, g) ∈ G × G | g ∈ G}. Let A and A′ be block algebras of OG and OG′, respectively. Then, we say that A and A′ are Puig equivalent if A and A′ have a common defect group P (and hence P ⊆ G ∩ G′) and if there is a Morita equivalence between A and A′ which is induced by an (A, A′)- bimodule 𝔐 such that, as a right O[G × G′]-module, 𝔐 is a p-permutation (trivial source) module and ΔP -projective. Similar for blocks of kG and kG′. Due to a result of Puig (and independently of Scott), see [55, Remark 7.5], this is equivalent to a condition that A and A′ have source algebras which are isomorphic as interior P -algebras, see [40, Theorem 4.1]. For an (OG, OG′)-bimodule V and a common subgroup Q of G and G′, we set V^{Q} = {v ∈ V | qv = vq, ∀q ∈ Q}. If Q is a p-group, the Brauer construction is defined to be a quotient V (Q) = V^{Q}/[∑_{R≨Q} Tr↑_{R}^{Q}(V^{R})+radO·V^{Q}] where Tr↑_{R}^{Q} is the usual trace map. The Brauer homomorphism Br_{Q} : (OG)^{Q} → kC_{G}(Q) is obtained from composing the canonical epimorphism (OG)^{Q} ↠ (OG)(Q) and a canonical isomorphism (OG)(Q) →^{≈} kC_{G}(Q).

We say that A and A′ are stably equivalent of Morita type if there exists an (A, A′)- bimodule 𝔐 such that _{A}(𝔐⊗_{A′} 𝔐^{∨})_{A} ≅ _{A}A_{A}⊕(projective (A, A)-bimodule) and _{A′}(𝔐^{∨}⊗_{A} 𝔐)_{A′} ≅ _{A′}A′_{A′} ⊕ (projective (A′, A′)-bimodule). We say that A and A′ are splendidly stably equivalent of Morita type if A and A′ have a common defect group P and the stable equivalence of Morita type is induced by an (A, A′)-bimodule 𝔐 which is a p-permutation (trivial source) O[G × G′]-module and is ΔP -projective, see [40, Theorem 3.1]. We say that A and A′ are Rickard equivalent if A and A′ are derived equivalent, namely, D^{b}(mod-A) and D^{b}(mod-A′) are equivalent as triangulated categories. We say that A and A′ are splendidly Rickard equivalent if A and A′ are derived equivalent by a complex M^{•} ∈ C^{b}(A-mod-A′) and its dual (M^{•})^{∨} such that each term M^{n} of M^{•} is a Δ(P)-projective and p-permutation module as an O[G×G′]-module, where C^{b}(A-mod-A′) is the category of bounded complexes of finitely generated (A, A′)-bimodules.

For a positive integer n, A_{n} and S_{n} denote the alternating and symmetric group on n letters, M_{n} denotes the Mathieu group, and C_{n}, D_{n} and SD_{n} denote the cyclic group, the dihedral group and the semi-dihedral group of order n, respectively. For a subgroup E of Aut(G), G ⋊ E denotes a semi-direct product such that G is normal in G ⋊ E and E acts on G canonically. For g ∈ G and a subset S of G, we denote g^{−1}Sg by S^{g}, and similarly, x^{g} = g^{−1}xg for x ∈ G. For non-empty subsets S and T of G, we write S =_{G} T if T = S^{g} for an element g ∈ G.

For other notation and terminology, see the books of Nagao-Tsushima [48] and Thévenaz [62].

2. Preliminaries

In this section we list many lemmas, some of which are theorems due to other people. These lemmas are so useful and powerful to prove our main results.

2.1.Lemma ([24, (1.1)Lemma]). Let A be a finite-dimensional algebra over a field and X an A-module. Assume that Y is a non-zero uniserial A-submodule of X with Loewy layers

rad^{i−1}(Y)/rad^{i}(Y) ≅ S_{i} for i = 1, ⋯, n

where S_{i} is a simple A-module. Set X̄ = X/Y. Then, we get the following:

(i) For each j = 1, ⋯, j(X), rad^{j−1}(X)/rad^{j}(X) ≅ rad^{j−1}(X̄)/rad^{j}(X̄) or

rad^{j−1}(X)/rad^{j}(X) ≅ rad^{j−1}(X̄)/rad^{j}(X̄) ⨁ S_{i} for some S_{i}.

(ii) For each i = i, ⋯, n, there is a positive integer m_{i} such that m_{1} < m_{2} < ⋯ < m_{n}

and that rad^{m_{i}−1}(X)/rad^{m_{i}}(X) ≅ (rad^{m_{i}−1}(X̄)/rad^{m_{i}}(X̄)) ⨁ S_{i}.

2.2.Lemma (Okuyama [50, Lemma 2.2]). Let S be a simple kG-module with vertex P, and let f be the Green correspondence with respect to (G, P, N_{G}(P)). If S is a trivial source module, then its Green correspondent f(S) is again simple as kN_{G}(P)-module.

2.3.Lemma (Scott [35, II Theorem 12.4 and I Proposition 14.8] and [5, Corollary 3.11.4]).

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

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

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

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

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

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

between IBr(Ã) and IBr(A),

Proof. See [30, 1.5.Theorem] and [48, Chapter 5 Theorem 5.10]. ■

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

Proof. See [31, Lemma 2.4] and [48, Chapter 5, Theorem 5.10]. ■

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

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

Proof. See [31, Lemma 2.7]. ■

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

Proof. Follows by [13, (3.2), (3.2*), (3.3)], see [35, II, Lemma 2.7, Corollary 2.8]. ■

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

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

(ii) (dual of (i)) Let X be a projective-free A-module such that X has an A-submodule X′ satisfying that X/X′ is simple. Set T = F (X/X′). Then, we can write F (X) = Y ⊕ R for a projective-free B-module Y and a projective B-module R. Now, if T is a simple B-module, then we may assume that T is an epimorphic image of Y and that Ker(F (X) ↠ T) = Ker(Y ↠ T) ⊕ (proj).

Proof. We get (i) from 2.7 and [30, 1.11.Lemma], just as in the proof of [30, 3.25.Lemma and 3.26.Lemma], see [34, Proposition 2.2]. (ii) is just the dual of (i). ■

2.9.Lemma (Linckelmann [37, Theorem 2.1(ii)]). Let A and B be finite-dimensional kalgebras for a field k such that A and B are both self-injective and indecomposable as algebras, and none of them are simple algebras. Suppose that there is an indecomposable (A, B)- bimodule M such that a pair (M, M^{∨}) induces a stable equivalence between A and B. If S is a simple A-module, then (S ⊗_{A} M)_{B} is a non-projective indecomposable B-module.

The next lemma is a new result due to Kunugi and the first author. This is actually so useful and convenient when we want to apply so-called ”Rouquier’s glueing” to our inductive argument in order to get a stable equivalence between two block algebras which we are looking at.

2.10.Lemma (Koshitani-Kunugi [28, Theorem 1.2]). Let A be a block algebra of OG with a cyclic defect group P ≠ 1. Let H = N_{G}(P), and let B be a block algebra of OH such that B is the Brauer correspondent of A. Then, we get the following:

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

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

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

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

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

2.11.Lemma. Let A be a block algebra of OG with defect group P. Set H = N_{G}(P), and let B be a block algebra of OH such that B is the Brauer correspondent of A. Assume that Q is a subgroup of P with Q ⊆ Z(G). Set Ḡ = G/Q, H̄ = H/Q and P̄ = P/Q. It is well-known that there exist block algebras Ā and B̄ of OḠ and OH̄, respectively, such that Ā and B̄ dominate A and B, namely Irr(Ā) ⊆ Irr(A) and Irr(B̄) ⊆ Irr(B), and that both Ā and B̄ have P̄ as defect groups, see [48, Chapter 5 Theorems 8.10 and 8.11].

(i) It holds that H̄ = NḠ(P̄) and that B̄ is the Brauer correspondent of Ā. In the rest of the lemma, assume in addition that P is elementary abelian of order p^{2}, namely, P = Q × R with Q ≅ R ≅ C_{p}.

(ii) It holds that

Ā ⊗_{OH̄} B̄ = Ā(Ā·1B̄)B̄ = ĀXB̄ ⊕ (proj)

for an indecomposable (Ā, B̄)-bimodule X with vertex ΔP̄. (iii) In particular, if X realizes a Morita equivalence between Ā and B̄, then there ex-

ists an (A, B)-bimodule M such that M is an indecomposable direct summand of A(A·1_{B})_{B} with vertex ΔP, and hence M induces a Puig equivalence between A and B.

Proof. (i) The first part is easy. The second part follows from [49, (3.2)Lemma], see [41, ℓ.10 on p.1314].

(ii) This follows by [38, Proposition 6.1] since P̄ ≅ C_{p}.

(iii) This is obtained from (ii) and [27, Theorem], see [41, ℓ. − 7 ∼ ℓ. − 4 on p.1314] and [40, Theorem 4.1]. ■

2.12.Lemma. Suppose that p = 3 and G = A_{9}.

(i) There uniquely exists a non-principal block algebra A of OG with defect group P ≅ C_{3}. In addition we can write Irr(A) = {χ_{5}, χ_{17}, χ_{18}} such that χ_{5}(1) = 27, χ_{17}(1) = 189, χ_{18}(1) = 216 and χ_{5}(u) = χ_{17}(u) = 9 for any element u ∈ P − {1}, and that a part of the 3-decomposition matrix is

\begin{aligned}& 27 & 189 \\ χ_{5} & 1 & 0 \\ χ_{17} & 0 & 1 \\ χ_{18} & 1 & 1 \end{aligned}

where the indices of χ_{i} are the same as in [10, p.37]. (In the following, we use the notation A and P as in (i)).

(ii) Set H = N_{G}(P). Then H = (P × A_{6}).C_{2}, where the action on P × A_{6} by C_{2} is the

diagonal one, extending A_{6} to S_{6}. (iii) Let H be as in (ii), and let B be a block algebra of OH, which is the Brauer cor-

respondent of A. Then, A and B are Morita equivalent via an (A, B)-bimodule M such that M is (up to isomorphism) the unique indecomposable direct summand of A(A·1_{B})_{B} with vertex ΔP, and hence it holds that M induces a Puig equivalence between A and B, and that the simples 27 and 189 in A are both trivial source kG-modules.

Proof. (i) This follows from [10, p.37], [67, A_{9} (mod 3)] and [19].

(ii) Easy by inspection.

(iii) This is obtained from (i) and 2.10. ■

2.13.Lemma. Let A and B be finite dimensional k-algebras. Assume that there exists a functor F : \underline{mod}-A → \underline{mod}-B realizing a stable equivalence between A and B. Assume, in addition, that there is a simple A-module S_{0} such that S_{0} is sent to a simple B-module T_{0}, namely, F (S_{0}) = T_{0}. Then, for any simple A-module S with S ≅ S̸_{0}, it holds [F (S), T_{0}]^{B} = [T_{0}, F (S)]^{B} = 0.

Proof. We get by 2.7 and the assumptions that

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

Hence [F (S), T_{0}]^{B} = 0. The rest is similar. ■

2.14.Lemma. Let A be a finite-dimensional k-algebra, and assume that X is an A-module satisfying that X = P_{1} ⊕ ⋯ ⊕ P_{n} ⊕ Y ⊇ Z for an integer n ⩾ 2, A-submodules P_{1}, ⋯, P_{n} and Y of X such that P_{1} ≅ ⋯ ≅ P_{n} ≅ P (S) for a simple A-module S, and soc(Z) ≅ S, and Y_{A} is projective-free. Assume moreover that j(P (S)) = j(A). Then, X/Z has a direct summand isomorphic to P (S).

Proof. Set j = j(A) = j(P (S)), and X̄ = X/Z. Assume that X̄ is projective-free. Then, rad^{j−1}(X̄) = 0, and hence rad^{j−1}(P_{1} ⊕ ⋯ ⊕ P_{n} ⊕ Y) ⊆ Z. Clearly, rad^{j−1}(P_{1} ⊕ ⋯ ⊕ P_{n} ⊕ Y) = soc(P_{1}) ⊕ ⋯ ⊕ soc(P_{n}) ≅ S ⊕ ⋯ ⊕ S (n times) since rad^{j−1}(Y) = 0. This means that (S ⊕ S ⊕ ⋯ ⊕ S) | soc(Z), a contradiction. Thus, P (T)|X̄ for a simple A-module T. This implies that there is an epimorphism X ↠ P (T), and hence P (T)|X. Then, we get P (T) ≅ P (S) by Krull-Schmidt theorem. ■

2.15.Lemma. Let G, H and L be finite groups such that all of them contain a common subgroup P, namely, P ⊆ G ∩ H ∩ L. Let M be a k[G × H]-module such that M | k_{ΔP} ↑^{G×H}, and let N be a k[H ×L]-module such that N | k_{ΔP} ↑^{H×L}. Then, it follows that

M ⊗_{kH} N | k_{ΔP} ↑^{G×L}.

Proof. This is a special case of [16, 2.5.Proposition]. ■

2.16.Lemma. Let A be a finite-dimensional k-algebra, and assume that X is an indecomposable non-simple A-module. Then, it holds soc(X) ⊆ rad(X).

Proof. Assume that soc(X) ⊈ rad(X). Then, X has a simple A-submodule S with S ⊈ rad(X). Hence, X has a maximal A-submodule M with S ⊈ M. These imply that S∩M = 0 and S + M = X. Namely, X = S ⊕ M. Since M is indecomposable, X = S. This is a contradiction. ■

3. 3-Local structure for HN

3.1.Notation and assumption. From now on, we assume that G is the Harada-Norton simple group HN, and hence |G| = 2^{14}·3^{6}·5^{6}·7·11·19 ≒ 2.7 × 10^{14}, see [10, p.164–166] and [14].

3.2.Lemma.

(i) In order to prove Broué’s abelian defect group conjecture for G = HN, it suffices to prove it for the case p = 3.

(ii) There exists a unique 3-block A with non-cyclic abelian defect group P, and P is

elementary abelian of order 9. (iii) P is the Sylow 3-subgroup of the second largest maximal subgroup 2.HS.2 of G, a

two-fold cover of the automorphism group of the Higman-Sims simple group HS.

Proof. (i) We may assume p ∈ {2, 3, 5} by 3.1 just as in the proof of [31, Lemma 3.2]. Assume that p = 2. Then, G has only two 2-blocks B_{0} and B_{1} with positive defect by [67], where B_{0} is the principal 2-block. Then, the non-principal 2-block B_{1} has a defect group D with D ≅ SD_{16}, see [4, Lemma 4.2(c)]. Thus, B_{0} and B_{1} both have non-abelian defect groups. Next, suppose p = 5. By [67], G has only a unique 5-block B_{0} which has defect ⩾ 2, and hence B_{0} is the principal 5-block. Then, B_{0} has non-abelian defect group 5_{+}^{1+4}.5 by [10, p.164–166].

(ii) Finally, assume p = 3. Sylow 3-subgroups of G are non-abelian by [10, p.164–166]. Thus, G has a unique non-principal 3-block A such that A has a defect group P with |P | ⩾ 3^{2}, and actually P ≅ C_{3} × C_{3}, see [4, Lemma 4.2(b)].

(iii) Using the character table of G, calculations with GAP [12] show that the conjugacy class 2A of G is a defect class of A, where we follow the notation in [10, p.164–166]. Hence P is a Sylow 3-subgroup of the centralizer C_{G}(2A) ≅ 2.HS.2. ■

3.3.Notation. From now on, we assume p = 3, and we use the notation A and P as in 3.2, namely, A is a block algebra of kG with defect group P ≅ C_{3} × C_{3}. Set H = N_{G}(P), and let B be a block algebra of kH that is the Brauer correspondent of A. Let (P, e) be a maximal A-Brauer pair in G, that it, e is a block idempotent of kC_{G}(P) such that Br_{P} (1_{A})·e = e, see [2], [9] and [62, §40]. Set H̃ = N_{G}(P, e), namely, H̃ = {g ∈ N_{G}(P) | e^{g} = e}, where e^{g} = g^{−1}eg. Finally set E = H/C_{G}(P), and let Q be a subgroup of P of order 3.

3.4.Lemma. It holds the following:

(i) H = H̃ = (P × A_{6}).SD_{16}.

(ii) C_{G}(P) = C_{H}(P) = P × A_{6}.

(iii) E = H̃/C_{G}(P) ≅ SD_{16}, where the action of E on P is given by the embedding of

SD_{16} as a Sylow 2-subgroup of Aut(P) ≅ GL_{2}(3). (iv) All elements in P − {1} are conjugate in H, and hence in G.

(v) P − {1} ⊆ 3A, where 3A is a conjugacy class of G following the notation in [10, p.164–166].

(vi) All subgroups of P of order 3 are conjugate in H, and hence in G.

(vii) Recall the subgroup Q of P in 3.3. Then, we have C_{G}(Q) = Q × A_{9}, N_{G}(Q) =

(Q × A_{9}).2 ⩽ A_{12}, C_{H}(Q) = (P × A_{6}).2, and N_{H}(Q) = (P × A_{6}).2^{2}. (viii) C_{G}(Q)/Q ≅ A_{9}, C_{H}(Q)/Q ≅ (C_{3} × A_{6}).2, N_{G}(Q)/Q ≅ A_{9}.2, and N_{H}(Q)/Q ≅

(C_{3} × A_{6}).2^{2}.

Proof. This is found using explicit computation with GAP [12]. The starting point is the smallest faithful permutation representation of G on 1140000 points, available in terms of so-called standard generators [65] in [68]. The associated one-point stabiliser is the largest maximal subgroup A_{12} of G, which hence can be found explicitly by a randomised Schreier-Sims technique. Having completed that, all the following computations can be done using this permutation representation of G.

Actually, one of the standard generators is an element of the 2A conjugacy class of G, where we use the notation in [10, p.164–166]. Hence the second largest maximal subgroup 2.HS.2 ≅ C_{G}(2A) can be found be a centraliser computation. In turn, by 3.2(iii) P can be computed explicitly as a Sylow 3-subgroup of 2.HS.2.

(i)–(ii) The normaliser H = N_{G}(P) and the centraliser C_{G}(P) of P can be computed explicitly, and as these are fairly small groups their structure is easily revealed.

(iii) It follows from [67, A_{6} (mod 3)] and [19] that A_{6} has exactly two 3-blocks. Let β be the non-principal block algebra of kA_{6}, and hence β is of defect zero. Then, e = 1_{β}. Since β is a unique block algebra of kA_{6} of defect zero, this shows H = H̃.

(iv) Easy by (iii) and inspection.

(v) We use the notation 3A and 3B as in [10, p.164–166]. By (iv), P − {1} ⊆ 3A or 3B. Assume P −{1} ⊆ 3B. Then, χ(u) = 0 for any χ ∈ Irr(A) and any u ∈ 3A by [48, Chapter 5 Corollary 1.10(i)]. But we know that χ_{8} ∈ Irr(A) by [4, Lemma 4.2(b)], see also 4.1, and that χ_{8}(u) = 27 for any u ∈ 3A. This is a contradiction.

(vi) Easy by (iv).

(vii)–(viii) It is easy to see that N_{G}(Q) < A_{12}, the largest maximal subgroup of G, which is the one-point stabiliser in the given permutation representation of G. Hence again the normaliser N_{A_{12}}(Q) and the centraliser C_{A_{12}}(P) of P can be computed explicitly and their structure determined. ■

3.5.Lemma. We get the following diagram:

[figure: G = HN; A_{12}; N_{G}(Q) = N_{A_{12}}(Q) = (Q × A_{9}).2; C_{G}(Q) = C_{A_{12}}(Q) = Q × A_{9}; H = N_{G}(P) = (P × A_{6}).SD_{16}; N_{A_{12}}(P) = (P × A_{6}).D_{8}; N_{H}(Q) = (P × A_{6}).2^{2}; C_{H}(Q) = (P × A_{6}).2; C_{G}(P) = C_{H}(P) = P × A_{6}; (A_{6} × A_{6}).D_{8}; (A_{6} × A_{6}).2^{2}; 84; 84; 1140000; 220; 2; 2; 2; 2; 2; 20; 20; 263340000; 462]

Proof. This follows from [10, p.164–166], 3.4 and calculations with GAP [12]. ■

3.6.Lemma. The following holds:

(i) B ≅ Mat_{9}(O[P ⋊ SD_{16}]) as O-algebras, (ii) The block algebra B has a source algebra jBj ≅ O[P ⋊SD_{16}], as interior P -algebras,

where j is a source idempotent of B with respect to P, namely, j is a primitive idempotent of B^{P} such that Br_{P} (j) ≠ 0 for the Brauer homomorphism Br_{P} for P, see [62, §§19 and 27].

(iii) We can write

Irr(B) = {χ_{9a}, χ_{9b}, χ_{9c}, χ_{9d}, χ_{18a}, χ_{18b}, χ_{18c}, χ_{72a}, χ_{72b}}

and

IBr(B) = {9a, 9b, 9c, 9d, 18a, 18b, 18c},

where the numbers mean the degrees of characters and the dimensions of simples, respectively. Note that χ_{18b} and χ_{18c} are dual each other, and so are 18b and 18c. The other characters and simples are self-dual.

(iv) The 3-decomposition matrix and the Cartan matrix of B are the following:

9a 9b 9c 9d 18a 18b 18c
χ_{9a} 1 ….  .  .
χ_{9b} . 1 …  .  .
χ_{9c} . . 1 . .  .  .
χ_{9d} … 1 .  .  .
χ_{18a} …. 1  .  .
χ_{18b} . . . . .  1  .
χ_{18c} . . . . .  .  1
χ_{72a} 1 1 . . 1 1 1
χ_{72b} . . 1 1 1 1 1
P (9a) P (9b) P (9c) P (9d) P (18a) P (18b) P (18c)
9a  2  1  0  0  1  1  1
9b  1  2  0  0  1  1  1
9c  0  0  2  1  1  1  1
9d  0  0  1  2  1  1  1
18a  1  1  1  1  3  2  2
18b  1  1  1  1  2  3  2
18c  1  1  1  1  2  2  3

(v) There are unique conjugacy classes 4A and 4B of H, consisting of elements of order 4, and having centralisers of order 40 and 48, respectively. A part of the character table of Irr(B) then is the following:

conjugacy class 4A 4B 12A
centraliser  40 48 24
χ_{9a}  1 −1 −1
χ_{9b}  −1 −1 −1
χ_{9c}  1 1  1
χ_{9d}  −1 1  1
χ_{18a}  0 0  0
χ_{18b}  0 0  0
χ_{18c}  0 0  0
χ_{72a}  0 −2  1
χ_{72b}  0 2 −1

Note that this identifies the characters χ_{9a}, χ_{9b}, χ_{9c}, χ_{9d}, χ_{72a} and χ_{72b} uniquely.

(vi) The radical and socle series of PIMs in B are the following:

\begin{matrix}9a \\ 18b \\ 9b 18a \\ 18c \\ 9a \end{matrix}  \begin{matrix}9b \\ 18c \\ 9a 18a \\ 18b \\ 9b \end{matrix}  \begin{matrix}9c \\ 18c \\ 9d 18a \\ 18b \\ 9c \end{matrix}  \begin{matrix}9d \\ 18b \\ 9c 18a \\ 18c \\ 9d \end{matrix}

\begin{matrix}18a \\ 18b 18c \\ 9b 9c 18a 9a 9d \\ 18c 18b \\ 18a \end{matrix}  \begin{matrix}18b \\ 9b 18a 9c \\ 18c 18b 18c \\ 9a 18a 9d \\ 18b \end{matrix}  \begin{matrix}18c \\ 9a 18a 9d \\ 18b 18c 18b \\ 9b 18a 9c \\ 18c \end{matrix}

Note that this identifies the simples 18b and 18c uniquely.

(vii) An Alperin diagram of the PIM P (18a) is given as follows:

P (18a) = [figure: 18a; 18b; 18c; 9b; 9c; 18a; 9a; 9d; 18c; 18b; 18a]

Proof. This again relies on computations with GAP [12]. Starting with the explicit restriction of the permutation representation of G to H obtained in 3.4, we find a faithful permutation representation of H on a small number of points. This then is used to compute the conjugacy classes of H, and its ordinary character table using the Dixon-Schneider algorithm.

(i) Since the Schur multiplier of SD_{16} is trivial, see e.g. [25, Proof of Lemma 1.3], we get the assertion by 3.4(i)-(iii), [33, A.Theorem].

(ii) This follows by a result of Puig [54, Proposition 14.6] and (i), see [3, Theorem 13] and [62, (45.12)Theorem].

(iii)–(v) Easy from the character table of H. (vi) The radical and socle series have been determined in [63].

(vii) To find the structure of P (18a), we have used the MeatAxe [60] to construct P (18a) explicitly as a matrix representation, from the permutation representation of H obtained above, and subsequently we have used the method described in [42] to compute the whole submodule lattice of P (18a), from which the result follows easily. ■

3.7.Notation. We use the notation χ_{9a}, χ_{9b}, χ_{9c}, χ_{9d}, χ_{18a}, χ_{18b}, χ_{18c}, χ_{72a}, χ_{72b}, 9a, 9b, 9c, 9d, 18a, 18b, 18c, and also the source idempotent j as in 3.6.

3.8.Lemma. The block algebra B and its source algebra k[P ⋊ SD_{16}] have exactly 18 trivial source modules. In fact, it holds the following:

(i) Seven PIMs: P (9a), P (9b), P (9c), P (9d), P (18a), P (18b), P (18c).

(ii) Seven trivial source modules with a vertex P : 9a, 9b, 9c, 9d, 18a, 18b, 18c.

(iii) Four trivial source modules with vertex Q ≅ C_{3}:

χ_{9a} + χ_{9b} + χ_{18a} + χ_{72a} ↔ V_{1} = [figure: 18a; 18b; 18a; 18c; 9b; 9a; 9b; 9a]

χ_{9c} + χ_{9d} + χ_{18a} + χ_{72b} ↔ V_{2} = [figure: 18a; 18c; 18a; 18b; 9d; 9c; 9d; 9c]

χ_{18b} + χ_{18c} + χ_{72a} ↔ V_{3} = [figure: 18c; 18b; 9a; 18a; 9b; 18b; 18c]

χ_{18b} + χ_{18c} + χ_{72b} ↔ V_{4} = [figure: 18c; 18b; 9d; 18a; 9c; 18b; 18c]

and all characters χ_{V_{i}} realized by V_{i} has values χ_{V_{i}}(u) = 27 for any u ∈ 3A, where 3A is the unique conjugacy class of H of elements of order 3 on which χ_{V_{i}} does not vanish, see [48, Chapter 5, Corollary 1.10(i)].

Proof. These follow from 3.4, a theorem of Green [48, Chapter 4, Problem 10, p.302] As for (iii), starting again with the permutation representation of H, using GAP [12] we compute N_{H}(Q), use the MeatAxe [60] and the methods described in [43] to find the PIMs of N_{H}(Q)/Q as direct summands of its regular representation, induce them to H, and find the submodule structure of the induced modules using the methods described in [42]. ■

3.9.Notation. We use the notation V_{1}, V_{2}, V_{3}, V_{4} as in 3.8.

3.10.Lemma. There are no kH-modules in B whose radical and socle series are the same and which have the following structure:

\begin{matrix}18a \\ 18b 18c \\ 18a \\ 18b \end{matrix} ; (i)  \begin{matrix}18a \\ 18b 18c \\ 18a \\ 18c \end{matrix} ; (ii)  \begin{matrix}9c \\ 18c \\ 9d \\ 18b \end{matrix} ; (iii)  \begin{matrix}9d \\ 18b \\ 9c \\ 18c \end{matrix} ; (iv)

Proof. (i) Assume that such a kH-module, which we call M, exists. There is an epimorphism π : P (18a) ↠ M. Set K = Ker(π). Then, 3.6(vi) and 1.1 imply that K has radical and socle series

\begin{matrix}9b 9c 9a 9d \\ 18c \\ 18a \end{matrix}

Since there does not exist a kH-module \begin{matrix}9a \\ 18c \end{matrix} by 3.6(vi), we have a contradiction.

(ii) Similar to (i).

(iii) Assume that such a kH-module, which we call M, exists. There is an epimorphism

π : P (9c) ↠ M. Set K = Ker(π). Then, by 3.6(vi) we get K = \begin{matrix}18a \\ 9c \end{matrix} . This contradicts the

structure of P (9c) in 3.6(vi).

(iv) Similar to (iii). ■

4. 3-Modular representations of HN

4.1.Theorem (Hiss-Müller-Noeske-Thackray [17]). The 3-decomposition matrix and the Cartan matrix of A are the following:

\begin{matrix}degree & [10, p.164–166] & S_{1} & S_{2} & S_{3} & S_{4} & S_{5} & S_{6} & S_{7} \\ 8910 & χ_{8} & 1 & . & . & . & . & . & . \\ 16929 & χ_{10} & . & 1 & . & . & . & . & . \\ 270864 & χ_{19} & . & . & 1 & . & . & . & . \\ 1185030 & χ_{32} & 1 & 1 & . & 1 & . & . & . \\ 1354320 & χ_{33} & 1 & . & . & . & 1 & 1 & . \\ 1575936 & χ_{37} & . & . & 1 & . & . & 1 & . \\ 2784375 & χ_{43} & 1 & . & 1 & 1 & 1 & 1 & . \\ 4561920 & χ_{49} & . & . & . & 1 & 1 & . & 1 \\ 4809375 & χ_{50} & . & 1 & 1 & 1 & . & . & 1 \end{matrix}

P (S_{1}) P (S_{2}) P (S_{3}) P (S_{4}) P (S_{5}) P (S_{6}) P (S_{7})
S_{1}  4  1  1  2  2  2  0
S_{2}  1  3  1  2  0  0  1
S_{3}  1  1  4  2  1  2  1
S_{4}  2  2  2  4  2  1  2
S_{5}  2  0  1  2  3  2  1
S_{6}  2  0  2  1  2  3  0
S_{7}  0  1  1  2  1  0  2

where S_{1}, ⋯, S_{7} are non-isomorphic simple kG-modules in A whose degrees respectively are 8910, 16929, 270864, 1159191, 40338, 1305072, 3362391.

4.2.Notation. We use the notation χ_{8}, χ_{10}, χ_{19}, χ_{32}, χ_{33}, χ_{37}, χ_{43}, χ_{49}, χ_{50} and S_{1}, ⋯, S_{7} as in 4.1.

4.3.Lemma.

(i) All simples S_{1}, ⋯, S_{7} are self-dual.

(ii) (Knörr) All simples S_{1}, ⋯, S_{7} have P as vertices.

Proof. (i) Easy from 4.1.

(ii) This is a result of Knörr [22, 3.7.Corollary]. ■

4.4.Lemma.

(i) The heart H(P (S_{i})) = rad(P (S_{i}))/soc(P (S_{i})) is indecomposable as a kG-module for

any i = 1, ⋯, 7.

(ii) Ext_{kG}^{1}(S_{i}, S_{j}) = 0 for any pair (i, j) ∈ {(1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (2, 7), (3, 1), (3, 2), (3, 5), (3, 7), (4, 6), (5, 3), (5, 5), (5, 7), (6, 4), (6, 6), (7, 2), (7, 3), (7, 5), (7, 7)}.

Proof. (i) This follows by the Cartan matrix of A in 4.1 and results of Erdmann and Kawata, see [11, Theorem 1], [20, Theorem 1.5] and [29, 1.9.Lemma].

(ii) If Ext_{kG}^{1}(S_{1}, S_{2}) ≠ 0, then S_{2}|H(P (S_{1})) since c_{12} = 1 by 4.1, which contradicts to (i). Similar for the others. ■

4.5.Lemma.

(i) The simple S_{1} is a trivial source module with S_{1} ↔ χ_{8}. (ii) The simple S_{2} is a trivial source module with S_{2} ↔ χ_{10}. (iii) The simple S_{3} is a trivial source module with S_{3} ↔ χ_{19}.

Proof. (i)–(ii) Let M = 2.HS.2, where HS is the Higman-Sims simple group, be the second largest maximal subgroups of G, see by [10, p.164–166]. Now, a calculation with GAP [12], using the character tables of M and G, shows that 1_{M}↑^{G}·1_{A} = χ_{8} +χ_{10}. Set X = k_{M}↑^{G}·1_{A}. We then get X = S_{1} + S_{2} (as composition factors) by 4.1. Since X, S_{1}, S_{2} are all self-dual by 4.3(i), we obtain X = S_{1} ⊕ S_{2}.

(iii) Let M be the same as above. There uniquely exists a non-trivial linear character χ of M. Then, a calculation with GAP [12] shows that χ↑G M·1_{A} = χ_{19}. Hence, by 4.1, S_{3} is a trivial source module. ■

4.6.Lemma. There is a trivial source kG-module in A whose vertex is Q and whose structure is

[figure: S_{4}; S_{4}; S_{7}; S_{5}; S_{2}; S_{1}] ↔ χ_{32} + χ_{49}.

Proof. It follows from [10, p.164–166] that the fourth largest maximal subgroup of G is of the form M = 2_{+}^{1+8}.(A_{5} ×A_{5}).2. Let P_{M} ∈ Syl_{3}(M). Then P_{M} ≅ C_{3} ×C_{3}, but a calculation with GAP [12], using the character tables of G and M, shows that P_{M} contains elements belonging to the 3B conjugacy class of G, hence P_{M} ≠_{G} P by 3.4(v). Clearly, there is a non-trivial kM-module T with dim_{k} T = 1. Set X = T ↑_{M}^{G}·1_{A}. Then, X is a direct sum of trivial source kG-modules, and a calculation with GAP [12] shows that X ↔ χ_{32} + χ_{49}.

Since P is a defect group of A, any indecomposable kG-module Y with Y |X does not have P as its vertex.

Suppose that X is decomposable. Then, 2.3(i) implies that X = Y ⊕ Z such that Y ↔ χ_{32} and Z ↔ χ_{49}. Hence, 4.1 yields that Y = S_{1} + S_{2} + S_{4} (as composition factors). We know by 4.3(ii) and 4.1 that S_{1}, S_{2}, and Y are all self-dual. If [Y, S_{1}]^{G} ≠ 0, then the self-dualities imply S_{1}|Y, and hence 0 ≠ [S_{1}, Y]^{G} = (χ_{8}, χ_{32})^{G} from 2.3(ii) and 4.5(i), a contradiction. Hence, [Y, S_{1}]^{G} = [S_{1}, Y]^{G} = 0. Similarly, we obtain [Y, S_{2}]^{G} = [S_{2}, Y]^{G} = 0 by 2.3(ii) and 4.5(ii). This is a contradiction since Y has only three composition factors S_{1}, S_{2} and S_{4}.

Thus, X is indecomposable. By the decomposition matix of A in 4.1, X is not a PIM. Thus, the order of a vertex of X is 3, and hence Q is a vertex of X by 3.4(vi). Clearly, X is a trivial source kG-module in A. We know by 4.1 that X = S_{1} + S_{2} + 2 × S_{4} + S_{5} + S_{7} (as composition factors). Note that X, S_{1}, S_{2}, S_{4}, S_{5}, S_{7} are all self-dual from 4.3(i). Then, [S_{i}, X]^{G} = [X, S_{i}]^{G} = 0 for any i = 1, 2, 5, 7 since X is indecomposable. Thus, X/rad(X) ≅ Soc(X) ≅ S_{4}. Therefore, again by the self-dualities, it holds that rad(X)/Soc(X) ≅ S_{1} ⊕ S_{2} ⊕ S_{5} ⊕ S_{7}. ■

4.7.Lemma. There is a trivial source kG-module in A which has Q as a vertex and has

radical and socle series \begin{matrix}S_{3} \\ S_{6} \\ S_{3} \end{matrix} ↔ χ_{19} + χ_{37}.

(Note: We can prove that this module has Q as its vertex, but only later on in 7.2(ii)).

Proof. First, the third largest maximal subgroup of G is of shape M = U_{3}(8).3, see [10, p.164–166]. Then, a calculation with GAP [12], using the character tables of G and M, shows that

1_{M}↑^{G}·1_{A} = χ_{19} + χ_{32} + χ_{37} + χ_{49}. (1)

Set X = k_{M}↑^{G}·1_{A}, hence X is self-dual and is a direct sum of trivial source kG-modules. Then, by the decomposition matrix in 4.1, we know

(2) X = S_{1} + S_{2} + 2 × S_{3} + 2 × S_{4} + S_{5} + S_{6} + S_{7} (as composition factors).

If [X, S_{1}]^{G} ≠ 0, then 2.3(ii) and 4.5(i) imply that (χX̂, χ_{8})^{G} = [X, S_{1}]^{G} ≠ 0 where χX̂ is a character afforded by X (see 2.3(i)), which is a contradiction by (1). Hence, it holds [X, S_{1}]^{G} = [S_{1}, X]^{G} = 0 by the self-dualities in 4.3(i). Similarly, by 4.5(ii)-(iii) and 2.3(ii), we know also [X, S_{2}]^{G} = [S_{2}, X]^{G} = 0 and [X, S_{3}]^{G} = [S_{3}, X]^{G} = 1. If [X, S_{5}]^{G} ≠ 0, then (2) and the self-dualities imply that S_{5}|X, and hence S_{5} is liftable by 2.3(i), which contradicts 4.1. Hence, [X, S_{5}]^{G} = [S_{5}, X]^{G} = 0 by the self-dualities. Similarly, it holds also that [X, S_{i}]^{G} = [S_{i}, X]^{G} = 0 for i = 6, 7. If [X, S_{4}]^{G} = 2, then it follows from (2) and the self-dualities that (S_{4} ⊕ S_{4})|X, and hence S_{4} is liftable by 2.3(i), which contradicts 4.1. This shows [X, S_{4}]^{G} = [S_{4}, X]^{G} ≠ 2. Namely,

[S_{3}, X]^{G} = [X, S_{3}]^{G} = 1, (3)

[S_{4}, X]^{G} = [X, S_{4}]^{G} ≠ 2, (4)

[S_{i}, X]^{G} = [X, S_{i}]^{G} = 0 for i = 1, 2, 5, 6, 7. (5)

Now, 4.6 says that there is a trivial source kG-module Y that has radical and socle series

Y = [figure: S_{4}; S_{4}; S_{7}; S_{5}; S_{2}; S_{1}] ↔ χ_{32} + χ_{49}. (6)

in A. Then, by (1), (6) and 2.3(ii), we have

[Y, X]^{G} = [X, Y]^{G} = 2. (7)

Then, by (4) and (2), we know that

[S_{4}, X]^{G} = [X, S_{4}]^{G} ⩽ 1. (8)

Next, we want to claim that there is a homomorphism φ ∈ Hom_{kG}(Y, X) with 0 ≠ Im(φ) ≅ S̸_{4}. Suppose that any non-zero φ ∈ Hom_{kG}(Y, X) satisfies that Im(φ) ≅ S_{4}. By (7), let {φ_{1}, φ_{2}} be a k-basis of Hom_{kG}(Y, X). Then, it follows from Schur’s lemma that Im(φ_{1}) ≠ Im(φ_{2}), and hence that there exists a direct sum Im(φ_{1}) ⊕ Im(φ_{2}) ⊆ X. This means that [S_{4}, X]^{G} ⩾ 2, contradicting (8).

Therefore, there is a homomorphism φ ∈ Hom_{kG}(Y, X) with 0 ≠ Im(φ) ≅ S̸_{4}. Then, by (6), we know Ker(φ) = 0 since S_{i} |̸ soc(X) for i = 1, 2, 5, 7 by (5). That is, there is a monomorphism φ : Y ↣ X of kG-modules.

Then, just by the dual argument, we know also that there is an epimorphism ψ : X ↠ Y of kG-modules. It follows then by (2) and (6) that there is a direct sum Im(φ) ⊕ Ker(ψ) ⊆ X, and hence Im(φ) ⊕ Ker(ψ) = X. Set Z = Ker(ψ). We can write X = Y ⊕ Z. Since

Z = 2 × S_{3} + S_{6} (as composition factors), we get by (5) that Z = \begin{matrix}S_{3} \\ S_{6} \\ S_{3} \end{matrix} . Hence, it is easy to

know from (1) and (6) that Z is a trivial source kG-module with Z ↔ χ_{19} + χ_{37}. ■

4.8.Lemma. There is a trivial source kG-module in A whose structure is

[figure: S_{1}; S_{2}; S_{4}; S_{1}; S_{2}] ↔ χ_{8} + χ_{10} + χ_{32}.

(Note: We can prove that this module has Q as its vertex, but only later on in 7.2(i)).

Proof. By [10, p.91], we have 1_{A_{11}}↑^{A_{12}} = 1_{A_{12}} + χ̃_{11}, where χ̃_{11} ∈ Irr(A_{12}) is of degree 11. It follows from the 3-decomposition matrix of A_{12} in [67, A_{12} (mod 3)] and [19] that

(9)  k_{A_{11}}↑^{A_{12}} = \begin{matrix}k \\ 10 \\ k \end{matrix} ↔ 1_{A_{12}} + χ̃_{11},

where 10 is a simple kA_{12}-module of dimension 10. Set X = (k_{A_{11}}↑^{A_{12}})↑^{G}·1_{A}. Note that X is a direct sum of trivial source kG-modules. Then, we know from a calculation with GAP [12], using the character tables of G and A_{12}, that

(10)  X ↔ χ_{8} + χ_{10} + χ_{32}

and

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

By (7), 2.3(ii) and 4.5(ii), we obtain [X, S_{1}]^{G} = (χX̂, χ_{8})^{G} = 1. Hence, [X, S_{1}]^{G} = [S_{1}, X]^{G} = 1 by the self-dualities. Similarly, we have [X, S_{2}]^{G} = [S_{2}, X]^{G} = 1. Since S_{4} is not liftable by 4.1, S_{4} is not a trivial source module by 2.3(i). This implies that [X, S_{4}]^{G} = [S_{4}, X]^{G} = 0 by (8). These yield

(12)  X/rad(X) ≅ Soc(X) ≅ S_{1} ⊕ S_{2}.

Next, we want to claim that X is indecomposable. Suppose that X is decomposable. By (12), we can write X = X_{1} ⊕ X_{2} for A-submodules X_{1} and X_{2} of X with soc(X_{i}) ≅ S_{i} for i = 1, 2. If X_{1}/rad(X_{1}) ≅ S̸_{1}, then (12) shows that X_{1}/rad(X_{1}) ≅ S_{2}, and hence we

get by (12) and (11) that X = X_{1} ⊕ X_{2} = \begin{matrix}S_{2} \\ S_{4} \\ S_{1} \end{matrix} ⊕ \begin{matrix}S_{1} \\ S_{2} \end{matrix} or X = X_{1} ⊕ X_{2} = \begin{matrix}S_{2} \\ S_{1} \end{matrix} ⊕ \begin{matrix}S_{1} \\ S_{4} \\ S_{2} \end{matrix} ,

which is a contradiction by the self-dualities of X and each S_{i} in 4.4(i). This means that X_{i}/rad(X_{i}) ≅ S_{i} for i = 1, 2 by (12). If X_{1} is simple, then we get by (12) that X_{2} has radical and socle series which is one of the following three cases:

\begin{matrix}S_{2} \\ S_{1} S_{4} \\ S_{2} \end{matrix}  \begin{matrix}S_{2} \\ S_{1} \\ S_{4} \\ S_{2} \end{matrix}  \begin{matrix}S_{2} \\ S_{4} \\ S_{1} \\ S_{2} \end{matrix}

So we have a contradiction by 4.4(ii). Thus, X_{1} is not simple. Similarly, we know that X_{2} is not simple. Hence, 2.16 yields that soc(X_{i}) ⊆ rad(X_{i}) for i = 1, 2. Thus, X = X_{1} ⊕X_{2} =

\begin{matrix}S_{1} \\ S_{4} \\ S_{1} \end{matrix} ⊕ \begin{matrix}S_{2} \\ S_{2} \end{matrix} or X = X_{1} ⊕ X_{2} = \begin{matrix}S_{1} \\ S_{1} \end{matrix} ⊕ \begin{matrix}S_{2} \\ S_{4} \\ S_{2} \end{matrix} . This is a contradiction by (10), 2.3(i) and 4.1.

Therefore X is indecomposable. Hence, we get by (11), (12) and 2.16 that soc(X) ⊆ rad(X). Thus we get the structure of X as desired. ■

4.9.Notation. In the rest of paper let f be the Green correspondence from G to H with respect to P, see [48, Chapter 4 §4].

4.10.Lemma. It holds that f(S_{1}) = 9a.

Proof. It follows from 4.5(i), 4.3(ii) and 2.1 that f(S_{1}) is a simple kH-module in B, see 3.4(i). Using the ordinary characters afforded by the trivial source kH-modules in B, see 3.8, we get the following possible decompositions of S_{1}↓_{H}·1_{B}, by a calculation with GAP [12] using the character tables of G and H:

S_{1}↓_{H}·1_{B} = 9a ⨁ (7 × P (9a) ⊕ 7 × P (9b) ⊕ 5 × P (18a) ⊕ P (18b) ⊕ P (18c))

or

S_{1}↓_{H}·1_{B} = 9b ⨁ (8 × P (9a) ⊕ 6 × P (9b) ⊕ 5 × P (18a) ⊕ P (18b) ⊕ P (18c)).

In particular, f(S_{1}) = 9a or f(S_{1}) = 9b, and we have to decide which case actually occurs.

To this end, let M = 2.HS.2 be the second largest maximal subgroup of G, see 4.5. By [67, HS (mod 3)] and [19], let A^{−} be the block algebra of OM containing the unique non-trivial linear character χ of M. Hence letting A^{+} and A′, see 5.1, be the principal block algebras of OM and of OHS, respectively, we have A^{+} ≅ A′ and an isomorphism − ⊗ χ: A^{+} → A^{−}. Moreover, P being a Sylow 3-subgroup of M, it is the block defect group of A^{−}, and hence let B^{−} be the Brauer correspondent of A^{−} in N_{M}(P).

Using the smallest faithful permutation representation of M on 1408 points, available in [68], the normaliser N_{M}(P) and the centraliser C_{M}(P) of the Sylow 3-subgroup P is easily computed explicitly with GAP [12] and their structure determined, we find N_{M}(P) = (P × D_{8}).SD_{16} and C_{M}(P) = P × D_{8}. Now the conjugacy classes of N_{M}(P) can be computed, its ordinary character table is found using the Dixon-Schneider algorithm, from that its blocks are determined and B^{−} is identified.

Then a computation with GAP [12], using the character tables of G and M, shows that S_{1}↓_{M}·1_{A^{−}} = 22^{−}, where the latter denotes the unique simple A^{−}-module of that dimension. Moreover, using the character tables of M and N_{M}(P), GAP [12] shows that (22^{−})↓_{N_{M} (P)}·1_{B^{−}} = λ, where λ is a certain linear character; actually, λ is the Green correspondent of 22^{−} with respect to (M, P, N_{M}(P)), which must be linear in view of 5.7. Hence λ = (S_{1}↓_{M}·1_{A^{−}})↓_{N_{M} (P)}·1_{B^{−}} is a direct summand of

(S_{1}↓_{H})↓_{N_{M} (P)}·1_{B^{−}} = (f(S_{1}) ⊕ (R-proj))↓_{N_{M} (P)}·1_{B^{−}}
= f(S_{1})↓_{N_{M} (P)}·1_{B^{−}} ⊕ (Q-proj) ⊕ (proj),

where R consists of elementary-abelian 3-subgroups of H, of order at most 9, not Hconjugate to P, and Q ≅ C_{3} is as in 3.3. Since λ has P as a vertex, we conclude that λ is a direct summand of f(S_{1})↓_{N_{M} (P)}·1_{B^{−}}.

Now a computation with GAP [12], using the character tables of H and N_{M}(P), shows that f(S_{1})↓_{N_{M} (P)}·1_{B^{−}} = (9x)↓_{N_{M} (P)}·1_{B^{−}}, where x ∈ {a, b}, already is linear, where

(9a)↓_{N_{M} (P)}·1_{B^{−}} = λ ≠ (9b)↓_{N_{M} (P)}·1_{B^{−}}.

This shows that f(S_{1}) = 9a. ■

4.11.Lemma. It holds that f(S_{2}) = 9b.

Proof. It follows from 4.5(i), 4.3(ii) and 2.1 that f(S_{1}) is a simple kH-module in B, see 3.4(i). Using the ordinary characters afforded by the trivial source kH-modules in B, see 3.8, we get the following possible decompositions of S_{1}↓_{H}·1_{B}, by a calculation with GAP [12] using the character tables of G and H:

S_{2}↓_{H}·1_{B} = 9b ⨁ (9 × P (9a) ⊕ 8 × P (9b) ⊕ 7 × P (18a) ⊕ 5 × P (18b) ⊕ 5 × P (18c))

or

S_{2}↓_{H}·1_{B} = 9a ⨁ (8 × P (9a) ⊕ 9 × P (9b) ⊕ 7 × P (18a) ⊕ 5 × P (18b) ⊕ 5 × P (18c)).

In particular, f(S_{2}) = 9b or f(S_{2}) = 9a, hence the assertion follows from 4.10. ■

4.12.Lemma. It holds that f(S_{3}) = 9c.

Proof. It follows from 4.5(i), 4.3(ii) and 2.1 that f(S_{1}) is a simple kH-module in B, see 3.4(i). Using the ordinary characters afforded by the trivial source kH-modules in B, see 3.8, we get the following possible decompositions of S_{1}↓_{H}·1_{B}, by a calculation with GAP [12] using the character tables of G and H:

S_{3}↓_{H}·1_{B} = 9c ⨁ (54 × P (9a) ⊕ 54 × P (9b) ⊕ 40 × P (9c) ⊕ 41 × P (9d)
⊕ 94 × P (18a) ⊕ 95 × P (18b) ⊕ 95 × P (18c)) ⨁ V_{3}

or

S_{3}↓_{H}·1_{B} = 9c ⨁ (54 × P (9a) ⊕ 54 × P (9b) ⊕ 39 × P (9c) ⊕ 40 × P (9d)
⊕ 93 × P (18a) ⊕ 96 × P (18b) ⊕ 96 × P (18c)) ⨁ V_{2}

or

S_{3}↓_{H}·1_{B} = 9d ⨁ (54 × P (9a) ⊕ 54 × P (9b) ⊕ 41 × P (9c) ⊕ 40 × P (9d)
⊕ 94 × P (18a) ⊕ 95 × P (18b) ⊕ 95 × P (18c)) ⨁ V_{3}

or

S_{3}↓_{H}·1_{B} = 9d ⨁ (54 × P (9a) ⊕ 54 × P (9b) ⊕ 40 × P (9c) ⊕ 39 × P (9d)
⊕ 93 × P (18a) ⊕ 96 × P (18b) ⊕ 96 × P (18c)) ⨁ V_{2},

where V_{3} and V_{2} are the trivial source kH-modules in B with vertex Q given in 3.8. In particular, f(S_{3}) = 9c or f(S_{3}) = 9d, and we have to decide which case actually occurs.

Keeping the notation from 4.10, we by the proof of 4.5(iii) have S_{3} = χ↑^{G}·1_{A}, hence (χ↑^{G}·1_{A})↓_{H}·1_{B} = S_{3}↓_{H}·1_{B} = f(S_{3}) ⊕ (Q-proj) ⊕ (proj). Hence f(S_{3}) is a direct summand of

(χ↑^{G})↓_{H}·1_{B} = ⨁_{g} ((χ^{g})↓_{M^{g}∩H})↑^{H}·1_{B},

where g runs through a set of representatives of the M-H double cosets in G. Since f(S_{3}) has P as a vertex, and P is normal in H, we only have to look at summands coming from g ∈ G such that P ≤ M^{g} ∩ H. But for these g we have P, P^{g^{−1}} ≤ M, which since P is a Sylow 3-subgroup of M implies the existence of m ∈ M such that P^{m} = P^{g^{−1}}, hence h := mg ∈ H, and thus g = m^{−1}h ∈ MH, that is, we may assume g = 1.

Thus we conclude that f(S_{3}) is a direct summand of (χ↓_{M∩H})↑^{H}·1_{B} = (χ↓_{N_{M} (P)})↑^{H}·1_{B}. Now a computation with GAP [12], using the character tables of N_{M}(P) and H, shows that (χ↓_{N_{M} (P)})↑^{H}·1_{B} = 9c is indecomposable, showing that f(S_{3}) = 9c.

We just remark that it is possible, using GAP [12] and specially tailored programs to deal efficiently with permutations on millions of points, to construct the transitive permutation representation of G on 3078000 points, that is, the action of G on the cosets of 2.HS in G, where 2.HS is the derived subgroup of M, and to use the restriction of this representation to H to show that the first of the four possible decompositions of S_{3}↓_{H}·1_{B} listed above actually occurs. But we will not need this fact. ■

5. Green correspondence for HS

5.1.Notation and assumption. In the rest of this paper, we use the following notation, too. Let G′ be the Higman-Sims simple group HS. Since Sylow 3-subgroups of G′ are isomorphic to C_{3} × C_{3}, we by abuse of notation let P denote a Sylow 3-subgroup of HS as well. There is exactly one conjugacy class of G′ which contain elements of order 3, that is, P has exactly one G′-conjugacy class of subgroups of order 3, see [10, p.81]. Let H′ = N_{G′}(P), and hence H′ = (P ⋊ SD_{16}) × 2, where the action of SD_{16} on P is given by the embedding of SD_{16} as a Sylow 2-subgroup of Aut(P) ≅ GL_{2}(3). Let A′ and B′, respectively, be the principal block algebras of OG′ and OH′.

5.2.Lemma.

(i) The character table of P ⋊ SD_{16} is given as follows:

conjugacy class 1A 2A 2B 3A 4A 4B 6A  8A  8B
centraliser  144 16 12 18 8 4 6  8  8
χ_{1a}  1 1 1 1 1 1 1  1  1
χ_{1b}  1 1 1 1 1 −1 1  −1  −1
χ_{1c}  1 1 −1 1 1 1 −1  −1  −1
χ_{1d}  1 1 −1 1 1 −1 −1  1  1
χ_{2a}  2 2 0 2 −2 0 0  0  0
χ_{2b}  2 −2 0 2 0 0 0 \sqrt{−2} −\sqrt{−2}
χ_{2c}  2 −2 0 2 0 0 0 −\sqrt{−2} \sqrt{−2}
χ_{8a}  8 0 2 −1 0 0 −1  0  0
χ_{8b}  8 0 −2 −1 0 0 1  0  0

Note that this identifies the characters χ_{1a}, χ_{1b}, χ_{1c}, χ_{1d}, χ_{8a}, and χ_{8b} uniquely. (ii) B′ ≅ O[P ⋊ SD_{16}], as interior P -algebras and hence k-algebras, and we can write

that

Irr(B′) = {1_{H′} = χ_{1a}, χ_{1b}, χ_{1c}, χ_{1d}, χ_{2a}, χ_{2a}, χ_{2c} = χ_{2b}^{∨}, χ_{8a}, χ_{8b}},
IBr(B′) = {1a, 1b, 1c, 1d, 2a, 2b, 2c = 2b^{∨}},

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

(iii) The 3-decomposition and the Cartan matrices of B′, respectively, are the following:

1a 1b 1c 1d 2a 2b 2c
χ_{1a} 1 . . . . . .
χ_{1b} . 1 . . . . .
χ_{1c} . . 1 ….
χ_{1d} … 1 …
χ_{2a} …. 1 . .
χ_{2b} . . . . . 1 .
χ_{2c} . . . . . . 1
χ_{8a} 1 1 . . 1 1 1
χ_{8b} . . 1 1 1 1 1
P (1a) P (1b) P (1c) P (1d) P (2a) P (2b) P (2c)
1a  2  1  0  0  1  1  1
1b  1  2  0  0  1  1  1
1c  0  0  2  1  1  1  1
1d  0  0  1  2  1  1  1
2a  1  1  1  1  3  2  2
2b  1  1  1  1  2  3  2
2c  1  1  1  1  2  2  3

Proof. (i) This is found using explicit computation with GAP [12]. Using the smallest faithful permutation representation of G′ on 100 points, available in [68], P can be computed as a Sylow 3-subgroup of G′, and hence the normaliser H′ = N_{G′}(P) of P is easily determined explicitly. Now the conjugacy classes of H′ can be computed, and its ordinary character table is found using the Dixon-Schneider algorithm. Note that there are unique conjugacy classes 2B and 4B consisting of elements of order 2 and 4, respectively, and having centralisers of order 12 and 4, respectively.

(ii)–(iii) Easy from the character table. ■

5.3.Notation. We use the notation 1_{H′} = χ_{1a}, χ_{1b}, χ_{1c}, χ_{1d}, χ_{2a}, χ_{2a}, χ_{2c} = χ_{2b}^{∨}, χ_{8a}, χ_{8b} and 1a, 1b, 1c, 1d, 2a, 2b, 2c = 2b^{∨}, as in 5.2. Namely, we can write

Irr(B′) = Irr(H′) = {1_{H′} = χ_{1a}, χ_{1b}, χ_{1c}, χ_{1d}, χ_{2a}, χ_{2a}, χ_{2c} = χ_{2b}^{∨}, χ_{8a}, χ_{8b}},
IBr(B′) = IBr(H′) = {1a, 1b, 1c, 1d, 2a, 2b, 2c = 2b^{∨}}

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

5.4.Lemma.

(i) The radical and socle series of PIMs in B′ are the following:

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

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

Note that this identifies the simples 2b and 2c uniquely. (ii) An Alperin diagram of the PIM P (2a) is given as follows:

P (2a) = [figure: 2a; 2b; 2c; 1b; 1c; 2a; 1a; 1d; 2c; 2b; 2a]

Proof. Using the faithful permutation representation of H′ obtained in 5.2, we have used the MeatAxe [60] to construct the PIMs explicitly as matrix representations. Then we have used the method described in [45] to find the radical and socle series, and the method in [42] to compute the whole submodule lattice of P (2a). ■

5.5.Lemma.

(i) We can write that

Irr(A′) = {χ′_{1} = 1_{G′}, χ′_{154}, χ′_{22}, χ′_{1408}, χ′_{1925}, χ′_{770}, χ′_{3200}, χ′_{2750}, χ′_{1750}}
IBr(A′) = {1_{G′}, 154, 22, 1253, 1176, 748, 321}

(ii) All simples 1_{G′}, 154, 22, 1253, 1176, 748, 321 in A′ are self-dual, and have P as their

vertices. (iii) The simples 1_{G′}, 154, 22 are trivial source kG′-modules.

Proof. (i) This was first calculated by Humphreys [18, p.329]; see also [67, HS (mod 3)] and [19].

(ii) This is obtained by a result of Knörr [22, 3.7.Corollary].

(iii) It follows from [64] or [51, Example 4.8] that the Green correspondents f′(k_{G′}), f′(22) and f′(154) are k_{H′} = 1a, 1b and 1c, respectively, see 5.7 below. ■

5.6.Notation. We write χ′_{1} = 1_{G′}, χ′_{154}, χ′_{22}, χ′_{1408}, χ′_{1925}, χ′_{770}, χ′_{3200}, χ′_{2750}, χ′_{1750}, as well as 1_{G′}, 154, 22, 1253, 1176, 748, 321 as in 5.5.

5.7.Lemma. f′(k_{G′} = 1a) = k_{H′} = 1a  f′(154) = 1b  f′(22) = 1c

f′(1253) = [figure: 2a; 2c; 2b; 2a] f′(1176) = [figure: 2b; 1c; 1b; 2c]

f′(748) = [figure: 1d; 2a; 2b; 2c; 2a; 1d] f′(321) = [figure: 2c; 1d; 1a; 2b]

Proof. This follows from [64], see [51, Example 4.8, HS]. ■

5.8.Lemma. The Cartan matrix of A′ is the following:

P (k_{G′}) P (154) P (22) P (1253) P (1176) P (748) P (321)
k_{G′}  4  1  1  2  2  2  0
154  1  3  1  2  0  0  1
22  1  1  4  2  1  2  1
1253  2  2  2  4  2  1  2
1176  2  0  1  2  3  2  1
748  2  0  2  1  2  3  0
321  0  1  1  2  1  0  2

Proof. This was first calculated by Humphreys [18, p.329]; see also [67, HS (mod 3)] and [19]. ■

6. Stable equivalence between A and B for HN

6.1.Notation. First of all, recall the notation G, A, P, H, B, e, Q, E, f as in 3.3 and 4.9. Let i and j respectively be source idempotents of A and B with respect to P. As remarked in [40, pp.821–822], we can take i and j such that Br_{P} (i)·e = Br_{P} (i) ≠ 0 and that Br_{P} (j)·e = Br_{P} (j) ≠ 0. Set G_{P} = C_{G}(P) = C_{H}(P) = H_{P}, and set G_{Q} = C_{G}(Q) and H_{Q} = C_{H}(Q). By replacing e_{Q} and f_{Q} (if necessary), we may assume that e_{Q} and f_{Q} respectively are block idempotents of kG_{Q} and kH_{Q} such that e_{Q} and f_{Q} are determined by i and j, respectively. Namely, Br_{Q}(i)·e_{Q} = Br_{Q}(i) and Br_{Q}(j)·f_{Q} = Br_{Q}(j). Let A_{Q} = kC_{G}(Q)·e_{Q} and B_{Q} = kC_{H}(Q)·f_{Q}, so that e_{Q} = 1_{A_{Q}} and f_{Q} = 1_{B_{Q}}.

6.2.Lemma. Let 𝔐_{Q} be a unique (up to isomorphism) indecomposable direct summand of A_{Q}↓_{G_{Q}×H_{Q}}^{G_{Q}×G_{Q}}·1_{B_{Q}} with vertex ΔP (note that such an 𝔐_{Q} always exists by 2.5). Then, a pair (𝔐_{Q}, 𝔐_{Q}^{∨}) induces a Puig equivalence between A_{Q} and B_{Q}.

Proof. This follows from 3.4(viii), 2.12(iii) and 2.11(iii). ■

6.3.Lemma.

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

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

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

Proof. This follows just like in [31, Proof of Lemma 6.3]. Namely, we get the assertion by [2, Proposition 4.21] and [9, Theorem 1.8(i)] for the morphisms in the Brauer categories and also by [32, Theorem], 6.2, 3.4(vi) and [40, Theorem 3.1], see [41, Theorem A.1]. ■

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

7. Images of simples via the functor F

7.1.Lemma. F (S_{1}) = 9a, F (S_{2}) = 9b, F (S_{3}) = 9c.

Proof. These follow from 4.10, 4.11, 4.12, 2.9 and 6.3. ■

7.2.Lemma.

(i) The trivial source kG-module in 4.8 has Q ≅ C_{3} as its vertex. (ii) The trivial source kG-module in 4.7 has Q ≅ C_{3} as its vertex.

Proof. (i) Let X be the trivial source kG-module in 4.8. We get by 6.3(ii) that F (X) = Y ⊕ (proj) for a non-projective indecomposable B-module Y. Then, it follows from 2.7, 6.3(i) and 7.1 that

0 ≠ Hom_{A}(X, S_{1}) ≅ \underline{Hom}_{A}(X, S_{1}) ≅ \underline{Hom}_{B}(F (X), F (S_{1}))
= \underline{Hom}_{B}(F (X), 9a) = \underline{Hom}_{B}(Y, 9a) ≅ Hom_{B}(Y, 9a)

as k-spaces. Clearly, Y is a trivial source kH-module in B by 6.3(ii).

Suppose that X has P as a vertex. Then, so does Y by 6.3(ii). This yields that Y ∈ {9a, 9b, 9c, 9d, 18a, 18b, 18c} from 3.8, and hence Y ∈ {9d, 18a, 18b, 18c} by 7.1. But, the above computation shows that Hom_{B}(Y, 9a) ≠ 0, a contradiction.

Since X is non-projective, we know that Q is a vertex of X from 3.4(vi).

(ii) Let X′ be the trivial source kG-module in 4.7. We get by 6.3(ii) that F (X′) = Y′ ⊕ (proj) for a non-projective indecomposable B-module Y′. Then, it follows from 2.7, 6.3(i) and 7.1 that

0 ≠ Hom_{A}(X′, S_{3}) ≅ \underline{Hom}_{A}(X′, S_{3}) ≅ \underline{Hom}_{B}(F (X′), F (S_{3}))
= \underline{Hom}_{B}(F (X′), 9c) = \underline{Hom}_{B}(Y′, 9c) ≅ Hom_{B}(Y′, 9c)

as k-spaces. Clearly, Y′ is a trivial source kH-module in B by 6.3(ii).

Suppose that X′ has P as a vertex. Then, so does Y′ by 6.3(ii). This yields that Y′ ∈ {9a, 9b, 9c, 9d, 18a, 18b, 18c} from 3.8, and hence Y′ ∈ {9d, 18a, 18b, 18c} by 7.1. But, the above computation shows that Hom_{B}(Y′, 9c) ≠ 0, a contradiction.

Since X′ is non-projective, we know that Q is a vertex of X′ from 3.4(vi).

■

7.3.Lemma. Let X be the trivial source kG-module with vertex Q showing up in 4.8 and 7.2(i). Then, F (X) = V_{1} ⊕ (proj), where V_{1} is the trivial source kH-module in B with vertex Q given in 3.8(iii). Namely,

F ( [figure: S_{1}; S_{2}; S_{4}; S_{1}; S_{2}] ) = [figure: 18a; 18b; 18a; 18c; 9b; 9a; 9b; 9a] ⨁ (proj).

Proof. We know from the proof of 7.2(i) that [Y, 9a]^{B} ≠ 0. Hence we get the assertion by 3.8(iii). ■

7.4.Lemma. Let X′ be the trivial source kG-module with vertex Q showing up in 4.7 and 7.2(ii). Then, F (X′) = V_{2} ⊕ (proj), where V_{2} is the trivial source kH-module in B with vertex Q given in 3.8(iii). Namely,

F ( \begin{matrix}S_{3} \\ S_{6} \\ S_{3} \end{matrix} ) = [figure: 18a; 18c; 18a; 18b; 9d; 9c; 9d; 9c] ⨁ (proj).

Proof. We know from the proof of 7.2(ii) that [Y′, 9c]^{B} ≠ 0. Hence we get the assertion by 3.8(iii). ■

7.5.Lemma. It holds that F (S_{4}) = [figure: 18a; 18c; 18b; 18a]

Proof. Let X be the trivial source kG-module in A with vertex Q given in 4.8 and 7.2(i). By 7.3, we can write F (X) = V_{1} ⊕ (proj), where V_{1} is the trivial source kH-module in B given in 3.8(iii). Then, since F (S_{1}) = 9a by 7.1, it follows from 2.8 that

F ( [figure: S_{2}; S_{1}; S_{4}; S_{2}] ) = F (X/S_{1}) = (V_{1}/9a) ⨁ (proj) = [figure: 18a; 18b; 18a; 18c; 9b; 9a; 9b] ⨁ (proj).

Similarly, we get by 2.8 that

F ( \begin{matrix}S_{2} \\ S_{4} \\ S_{2} \end{matrix} ) = F (Ker[ [figure: S_{2}; S_{1}; S_{4}; S_{2}] ↠ S_{1}])

≅ Ker( [figure: 18a; 18b; 18a; 18c; 9b; 9a; 9b] ↠ 9a) ⨁(proj) = [figure: 18a; 18b; 18a; 18c; 9b; 9b] ⨁ (proj).

Then, since F (S_{2}) = 9b by 7.1, we similarly obtain by 2.8 that

F (S_{4}) = [figure: 18a; 18c; 18b; 18a] ⨁ (proj).

Therefore, 6.3(i) and 2.9 imply the assertion. ■

7.6.Lemma. It holds that F (S_{6}) = [figure: 18a; 18c; 18a; 18b; 9d; 9d] .

Proof. Let X′ = \begin{matrix}S_{3} \\ S_{6} \\ S_{3} \end{matrix} in 4.7, that is, X′ is a trivial source kG-module in A with vertex Q.

Then, 7.4 yields that

F (X′) = [figure: 18a; 18c; 18a; 18b; 9d; 9c; 9d; 9c] ⨁ (proj).

Since F (S_{3}) = 9c by 7.1, we obtain the assertion from 6.3(i) and 2.9 just as in the proof of 7.5. ■

7.7.Notation. We use the notation W = F (S_{5}) ⊕ F (S_{7}) in the rest of this paper.

7.8.Lemma. We get the following.

(i) The module W is self-dual.

(ii) The module W is a direct sum of exactly two non-projective non-simple indecompos-

able B-modules, and both of them are self-dual. (iii) It holds that F (S_{5}) and F (S_{7}) are neither simple B-modules, and 2 ⩽ j(W) ⩽ 4. (iv) [9x, W]^{B} = [W, 9x]^{B} = 0 for any x ∈ {a, b, c}.

(v) [18a, W]^{B} = [W, 18a]^{B} = 0.

Proof. (i) This follows from 4.3(i) and 6.3(i).

(ii) This follows from 2.9, 6.3(i), 4.3(i), 2.3(i) and 4.1.

(iii) By (ii) and 3.6(vi), we get j(W) ⩽ 4. Assume that F (S_{5}) is simple. Then, we know by 3.8(ii) and 6.3(ii) that S_{5} is a trivial source module, and hence S_{5} lifts to a trivial source OG-module by 2.3(i). This contradicts the 3-decomposition matrix in 4.1. Hence, F (S_{5}) is not simple. Similarly, we know that F (S_{7}) is not simple. These imply j(W) ⩾ 2.

(iv) This is obtained by 7.1 and 2.13.

(v) Set X = F (S_{4}). By 7.5, there is an epimorphism X ↠ 18a. Hence, we get from 2.7 and 7.5 that

Hom_{B}(18a, W) ≅ \underline{Hom}_{B}(18a, W) ≅ \underline{Hom}_{A}(F^{−1}(18a), F^{−1}(W))
= \underline{Hom}_{A}(F^{−1}(18a), S_{5} ⊕ S_{7}) ≅ Hom_{A}(F^{−1}(18a), S_{5} ⊕ S_{7})
⊆ Hom_{A}(F^{−1}(X), S_{5} ⊕ S_{7}) = Hom_{A}(S_{4}, S_{5} ⊕ S_{7}) = 0. ■

7.9.Notation. Let M = [figure: S_{4}; S_{4}; S_{7}; S_{5}; S_{2}; S_{1}] be the trivial source kG-module in A showing

up in 4.6, and set 𝔛_{B} = F (M) and we use the notation 𝔛 in the rest of this paper.

7.10.Lemma.

(i) The module 𝔛 has a filtration

𝔛 = [figure: 18a; 18c; 18b; 18a] | 9a ⊕ 9b ⊕ W | [figure: 18a; 18c; 18b; 18a]

namely, 𝔛 has submodules 𝔛 ⫌ Y ⫌ Z such that 𝔛/Y ≅ Z ≅ [figure: 18a; 18c; 18b; 18a] and

Y/Z ≅ 9a ⊕ 9b ⊕ W. (ii) It holds 𝔛 = V ⊕ P (18a) where V ∈ {V_{3}, V_{4}}.

Proof. (i) This follows from 4.6, 7.1 and 7.5.

(ii) We know by 6.3(ii) that 𝔛 = V ⊕ L for an indecomposable kH-module V in B with vertex Q and a projective kH-module L in B. Note that V_{i} |̸ 𝔛 for i = 1, 2 by 7.3 and 7.4. Thus, V ∈ {V_{3}, V_{4}} by 3.8(iii). Moreover, since [V_{3}, 18a]^{B} = [V_{4}, 18a]^{B} = 0 by 3.8(iii), we know that [V, 18a]^{B} = [V, 18a]^{B} = 0 again by 3.8(iii). Thus, we have P (18a)|L by (i), and hence P (18a)|𝔛.

Next, assume that P (T)|L for a simple kH-module T in B with T ≅ 1̸8a. Since Z has a unique minimal submodule, and which is isomorphic to 18a, we have that P (T) ∩ Z = 0 in 𝔛, and hence that there is a direct sum P (T) ⊕ Z in 𝔛. Set 𝔛̄ = 𝔛/Z. Clearly, 𝔛̄ ⊇ (P (T) ⊕ Z)/Z ≅ P (T). Since P (T) is injective, it holds P (T)|𝔛̄. Set U = (𝔛̄)^{∨}. Then, by the dualities, we know P (T^{∨})|U. Now, by the filtration of 𝔛, U has a filtration

U = 9a ⊕ 9b ⊕ W | [figure: 18a; 18c; 18b; 18a]

Namely, U has a submodule Z′ such that

Z′ ≅ [figure: 18a; 18c; 18b; 18a] and U/Z′ ≅ 9a ⊕ 9b ⊕ W.

We have T^{∨} ≅ 1̸8a by 3.6(iii). Hence, we get P (T^{∨}) ∩ Z′ = 0 in U, and hence there is a direct sum P (T^{∨}) ⊕ Z′ ⊆ U. Then, we have

P (T^{∨}) ≅ (P (T^{∨}) ⊕ Z′)/Z′ ⊆ U/Z′ ≅ 9a ⊕ 9b ⊕ W.

Since P (T^{∨}) is injective, it holds that P (T^{∨})|(9a ⊕ 9b ⊕ W), so that P (T^{∨})|W by 3.6(vi). This is a contradiction by 7.8(ii).

Now, assume that (P (18a) ⊕ P (18a))|𝔛. Then, since soc(Z) ≅ 18a, it follows from 2.14 that

P (18a) | 𝔛/Z = ( [figure: 18a; 18c; 18b; 18a] | 9a ⊕ 9b ⊕ W ).

Then, by taking its dual, we get also that

P (18a) | (𝔛/Z)^{∨} = (9a ⊕ 9b ⊕ W | Z)

where the right-hand-side is a filtration, by using 7.8(i) and 3.6(iii). Set N = (𝔛/Z)^{∨}. Then, we may consider that N has a B-submodule Z such that N/Z ≅ 9a ⊕ 9b ⊕ W and N = P (18a) ⊕ N′ for a B-submodule N′ of N. Since j(Z) = 3, it holds Z ⊆ soc_{3}(N) = soc_{3}(P (18a)) ⊕ soc_{3}(N′). This implies that there exists a B-epimorphism π : N/Z ↠ N/soc_{3}(N). Clearly,

N/soc_{3}(N) = [P (18a) ⊕ N′]/[soc_{3}(P (18a)) ⊕ soc_{3}(N′)]
≅ [P (18a)/soc_{3}(P (18a)] ⊕ [N′/soc_{3}(N′)].

Since P (18a)/soc_{3}(P (18a)) = \begin{matrix}18a \\ 18b 18c \end{matrix} by 3.6(vi), we get that 18a | [(N/Z)/rad(N/Z)] ≅

9a ⊕ 9b ⊕ [W/rad(W)]. This shows that [W, 18a]^{B} ≠ 0, which is a contradiction by 7.8(v). Thus, we know [P (18a)|L]^{B} = 1. Therefore, we get L ≅ P (18a). We are done. ■

7.11.Lemma. W/rad(W) ≅ soc(W) ≅ 18b ⊕ 18c.

Proof. By 7.8(i) and 3.6(iii), it suffices to show only W/rad(W) ≅ 18b⊕18c. By 7.10(ii), we have

(13) 𝔛 = V ⊕P (18a) = [figure: 18c; 18b; 9y; 18a; 9x; 18b; 18c] ⊕ P (18a), where (x, y) ∈ {(b, a), (c, d)}.

By 7.10(i), 𝔛 has a filtration

(14)  𝔛 = [figure: 18a; 18c; 18b; 18a] | 9a ⊕ 9b ⊕ W | [figure: 18a; 18c; 18b; 18a]

Set L_{i}(W) = rad^{i}(W)/rad^{i+1}(W) for each i = 0, 1, .... Then (13) and (14) show that (18b ⊕ 18c)|L_{1}(W). Recall (18b)^{∨} ≅ 18c by 3.6(iii).

Suppose that (18b ⊕ 18b) | L_{1}(W). Then, by (14), 𝔛 has a factor module 𝔛̄ which has a filtration

[figure: 18a; 18c; 18b; 18a] | 18b ⊕ 18b

Since [𝔛, 18b]^{B} = 1 by (13), and since there do not exist modules of forms \begin{matrix}18b \\ 18b \end{matrix} nor \begin{matrix}18c \\ 18b \end{matrix}

by 3.6(vi), there must be a kH-module having radical and socle series

\begin{matrix}18a \\ 18b 18c \\ 18a \\ 18b \end{matrix}

But this is a contradiction by 3.10(i).

Similarly, we get a contradiction by using 3.10(ii) if (18c ⊕ 18c)|L_{1}(W).

Thus it holds that [W, 18b]^{B} = [W, 18c]^{B} = 1 and [W, T]^{B} = 0 for any T ∈ {9a, 9b, 9c, 18a} by 3.6(iii) and 7.8(iv)-(v). However, we have to investigate for 9d.

Assume, first, that the case (x, y) = (b, a) happens in (13). Then, (14) and (13) imply that W = 9a + 9b + 9c + 9d + 2 × 18b + 2 × 18c, as composition factors.

Suppose that 9d | L_{1}(W). Then, since c_{W} (9d) = 1 and since W and 9d are both self-dual by 3.6(iii) and 7.8(i), we get that 9d | W = F (S_{5}) ⊕ F (S_{7}). Recall that F (S_{5}) and F (S_{7}) are both non-projective indecomposable kH-modules by 2.9 and 6.3(i)-(ii). Since 9d is a trivial source kH-module by 3.8(ii), we know by 6.3(ii) that S_{5} or S_{7} is a trivial source module, and hence that S_{5} or S_{7} lifts from k to O by 2.3(i). This is a contradiction by the 3-decomposition matrix in 4.1.

Hence, 9d |̸ L_{1}(W). This yields L_{1}(W) ≅ 18b ⊕ 18c.

Next, assume that the case (x, y) = (c, d) in (13) happens. Then, (13) and (14) imply that

(15)  W = 2 × 9c + 2 × 9d + 2 × 18b + 2 × 18c,  as composition factors.

Suppose that (9d⊕9d) | L_{1}(W). Then, the self-dualities of 9d and W in 3.6(iii) and 7.8(i) imply that (9d ⊕ 9d) | W = F (S_{5}) ⊕ F (S_{7}). Hence, W ≅ 9d ⊕ 9d by 7.8(ii), contradicting 7.7 and 7.8.

Thus,

[W, 9d]^{B} ⩽ 1. (16)

Assume, next, that [W, 9d]^{B} = 1. Hence, by the dualities in 3.6(iii), we have

L_{1}(W) ≅ soc(W) ≅ 18b ⊕ 18c ⊕ 9d. (17)

We get by 7.8 that W = W_{1} ⊕ W_{2} where W_{i} is a non-simple non-projective indecomposable self-dual B-module for i = 1, 2. Thus, by (17) and by interchanging W_{1} and W_{2}, we may assume that L_{1}(W_{1}) ≅ 18b, 18c or 9d.

Case 1: L_{1}(W_{1}) ≅ 18b. Then, soc(W_{1}) ≅ 18c since (18b)^{∨} ≅ 18c by 3.6(iii) and since

W_{1} is self-dual. Hence, the structure of P (18b) in 3.6(vi) yields that W_{1} = \begin{matrix}18b \\ 9c \\ 18c \end{matrix} . Hence,

(15) and (17) imply that L_{1}(W_{2}) ≅ 18c ⊕ 9d and L_{2}(W_{2}) ≅ 9c. But this is a contradiction since Ext_{B}^{1}(18c, 9c) = 0 = Ext_{B}^{1}(9d, 9c) by 3.6(vi).

Case 2: L_{1}(W_{1}) ≅ 18c. As in Case 1, we know that W_{1} = \begin{matrix}18c \\ 9c \\ 18b \end{matrix} . Then we get a

contradiction by 3.6(vi) as in Case 1.

Case 3: L_{1}(W_{1}) ≅ 9d. By the self-dualities of W_{1} in 7.8(ii) and simple B-modules in 3.6(iii), we get that soc(W_{1}) ≅ 9d. It follows by 2.16 that soc(W_{1}) ⊆ rad(W_{1}). Hence c_{W_{1}}(9d) = 2 by (15). Thus, the structure of P (9d) in 3.6(vi) yields that W_{1} ≅ P (9d), a contradiction.

Therefore [W, 9d]^{B} ≠ 1, and hence [W, 9d]^{B} = 0 by (16). So that we have L_{1}(W) ≅ 18b ⊕ 18c. ■

7.12.Lemma. 𝔛 = V_{3} ⊕ P (18a).

Proof. Suppose that 𝔛 = V_{4} ⊕ P (18a). Then, we get by 7.10(i)-(ii) and 3.6(iv) that W = 2 × 9c + 2 × 9d + 2 × 18b + 2 × 18c, as composition factors. We use the same notation L_{i}(W) as in the proof or 7.11. By 7.11, L_{1}(W) ≅ 18b ⊕ 18c. Since c_{W} (9c) = 2, it follows from 3.6(vi) and 7.8(iii) that j(W) = 4 and 9c | L_{4}(W). This means 9c | soc(W), contradicting 7.11. Therefore, we get the assertion by 7.10(ii). ■

7.13.Lemma. W = [figure: 18b; 9c; 9b; 18c] ⊕ [figure: 18c; 9d; 9a; 18b] = F (S_{5}) ⊕ F (S_{7}).

Namely, either one of the following two cases occurs:

Case (a) :  F (S_{5}) = [figure: 18b; 9c; 9b; 18c] and F (S_{7}) = [figure: 18c; 9d; 9a; 18b]

Case (b) :  F (S_{5}) = [figure: 18c; 9d; 9a; 18b] and F (S_{7}) = [figure: 18b; 9c; 9b; 18c]

Proof. Here as well we use the notation L_{i}(W) for i = 1, 2, ... just as in the proof of 7.11. It follows from 7.10(i) that 𝔛 has a filtration

𝔛 = [figure: 18a; 18c; 18b; 18a] | 9a ⊕ 9b ⊕ W | [figure: 18a; 18c; 18b; 18a] (18)

namely, 𝔛 has submodules Y and Z such that 𝔛 ⫌ Y ⫌ Z, 𝔛/Y ≅ Z ≅ [figure: 18a; 18c; 18b; 18a] and

Y/Z ≅ 9a ⊕ 9b ⊕ W. On the other hand, 7.12 says that

(19)  𝔛 = [figure: 18c; 18b; 9a; 18a; 9b; 18b; 18c] ⨁ P (18a).

Then, we know by (18), (19) and 3.6(iv) that

(20)  W = 9a + 9b + 9c + 9d + 2 × 18b + 2 × 18c, as composition factors.

By 7.11 and (20), we know j(W) ⩾ 3.

Assume that j(W) ⩾ 4. Then, j(W) = 4 by 7.8(iii). Since L_{1}(W) ≅ 18b ⊕ 18c by 7.11, we get by 3.6(vi) that

L_{4}(W) | L_{4}(P (18b)) ⨁ L_{4}(P (18c)) = (9a ⊕ 18a ⊕ 9d) ⨁(9b ⊕ 18a ⊕ 9c)

and

L_{4}(W) | soc(W) = 18b ⊕ 18c.

This is a contradiction.

Hence j(W) = 3. Thus, again by 7.11, (20) and 3.6(vi), we know that W has radical and socle series

(21)  \begin{matrix}18b 18c \\ 9b 9c 9a 9d \\ 18c 18b \end{matrix}

Now, as in the proof of 7.11, we get by 7.8 that W = W_{1}⊕W_{2} where W_{i} is a non-simple nonprojective indecomposable self-dual B-module for i = 1, 2. Then, by (21), we may assume that L_{1}(W_{1}) ≅ 18b, soc(W_{1}) ≅ 18c, L_{1}(W_{2}) ≅ 18c and soc(W_{2}) ≅ 18b since (18b)^{∨} ≅ 18c by 3.6(iii). Hence the structures of P (18b) and P (18c) in 3.6(vi) yield that

W_{1} = \begin{matrix}18b \\ 9b 9c \\ 18c \end{matrix}  and  W_{2} = \begin{matrix}18c \\ 9a 9d \\ 18b \end{matrix}  ■

8. Proof of main results

8.1.Notation. We still keep the notation F, j, B′, f′ and g′, see 6.4, 3.7 and 5.2–5.4. Set E = SD_{16}, and let P ⋊ E be the canonical semi-direct product such that E acts on P faithfully. Recall that Aut(P) ≅ GL_{2}(3) since P = C_{3} × C_{3}, and hence SD_{16} is a Sylow 2-subgroup of GL_{2}(3).

8.2.Lemma. The non-principal block algebra A of HN and the principal block algebra A′ of HS are Puig equivalent.

Proof. Let j be the same as in 3.6(ii). Since jBj ≅ O[P ⋊ E] = B′ as interior P - algebras by 3.6(ii), we can identify jBj and B′. Define a functor F′ : mod-B → mod-B′ via F′(−) = − ⊗_{B} Bj. By 3.6(ii), F′ induces a Puig equivalence (which is stronger than a Morita equivalence) between B and B′. In the following we use the information on the structures of PIMs in B and B′ described in 3.6(vi) and 5.2(iii), respectively, without quoting these statements.

Then, first of all, we know that F′(18a) = 2a by looking at the PIMs P (18a) and P (2a). Similarly, we know at least that {F′(9a), F′(9b), F′(9c), F′(9d)} = {1a = k_{H′}, 1b, 1c, 1d}. It follows from 5.4 that 1x ⊗ 1x = 1a for any x ∈ {a, b, c, d} since they are just in Irr(E). Hence a technique of self-Puig equivalence in [31, 2.8.Lemma] can be used just as in the proof of [31, 6.8.Lemma]. Namely, we can assume that F′(9a) = 1a. Hence, by comparing the second Loewy layers of P (9a) and P (1a), we get F′(18b) = 2b. Similarly, by looking at the third Loewy layers of P (9a) and P (1a), we have F′(9b) = 1b. If we look at the fourth Loewy layers of these PIMs, then we know F′(18c) = 2c. Thus, by looking at the second Loewy layers of P (18c) and P (2c), we know also that F′(9d) = 1d. These mean that F′(9c) = 1c. Namely, we can assume that

\begin{matrix}F′(9a) = 1a, F′(9b) = 1b, & F′(9c) = 1c, F′(9d) = 1d, \\ F′(18a) = 2a, F′(18b) & = 2b, F′(18c) = 2c. \end{matrix} (22)

Table 1. Case(b).

mod-A ⟶^{F}  mod-B  ⟶^{F′}  mod-B′  ⟶^{f^{′−1}} mod-A′
S_{1}  ↦  9a  ↦  1a  ↦  k_{G′}
S_{2}  ↦  9b  ↦  1b  ↦  154
S_{3}  ↦  9c  ↦  1c  ↦  22

[figure: S_{4}; ↦; 18a; 18c; 18b; 18a; ↦; 2a; 2c; 2b; 2a; ↦; 1253]

[figure: S_{5}; ↦; 18c; 9d; 9a; 18b; ↦; 2c; 1d; 1a; 2b; ↦; 321]

[figure: S_{6}; ↦; 18a; 18c; 18a; 18b; 9d; 9d; ↦; 2a; 2c; 2a; 2b; 1d; 1d; ↦; 748]

[figure: S_{7}; ↦; 18b; 9c; 9b; 18c; ↦; 2b; 1c; 1b; 2c; ↦; 1176]

We know by 7.13 that Case(a) or Case(b) happens.

Assume, first, that Case(b) occurs. Then, by bunching up 2.2, 7.1, 7.5, 7.6, 7.13 and 5.7, we get the diagram shown in Table 1.

First, all the three functors above are given by bimodules which are p-permutation modules over O[G_{1}×H_{1}] for corresponding two finite groups G_{1} and H_{1}, which are ΔP -projective, and also which induce a stable equivalence of Morita type at each step, if we indentify the source algebra jBj as O[P ⋊ E].

Secondly, it has to be noted that all non-simple modules in the above diagram are uniquely determined (up to isomorphism) by just the diagrams given in the above boxes: This is clear for F (S_{1}), F (S_{2}), F (S_{1}), f′(k_{G′}), f′(154), and f′(22) anyway, as well as for F (S_{4}) and f′(1253) by the structure of P (18a) and P (2a) given in 3.6(vi) and 5.2(iii).

To tackle F (S_{6}), the structure of P (18a) specified in 3.6(vii) shows that P (18a) has a unique quotient with composition factors 9d + 2 × 18a + 18b + 18c. Moreover, P (9d) has a unique quotient with composition factors 9d + 18a + 18b. Since they both have a unique submodule with composition factors 18a + 18b, the glueing to yield F (S_{6}) also is uniquely defined, and thus F (S_{6}) is uniquely determined by the diagram given. For f′(748) we argue similarly using 5.2(iv).

We consider F (S_{7}): Note first that for P (18b) there is no Alperin diagram defined. By 3.6(vi), let X be the unique quotient module of P (18b) having radical and socle series \begin{matrix}18b \\ 9b 9c \end{matrix} . By the structure of P (18b) given in 3.6(vi) we have [Ω(X), 18a]^{B} = 1, hence using

3.6(vii) there is a homomorphism φ ∈ Hom_{B}(P (18a), Ω(X)) such that

Im(φ) = [figure: 18a; 18c; 18b; 9a; 9d; 18a; 18b]

This implies Ω(X)/Im(φ) ≅ 18c. Since 18c occurs exactly twice as a composition factor of Ω(X), and also is a composition factor of Im(φ), we conclude that [Ω(X), 18c]^{B} = 1, thus

dim_{k}[Ext_{B}^{1}(X, 18c)] = 1. Hence a module having radical and socle series \begin{matrix}18b \\ 9b 9c \\ 18c \end{matrix} is uniquely

defined. For F (S_{5}), f′(1176), and f′(321) we argue similarly.

Then, it follows from 2.15 that A and A′ are splendidly stable equivalent of Morita type, that is, A and A′ are stable equivalent which is realized by an O[G × G′]-bimodule which is a p-permutation module and ΔP -projective. Hence, first of all, the stable equivalence actually gives a Morita equivalence by a result of Linckelmann [37, Theorem 2.1(iii)]. Then, if we look at the proof of [37, Theorem 2.1(iii)] which is actually given in [37, Remark 2.7], we know that the Morita equivalence between A and A′ gives a bijection such as S_{5} ↔ 321. Hence, we must have equalities between the corresponding Cartan invariants, namely, c(S_{5}, S_{5}) = c(321, 321). However, we get that c(S_{5}, S_{5}) = 3 by 4.1, and on the other hand, that c(321, 321) = 2 by 5.8. This is a contradiction. Thus, Case(b) cannot happen.

This means that only Case(a) occurs, as is shown in Table 2. Then, again the same argument given above still works. Namely, we have a Morita equivalence between A and A′, and hence the Morita equivalence is a Puig equivalence by a result of Puig (and, independently, of Scott) [55, Remark 7.5], see [40, Theorem 4.1]. ■

8.3.Proofs of 1.3 and 1.4. Recall that a Puig equivalence lifts from k to O by a result of Puig [53, 7.8.Lemma] (see [62, (38.8)Proposition], and that so does a splendid Rickard equivalence by a result of Rickard [57, Theorem 5.2], see [15, P.75, lines −17 ∼ −16]. Thus, it is enough to consider blocks A, B, A′ and B′ only over k. Thus, we get 1.4 by 8.2.

By results of Okuyama [51, Example 4.8] and [52, Corollary 2], the conjectures 1.1 and 1.2 hold for A′. Namely, we get the following diagram:

A \underrightarrow{Puig equiv.} A′
↓splendid Rickard equiv.
B \overleftarrow{Puig equiv.} B′

Therefore, we finally get that A and B are splendidly Rickard equivalent. That is, the proof of 1.3 is completed. ■

8.4.Proof of 1.5. We get 1.5 from 3.2 and 1.3. ■

Table 2. Case(a).

mod-A ⟶^{F}  mod-B  ⟶^{F′}  mod-B′  ⟶^{f^{′−1}} mod-A′
S_{1}  ↦  9a  ↦  1a  ↦  k_{G′}
S_{2}  ↦  9b  ↦  1b  ↦  154
S_{3}  ↦  9c  ↦  1c  ↦  22

[figure: S_{4}; ↦; 18a; 18c; 18b; 18a; ↦; 2a; 2c; 2b; 2a; ↦; 1253]

[figure: S_{5}; ↦; 18b; 9c; 9b; 18c; ↦; 2b; 1c; 1b; 2c; ↦; 1176]

[figure: S_{6}; ↦; 18a; 18c; 18a; 18b; 9d; 9d; ↦; 2a; 2c; 2a; 2b; 1d; 1d; ↦; 748]

[figure: S_{7}; ↦; 18c; 9d; 9a; 18b; ↦; 2c; 1d; 1a; 2b; ↦; 321]

Acknowledgements

The first author thanks Gabriel Navarro for informing [49, (3.2)Lemma].

A part of this work was done while the first author was staying in Braunschweig Technical University and RWTH Aachen University in 2007 and 2009. He is grateful to Bettina Eick and Gerhard Hiss for their kind hospitality. For this research the first author was partially supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C)17540010, 2005–2007, and (C)20540008, 2008–2010.

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