BROUÉ’S CONJECTURE FOR THE HALL-JANKO GROUP AND ITS DOUBLE COVER

MILES HOLLOWAY

Abstract. Broué’s abelian defect conjecture suggests a deep link between the module categories of a block of a group algebra and its Brauer correspondent, viz that they should be derived equivalent. We are able to verify Broué’s conjecture for the Hall-Janko group, even its double cover 2.J_{2}, as well as for U_{3}(4) and Sp_{4}(4). In fact we can verify Rickard’s refinement to Broué’s conjecture and show that the derived equivalence can be chosen to be a splendid equivalence for these examples.

1. Introduction

The representations of a finite group over an algebraically closed field of characteristic p dividing the order of the group and the representations of its local subgroups, the normalizers of non-trivial p-subgroups, are related. Amongst the many attempts to tease out the relationship, Broué’s abelian defect conjecture (see [2]) provides, albeit only for abelian defect groups, one of the most complete and deepest links to date.

In this paper we verify Broué’s conjecture for the groups 2.J_{2}, U_{3}(4) and Sp_{4}(4) with the aid of the computer algebra packages GAP [3] and MAGMA [6].

Let us recall Broué’s conjecture:

Conjecture 1.1 (Broué). Let G be a finite group, k an algebraically closed field of characteristic p, and ekG a block of kG with abelian defect group D. Let H = N_{G}(D) and fkH be the Brauer correspondent of ekG, then ekG and fkH should be derived equivalent.

Conjecture 1.1 is known to be true for cyclic defect groups, see for example [12] and for defect groups of principal blocks that are elementary abelian of order 9, see [4], as well as for certain other specific groups. Apart from these cases the only other prime p for which conjecture 1.1 is applicable to 2.J_{2}, U_{3}(4) or Sp_{4}(4) is p = 5. When p = 5 the defect groups of blocks of k[2.J_{2}], kU_{3}(4) and kSp_{4}(4) that are abelian and non-cyclic are all elementary abelian of order 25; in fact the defect groups are all Sylow 5-subgroups.

Our approach to verifying conjecture 1.1 for our examples will be to first establish a stable equivalence between ekG and fkH in Section 3. The stable equivalence will be of a particular kind - a splendid stable equivalence and is based upon a construction in [11]. Actually we will calculate the images of simple ekG-modules under this equivalence.

In Section 4 we lift the stable equivalence of Section 3 to a derived equivalence. In fact we will find the images of simple ekG-modules under a derived equivalence that agree, upon passing to the stable module category, with the images calculated in Section 3. In Subsection 4.1 we show how, under certain circumstances which include our examples, the derived equivalence can be refined to a splendid equivalence. In Subsection 4.2 we complete the verification of Conjecture 1.1 for our examples by constructing complexes that are images of simple ekG-modules under a splendid equivalence. Finally in Subsection 4.3 we show that the complexes constructed in Subsection 4.2 are independent, up to isomorphism in the homotopy category, of the choice of differentials.

1991 Mathematics Subject Classification. 20C20.

In Section 2 we record some relevant information about our groups and the blocks for which conjecture 1.1 is applicable.

Let us introduce some notation. Unless otherwise stated we’ll take k to be an algebraically closed field with char(k) = 5 (in fact as far as the computer calculations are concerned, GF(5) is a suitable splitting field for our groups). We’ll write ekG and fkH to denote blocks of full defect as in conjecture 1.1, with G being one of our groups. Which blocks and which group should be clear from the context.

Given a finite dimensional k-algebra Λ, mod-Λ denotes the category of finitely generated Λ-modules and proj- Λ the subcategory of finitely generated projective Λ-modules. Unless stated otherwise all Λ-modules will be right Λ-modules. All complexes will be cochain complexes and we’ll write D^{b}(Λ) for the bounded derived category D^{b}(mod- Λ). Similarly K^{b}(Λ) denotes the bounded homotopy category K^{b}(mod- Λ) and K^{−}(Λ) denotes the bounded to the right homotopy category K^{−}(mod- Λ). We’ll write Λ^{∘} to denote the opposite algebra, equal to Λ as a k-vector space but with the opposite multiplication to that in Λ. We write ⊗ to mean ⊗_{k}. If M is a Λ-module, M^{∨} denotes the dual module Hom_{k}(M, k).

For Q a p subgroup of a finite group G, we’ll denote the Brauer functor as

−(Q) : mod- kG → mod- kN_{G}(Q)

Also if Q is a subgroup of a group H as well, we’ll write the diagonal embedding of Q in G × H as ΔQ = {(q, q)| q ∈ Q}.

At least as far as Section 2, the defect group of our blocks will be written P, a Sylow 5-subgroup of G. We will use ATLAS notation, where suitable, and write H.G to mean a group having a normal subgroup H. The semidirect product H ⋊ G will be written H : G and p^{n} will be used to denote the elementary abelian group of that order.

Simple ekG-modules will be denoted by {S_{i}}, while simple fkH-modules will be indexed by integers except for the trivial module which will always be written as k. The ordering used can be determined with reference to the Loewy structure of the Green correspondents listed in Appendix A.

Having established notation our main result is:

Theorem 1.2. Conjecture 1.1 is true for ekG and fkH.

The work contained in this paper was carried out while I was a PhD student at the University of Bristol and was financially supported by the Engineering and Physical Sciences Research Council. I would like to express my thanks to the Research Council and also to my supervisor, Jeremy Rickard, for his advice and encouragement as well as for many helpful conversations.

2. The representation theory of 2.J_{2}, U_{3}(4) and Sp_{4}(4)

\underline{G = U_{3}(4)}. |G| = 62400 = 2^{6}.3.5^{2}.13. The principal block ekG is the only block of full defect of kG. H = P : Σ_{3}. Up to conjugacy in H there are two cyclic 5-subgroups of P. Only one, Q, has distinct centralizers in G and H, with C_{G}(Q) = Q × A_{5} and C_{H}(Q) = Q × D_{10}.

simple ekG-modules S_{1} = k S_{2} S_{3}
dimension 1 12 39
simple fkH-modules k  2 3
dimension  1  1 2

\underline{G = 2.J_{2}}. |G| = 1209600 = 2^{8}.3^{3}.5^{2}.17. There are two blocks of full defect. The principal block is of course the same as the principal block of kJ_{2} and we’ll always view it as such in the sequel.

For the principal block ekG with G = J_{2} and H = P : D_{12} there are, up to conjugacy in H, two cyclic 5-subgroups. Only one, Q, has distinct centralizers in G and H, with C_{G}(Q) = Q × A_{5} and C_{H}(Q) = Q × D_{10}.

simple ekG-modules S_{1} = k S_{2} S_{3} S_{4} S_{5} S_{6}
dimension 1 14 21 41 85 189
simple fkH-modules k  2 3 4 5 6
dimension 1 1 1 1 2 2

For the non-principal block ekG with G = 2.J_{2} and H = P : (2.D_{12}), There is only one cyclic 5-subgroup Q, up to conjugacy in H, with distinct centralizers. C_{G}(Q) = Q×SL_{2}(5) and C_{H}(Q) = Q × (5 : 4).

simple ekG-modules S_{1} S_{2} S_{3} S_{4} S_{5} S_{6}
dimension  6 14 56 64 190 202
simple fkH-modules 1 2 3 4 5 6
dimension  1 1 1 1 2 2

\underline{G = Sp_{4}(4)}. |G| = 979200 = 2^{8}.3^{2}.5^{2}.17. The principal block ekG is the only block of full defect and H = P : D_{8}. Up to conjugacy in H there are only two cyclic 5-subgroups with distinct centralizers in G and H. Denoting these two groups as Q_{1} and Q_{2}, we have C_{G}(Q_{i}) = Q_{i} × A_{5} and C_{H}(Q_{i}) = Q_{i} × D_{10} for i = 1, 2.

simple ekG-modules S_{1} = k S_{2} S_{3} S_{4} S_{5}
dimension 1 18 33 33 153
simple fkH-modules k  2 3 4 5
dimension  1  1 1 1 2

3. Splendid stable equivalence

We will use the results in [11] to produce a splendid stable equivalence between ekG and fkH for our examples. If ekG and fkH are arbitrary blocks as in conjecture 1.1 then we have the following definition in [11] Definition 3.1. Let C be a complex of ekG-fkH-bimodules all of whose terms as k[G × H]-modules are direct summands of relatively ΔD-projective permutation modules. Then C induces a splendid stable equivalence between ekG and fkH if

C^{∨} ⊗_{ekG} C ≅ ekG ⊕ P in K^{b}(ekG ⊗ ekG^{∘})
C ⊗_{fkH} C^{∨} ≅ fkH ⊕ Q in K^{b}(fkH ⊗ fkH^{∘})

where P and Q are bounded complexes of projective bimodules.

The stable equivalence induced by C is given by, for N in \underline{mod}- ekG, taking the image of N ⊗_{ekG} C under the composition of exact functors

K^{b}(fkH) → D^{b}(fkH) → D^{b}(fkH)/K^{b}(proj- fkH) ≅ \underline{mod}- fkH

where the last equivalence is given in [8].

By using the same argument as used in Corollary 5.5 of [7], it is easy to see that the stable equivalence induced by C is of Morita type, that is a stable equivalence induced by an exact functor between the module categories.

If in Definition 3.1 the complexes P and Q were required to be contractible, so that the functor − ⊗_{ekG} C induced an equivalence between K^{b}(ekG) and K^{b}(fkH) (hence also an equivalence between D^{b}(ekG) and D^{b}(fkH)) then C would be a splendid tilting complex (see [9]). The derived equivalence induced by a splendid tilting complex is called a splendid equivalence.

For the situation when the defect group D ≅ C_{p} × C_{p}, an explicit two term complex C is constructed in [11] that induces a splendid stable equivalence between ekG and fkH.

Let us briefly recall the details of this construction before describing the terms of C for our examples and finally calculating the complexes S_{i} ⊗_{ekG} C.

The complex C is of the form

⨁_{Q} Ñ_{Q} ↑_{N_{G×H}(ΔQ)}^{G×H} ⟶^{d} ekGf,

where the sum is taken over all cyclic p-subgroups Q < P, up to conjugacy in H. The complex

C̃_{Q} = Ñ_{Q} ⟶^{d̃} e_{Q}kC_{G}(Q)f_{Q}

is a complex of kN_{G×H}(ΔQ)-modules such that C_{Q} = C̃_{Q} ↓_{C_{G×H}(ΔQ)}^{N_{G×H}(ΔQ)} induces a splendid equivalence between e_{Q}kC_{G}(Q) and f_{Q}kC_{H}(Q). Here e_{Q} and f_{Q} are the images of e and f respectively under the respective Brauer maps br_{Q} : Z(kG) → Z(kC_{G}(Q)) and br_{Q} : Z(kH) → Z(kC_{H}(Q)). When e and f are principal block idempotents, so are e_{Q} and f_{Q}.

What are the modules Ñ_{Q} for our examples? To keep matters simple we’ll start with the principal blocks only. For principal blocks the information from Section 2 shows that the local situation in each of our cases is essentially the same. If Q denotes a cyclic 5-subgroup with distinct centralizers in G and H, then we have C_{G}(Q) = Q × A_{5} and C_{H}(Q) = Q × D_{10}.

A splendid equivalence between kQ and kQ is induced by kQ viewed as a kQ-kQ-bimodule. A splendid equivalence between the principal blocks e_{Q}kA_{5} and f_{Q}kD_{10} can be calculated using [12]. It is induced by the splendid tilting complex

P_{1} ⊗ R_{1} → e_{Q}kA_{5}f_{Q},

where P_{1} and R_{1} are the unique indecomposable projective modules of e_{Q}kA_{5} and f_{Q}kD_{10} respectively that are not the projective covers of the trivial module. So the required local splendid equivalence between e_{Q}kC_{G}(Q) and f_{Q}kC_{H}(Q) is induced by the splendid tilting complex

kQ ⊗ P_{1} ⊗ R_{1} → e_{Q}kC_{G}(Q)f_{Q}

We need to lift this to a complex of kN_{G×H}(ΔQ)-modules. For G = U_{3}(4), N_{G×H}(ΔQ) = C_{G×H}(ΔQ), and so it is already a complex of kN_{G×H}(ΔQ)-modules. For the other cases, we always have N_{G×H}(ΔQ) = C_{G×H}(ΔQ).2 and so we need to lift kQ ⊗ P_{1} ⊗ R_{1} through a cyclic 2-group. In fact N_{G×H}(ΔQ) = A_{5} × (Q × C_{H}(Q)).2, so that P_{1} already lifts to a kN_{G×H}(ΔQ)-module. To lift R_{1} and kQ we have that:

Q × Q ◁ (Q × C_{H}(Q)).2 with (Q × C_{H}(Q)).2/(Q × Q) ≅ D_{10}.2
1 × D_{10} ◁ (Q × C_{H}(Q)).2 with (Q × C_{H}(Q)).2/(1 × D_{10}) ≅ (Q × Q).2

So it is enough to lift R_{1} to a k[D_{10}.2]-module and kQ to a k(Q × Q).2-module. Now k[D_{10}.2] has 2 blocks each containing two indecomposable projective modules that are lifts of the 2 indecomposable projective kD_{10}-modules. We want the projective module R̄_{1} lying in the principal block of k[D_{10}.2] that lifts R_{1}. For kQ we have kQ = k ↑_{ΔQ}^{Q×Q}, as a

kQ-kQ-bimodule. It follows easily from the Mackey decomposition formula that k ↑_{ΔQ.2}^{(Q×Q).2}

is the required lift. So the module Ñ_{Q} = k ↑_{ΔQ.2}^{(Q×Q).2} ⊗P_{1} ⊗ R̄_{1}. Note that in the case of G = Sp_{4}(4) we have two non-zero contributions Ñ_{Q_{1}} ↑^{G×H} and Ñ_{Q_{2}} ↑^{G×H} to the degree −1 term of the complex C.

For the non-principal block ekG (G = 2.J_{2}), there are three blocks of kC_{G}(Q), two with full defect, and it’s easy to see that e_{Q} = br_{Q}(e) must be the non-principal block of full defect. For kC_{H}(Q) there are only two blocks and so f_{Q} = br_{Q}(f) is the unique non-principal block. The local splendid tilting complex is given by

kQ ⊗ P_{2} ⊗ R_{2} → e_{Q}kC_{G}(Q)f_{Q}

where P_{2} is the projective lying in the non-principal block of kSL_{2}(5) that is the projective cover of the 4 dimensional simple module T. R_{2} is the projective cover of the top of T ↓_{2.D_{10}}. Again N_{G×H}(ΔQ) = C_{G×H}(ΔQ).2 and as in the principal block case, P_{2} already lifts. We take R̄_{2} to be the lift of R_{2} lying in the non-principal block of k[2.(D_{10}.2)]. So Ñ_{Q} = k ↑_{ΔQ.2}^{(Q×Q).2} ⊗P_{2} ⊗ R̄_{2}.

As far as performing calculations are concerned we won’t want to calculate the complex C as a complex of bimodules, rather we will only need to calculate the images under −_{ekG} ⊗ C of simple ekG-modules. In particular, although the differential in C is described explicitly in [11], for the calculations we perform it is enough to know that such a differential exists giving a stable equivalence, in order to calculate it.

To help with the calculations, the character of the complex C was computed using GAP. More precisely the character of each term of C was calculated and so the composition factors of each term of the complexes S_{i} ⊗_{ekG} C were calculated (this was not done for the non-principal block of k[2.J_{2}] due to limitations in memory space available to GAP).

We’ll want to calculate the terms of the complexes S_{i} ⊗_{ekG} C, using functions in MAGMA’s toolkit. By way of illustration we only consider the principal block cases and the contribution from only one cyclic 5-subgroup Q. The degree zero term is easy as this is just restriction S_{i} ↓_{H}^{G}. The degree −1 term is

S_{i} ⊗_{ekG} Ñ_{Q} ↑_{N_{G×H}(ΔQ)}^{G×H}= S_{i} ⊗_{e_{Q}kN_{G}(Q)} (Ñ_{Q} ↑_{N_{G×H}(ΔQ)}^{N_{G}(Q)×N_{H}(Q)}) ⊗_{f_{Q}kN_{H}(Q)} kHf

= S_{i} ⊗_{e_{Q}kN_{G}(Q)} P_{1} ⊗ k(Q.2) ⊗_{k[(ΔQ).2]} k(Q.2) ⊗ R̄_{1} ⊗_{f_{Q}kN_{H}(Q)} kfH

In order to use MAGMA’s functions, we re-interpret this term as a summand of S_{i} ↓_{Q.2}^{G}↑_{Q.2}^{H}. Although we cannot determine the summand of S_{i} ↓_{Q.2}^{G} we want, we can at least calculate its dimension as dim Hom_{kA_{5}}(P_{1}, S_{i} ↓_{A_{5}}^{G}). Once this summand has been identified, to calculate the summand of ? ↑_{Q.2}^{H} required, one should calculate ? ↑_{Q.2}^{Q.2×D_{10}.2}↓_{N_{H}(Q)}^{Q.2×D_{10}.2}

taking only that summand of ? ↑_{Q.2}^{Q.2×D_{10}.2} that restricts to D_{10}.2 to give nR̄_{1} (for some n). As Q.2 × D_{10}.2 ≰ H, one would have to carefully construct this group in addition to the others, but it’s simpler to proceed as follows. We can split ? ↑_{Q.2}^{N_{H}(Q)} into 2 summands say M_{0} and M_{1}, where M_{1} ↓_{D_{10}} is nR_{1} (for some n) and M_{0} restricts to give a sum of copies of the other projective D_{10}-module. Let M be an indecomposable summand of M_{1}, and let M ↑^{Q.2×D_{10}.2}= M̄ ⊕ ¯ M̄ where M̄ ↓_{D_{10}.2} is nR̄_{1} (for some n) and ¯ M̄ restricts to give a sum of copies of the other lift of R_{1}. As the identity functor is a summand of ↑_{N_{H}(Q)}^{Q.2×D_{10}.2}↓_{N_{H}(Q)} we have M|(M̄ ↓_{N_{H}(Q)} ⊕ ¯M̄ ↓_{N_{H}(Q)}). There are two possibilities; either

M̄ ↓_{N_{H}(Q)}≅ ¯ ¯M̸ ↓_{N_{H}(Q)}, in which case we would expect M ↓_{D_{10}.2} to be a sum of copies of R̄_{1} or a sum of copies of the other lift, or M̄ ↓_{N_{H}(Q)}≅ ¯ M̄ ↓_{N_{H}(Q)}, in which case M ↓_{D_{10}.2} will be a sum possibly involving both lifts of R_{1}. In our calculations this latter situation always arose and so the correct summand we want, viz M̄ ↓_{N_{H}(Q)}≅ M, can be identified by examining the summands of ? ↑_{Q.2}^{N_{H}(Q)}.

These calculations were carried through for our examples and the Loewy structures of the non-projective summands of the degree −1 terms of the complexes S_{i} ⊗_{ekG} C are listed in Appendix B. Where ambiguities arose as to which summand of ? ↓_{Q.2} to take, most were resolved with reference to the composition factors of the module required (for G = J_{2} and S_{6} there were still two possible candidates and one was eliminated as no suitable differential exists, see below).

Having determined the terms of the complexes {S_{i} ⊗_{ekG} C} we now need to calculate the differentials. The general situation we’re faced with is having a complex

Z = 0 ⟶ X ⟶^{d} Y ⟶ 0

with X, Y non-projective modules, whose image in \underline{mod}- fkH, under an equivalence, is the image of a simple ekG-module. In particular \underline{End}_{fkH}(Z) ≅ k (we identify the complex Z with its image in \underline{mod}- fkH). Clearly we must have a choice of d that is non-zero in \underline{Hom}_{fkH}(X, Y). In most of our cases dim \underline{Hom}_{fkH}(X, Y) = 2. By careful choice of coset representatives, maps f_{1}, f_{2} ∈ Hom_{fkH}(X, Y) were found whose images formed a basis for \underline{Hom}_{fkH}(X, Y), and such that f_{2} = ϕf_{1} with ϕ ∈ End_{fkH}(X) a nilpotent endomorphism. Up to a scalar multiple an element of Hom_{fkH}(X, Y) spanned by f_{1}, f_{2} will be of the form f_{1} + λϕf_{1} (λ ∈ k) or ϕf_{1}. The isomorphism in Ch(mod- fkH):

X ⟶^{f_{1}+λϕf_{1}} Y
↓^{1+λϕ}  ∥
X ⟶^{f_{1}} Y

shows that up to isomorphism there are only two choices for the differential, viz f_{1} or ϕf_{1}. For each choice dim \underline{End}_{fkH}(Z) was calculated and was equal to 1 only for d = f_{1}. To calculate dim \underline{End}_{fkH}(Z) using MAGMA functions, it is much easier to work with the module in \underline{mod}- fkH represented by Z. This module can be calculated by noting that we have the triangle

X ⟶^{d} Y ⟶ Z ⟶ X[1]

in D^{b}(fkH) and hence the same triangle in \underline{mod}- fkH, whose triangulated structure can be used to calculate the module representing Z.

4. Derived equivalence

Having described the stable equivalence between ekG and fkH in Section 3 we describe in this section how the stable equivalence was lifted to a derived equivalence and even a splendid equivalence for our cases.

The central result that we’ll use is a theorem due to Rickard (see [10])

Theorem 4.1 (Rickard). Let Λ and Γ be finite-dimensional symmetric k-algebras, let

F : mod-Γ → mod-Λ

be an exact functor inducing a stable equivalence (i.e. a stable equivalence of Morita type), and let {S_{1}, …, S_{r}} be a set of representatives for the isomorphism classes of simple Γmodules.

If there are objects X_{1}, …, X_{r} of D^{b}(Λ) such that, for each 1 ≤ i ≤ r, X_{i} is stably isomorphic to F (S_{i}), and such that

(a) Hom_{D^{b}(Λ)}(X_{i}, X_{j}[m]) = 0 for m < 0,

(b) Hom_{D^{b}(Λ)}(X_{i}, X_{j}) = \{ \begin{matrix}0 & if i ≠ j \\ k & if i = j \end{matrix} , and

(c) X_{1}, …, X_{r} generate D^{b}(Λ) as a triangulated category, then D^{b}(Λ) and D^{b}(Γ) are equivalent as triangulated categories.

Theorem 4.1 requires knowledge of homomorphisms in derived categories which are hard to compute in general. Fortunately it will be sufficient to calculate homomorphisms in the homotopy category instead.

Lemma 4.2. Let Λ be a self-injective algebra and X, Y objects in K^{−}(Λ) (we’ll identify X, Y with their images in D^{b}(Λ)) such that the natural map Hom_{K^{−}(Λ)}(X, Y [m]) → Hom_{D^{b}(Λ)}(X, Y [m]) is an isomorphism for m ≤ 0 and injective for m = 1.

If X′ and Y′ are objects in K^{−}(Λ) that fit into triangles

X → X′ → Q → X[1] (4.1)

Y → Y′ → P → Y [1] (4.2)

where Q and P are objects in K^{b}(proj- Λ), then the natural map Hom_{K^{−}(Λ)}(X′, Y′[m]) → Hom_{D^{b}(Λ)}(X′, Y′[m]) is an isomorphism for m ≤ 0 and injective for m = 1.

Proof. To avoid unnecessary notational clutter, we’ll write Hom_{K^{−}(Λ)}(−, −) as _{K}(−, −), and Hom_{D^{b}(Λ)}(−, −) as _{D}(−, −).

Applying the contravariant functors _{K}(−, Y [m]) and _{D}(−, Y [m]) to triangle (4.1) we obtain the following ladder diagram with exact rows:

K(X[1], Y [m]) ⟶ _{K}(Q, Y [m]) ⟶ _{K}(X′, Y [m]) ⟶ _{K}(X, Y [m]) ⟶ _{K}(Q[−1], Y [m])
↓^{α}  ↓^{β}  ↓^{γ}  ↓^{δ}  ↓^{β}
D(X[1], Y [m]) ⟶ _{D}(Q, Y [m]) ⟶ _{D}(X′, Y [m]) ⟶ _{D}(X, Y [m]) ⟶ _{D}(Q[−1], Y [m])

The maps β are isomorphisms for all m by standard properties of derived categories, while for m ≤ 0, α and δ are isomorphisms by assumption. Hence by the 5-lemma γ is an isomorphism for m ≤ 0. When m = 1, α is an isomorphism and δ is injective by assumption and a diagram chase shows γ is injective.

We can now apply the covariant functors _{K}(X′, −) and _{D}(X′, −) to triangle (4.2) giving the ladder diagram with exact rows:

K(X′, P [m − 1]) ⟶ _{K}(X′, Y [m]) ⟶ _{K}(X′, Y′[m]) ⟶ _{K}(X′, P [m]) ⟶ _{K}(X′, Y [m + 1])
↓^{ζ}  ↓^{γ}  ↓^{η}  ↓^{ζ}  ↓^{γ}
D(X′, P [m − 1]) ⟶ _{D}(X′, Y [m]) ⟶ _{D}(X′, Y′[m]) ⟶ _{D}(X′, P [m]) ⟶ _{D}(X′, Y [m + 1])

As Λ is self-injective, it follows from standard properties of derived categories that the maps ζ are isomorphisms for all m. For m < 0 the γ are isomorphisms and hence by the 5-lemma so is η. When m = 0 we can replace _{D}(X′, Y [1]) by im(γ) and again apply the 5-lemma to conclude that η is an isomorphism. For m = 1 the injectivity of the γ on the left in the ladder diagram, together with a diagram chase, shows η is injective.

We will apply Lemma 4.2 to parts (a) and (b) of Theorem 4.1 when Γ = ekG and Λ = fkH are as in Conjecture 1.1 and the stable equivalence is a splendid stable equivalence, induced by a complex C. In the notation of Lemma 4.2 we’ll take X = S_{i} ⊗_{ekG} C and Y = S_{j} ⊗_{ekG} C.

In applying Lemma 4.2 to Theorem 4.1 one must be careful as to which choice of complex X_{i}, up to isomorphism in the homotopy category, one chooses as a representative from the isomorphism class of X_{i} in the derived category.

In all our cases the complexes {X_{i}} we construct will fit into triangles (4.1) and (4.2) with X′ = X_{i} and Y′ = X_{j} (see triangle (4.5) in Subsection 4.2). That the remaining conditions of Lemma 4.2 hold follows from the commutative diagram

[figure: Hom_{K^{b}(fkH)}(S_{i} ⊗_{ekG} C, S_{j} ⊗_{ekG} C[m]); f; Hom_{D^{b}(fkH)}(S_{i} ⊗_{ekG} C, S_{j} ⊗_{ekG} C[m]); Hom_{K^{b}(ekG)}(S_{i} ⊗_{ekG} C ⊗_{fkH} C^{∨}, S_{j}[m]); g; Hom_{D^{b}(ekG)}(S_{i} ⊗_{ekG} C ⊗_{fkH} C^{∨}, S_{j}[m])]

The vertical maps are the adjunction isomorphisms and as C ⊗_{fkH} C^{∨} ≅ ekG ⊕ P in K^{b}(ekG ⊗ ekG^{∘}), where P is a bounded complex of projective bimodules, g is trivially an isomorphism for m ≤ 0 and injective for m = 1 and so the same is true for f.

This observation together with, for X, Y ∈ K^{b}(Λ),

Hom_{K^{b}(Λ)}(X, Y [m]) = H^{m}(Hom_{Λ}^{•}(X, Y))

provides a computational approach to verifying conditions (a) and (b) of Theorem 4.1.

Routines were written in MAGMA to take complexes X, Y stored as lists of differentials and compute the total complex Hom_{Λ}^{•}(X, Y), and its cohomology. In this way conditions (a) and (b) of Theorem 4.1 were checked. In fact for all the complexes {X_{i}} found that satisfied the conditions in Theorem 4.1, the stronger condition

(4.3) Hom_{K^{b}(fkH)}(X_{i}, X_{j}[m]) = \{ \begin{matrix}k & if i = j and m = 0 \\ 0 & otherwise \end{matrix}

was found to hold. This would suggest that we have found the images of simple modules under a splendid equivalence or at least a homotopy equivalence. That this is indeed the case follows from the results in Subsection 4.1.

4.1. Splendid equivalence. There is a refinement to Conjecture 1.1 due to Rickard (see [9]) that, for ekG and fkH as in Conjecture 1.1, requires the derived equivalence between ekG and fkH to be a splendid equivalence.

For our examples we’ll construct suitable objects {X_{i}} and apply Theorem 4.1, with Γ = ekG and Λ = fkH, to produce a derived equivalence. In order to do so we’ll use the stable equivalence from Section 3, but that stable equivalence is actually a splendid stable equivalence and this is sufficient to show that, by successfully applying Theorem 4.1, we’ll also produce a splendid equivalence, see Theorem 4.4, and thus also verify Rickard’s refinement to Conjecture 1.1 for our examples.

Proposition 4.3. Suppose that Λ and Γ are finite-dimensional symmetric k-algebras and C is a complex of exact Λ-Γ-bimodules such that

C^{∨} ⊗_{Λ} C ≅ Γ ⊕ Z in K^{b}(Γ ⊗ Γ^{∘}),
C ⊗_{Γ} C^{∨} ≅ Λ ⊕ W in K^{b}(Λ ⊗ Λ^{∘})

where Z and W are bounded complexes of projective bimodules, so that − ⊗_{Λ} C induces a stable equivalence between Λ and Γ.

Suppose F : D^{b}(Λ) → D^{b}(Γ) is an equivalence that induces a stable equivalence that agrees on each simple Λ-module, up to isomorphism, with that induced by C, then there is complex X of Λ-Γ-bimodules that induces a homotopy equivalence between Λ and Γ and such that X^{i} = C^{i} ⊕ P^{i} for each i where P^{i} is a projective bimodule. Moreover, for each Λ-module Y, Y ⊗_{Λ} X ≅ F Y in D^{b}(Γ).

Proof. By the results in [7] we can replace the equivalence F by an equivalence induced by a standard two-sided tilting complex Δ′ and , by taking a truncated projective resolution if necessary, we can assume that

Δ′ = 0 → N → R^{−n+1} → R^{−n+2} → ⋯

where the R^{i} are projective bimodules and N is an exact bimodule. The exact functor − ⊗_{Λ} Ω^{−n}(N) induces a stable equivalence such that F Y is stably isomorphic to Y ⊗_{Λ} Ω^{−n}(N) for each Λ-module Y.

Let T be a projective bimodule resolution of C. Truncating T, for sufficiently large m we have the complex

Σ = 0 → M → T^{−m+1} → T^{−m+2} → ⋯

where M is an exact bimodule and the T^{i} are projective bimodules. The exact functor − ⊗_{Λ} Ω^{−m}(M) induces a stable equivalence isomorphic to that induced by C.

By taking m = n sufficiently large we have a stable auto-equivalence on \underline{mod}- Λ induced by − ⊗_{Λ} Ω^{−n}(M) ⊗_{Γ} Ω^{n}(N^{∨}). By assumption this sends simple Λ-modules to isomorphic copies of themselves. By a theorem due to Linckelmann (see [5]), this stable equivalence is actually a Morita auto-equivalence and so is induced by the Λ-bimodule _{ϕ}Λ, which is Λ with Λ action on the left twisted by ϕ ∈ Aut(Λ). So we have M ≅ _{ϕ}N and for each Λ-module Y, Y ⊗_{Λ} M ≅ Y ⊗_{Λ} N. Replacing Δ′ by the two-sided tilting complex Δ = _{ϕ}Λ ⊗_{Λ} Δ′, we have complexes

Δ = 0 → M → L^{−n+1} → L^{−n+2} → ⋯
Σ = 0 → M → T^{−n+1} → T^{−n+2} → ⋯

with L^{i} projective bimodules and for each Λ-module Y, F Y ≅ Y ⊗_{Λ} Δ.

As Σ is a truncated resolution of C, we have a map f : Σ → C in K^{b}(Γ ⊗ Λ^{∘}). By the axioms of a triangulated category we have the following map of triangles

[figure: T̃; i; Σ; f; M[n]; h; T̃[1]; T̃; fi; C; R; T̃[1]]

Here R is the cone of the map fi and T̃ →^{i} Σ is the obvious map where

T̃ = 0 → T^{−n+1} → T^{−n+2} → ⋯

Viewed as a map of triangles in D^{b}(Γ ⊗ Λ^{∘}), f is an isomorphism and, again by the axioms of a triangulated category, so is h. In other words R[−n] is a right resolution of M.

If we set

L̃ = 0 → L^{−n+1} → L^{−n+2} → ⋯

we have a complex of injective bimodules. The identity map M → M lifts to a map r : R[−1] → L̃ in K^{b}(Γ ⊗ Λ^{∘}) which we can complete to a triangle:

R[−1] →^{r} L̃ → X → R

defining an object X in K^{b}(Γ ⊗ Λ^{∘}). By construction X^{n} = C^{n} ⊕ P^{n} for some projective bimodule P^{n}, for each n and X ≅ C in K^{b}(Γ ⊗ Λ^{∘})/K^{b}(proj- Γ ⊗ Λ^{∘}).

We have to check that X induces a homotopy equivalence between Λ and Γ. Taking − ⊗_{Γ} X^{∨} as left adjoint to − ⊗_{Λ} X we can consider the cone V of the counit ϵ,

X^{∨} ⊗_{Λ} X →^{ϵ} Γ → V (4.4)

InK^{b}(Γ ⊗ Λ^{∘})/K^{b}(proj- Γ ⊗ Λ^{∘}) we have X^{∨} ⊗_{Λ} X ≅ C^{∨} ⊗_{Λ} C ≅ Γ and so V is homotopic to a complex of projective Γ-bimodules. So we only have to show V is acyclic, but X ≅ Δ in D^{b}(Γ ⊗ Λ^{∘}) and so the long exact sequence of cohomology applied to (4.4) now shows V is acyclic as required.

We are particularly interested in the situation where Λ = ekG and Γ = fkH are as in Conjecture 1.1 and for which we have a splendid stable equivalence. Combining Theorem 4.1 with Proposition 4.3 we have:

Theorem 4.4. Suppose that C is a complex of ekG-fkH-bimodules that induces a splendid stable equivalence between ekG and fkH and let {S_{1}, …, S_{r}} be a set of representatives for the isomorphism classes of simple ekG-modules. If there are objects X_{1}, …, X_{r} of D^{b}(fkH) such that, for each 1 ≤ i ≤ r, X_{i} is stably isomorphic to S_{i} ⊗_{ekG} C, and such that:

(a) Hom_{D^{b}(fkH)}(X_{i}, X_{j}[m]) = 0 for m < 0,

(b) Hom_{D^{b}(fkH)}(X_{i}, X_{j}) = \{ \begin{matrix}0 & if i ≠ j \\ k & if i = j \end{matrix} , and

(c) X_{1}, …, X_{r} generate D^{b}(fkH) as a triangulated category, then there is a splendid tilting complex X that lifts C and induces a splendid equivalence between ekG and fkH such that, for each 1 ≤ i ≤ r, S_{i} ⊗_{ekG} X ≅ X_{i} in D^{b}(fkH).

For our examples the construction of objects {X_{i}} satisfying the conditions of Theorem 4.4 will be such that each X_{i} will not only be stably isomorphic to S_{i} ⊗_{ekG} C but also isomorphic in K^{b}(fkH)/K^{b}(proj- fkH), see triangle (4.5) of Subsection 4.2. Under these circumstances we have the following.

Corollary 4.5. With the notation of Theorem 4.4, let objects {X_{i}} satisfy the conditions of Theorem 4.4. If X_{i} ≅ S_{i} ⊗_{ekG} C in K^{b}(fkH)/K^{b}(proj- fkH) then S_{i} ⊗_{ekG} X ≅ X_{i} in K^{b}(fkH).

Proof. We’re given that:

(a) X_{i} ≅ S_{i} ⊗_{ekG} X in D^{b}(fkH),

(b) X_{i} ≅ S_{i} ⊗_{ekG} C in K^{b}(fkH)/K^{b}(proj- fkH)

Setting Y_{i} = X_{i} ⊗_{fkH} X^{∨} and using X ≅ C in K^{b}(fkH ⊗ ekG^{∘})/K^{b}(proj- fkH ⊗ ekG^{∘}) these become: (a′) Y_{i} ≅ S_{i} in D^{b}(ekG), (b′) Y_{i} ≅ S_{i} in K^{b}(ekG)/K^{b}(proj- ekG) The latter condition says that we have, in K^{b}(ekG), triangles:

Z → Y_{i} → P → Z[1],

Z → S_{i} ⟶^{u} Q → Z[1]

where P and Q are bounded complexes of projectives. The second triangle shows that Z ≅ cone(u)[−1]. The first triangle now shows that Y_{i}, as a complex, has projective modules in non-zero degrees. It now follows from (a′) that Y_{i} ≅ S_{i} in K^{b}(ekG) and hence X_{i} ≅ S_{i} ⊗_{ekG} X in K^{b}(fkH).

4.2. The calculation. Each complex X_{i} constructed for our examples will fit into a triangle, in K^{b}(fkH),

P_{i} → S_{i} ⊗ ekGC → X_{i} → P_{i}[1] (4.5)

In all cases the complex P_{i} and map P_{i} → S_{i} ⊗_{ekG} C were computed as a trimmed projective resolution of S_{i} ⊗_{ekG} C. That is, having already constructed the degree −n + 1 term, the degree −n term, P_{i}^{−n}, will be a certain summand i : P_{i}^{−n} → P C where θ : P C → K_{i}^{−n+1} is the projective cover of K_{i}^{−n+1} = ker(d^{−n+1}). The differential d^{−n} will then be the composite

P_{i}^{−n} ⟶^{i} P C ⟶^{θ} K_{i}^{−n+1} ⟶^{j} P_{i}^{−n+1}

where j is the natural inclusion. There will of course be no unique choice for θ, see Subection 4.3.

In trying to seek the correct summands of X_{i}, or equivalently the terms of the complex P_{i} in triangle (4.5), character calculations using GAP proved a useful guide.

As described in Section 3 the character χ_{C} of the complex C was calculated. In addition an isotypy for each of our cases was calculated. In particular the perfect isometry I : CF ((G, e), K) → CF ((H, f), K) (where (K, O, k) is a p modular system) coming from the isotypy is believed to arise as the map induced by the character of a complex K ⊗_{O} X, where X is a splendid tilting complex of eOG-fOH-bimodules. Using decomposition matrices, the map on Grothendieck groups K_{0}(mod-ekG) → K_{0}(mod-fkH) induced by the character χ of k ⊗_{O} X can be calculated. Assuming the splendid tilting complex X lifts our complex C, then we would have a triangle

Q → C → k ⊗_{O} X → Q[1]

where Q is a bounded complex of projective ekG-fkH-bimodules. If this is the case then χ − χ_{C} should induce a map K_{0}(mod-ekG) → K_{0}(proj- fkH). The image of the simple ekG module S_{i} will then be the image in K_{0}(proj- fkH) of the complex P_{i} required to define X_{i} and this can act as a guide in choosing the terms of P_{i}.

After suitable candidate objects {X_{i}} were found that agreed, in the Grothendieck group, with our character calculations, the checking of conditions (a) and (b) of Theorem 4.4 ( in fact the verification of (4.3)) was done using written routines in MAGMA, as described earlier. The complexes {X_{i}}, whose terms are described in Appendix C and whose cohomology is described in Appendix D, were found to satisfy conditions (a) and (b) of Theorem 4.4.

All that remains is to check condition (c) of Theorem 4.4 for our objects {X_{i}}. In doing so we’ll make use of the results in Appendix D without explicit reference. Appendix D describes the Loewy layers of the non-zero cohomology groups of the X_{i}, together with some useful short exact sequences. The short exact sequences together with the isomorphisms of some of the cohomology modules were verified using standard functions in MAGMA.

proof of condition (c) of Theorem 4.4. If we let T denote the full triangulated subcategory of D^{b}(fkH) generated by the {X_{i}} then, for each of our examples, we’ll show that T = D^{b}(fkH) by showing that T contains all the simple fkH-modules.

\underline{G = U_{3}(4)}

Clearly k ∈ T and as H^{−1}(X_{2}) = k, we must have H^{−2}(X_{2}) ∈ T and hence 3 ∈ T . Finally as H^{−3}(X_{3}) ∈ T and has 2 occuring as a composition factor only once, 2 ∈ T .

All simple fkH-modules have now been shown to be lie in T .

\underline{G = J_{2}}

We have H^{0}(X_{1}) = k ∈ T . As k = H^{−1}(X_{3}) ∈ T we must have H^{−2}(X_{3}) ∈ T . Also, as X_{5} only has one non-zero cohomology module, H^{−4}(X_{5}) ≅ H^{−3}(X_{4}) ∈ T , and hence 2 = H^{−2}(X_{4}) ∈ T .

The short exact sequence (D.2) now shows {2 \atop 5} ∈ T and hence 5 ∈ T . The short exact

sequence (D.3) now implies H^{−3}(X_{6}) ∈ T . As k = H^{−2}(X_{6}) the only other non-zero cohomology for X_{6}, viz 3 = H^{−4}(X_{6}), must be in T .

Finally as H^{−3}(X_{2}) ∈ T and has only one composition factor isomorphic to 6, with multiplicity one, 6 ∈ T . So all simple fkH-modules belong to T .

\underline{G = Sp_{4}(4)}

Again we have k = H^{0}(X_{1}) ∈ T . As H^{−2}(X_{2}) = k, we must have H^{−3}(X_{2}) ∈ T and thus 5 ∈ T . Similarly, as H^{−1}(X_{3}) = H^{−1}(X_{4}) = k, we also have H^{−3}(X_{3}) = 2 ∈ T and H^{−3}(X_{4}) = 4 ∈ T .

Finally X_{5} has only one non-zero cohomology and so H^{−4}(X_{5}) ∈ T . The simple fkHmodule 3 occurs in the composition factors of H^{−4}(X_{5}) with multiplicity one, so 3 ∈ T . All simple fkH-modules have now been shown to be in T .

G = 2.J_{2}, non-principal block

As X_{2} and X_{5} both only have one non-zero cohomology module, 2 = H^{−3}(X_{2}) ∈ T and H^{−5}(X_{5}) ∈ T . The short exact sequence (D.4) shows H^{−2}(X_{6}) ∈ T and so we must have H^{−1}(X_{6}) ∈ T . The short exact sequence (D.5) now implies that 1 ∈ T .

As H^{−3}(X_{3}) ≅ H^{−4}(X_{3}) ≅ H^{−5}(X_{5}) ∈ T , the only other non-zero cohomology module for X_{3}, viz H^{−2}(X_{3}), must belong to T . The short exact sequence (D.6) now implies that H^{−2}(X_{4}) ∈ T and hence 6 ∈ T and H^{−3}(X_{4}) ∈ T . Using H^{−2}(X_{1}) ≅ H^{−3}(X_{4}) ∈ T , we must have H^{−1}(X_{1}) ∈ T and hence 4 ∈ T .

The simple fkH-module 5 occurs only once in the Loewy structure of H^{−3}(X_{4}) ∈ T , thus 5 ∈ T . Similarly the simple fkH-module 3 occurs only once in the Loewy structure of H^{−2}(X_{3}) ∈ T and so 3 ∈ T . All simple fkH-modules have now been shown to belong to T .

So all the conditions on the {X_{i}} in Theorem 4.4 have been verified, thus proving Theorem 1.2.

4.3. Independence of choice of differentials. In this subsection we’ll show that, given the information in Appendices C and D, the complexes {X_{i}} constructed in Subsection 4.2 are independent, up to isomorphism in K^{b}(fkH), of the choice of differentials.

Suppose we have constructed two trimmed resolutions of S_{i} ⊗_{ekG} C, both consistent with the information in Appendices C and D, up to degree −n + 1. Suppose also that we have a cochain isomorphism between the two complexes, which we may as well take to be an identification.

In constructing the degree −n differential suppose we make two choices; ψ, ψ′ ∈ Hom_{fkH}(P_{i}^{−n}, K_{i}^{−n+1}), where K_{i}^{−n+1} = ker(d^{−n+1}), and hence two choices for the degree −n differential. Can we find a cochain isomorphism between the two new complexes?

We’ll show that we can for all the X_{i} described in Appendices C and D except for one. However, for that exceptional case we’ll still be able to show, with the help of computer calculations, that it is independent, up to isomorphism in K^{b}(fkH), of the choice of differentials.

Returning to our two partially constructed trimmed resolutions, we have the following diagram,

[figure: P_{i}^{−n}; ψ; P_{i}^{−n+1}; P_{i}^{−n+2}; …; K_{i}^{−n+1}; j; P_{i}^{−n}; ψ′; P_{i}^{−n+1}; P_{i}^{−n+2}; …; K_{i}^{−n+1}; j]

where j : K^{−n+1} → P_{i}^{−n+1} is the natural inclusion.

If Hom_{fkH}(P_{i}^{−n}, H^{−n+1}(X_{i})) = 0, then im(ψ) can be described as the largest submodule of K_{i}^{−n+1} for which there are no maps from P_{i}^{−n} to the quotient. Hence im(ψ) = im(ψ′) and so we have an isomorphism ϕ such that

[figure: P_{i}^{−n}; ϕ; d^{−n}=jψ; P_{i}^{−n+1}; P_{i}^{−n+2}; …; P_{i}^{−n}; d′^{−n}=jψ′; P_{i}^{−n+1}; P_{i}^{−n+2}; …]

is a cochain map isomorphism.

From the information in Appendices C and D the only complexes not covered by the above argument are: X_{6} (n = 4) for J_{2}, X_{6} (n = 2) and X_{3} (n = 3, 4) for 2.J_{2}, nonprincipal block. For these cases Hom_{fkH}(P_{i}^{−n}, H^{−n+1}(X_{i})) ≠ 0.

We now fix a map ψ, with q : K^{−n+1} → K^{−n+1}/ im(ψ) = H^{−n+1}(X_{i}) the natural quotient, and suppose there is a map η : P_{i}^{−n+1} → K_{i}^{−n+1} such that qη ≠ 0. A choice of map ψ′ not covered by the above argument is to take ψ′ = ψ + η. If the map η factors as η = γd^{−n} = γjψ, where γ : P_{i}^{−n+1} → K_{i}^{−n} then we can adjust our degree −n + 1 identification to an isomorphism, 1 + jγ, and so extend the cochain isomorphism to

[figure: P_{i}^{−n}; d^{−n}=jψ; P_{i}^{−n+1}; 1+jγ; P_{i}^{−n+2}; …; P_{i}^{−n}; d′^{−n}=j(ψ+η); P_{i}^{−n+1}; P_{i}^{−n+2}; …]

By using standard MAGMA routines it was checked that, in the cases X_{6} for J_{2} and X_{6} for 2.J_{2}, all possible such maps η do factor as described above. So these complexes are independent, up to isomorphism, of the choice of differential.

We’re left with X_{3} (n = 3, 4) for 2.J_{2}, non-principal block. In constructing the degree −3 differential, there is a map η : X^{−3} = P_{3}^{−3} ⟶^{p} P_{5} → K_{3}^{−2}, with qη ≠ 0 where p : P_{3}^{−3} → P_{5} is the projection onto P_{5}. From Appendix C it’s clear that η cannot factor through d^{−3} and so a choice of ψ′ = ψ + λη, λ ∈GF(5)^{∗}, cannot be dealt with as above. The map η was constructed in MAGMA and coker(ψ + λη) computed, for λ ∈GF(5)^{∗}. In all cases, the cokernel was found to have dimension less than dim H^{−2}(X_{3}) i.e. a choice ψ′ = ψ +λη is inconsistent with the information about X_{3} in Appendices C and D. We can therefore only change ψ by adding η : X^{−3} ⟶^{p} P_{3} → K_{3}^{−2}, where qη ≠ 0 and p is the projection onto P_{3}. Using MAGMA, this map η was found to factor through X_{3}^{−2} and so a choice of ψ′ = ψ + λη can be dealt with as above.

All that remains is the construction of the degree −4 differential. There are two maps:

η_{1} : X_{3}^{−4} ⟶^{p_{1}} P_{1} → K_{3}^{−3} and η_{3} : X_{3}^{−4} ⟶^{p_{3}} P_{3} → K_{3}^{−3}, where p_{1} and p_{3} are the natural

projections and qη_{1} ≠ 0 and qη_{3} ≠ 0. Again, using standard routines in MAGMA, both maps can be shown to factor through X_{3}^{−4}. In general, we may choose ψ′ = λ_{1}η_{1}+λ_{3}η_{3}, λ_{1}, λ_{3} ∈GF(5). By choosing to factorise say η_{1} through X_{3}^{−4} and using the above arguments we can assume that λ_{1} = 0, but now η_{3} will no longer factorise through X_{3}^{−4}.

So there are, up to isomorphism, potentially four different choices of degree −4 differential and hence four different complexes: X_{3_{1}}, X_{3_{2}}, X_{3_{3}} and X_{3_{4}}, corresponding to each λ ∈GF(5)^{∗}, in addition to the complex X_{3} already constructed for Subsection 4.2.

These four complexes were constructed and, using the routines written in MAGMA for calculating homomorphisms in the homotopy category described earlier, the following calculations, for 1 ≤ r ≤ 4, were verified.

Hom_{K^{b}(fkH)}(X_{3_{r}}, X_{j}[m]) = Hom_{K^{b}(fkH)}(X_{j}, X_{3_{r}}[m]) = 0  for m ≠ 0, 1 ≤ j ≤ 6,
Hom_{K^{b}(fkH)}(X_{3_{r}}, X_{j}) = Hom_{K^{b}(fkH)}(X_{j}, X_{3_{r}}) = 0  for j ∈ {1, 2, 4, 5, 6},

Hom_{K^{b}(fkH)}(X_{3_{r}}, X_{3}) = Hom_{K^{b}(fkH)}(X_{3}, X_{3_{r}}) = k

Under the derived equivalence between ekG and fkH obtained from the proof of Theorem 1.2 in Subsection 4.2 it follows from the above calculations that the images Y_{r} in D^{b}(ekG) of X_{3_{r}}, for 1 ≤ r ≤ 4, satisfy

Hom_{D^{b}(ekG)}(Y_{r}, S_{j}[m]) = Hom_{D^{b}(ekG)}(S_{j}, Y_{r}[m]) = 0  for m < 0, 1 ≤ j ≤ 6,
Hom_{D^{b}(ekG)}(Y_{r}, S_{j}) = Hom_{D^{b}(ekG)}(S_{j}, Y_{r}) = 0  for j ∈ {1, 2, 4, 5, 6},

Hom_{D^{b}(ekG)}(Y_{r}, S_{3}) = Hom_{D^{b}(ekG)}(S_{3}, Y_{r}) = k

The first of these shows that each Y_{r} has non-zero cohomology only in degree 0. The remaining conditions now show Y_{r} ≅ S_{3} in D^{b}(ekG) and so, for 1 ≤ r ≤ 4, X_{3_{r}} ≅ X_{3} in D^{b}(fkH). By construction X_{3_{r}} ≅ S_{3} ⊗_{ekG} C in K^{b}(fkH)/K^{b}(proj- fkH) and hence, by the same argument as used in the proof of Corollary 4.5, we must have, for 1 ≤ r ≤ 4, X_{3_{r}} ≅ S_{3} ⊗_{ekG} X ≅ X_{3} in K^{b}(fkH).

BROUÉ’S CONJECTURE FOR THE HALL-JANKO GROUP AND ITS DOUBLE COVER 17

Appendix A. Loewy layers of green correspondents

We let R_{i} = S_{i} ↓_{H} and denote by RG_{i} the Green correspondent of S_{i}. P_{i} denotes the projective cover of the simple fkH-module indexed by i. Principal blocks. \underline{G = U_{3}(4)}.

R_{1} = RG_{1},  R_{2} = RG_{2},  R_{3} = RG_{3} ⊕ P_{1},
RG_{1} = k,  RG_{2} = 2 3 2  3 2  3 3 2 ,  RG_{3} = \begin{aligned}& & 3 \\ & k & & 3 \\ k & & 2 & & 3 \\ & k & & 3 \\ & & 3 \end{aligned}

\underline{G = J_{2}}.

R_{1} = RG_{1},  R_{2} = RG_{2},  R_{3} = RG_{3},
R_{4} = RG_{4} ⊕ P_{1},  R_{5} = RG_{5} ⊕ P_{1} ⊕ P_{5},  R_{6} = RG_{6} ⊕ 2P_{6} ⊕ P_{5},
RG_{1} = k,  RG_{2} = \begin{aligned}& & 5 \\ & 2 & & 6 \\ k & & 4 & & 5 \\ & 2 & & 6 \\ & & 5 \end{aligned} , RG_{3} = \begin{aligned}& 3 & & 6 \\ & k & & 5 \\ & 3 & & 6 \\ k & & 4 & & 5 \\ 6 & & 3 & & 6 \\ & & 5 \\ & & 3 \end{aligned} ,
RG_{4} = \begin{aligned}& & 3 \\ & & 5 \\ & 3 & & 6 \\ 4 & & k & & 5 \\ & 3 & & 6 \\ & & 5 \\ & & 3 \end{aligned} , RG_{5} = \begin{aligned}& 5 \\ & 6 \\ k & & 4 \\ & 6 \\ & 5 \end{aligned} ,  RG_{6} = \begin{aligned}& & & 4 & & 3 & & 2 \\ & & & 5 & & 6 & & 5 \\ & 4 & & 6 & & 2 & & 6 & & 3 \\ 5 & & 4 & & k & & k & & 4 & & 5 \\ & 6 & & 3 & & 6 & & 2 & & 6 \\ & & & 5 & & 4 & & 5 \\ & & & & 2 & & 3 \end{aligned}

18 MILES HOLLOWAY

\underline{G = Sp_{4}(4)}.

R_{1} = RG_{1}, R_{2} = RG_{2}, R_{3} = RG_{3} ⊕ P_{4},
R_{4} = RG_{4} ⊕ P_{2}, R_{5} = RG_{5} ⊕ P_{1} ⊕ 2P_{5},
RG_{1} = k,  RG_{2} = 3  3 5 4  3  2 5  5 3  4  2  3 5 3 , RG_{3} = \begin{aligned}& 4 \\ & 5 \\ k & & 4 \\ & 5 \\ & 4 \end{aligned} ,

RG_{4} = \begin{aligned}& 2 \\ & 5 \\ k & & 2 \\ & 5 \\ & 2 \end{aligned} ,  RG_{5} = \begin{aligned}& & 5 & & 5 \\ k & & 2 & & 4 & & k \\ & & 5 & & 5 \\ k & & 2 & & 3 & & k \\ & 5 & & 5 & & 5 \\ & 2 & & 4 & & k \\ & & & 5 \end{aligned}

Non-principal blocks.

\underline{G = 2.J_{2}}.

R_{1} = RG_{1}, R_{2} = RG_{2}, R_{3} = RG_{3}
R_{4} = RG_{4},  R_{5} = RG_{5} ⊕ P_{1} ⊕ P_{2} ⊕ P_{5} ⊕ P_{6}, R_{6} = RG_{6} ⊕ P_{3} ⊕ P_{4} ⊕ P_{5} ⊕ P_{6},

RG_{1} = 6 5 3 1 ,  RG_{2} = 3  4 6 4  5 3  6 4  5 3 ,  RG_{3} = 2 1 5 3 6 6 1 2 2 4 5 5 5 1 1 3 6 6 6 6 2 2 2 4 4 5 5 5 1 1 3 6 6 2 5 5 1 6 ,

RG_{4} = 5 6 1 2 3 4 5 6 1 2 3 4 5 5 6 6 2 4 5 5 5 6 6 1 1 2 3 3 4 6 6 6 2 4 5 5 5 1 3 6 6 5 , RG_{5} = 6 2 4 5 1 3 6 6 2 4 5 5 5 1 3 6 6 6 2 4 5 5 1 3 6 5 ,  RG_{6} = 3 5 6 4 5 5 1 3 3 6 6 2 4 4 5 5 5 1 3 3 6 6 6 6 2 4 4 5 5 3 6 6 4 5

Appendix B. Loewy layers of degree -1 terms of complexes S_{i} ⊗_{ekG} C

Let M_{i} denote the module in degree -1 in the complex S_{i} ⊗_{ekG} C. Let Mst_{i} denote the non-projective summand of M_{i}. P_{i} denotes the projective cover of the simple fkH-module indexed by i.

Principal blocks.

\underline{G = U_{3}(4)}.

M_{1} = 0,  M_{2} = Mst_{2},  M_{3} = Mst_{3},

Mst_{2} = \begin{matrix}2 & 3 \\ k & 2 & 3 & 3 \\ k & 2 & 2 & 3 & 3 & 3 \\ k & k & 2 & 2 & 3 & 3 & 3 & 3 \\ k & k & 2 & 2 & 3 & 3 & 3 & 3 \\ k & 2 & 2 & 3 & 3 & 3 \\ k & 2 & 3 & 3 \\ 2 & 3 \end{matrix} , Mst_{3} = \begin{matrix}2 & 3 \\ k & 2 & 3 & 3 \\ k & 2 & 2 & 3 & 3 & 3 \\ k & k & 2 & 3 & 3 & 3 \\ k & 2 & 2 & 3 & 3 & 3 \\ k & 2 & 3 & 3 \\ 2 & 3 \end{matrix}

\underline{G = J_{2}}.

M_{1} = 0,  M_{2} = Mst_{2},  M_{3} = Mst_{3} ⊕ P_{3} ⊕ P_{6},
M_{4} = P_{3} ⊕ P_{6}  M_{5} = 2P_{4} ⊕ 2P_{5},  M_{6} = Mst_{6} ⊕ 2P_{4} ⊕ 2P_{5} ⊕ 4P_{3} ⊕ 4P_{6}

Mst_{3} = \begin{matrix}3 & 6 \\ k & 5 \\ 3 & 6 \\ k & 5 \\ 3 & 6 \end{matrix} , Mst_{2} ≅ Mst_{6} = \begin{matrix}4 & 5 \\ 3 & 2 & 6 & 6 \\ k & 4 & 4 & 5 & 5 & 5 \\ 2 & 2 & 3 & 6 & 6 & 6 \\ k & 4 & 4 & 5 & 5 \\ 3 & 2 & 6 & 6 \\ 4 & 5 \end{matrix}

\underline{G = Sp_{4}(4)}. As there are two conjugacy classes in H of cyclic 5-subgroups, Q_{1}, Q_{2} with different centralizers, let M_{i}^{r} (resp. Mst_{i}^{r}) denote the M_{i} (resp. Mst_{i}) associated with Q_{r}.

M_{1}^{1} = 0,  M_{2}^{1} = Mst_{2}^{1} ⊕ P_{2} ⊕ P_{3},  M_{3}^{1} = Mst_{3}^{1},
M_{4}^{1} = 2P_{2} ⊕ 2P_{3},  M_{5}^{1} = Mst_{5}^{1} ⊕ 4P_{5},
Mst_{2}^{1} = 2  3 5 3  2 5 3  2 ,  Mst_{3}^{1} ≅ Mst_{5}^{1} = \begin{aligned}& 5 \\ k & & 4 \\ & 5 \\ k & & 4 \\ & 5 \end{aligned}
M_{1}^{2} = 0,  M_{2}^{2} = Mst_{2}^{2} ⊕ P_{3} ⊕ P_{4},  M_{3}^{2} = 2P_{3} ⊕ 2P_{4},
M_{4}^{2} = Mst_{4}^{2},  M_{5}^{2} = Mst_{5}^{2} ⊕ 4P_{5},
Mst_{2}^{2} = 4  3 5 4  3 5 4  3 ,  Mst_{4}^{2} ≅ Mst_{5}^{2} = \begin{aligned}& 5 \\ k & & 2 \\ & 5 \\ k & & 2 \\ & 5 \end{aligned}

Non-principal blocks. \underline{G = 2.J_{2}}. Only Mst_{i} were calculated. The non-zero ones were:

Mst_{2} = 6 2 4 5 1 3 6 6 2 4 5 5 5 1 3 6 6 6 2 4 5 5 1 3 6 5 ,  Mst_{4} = 2 5 1 3 6 6 2 2 4 5 5 5 1 1 3 6 6 6 2 2 4 5 5 5 1 3 6 6 2 5

BROUÉ’S CONJECTURE FOR THE HALL-JANKO GROUP AND ITS DOUBLE COVER 21

Appendix C. Images of simple modules under derived equivalence As described in Subection 4.3 the information contained here and in Appendix D is

sufficient to determine the {X_{i}}, up to isomorphism in K^{b}(fkH).

The degree zero term of the following cochain complexes is the first non-zero module reading from the right. P_{i} denotes the projective cover of the simple fkH-module indexed by i. Principal blocks. \underline{G = U_{3}(4)}.

X_{1} =  ⋯ → 0 ⟶ k ⟶ 0 → ⋯
X_{2} =  ⋯ → 0 ⟶ P_{3} ⟶ Mst_{2} ⟶ RG_{2} ⟶ 0 → ⋯
X_{3} =  ⋯ → 0 ⟶ P_{2} ⟶ 2P_{2} ⟶ Mst_{3} ⟶ RG_{3} ⟶ 0 → ⋯

\underline{G = J_{2}}.

X_{1} =  ⋯ → 0 ⟶ k ⟶ 0 → ⋯
X_{2} =  ⋯ → 0 ⟶ P_{3} ⟶ P_{3} ⊕ P_{4} ⟶ Mst_{2} ⟶ RG_{2} ⟶ 0 → ⋯
X_{3} =  ⋯ → 0 ⟶ P_{3} ⟶ P_{3} ⊕ Mst_{3} ⟶ RG_{3} ⟶ 0 → ⋯
X_{4} =  ⋯ → 0 ⟶ P_{5} ⟶ P_{6} ⟶ P_{3} ⟶ RG_{4} ⟶ 0 → ⋯
X_{5} =  ⋯ → 0 ⟶ P_{5} ⟶ P_{6} ⟶ P_{2} ⊕ P_{3} ⟶ P_{5} ⟶ RG_{5} ⟶ 0 → ⋯
X_{6} =  ⋯ → 0 ⟶ P_{3} ⟶ P_{3} ⊕ P_{5} ⟶ 2P_{5} ⟶ Mst_{6} ⊕ P_{2} ⊕ P_{3} ⟶ RG_{6} ⟶ 0 → ⋯

\underline{G = Sp_{4}(4)}.

X_{1} =  ⋯ → 0 ⟶ k ⟶ 0 → ⋯
X_{2} =  ⋯ → 0 ⟶ P_{5} ⟶ P_{2} ⊕ P_{3} ⊕ P_{4} ⟶ P_{3} ⊕ Mst_{2}^{1} ⊕ Mst_{2}^{2} ⟶ RG_{2} ⟶ 0 → ⋯
X_{3} =  ⋯ → 0 ⟶ P_{2} ⟶ P_{5} ⟶ P_{4} ⊕ Mst_{3}^{1} ⟶ RG_{3} ⟶ 0 → ⋯
X_{4} =  ⋯ → 0 ⟶ P_{4} ⟶ P_{5} ⟶ P_{2} ⊕ Mst_{4}^{2} ⟶ RG_{4} ⟶ 0 → ⋯
X_{5} = ⋯ → 0 ⟶ P_{3} ⟶ 2P_{3} ⟶ P_{3} ⊕ P_{5} ⟶ P_{5} ⊕ Mst_{5}^{1} ⊕ Mst_{5}^{2} ⟶ RG_{5} ⟶ 0 → ⋯

Non-principal blocks. \underline{G = 2.J_{2}}.

X_{1} =  ⋯ → 0 ⟶ P_{1} ⟶ P_{1} ⟶ RG_{1} ⟶ 0 → ⋯
X_{2} =  ⋯ → 0 ⟶ P_{2} ⟶ P_{6} ⟶ P_{4} ⊕ Mst_{2} ⟶ RG_{2} ⟶ 0 → ⋯
X_{3} = ⋯ → 0 ⟶ P_{1} ⊕ P_{3} ⟶ P_{1} ⊕ P_{3} ⊕ P_{5} ⟶ 2P_{5} ⟶ P_{1} ⊕ P_{2} ⊕ P_{5} ⟶ RG_{3} ⟶ 0 → ⋯
X_{4} =  ⋯ → 0 ⟶ P_{1} ⟶ P_{1} ⊕ P_{2} ⟶ P_{6} ⊕ Mst_{4} ⟶ RG_{4} ⟶ 0 → ⋯
X_{5} =  ⋯ → 0 ⟶ P_{1} ⊕ P_{3} ⟶ P_{5} ⟶ P_{5} ⟶ P_{1} ⊕ P_{3} ⟶ P_{6} ⟶ RG_{5} ⟶ 0 → ⋯
X_{6} =  ⋯ → 0 ⟶ P_{1} ⊕ P_{2} ⊕ P_{3} ⟶ P_{3} ⊕ P_{5} ⊕ P_{6} ⟶ RG_{6} ⟶ 0 → ⋯

22 MILES HOLLOWAY

Appendix D. Loewy layers of cohomology groups of complexes in

appendix C

Principal blocks.

\underline{G = U_{3}(4)}.

H^{0}(X_{1}) = k,  H^{−1}(X_{2}) = k,  H^{−2}(X_{2}) = \begin{matrix}k \\ 3 \end{matrix} ,
H^{−3}(X_{3}) = \begin{matrix}k \\ 3 \\ 3 \\ 2 \end{matrix}

\underline{G = J_{2}}.

H^{0}(X_{1}) = k,  H^{−3}(X_{2}) = \begin{matrix}k \\ 6 \\ 5 \\ 3 \end{matrix} ,  H^{−1}(X_{3}) = k,

H^{−2}(X_{3}) = \begin{aligned}4 & & k \\ & 6 \\ & 5 \\ & 3 \end{aligned} , H^{−2}(X_{4}) = 2,  H^{−3}(X_{4}) ≅ H^{−4}(X_{5}) = \begin{aligned}k & & 4 \\ & 6 \\ & 5 \\ 2 & & 3 \\ & 5 \end{aligned} ,

H^{−2}(X_{6}) = k,  H^{−3}(X_{6}) = \begin{aligned}k & & 4 \\ & 6 \\ & 5 \\ 2 & & 3 \end{aligned} ,  H^{−4}(X_{6}) = 3

We have the following short exact sequences:

0 ⟶ H^{−3}(X_{2}) ⟶ H^{−2}(X_{3}) ⟶ 4 ⟶ 0 (D.1)

0 ⟶ {2 \atop 5} ⟶ H^{−3}(X_{4}) ⟶ H^{−2}(X_{3}) ⟶ 0 (D.2)

0 ⟶ 5 ⟶ H^{−3}(X_{4}) ⟶ H^{−3}(X_{6}) ⟶ 0 (D.3)

\underline{G = Sp_{4}(4)}.

H^{0}(X_{1}) = k,  H^{−2}(X_{2}) = k,  H^{−3}(X_{2}) = \begin{matrix}k \\ 5 \end{matrix} ,
H^{−1}(X_{3}) = k,  H^{−3}(X_{3}) = 2,  H^{−1}(X_{4}) = k,

H^{−3}(X_{4}) = 4,  H^{−4}(X_{5}) = \begin{aligned}& k \\ & 5 \\ 4 & & 2 \\ & 5 \\ & 3 \end{aligned}

Non-principal blocks.

\underline{G = 2.J_{2}}.

H^{−1}(X_{1}) = {4 \atop 6} ,  H^{−3}(X_{2}) = 2,  H^{−2}(X_{1}) ≅ H^{−3}(X_{4}) = 6 2  4 6 5 1 ,

H^{−2}(X_{3}) = 4 6 5 3 ,  H^{−3}(X_{3}) ≅ H^{−4}(X_{3}) ≅ H^{−5}(X_{5}) = 6 2  4 6 5 1  3 ,  H^{−2}(X_{4}) = {6 \atop 2} ,

H^{−1}(X_{6}) = 6 2  4 6 5 3 ,  H^{−2}(X_{6}) = 6  2 2  4 6 5 1  3

We have the following short exact sequences:

0 ⟶ H^{−5}(X_{6}) ⟶ H^{−2}(X_{6}) ⟶ 2 ⟶ 0 (D.4)

0 ⟶ 1 ⟶ H^{−5}(X_{5}) ⟶ H^{−1}(X_{6}) ⟶ 0 (D.5)

0 ⟶ H^{−1}(X_{3}) ⟶ H^{−1}(X_{6}) ⟶ H^{−2}(X_{4}) ⟶ 0 (D.6)

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University of Bristol, School of Mathematics, University Walk, Bristol BS8 1TW, ENG-LAND

E-mail address: M.L.Holloway@bristol.ac.uk