Broué’s abelian defect group conjecture and 3-decomposition numbers of the sporadic simple Conway group Co_{1}

Shigeo Koshitani^{a,∗}, Jürgen Müller^{b}, Felix Noeske^{c}

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

Abstract In the representation theory of finite groups, Broué’s abelian defect group conjecture says 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. We prove that Broué’s conjecture, and even Rickard’s splendid equivalence conjecture, are true for the unique 3-block A of defect 2 of the sporadic simple Conway group Co_{1}, implying that both conjectures hold for all 3-blocks of Co_{1}. To do so, we determine the 3-decomposition numbers of A, and we actually show that A is Puig equivalent to the principal 3-block of the symmetric group 𝔖_{6} of degree 6.

Keywords: Broué’s conjecture; abelian defect group; splendid derived equivalence; sporadic simple Conway group; 3-decomposition numbers.

MSC: 20C, 20D

1. Introduction and notation

In the 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 Broué around 1988 [4]. He actually conjectures the following, where the various notions of equivalences used are recalled more precisely in 1.8:

Conjecture 1.1 (Broué’s Abelian Defect Group Conjecture [4]). Let (K, O, k) be a splitting pmodular system, where p is a prime, 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 provided P is abelian.

In fact, a stronger conclusion is expected:

Conjecture 1.2 (Rickard’s Splendid Equivalence Conjecture [52, 53]). Keeping the notation of 1.1, and still supposing P to be abelian, then there should be a splendid derived equivalence between the block algebras A of OG and B of ON_{G}(P).

Conjectures 1.1 and 1.2 have been verified for several cases, albeit the general conjecture is widely open still. For an overview see [6]; in particular, by [32, 51, 55, 56] 1.1 and 1.2 are proved for blocks with cyclic defect groups in arbitrary characteristic.

Moreover, it is shown in [18, (0.2)Theorem] that conjectures 1.1 and 1.2 hold for the principal block algebra of an arbitrary finite group when the defect group is elementary abelian of order 9. In view of the strategy used in [18], and of a possible future theory reducing conjectures 1.1 and 1.2 to the quasi-simple groups, it seems worthwhile to proceed with this class of groups, as far as non-principal 3-blocks with elementary abelian defect group of order 9 are concerned. Indeed, for these cases there are partial results already known, see [14, 21, 22, 23, 25, 30, 27, 28, 41] for instance; all the non-principal 3-blocks with elementary abelian defect group of order 9 for the sporadic simple groups and their covering groups are given in [43].

∗ Corresponding author.

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

The present paper is another step in that programme, our main result being the following:

Theorem 1.3. Let G be the sporadic simple Conway group Co_{1}, and let (K, O, k) be a splitting 3-modular system for all subgroups of G. Suppose that A is the unique block algebra of OG with elementary abelian defect group P = C_{3}×C_{3} of order 9, and that B is the block algebra of ON_{G}(P) such that B is the Brauer correspondent of A. Then, A and B are splendidly derived equivalent, hence Conjectures 1.1 and 1.2 of Broué and Rickard hold for the block algebra A.

As an immediate corollary we get:

Corollary 1.4. Broué’s abelian defect group conjecture 1.1, and Rickard’s splendid equivalence conjecture 1.2 are true for the prime p = 3 and for all block algebras of OG.

Actually, we prove 1.3 by proving the following:

Theorem 1.5. The block algebra A of G and the principal 3-block algebra A′ of the symmetric group 𝔖_{6} of degree 6 are Puig equivalent.

Hence, recalling the notion from [27, Remark 1.6(c)], this shows that A is pseudo-principal. Moreover, as an immediate corollary we get:

Corollary 1.6. The block algebra A of G and the principal 3-block algebra of the symplectic group Sp_{4}(q), for any prime power q such that q ≡ 2 or 5 (mod 9), are Puig equivalent.

Typically, the starting point to proving theorems like those above is the decomposition matrix of the block in question. Alone, for the block algebra A these have not been known before. So we first set out to tackle this problem, and arrived at a fairly good approximation to the decomposition matrix of A, but we have not been able to determine it completely. Still, the approximation was good enough to compare the result with the data in [18] to arrive at the sensible guess that A might be closely related to the principal block algebra A′ of the symmetric group 𝔖_{6}, whose decomposition matrix is well-known of course. Now, using the partial results on decomposition numbers, it was possible to activate the sophisticated block-theoretic machinery needed to actually compare the block algebras A and A′, which in turn finally paved the way to complete the decomposition matrix of A. Here it is:

Theorem 1.7. The 3-decomposition matrix of A and of A′ is the following, where we denote ordinary characters by their degrees, but also give the ATLAS notation for the characters in A:

\begin{aligned}& A & A′ & S_{1} & S_{2} & S_{3} & S_{4} & S_{5} \\ χ_{29} & 2 816 856 & 1 & 1 & . & . & . & . \\ χ_{38} & 16 347 825 & 5′ & 1 & 1 & . & . & . \\ χ_{51} & 44 013 375 & 1^{−} & . & . & 1 & . & . \\ χ_{55} & 57 544 344 & 5^{−} & . & 1 & 1 & . & . \\ χ_{62} & 91 547 820 & 10^{−} & . & 1 & . & 1 & . \\ χ_{80} & 251 756 505 & 5 & 1 & . & . & . & 1 \\ χ_{85} & 292 953 024 & 5′^{−} & . & . & 1 & . & 1 \\ χ_{89} & 326 956 500 & 10 & . & . & . & 1 & 1 \\ χ_{91} & 387 317 700 & 16 & 1 & 1 & 1 & 1 & 1 \end{aligned}

dimS_{1} = 2 816 856
dimS_{2} = 13 530 969
dimS_{3} = 44 013 375
dimS_{4} = 78 016 851
dimS_{5} = 248 939 649

We just remark that it is an ongoing project, see [60], to determine all the decomposition numbers of all the finite simple and closely related groups occurring in the Atlas [7]. The above result is also a contribution to this project. □

Following the general recipe indicated above, the present paper is organised as follows: In §2 we set out to find an approximation to the decomposition matrix of A, by using techniques from computational modular character theory. The result, where only a parameter α ∈ {0, 1, 2} is yet undetermined, is given in 2.7, In §3 we collect the necessary facts for the symmetric group 𝔖_{6} and its principal block A′. In particular, we determine the trivial-source modules in A′, which play a particularly important role in the sequel. In §4 we prove that the block A and its Brauer correspondent B in N_{G}(P) are splendidly stably equivalent of Morita type. This immediately implies that A and A′ also are splendidly stably equivalent of Morita type. In §5 we finally show that the stable equivalence between A and A′ respects simple modules, by a thorough consideration of trivial-source modules in A. This implies that A and A′ actually are Puig equivalent, from which the other assertions follow immediately. In order to facilitate the necessary computations, we make use of the computer algebra system GAP [9], to deal with finite groups, in particular permutation and matrix groups, and with ordinary and Brauer characters of finite groups. In particular, we make use of the character table library [3], which provides electronic access to the data collected in the Atlas [7] and in the ModularAtlas [13, 60], of the interface [59] to the database [61], and of the package [40] providing the necessary tools from computational modular character theory.

Moreover, we use the computer algebra system MeatAxe [54], and its extensions [36, 37, 38, 39] to deal with matrix representations over finite fields. Here, we use ‘small’ finite fields, but we always make sure, silently, that these are chosen such that the computational results thus obtained remain valid without change after scalar extension to the fixed field of positive characteristic which is ‘large enough’ in the sense of 1.8 below.

Notation/Definition 1.8. Throughout this paper, we use the standard notation and terminology as is used in the Atlas [7] and textbooks like [42, 57]. We recall a few for convenience:

(i) If A and B are finite dimensional k-algebras, where k is a field, 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. A module always refers to a finitely generated right module, unless stated otherwise. Let M be an A-module. We let M^{∨} = Hom_{A}(M_{A}, A_{A}) be the A-dual of M, so that M^{∨} becomes a left A-module. We denote by soc(M) and rad(M) the socle and the radical of M, respectively. For non-isomorphic simple A-modules S_{1}, …, S_{n}, and positive integers a_{1}, …, a_{n}, we write that ”M = a_{1} · S_{1} + ⋯ + a_{n} · S_{n}, as composition factors” when the composition factors are a_{1} times S_{1},…, a_{n} times S_{n}. For another A-module N, we write M|N if M is isomorphic to a direct summand of N as an A-module; and we set [M, N]^{A} = dim_{k}(Hom_{A}(M, N)).

(ii) By G we always denote 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, O is a complete discrete valuation ring of rank one such that its quotient field K is of characteristic zero, and its residue field k = O/rad(O) is of characteristic p, and that K and k are splitting fields for all subgroups of G.

We denote by Irr(G) and IBr(G) the sets of all irreducible ordinary and Brauer characters of G, respectively. If A is a block algebra 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 say that a kG-module M is a trivial-source module if M is indecomposable and M has a trivial source, see [31, II Definition 12.1]; note that the definition here is slightly different from [57, §27 p.218] where indecomposability is not assumed. Thus, in particular, projective indecomposable modules are trivial-source modules. We recall the following facts, see [31, II Theorem 12.4, I Proposition 14.8]: If M is a trivial-source module, then M lifts uniquely (up to isomorphism) to a trivial source OG-lattice M̂, in particular this associates an ordinary character χ_{M} to M. Moreover, if N is another trivial-source kG-module, then [M, N]^{G} = ⟨χ_{M}, χ_{N}⟩_{G}, where ⟨−, −⟩_{G} denotes the usual scalar product on ordinary characters.

(iii) 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 and g, g′ ∈ G. Let A and A′ be block algebras of OG and OG′, respectively, such that A and A′ have a defect group P in common. Then we say that A and A′ are Puig equivalent 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 trivialsource module and ΔP -projective. This is equivalent to a condition that A and A′ have source algebras which are isomorphic as interior P -algebras, see [50, Remark 7.5] and [35, Theorem 4.1]. We say that A and A′ are stably equivalent of Morita type if there exists an (A, A′)-bimodule 𝔐 such that both _{A}𝔐 and 𝔐_{A′} are projective and that _{A}(𝔐⊗_{A′} 𝔐^{∨})_{A} ≅ _{A}A_{A} ⊕(proj (A, A)-bimod) and _{A′}(𝔐^{∨} ⊗_{A} 𝔐)_{A′} ≅ _{A′}A′_{A′} ⊕(proj (A′, A′)-bimod). We say that A and A′ are splendidly stably equivalent of Morita type if the stable equivalence of Morita type is induced by an indecomposable (A, A′)-bimodule 𝔐 which is a trivial-source O[G × G′]-module and is ΔP -projective, see [35, Theorem 3.1].

We say that A and A′ are derived equivalent if D^{b}(mod-A) and D^{b}(mod-A′) are equivalent as triangulated categories, where D^{b}(mod-A) is the bounded derived category of mod-A. In that case, there even is a Rickard complex M^{•} ∈ C^{b}(A-mod-A′), where the latter is the category of bounded complexes of finitely generated (A, A′)-bimodules, all of whose terms are projective both as left A-modules and as right A′-modules, such that M^{•} ⊗_{A′} (M^{•})^{∨} ≅ A in K^{b}(A-mod-A) and (M^{•})^{∨} ⊗_{A} M^{•} ≅ A′ in K^{b}(A′-mod-A′), where K^{b}(A-mod-A) is the homotopy category associated with C^{b}(A-mod-A). We say that A and A′ are splendidly derived equivalent if K^{b}(mod-A) and K^{b}(mod-A′) are equivalent via a Rickard complex M^{•} ∈ C^{b}(A-mod-A′) as above, such that additionally each of its terms is a direct sum of ΔP -projective trivial-source modules as an O[G × G′]- module; see [34, 35]. Note that a Morita equivalence entails a derived equivalence, and that a Puig equivalence entails a splendid derived equivalence.

2. Decomposition numbers of Co_{1}

In this section we use methods from computational modular character theory to find an approximation to the decomposition matrix of the block in question. To do so, we make heavy use of the data on decomposition numbers known for various maximal subgroups of G, as is contained in [13, 60]. We stress that, to find the approximate decomposition matrix exhibited below in the first place, we make use of the full machinery laid out in [11], in combination with substantial computation, using GAP [9] and the package [40], while the proof presented here is subsequently derived from the computer-based one by straightening it out by hand. For convenience we recall a few of the basic notions used in this approach, while for full details we refer the reader to [11]:

Definition 2.1. We keep the notation in 1.8(ii). Let ℤIBr(A) be the lattice of generalised Brauer characters belonging to A, and ℕ_{0}IBr(A) ⊂ ℤIBr(A) be the non-negative cone of genuine Brauer characters spanned by IBr(A). Then a ℤ-basis BS of ℤIBr(A) which is contained in ℕ_{0}IBr(A) is called a basic set of Brauer characters; in particular, IBr(A) is such a basic set.

Moreover, any projective A-module has an ordinary character associated with it. Thus running through the projective indecomposable A-modules, this gives rise to the set IPr(A) ⊂ ℕ_{0}Irr(A), spanning the lattice ℤIPr(A) of generalised projective characters belonging to A. Then, similarly, a ℤ-basis PS of ℤIPr(A) which is contained in ℕ_{0}IPr(A) is called a basic set of projective characters; in particular, IPr(A) is such a basic set.

Now let Îrr(A) be the set of restrictions of the characters in Irr(A) to the p-regular conjugacy classes of G, thus we have Îrr(A) ⊂ ℕ_{0}IBr(A). Similarly we obtain ÎPr(A), and thus we get ℤÎPr(A) ⊆ ℤIBr(A) such that ÎPr(A) ⊂ ⟨Îrr(A)⟩_{ℕ_{0}} ⊆ ℕ_{0}IBr(A). The restriction ⟨−, −⟩′_{G} of the character-theoretic scalar product to the p-regular classes induces a non-degenerate pairing between ℤIBr(A) and ℤÎPr(A), such that IBr(A) and ÎPr(A) are a pair of mutually dual ℤ-bases.

More generally, given basic sets BS and PS as above, the associated matrix of mutual scalar products is unimodular and has non-negative entries. Now the general strategy is to use character theory to find sequences of basic sets of Brauer characters and of projective characters, which better and better approximate the sets IBr(A) and IPr(A), respectively, the guiding principle being the above-mentioned positivity properties.

Notation 2.2. ¿From now on we use the following notation. Let G := Co_{1} be the first sporadic simple Conway group, and hence |G| = 2^{21}·3^{9}·5^{4}·7^{2}·11·13·23, see [7, p.180]. Assume p = 3, and let (K, O, k) be a splitting 3-modular system for all finite groups we shall treat with. It follows from the character table of G, see [7, p.180], that there is a unique block algebra A of kG with elementary abelian defect group P of order 9; see also [43]. (We will specify P as a subgroup of G more precisely in 4.1 and 4.2 below.)

Moreover, it turns out that A contains k(A) = 9 irreducible ordinary characters and l(A) = 5 irreducible Brauer characters. According to the ordering in [7, p.180], the ordinary characters in A are {χ_{29}, χ_{38}, χ_{51}, χ_{55}, χ_{62}, χ_{80}, χ_{85}, χ_{89}, χ_{91}}, for convenience their degrees are indicated in 1.7.

2.3. We begin by choosing basic sets of Brauer characters and projective characters.

Let χ̂_{i} denote the i-th irreducible character of G, with respect to the ordering given in [7, p.180], restricted to its 3-regular conjugacy classes. Moreover, for a subgroup H ≤ G let ψ(H)_{i} denote the i-th irreducible Brauer character of H with respect to the ordering given in the databases mentioned, and let Ψ(H)_{i} denote the ordinary character of the corresponding projective indecomposable module. We remark that the irreducible Brauer characters and the projective indecomposable characters of the 2-local maximal subgroup 2^{11} : M_{24} are easily computed using Fong-Reynolds correspondence, but we only need those of the Mathieu group M_{24}, inflated along the natural map 2^{11} : M_{24} → M_{24}.

Then our initial basic set BS = {φ_{1}, …, φ_{5}} of Brauer characters constitutes of the A-parts of the Brauer characters

χ̂_{29}, ψ(3.Suz.2)_{7}↑^{G}, χ̂_{51}, ψ(3.Suz.2)_{11}↑^{G}, χ̂_{80},

and as an initial basic set PS = {Φ_{1}, …, Φ_{5}} of projective characters we choose the A-parts of

Ψ(Co_{2})_{18}↑^{G}, Ψ(Co_{2})_{22}↑^{G}, Ψ(Co_{2})_{29}↑^{G}, Ψ(2^{11} : M_{24})_{12}↑^{G},

These indeed are suitable basic sets, as the matrix (⟨φ_{i}, Φ_{j}⟩′_{G})_{1≤i,j≤5} shows:

\begin{matrix}⟨−, −⟩′_{G} & Φ_{1} & Φ_{2} & Φ_{3} & Φ_{4} & Φ_{5} \\ φ_{1} & . & . & . & . & 1 \\ φ_{2} & . & . & . & 1 & 1 \\ φ_{3} & . & . & 1 & . & . \\ φ_{4} & . & 1 & . & 2 & 1 \\ φ_{5} & 1 & 1 & 3 & 1 & 3 \end{matrix}

Note that in particular this initial matrix already turns out to be unitriangular, which will simplify the arguments to follow.

2.4. it is now immediate that φ_{1} and φ_{3} are irreducible Brauer characters, and that Φ_{1} is the character of a projective indecomposable module. We denote the known irreducible Brauer characters and projective indecomposable characters in bold face in

Next we prove that φ_{2} and φ_{4} actually are irreducible Brauer characters as well. To do so, we consider the A-parts of the projective characters Ψ(3.Suz.2)_{27}↑^{G} and Ψ(2^{11} : M_{24})_{2}↑^{G}, whose decompositions into PS are given as follows:

Φ_{1} Φ_{2} Φ_{3} Φ_{4} Φ_{5}
Ψ(3.Suz.2)_{27}↑^{G} −9 21 16 −1 1
Ψ(2^{11} : M_{24})_{2}↑^{G} −4 −1 0 0 4

Now assume that φ_{2} is reducible. Then φ_{2} − φ_{1} is a Brauer character, whose scalar product with Ψ(3.Suz.2)_{27}↑^{G} equals [0, 0, 0, 1, 0] · [−9, 21, 16, −1, 1]^{⊤} = −1, a contradiction. Hence φ_{2} is an irreducible Brauer character.

Next, assume that φ_{4} is reducible. Then either φ_{4} −φ_{2} or φ_{4} −φ_{1} is a Brauer character. Taking scalar products with Ψ(2^{11} : M_{24})_{2}↑^{G} yields [0, 1, 0, 1, 0] · [−4, −1, 0, 0, 4]^{⊤} = −1 and [0, 1, 0, 2, 0] · [−4, −1, 0, 0, 4]^{⊤} = −1, respectively, a contradiction. Thus φ_{4} is an irreducible Brauer character.

2.5. We now set out to improve our basic sets BS and PS.

that φ_{4} is not contained in φ_{5}, hence we conclude that Φ′_{2} := Φ_{2} − Φ_{1} is a projective character, and then as such is indecomposable.

Next, using the facts that φ_{4} and φ_{2} are irreducible, we conclude that Φ′_{4} := Φ_{4} − 2(Φ_{2} − Φ_{1}) and Φ′_{5} := Φ_{5} − (Φ_{2} − Φ_{1}) − (Φ_{4} − 2Φ_{2} + Φ_{1}) = Φ_{5} − Φ_{4} + Φ_{2} are projective characters.

Finally, it turns out that χ̂_{85} decomposes into BS as follows:

φ_{1} φ_{2} φ_{3} φ_{4} φ_{5}
χ̂_{85} −1 . 1 . 1

As φ_{3} is irreducible it follows that φ′_{5} := φ_{5} − φ_{1} is a Brauer character.

Thus, we have obtained new basic sets BS′ := {φ_{1}, …, φ_{4}, φ′_{5}} and PS′ := {Φ_{1}, Φ′_{2}, Φ_{3}, Φ′_{4}, Φ′_{5}}, whose matrix of mutual scalar products is given as follows:

\begin{matrix}⟨−, −⟩′_{G} & Φ_{1} & Φ′_{2} & Φ_{3} & Φ′_{4} & Φ′_{5} \\ φ_{1} & . & . & . & . & 1 \\ φ_{2} & . & . & . & 1 & . \\ φ_{3} & . & . & 1 & . & . \\ φ_{4} & . & 1 & . & . & . \\ φ′_{5} & 1 & . & 3 & 1 & 2 \end{matrix}

2.6. With respect to PS′ we now have:

Φ_{1} Φ′_{2} Φ_{3} Φ′_{4} Φ′_{5}
Ψ(2^{11} : M_{24})_{2}↑^{G} −9 3 0 4 4

Since Φ_{1} is already known to be indecomposable, show that it is contained at most once in Φ′_{4} and at most twice in Φ′_{5}, we from this conclude that Φ′′_{5} := Φ′_{5} − 2Φ_{1} and Φ′′_{4} := Φ′_{4} − Φ_{1} are projective characters. Then, as such, they both are indecomposable.

Finally, the A-part of the tensor product Φ′′_{4} ⊗ χ̂_{2} decomposes into the newly found projective characters as follows:

Φ_{1} Φ′_{2} Φ_{3} Φ′′_{4} Φ′′_{5}
Φ′′_{4} ⊗ χ̂_{2} −4 2 4 2 0

Hence Φ′_{3} := Φ_{3} − Φ_{1} is a projective character.

Thus we have obtained the new basic set PS′′ := {Φ_{1}, Φ′_{2}, Φ′_{3}, Φ′′_{4}, Φ′′_{5}}, whose matrix of scalar products with BS′ is given as follows:

\begin{matrix}⟨−, −⟩′_{G} & Φ_{1} & Φ′_{2} & Φ′_{3} & Φ′′_{4} & Φ′′_{5} \\ φ_{1} & . & . & . & . & 1 \\ φ_{2} & . & . & . & 1 & . \\ φ_{3} & . & . & 1 & . & . \\ φ_{4} & . & 1 & . & . & . \\ φ′_{5} & 1 & . & 2 & . & . \end{matrix}

Hence, only three possible decomposition matrices remain: one of Φ′_{3} − α · Φ_{1} for 0 ≤ α ≤ 2 is a projective indecomposable character. Thus we have proved:

Lemma 2.7. The 3-decomposition matrix of A is the following:

S_{1} S_{2} S_{3} S_{4} S_{5}
χ_{29} 1 .  .  . .
χ_{38} 1 1  .  . .
χ_{51} . .  1  . .
χ_{55} . 1  1  . .
χ_{62} . 1  .  1 .
χ_{80} 1 .  α  . 1
χ_{85} . . 1 + α . 1
χ_{89} . . α 1 1
χ_{91} 1 1 1 + α 1 1

where α is a certain integer such that α ∈ {0, 1, 2}.

Proof. This follows from 2.3–2.6, where here we reverse the order of the projective characters in the final basic set, and already indicate the notation for the associated simple A-modules which will be used later. □

3. The group 𝔖_{6}

Notation 3.1. Set G′ := 𝔖_{6}, the symmetric group of degree 6. Then the Sylow 3-subgroup of G′ are elementary abelian of order 9. Hence fixing a block defect group P of A, see 2.2, we may identify P with a Sylow 3-subgroup of G′. Doing so, we have P = Q × R with Q ≅ C_{3} ≅ R, where, with respect to the tautological permutation representation of G′, we may assume that the non-trivial elements of Q are 3-cycles, while those of R are fixed-point free.

Let A′ be the principal block algebra of kG′. Then we have IBr(A′) = {1a, 1b, 4a, 4b, 6} and Irr(A′) = {χ_{1}, χ_{1^{−}}, χ_{5}, χ_{5^{−}}, χ_{5′}, χ_{5′}^{−}, χ_{10}, χ_{10^{−}}, χ_{16}}, where we use the following notation:

As usual, we denote by χ_{1} := 1_{A′} ∈ Irr(A′) and χ_{1^{−}} := 1^{−} ∈ Irr(A′) the trivial and the sign character, respectively. Moreover, we let χ_{5} ∈ Irr(A′) be the non-trivial constituent of the tautological permutation character of A′, which is characterised by having positive values on both the 3-cycles and the transpositions, and we let χ_{5′} ∈ Irr(A′) be the irreducible character having positive values on both the fixed-point free elements of order 3 and 2; note that χ_{5} and χ_{5′} are interchanged by the non-trivial outer automorphism of A′. Finally, we let χ_{10} ∈ Irr(A′) be the irreducible character having positive value on the transpositions. Then we get χ_{5^{−}} := χ_{5} ⊗ 1^{−} ∈ Irr(A′) and χ_{5′}^{−} := χ_{5′} ⊗ 1^{−} ∈ Irr(A′) as well as χ_{10^{−}} := χ_{10} ⊗ 1^{−} ∈ Irr(A′).

As for the simple modular representations occurring, we let, again as usual, 1a := k_{G′} ∈ IBr(A′) and 1b := 1^{−} ∈ IBr(A′) be the trivial and the sign representation, respectively. Moreover, we let 4a be the non-trivial modular constituent of the tautological permutation representation of G′, then we get 4b := 4a ⊗ 1^{−} ∈ IBr(A′). Note that the outer automorphism of G′ mentioned earlier interchanges 4a ↔ 4b, and leaves 1a, 1b, and 6 fixed.

Lemma 3.2. The 3-decomposition matrix of A′ is given as follows:

1a 1b 4a 4b 6
χ_{1}  1 ….
χ_{1^{−}} . 1 …
χ_{5}  1 . 1 . .
χ_{5^{−}} . 1 . 1 .
χ_{5′}  1 . . 1 .
χ_{5′}^{−} . 1 1 . .
χ_{10}  . . 1 . 1
χ_{10^{−}} … 1 1
χ_{16} 1 1 1 1 1

Proof. This is taken from [13, p. 4]. □

Lemma 3.3. Using the notation introduced in 3.1, the Loewy and socle series of the trivial-source kG′-modules in A′ are given as follows, where for the non-projective ones we also include their associated ordinary characters:

(i) The trivial-source kG′-modules in A′ with vertex P are the following:

\begin{matrix}1a & 1b & \begin{matrix}4a \\ 1a 1b \\ 4a \end{matrix} & \begin{matrix}4b \\ 1a 1b \\ 4b \end{matrix} & \begin{matrix}6 \\ 4a 4b \\ 6 \end{matrix} \\ ↕ & ↕ & ↕ & ↕ & ↕ \\ χ_{1} & χ_{1^{−}} & χ_{5} + χ_{5′}^{−} & χ_{5^{−}} + χ_{5′} & χ_{10} + χ_{10^{−}}. \end{matrix}

(ii) The trivial-source kG′-modules in A′ with vertex Q are the following:

\begin{matrix}\begin{matrix}1a \\ 4a \\ 1a \end{matrix} & \begin{matrix}1b \\ 4b \\ 1b \end{matrix} & \begin{matrix}4a \\ 1a 6 \\ 4a \end{matrix} & \begin{matrix}4b \\ 1b 6 \\ 4b \end{matrix} \\ ↕ & ↕ & ↕ & ↕ \\ χ_{1} + χ_{5} & χ_{1^{−}} + χ_{5^{−}} & χ_{5} + χ_{10} & χ_{5^{−}} + χ_{10^{−}} \end{matrix}

(iii) The trivial-source kG′-modules in A′ with vertex R are the following:

\begin{matrix}\begin{matrix}1a \\ 4b \\ 1a \end{matrix} & \begin{matrix}1b \\ 4a \\ 1b \end{matrix} & \begin{matrix}4b \\ 1a 6 \\ 4b \end{matrix} & \begin{matrix}4a \\ 1b 6 \\ 4a \end{matrix} \\ ↕ & ↕ & ↕ & ↕ \\ χ_{1} + χ_{5′} & χ_{1^{−}} + χ_{5′}^{−} & χ_{5′} + χ_{10^{−}} & χ_{5′}^{−} + χ_{10} \end{matrix}

(iv) The projective indecomposable kG′-modules in A′ have the following structure:

\begin{matrix}1a \\ 4a 4b \\ 1a 1a 1b 6 \\ 4a 4b \\ 1a \end{matrix} \begin{matrix}1b \\ 4a 4b \\ 1a 1b 1b 6 \\ 4a 4b \\ 1b \end{matrix} \begin{matrix}4a \\ 1a 1b 6 \\ 4a 4a 4b \\ 1a 1b 6 \\ 4a \end{matrix} \begin{matrix}4b \\ 1a 1b 6 \\ 4a 4b 4b \\ 1a 1b 6 \\ 4b \end{matrix} \begin{matrix}6 \\ 4a 4b \\ 1a 1b 6 \\ 4a 4b \\ 6 \end{matrix}

Proof. (i)–(iii) This is found by explicit computation as follows: By Green correspondence and [42, Chap.4 Problem 10], the modules we are interested in are precisely the indecomposable direct summands with maximal vertex of the permutation kG′-modules on the cosets of P, Q, and R, respectively. Now, the relevant permutation actions of G′ can be computed using GAP [9], and going over to permutation kG′-modules, their indecomposable direct summands, together with their structure, are subsequently found using the MeatAxe [54]. Recall that the number of trivialsource kG′-modules with vertex as prescribed above can be determined a priority, using the facts that N_{G′}(P)/P ≅ D_{8} and N_{G′}(Q)/Q ≅ 2×𝔖_{3} ≅ N_{G′}(R)/R. Finally, knowing the structure of the trivial-source modules in question, the ordinary characters associated with them are determined using 3.2.

(iv) This is contained in [58], and can also be rechecked computationally by applying the MeatAxe [54] to the regular representation of kG′. □

Notation 3.4. Set H′ := N_{G′}(P). Then we have H′ ≅ P: D_{8}, where it follows from the description in 3.1 that P is simple, viewed as an 𝔽_{3}D_{8}-module, which hence determines H′ up to isomorphism. But note that, by its mere definition, H′ comes with a fixed embedding into G′.

Let B′ be the principal block algebra of kH′, being the Brauer correspondent of A′ in H′; note that we actually have B′ = kH′.

Lemma 3.5. Let M′ be the Scott k[G′ × H′]-module with vertex ΔP. (Note that this is the up to isomorphism unique indecomposable direct summand of the (A′, B′)-bimodule A′↓_{G′×H′}^{G′×G′}·1_{B′} with vertex ΔP.) Then the pair (M′, M′^{∨}) induces a splendid stable equivalence of Morita type between the block algebras A′ and B′.

Proof. This follows from [45, Example 4.4] and [46, Corollary 2]. □

4. Stable equivalence between A and B

Lemma 4.1. Recall the notation G, P, and A as in 2.2. Then the following holds: (i) We have H := N_{G}(P) = P.U_{4}(3).D_{8} and C_{G}(P) = P.U_{4}(3) . (ii) The non-trivial elements in P fall into two conjugacy classes of H. Thus P has exactly two H-conjugacy classes of subgroups of order 3. Moreover, the non-trivial elements of P belong to conjugacy classes 3A and 3B in [7, p.184].

Proof. We use the smallest faithful permutation representation of G on 98280 points, available in [61], and GAP [9] to find a defect group P ≤ G of A explicitly. In order to do so, using the character table of G, see [7, p.180], it turns out that conjugacy classes 4D, 5B 7A, and 8A are defect classes of A. By a random search we find an element of order 42 in G, whose 6-th power, x say, hence belongs to conjugacy class 7A. Thus we may let P be a Sylow 3-subgroup of C_{G}(x).

Having P found explicitly, it turns out that H = N_{G}(P) has order 235 146 240, and that C_{G}(P) has order 29 393 280 = |H|/8. Now a consideration of the orders of the maximal subgroups of G, see [7, p.180], shows that H necessarily is a maximal subgroup of G, of shape P.U_{4}(3).D_{8}. Then it is clear that C_{G}(P) is of shape P.U_{4}(3). This shows (i). The character table of the maximal subgroup H of G, available in [3], shows that P − {1} ⊂ H consists of two rational conjugacy classes. a consideration of character values shows, together with Brauer’s Second Main Theorem, that the latter conjugacy classes fuse into the conjugacy classes 3A and 3B of G. □

Notation 4.2. Set H := N_{G}(P). Then by 4.1(ii) we have P = Q × R with Q ≅ C_{3} ≅ R, such that Q and R are not conjugate in G, and we may assume that the non-trivial elements in Q and R belong to conjugacy classes 3A and 3B, respectively. Recall that we have already chosen an embedding of P into G′, see 3.1, but since the automorphism group of P acts transitively on the minimal generating sets of P, both choices are consistent, justifying the reuse of earlier notation.

Let B be a block algebra of kH which is the Brauer correspondent of A. Let (P, e) be a maximal A-Brauer pair in G, namely, e is a block idempotent of kC_{G}(P) such that Br_{P} (1_{A})·e = e, see [1],

[5] and [57, §40]. Let i and j respectively be source idempotents of A and B with respect to P. As remarked in [35, 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}. We 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} = kG_{Q}·e_{Q} and B_{Q} = kH_{Q}·f_{Q}, so that e_{Q} = 1_{A_{Q}} and f_{Q} = 1_{B_{Q}}. Similarly we define G_{R}, H_{R}, A_{R}, B_{R}, e_{R} and f_{R}.

Lemma 4.3. The following holds:

(i) We have H = N_{G}(P, e) := {g ∈ N_{G}(P)|g^{−1}eg = e}.

(ii) B = kHe ≅ Mat_{729}(k[P : D_{8}]), where P : D_{8} ≅ H′. Hence we may write IBr(B) :=

{729a, 729b, 729c, 729d, 1458}. (iii) The blocks B and B′ are canonically Puig equivalent.

Proof. (i) The group C_{G}(P)/P = U_{4}(3), being a simple group of Lie type in characteristic 3, has a unique irreducible character of defect zero, namely the Steinberg character of degree 729. Hence we conclude that C_{G}(P) has a unique block with defect group P, and thus we have N_{G}(P, e) = N_{G}(P) = H.

(ii) We argue similar to the lines of [27, Proof of Lemma 4.6(iv)]: By [29, A.Theorem] we have B ≅ Mat_{729}(k^{α}[P : D_{8}]), for some cocycle α ∈ Z^{2}(D_{8}, k^{×}), and where the action of D_{8} ≅ N_{G}(P)/C_{G}(P) in the semidirect product P : D_{8} is determined by the action of N_{G}(P) on P. Now it follows from 4.1(ii) that P is simple as an 𝔽_{3}D_{8}-module, hence by 3.4 we have P : D_{8} ≅ H′.

Since, by [12, Satz V.25.6] we have |H^{2}(D_{8}, k^{×})| = 2, it remains to show that α ≡ 1 (mod B^{2}(D_{8}, k^{×})). Indeed, using the character table of H, GAP [9] shows that B has nine irreducible characters and five irreducible Brauer characters, from which the last part of (ii) follows using [15, p.34, Tbl.1].

Finally, in (iii), the statement about source algebras follows from [49, Proposition 14.6], see [57, (45.12)Theorem] and [2, Theorem 13]. □

Lemma 4.4. The following holds:

(i) G_{Q} = Q.Suz, so that G_{Q}/Q ≅ Suz.

(ii) H_{Q} = P.U_{4}(3).2′_{3}, so that H_{Q}/Q ≅ 3_{2}.U_{4}(3).2′_{3}.

(iii) G_{R} = H_{R} = P.U_{4}(3).2′_{2}, so that G_{R}/R = H_{R}/R ≅ 3_{1}.U_{4}(3).2′_{2}.

Proof. By [7, p.183] we have N_{G}(Q) = Q.Suz.2, a maximal subgroup of G, implying (i).

To show (ii), a consideration of the 𝔽_{3}D_{8}-module shows that N_{H}(Q) has index 2 in H, and is of shape P.U_{4}(3).2^{2}. Thus H_{Q} is of shape P.U_{4}(3).2, and hence H_{Q}/Q is of shape 3.U_{4}(3).2. To find the precise structure, we note that H_{Q}/Q is a subgroup of G_{Q}/Q ≅ Suz, and a consideration of the orders of the maximal subgroups of Suz, see [7, p.131], shows that H_{Q}/Q necessarily is a maximal subgroup of Suz, namely the normaliser of a cyclic subgroup of order 3 whose non-trivial elements belong the conjugacy class 3A of Suz. Now there is a known typo in [7, p.131], the shape of the maximal subgroup in question being erroneously stated as 3_{2}.U_{4}(3).2_{3}. This has been corrected in the reprint of 2003, but can also be explicitly checked using GAP [9]: the character tables of Suz, see [7, p.131], and the various bicyclic extensions of U_{4}(3), see [7, pp.52–59], show that the shape is as asserted above. This implies (ii).

Finally, to show (iii), we use GAP [9] and the explicit choices of subgroups made in the proof of 4.1 and in 4.2, to determine N_{G}(R) explicitly. It turns out that N_{G}(R) is a subgroup of H of index 2. Hence we have N_{G}(R) = N_{H}(R), and as above we conclude that N_{G}(R) is of shape P.U_{4}(3).2^{2} and hence G_{R} is of shape P.U_{4}(3).2; but note that N_{H}(R) ≠ N_{H}(Q). Computing G_{R} and the character table of G_{R}/R explicitly, and comparing with those of the various bicyclic extensions of U_{4}(3), we conclude that G_{R}/R is of the shape asserted, thus (iii) follows. □

Lemma 4.5. Set G̃ = Suz.

(i) There is a unique block algebra Ã of kG̃ with a defect group D ≅ C_{3}; the non-trivial elements of D belong the conjugacy class 3A of G̃.

(ii) We can write Irr(Ã) = {χ_{16}, χ_{38}, χ_{41}} such that χ_{16}(1) = 18954, χ_{38}(1) = 189540, χ_{41}(1) = 208494 and χ_{16}(u) = χ_{38}(u) = 729 for any element u belonging to the conjugacy class 3A. Moreover, we can write IBr(Ã) = {φ_{1}, φ_{2}} such that the 3-decomposition matrix of Ã is as follows:

\begin{aligned}& φ_{1} & φ_{2} \\ χ_{16} & 1 & . \\ χ_{38} & . & 1 \\ χ_{41} & 1 & 1 \end{aligned}

Further, the simple kG̃-modules in Ã affording φ_{1} and φ_{2} are trivial-source kG̃-modules. (iii) Set H̃ = NG̃(D). Then H̃ = 3_{2}.U_{4}(3).2′_{3}. (iv) Let B̃ be the block algebra of kH̃ that is the Brauer correspondent of Ã. Let further 𝔣̃ be

the Green correspondence with respect to (G̃ × G̃, ΔD, G̃ × H̃). Then, 𝔣̃ induces a Puig equivalence between Ã and B̃.

Proof. (i)-(ii) follow from calculations by GAP [9], using the ordinary and Brauer character tables of Suz, see see [7, p.128ff.] and [13, Suz (mod 3)].

(iii) follows from [7, p.131], as was already remarked earlier in the proof of 4.4.

(iv) follows from (i)-(iii) and [20, Theorem 1.2]. □

Lemma 4.6. Let M_{Q} be the unique (up to isomorphism) indecomposable direct summand of A_{Q}↓_{G_{Q}×H_{Q}}^{G_{Q}×G_{Q}}·1_{B_{Q}} with vertex ΔP. Then, the pair (M_{Q}, M_{Q}^{∨}) induces a Puig equivalence between A_{Q} and B_{Q}.

Proof. Note first that M_{Q} exists by [23, 2.4.Lemma], and also that P is a defect group of A_{Q} and B_{Q} by [35, 7.6]. Now we follow the strategy already employed in [23, Proof of 6.2.Lemma]: Using 4.4(i) and (ii), as well as 4.5(iv), the assertion follows by going over to the central quotients G_{Q}/Q and H_{Q}/Q and their blocks Ã and B̃ dominating A_{Q} and B_{Q}, respectively, [19, Theorem]. □

Lemma 4.7. Let M be the unique (up to isomorphism) indecomposable direct summand of the (A, B)-bimodule A↓_{G×H}^{G×G}·1_{B} with vertex ΔP. Then the pair (M, M^{∨}) induces a splendid stable equivalence of Morita type between the block algebras A and B.

Proof. Note first that M exists, again, by [23, 2.4.Lemma]. Then, by [24, Theorem], we have M_{Q} ≅ e_{Q} ·M(ΔQ)·f_{Q} as (A_{Q}, B_{Q})-bimodules. Now by 4.6, and the fact from 4.4(iv) that G_{R} = H_{R}, the gluing theorem [35, 3.1.Theorem] implies the assertion; note that the fusion condition in the latter theorem is automatically satisfied by [22, 1.15.Lemma]. □

Lemma 4.8. The block algebras A and A′ are are splendidly stably equivalent of Morita type.

Proof. This follows immediately by 4.7, 4.3(iii) and 3.5. □

5. Image of the stable equivalence

Notation 5.1. Recall the notation in 2.2, 3.1, 3.4, and 4.2. We denote by f := f_{(G,P,H)} the Green correspondence with respect to (G, P, H). Moreover, let IBr(A) := {S_{1}, …, S_{5}} and α be as in 2.7. Finally, let M and M′ be the bimodules from 4.7 and 3.5, respectively.

Strategy 5.2. By 4.3(iii) there is a (B, B′)-bimodule 𝔐 inducing a Puig equivalence between B and B′. But there is a little bit of freedom in chosing 𝔐, which we are going to exploit. Anyway, for any admissible choice of 𝔐, a splendid stable equivalence between A and A′, as in 4.8, is afforded by the (A, A′)-bimodule M ⊗_{B} 𝔐 ⊗_{B′} M′^{∨}. This yields the functor

F : mod-A → mod-A′ : X ↦ X ⊗_{A} M ⊗_{B} 𝔐 ⊗_{B′} M′^{∨},

which induces an equivalence \underline{mod}-A → \underline{mod}-A′ between the respective stable module categories. Recall that hence, by [33, Theorem 2.1(ii)], the functor F maps each simple kG-module in A to an indecomposable kG′-module in A′.

Our practical aim in this section now is to prove that 𝔐 can be chosen such that F actually transfers each simple kG-module in A to a simple kG′-module. Our choice of 𝔐 is specified below in 5.6. If this has been achieved, then Linckelmann’s Theorem [33, Theorem 2.1(iii)] yields that F realizes a Morita equivalence between A and A′, and thus even a Puig equivalence between these block algebras. This will then also decide the missing entries in the 3-decomposition matrix of A, see 2.7.

Lemma 5.3. The following characters are afforded by direct sums of of trivial-source A-modules:

(i) χ_{29} (ii) χ_{29} + χ_{38} (iii) χ_{38} + χ_{62} (iv) χ_{29} + χ_{38} + χ_{55}

(v) χ_{29} + χ_{62} + χ_{89} Proof. Using GAP [9], and the character tables of G and its maximal subgroups, inducing suitable linear characters we determine various permutation characters of G, and their components belonging to the block A. Doing so we find

(1_{2_{+}^{1+8}.O_{8}^{+}(2)}) ↑^{G} ·1_{A} = χ_{29} and (1_{U_{6}(2).𝔖_{3}}) ↑^{G} ·1_{A} = χ_{29} + χ_{38},

where 1 denotes the respective trivial character, verifying (i) and (ii). Moreover, (iii) follows from

(λ_{2^{4+12}.(𝔖_{3}×3𝔖_{6})}) ↑^{G} ·1_{A} = χ_{38} + χ_{62},

where λ is the inflation to 2^{4+12}.(𝔖_{3} × 3𝔖_{6}) of the unique linear character of 𝔖_{3} × 3𝔖_{6} having kernel 3 × 3𝔖_{6}. Finally, for (iv) and (v) we observe

(1_{2^{2+12}:(𝔄_{8}×𝔖_{3})}^{−}) ↑^{G} ·1_{A} = χ_{29} + χ_{38} + χ_{55} and (1_{2^{2+12}:(𝔄_{8}×𝔖_{3})}) ↑^{G} ·1_{A} = χ_{29} + χ_{62} + χ_{89},

where 1^{−} denotes the unique non-trivial linear character of 2^{2+12} : (𝔄_{8} × 𝔖_{3}). □

Lemma 5.4. The characters in A have the following values at elements of order 3:

3A  3B
χ_{29} 18954 729
χ_{38} −18954 1458
χ_{51} 189540  729
χ_{55} 208494 −729
χ_{62} 18954 729
χ_{80} 208494 −729
χ_{85} −189540 1458
χ_{89} 189540  729
χ_{91} −208494 −1458

Proof. This follows from [7, pp.184–186]. □

Lemma 5.5. The following holds:

(i) S_{1} is a trivial-source kG-module with vertex P and associated character χ_{29}.

(ii) Using the notation in 4.3(ii) we have f(S_{1}) ∈ {729a, 729b, 729c, 729d}.

Proof. (i) From 5.3(i), we know that S_{1} is a trivial-source module with associated character χ_{29}, and, by Knörr’s result [16, 3.7.Corollary], S_{1} has P as its vertex.

(ii) Setting T_{1} := f(S_{1}), by (i) and [44, Lemma 2.2] we have T_{1} ∈ IBr(B). Set dim_{k}(T_{1}) = 729·d, where d ∈ {1, 2}. By the definition of Green correspondence, we have T_{1}↑^{G} = S_{1} ⊕ X, for a C_{3}- projective kG-module X. Then we have 3^{9−1}|dim_{k}(X) by [42, Theorem 4.7.5], implying that dim_{k}(T_{1}↑^{G}) ≡ dim_{k}(S_{1}) (mod 3^{8}). Now by (i) we have dim_{k}(S_{1}) = χ_{29}(1) = 2816586 ≡ 2187 (mod 3^{8}), and dim_{k}(T_{1}↑^{G}) = |G : H|·dim_{k}(T_{1}) = 17681664000 · 729 · d ≡ 2187 · d (mod 3^{8}), implying that d = 1. □

Lemma 5.6. The following holds:

(i) There is an (B, B′)-bimodule 𝔐 inducing a Puig equivalence between B and B′, such that S_{1} ⊗_{A} M ⊗_{B} 𝔐 = 1a.

(ii) Choosing 𝔐 like this we have F (S_{1}) = 1a.

Proof. (i) By 4.3(iii) there is a (B, B′)-bimodule 𝔐 inducing a Puig equivalence between B and B′. Hence the functor −⊗_{B} 𝔐 induces a bijection from {729a, 729b, 729c, 729d} to {1a, 1b, 1c, 1d}. Thus by 5.5(ii), using [26, Lemma A.3], we have S_{1} ⊗_{A} M ⊗_{B} 𝔐 = f(S_{1}) ⊗_{B} 𝔐 = 1x, for some x ∈ {a, b, c, d}. Now tensoring with 1x induces a Puig auto-equivalence of B′, see [23, Lemma 2.8]. Thus by replacing 𝔐 by 𝔐 ⊗ 1x, the assertion follows from observing that 1x ⊗ 1x = 1a as kH′-modules.

(ii) follows from F (S_{1}) = 1a ⊗_{B′} M′^{∨} = 1a, using [26, Lemma A.3], and the fact that Green correspondence maps the trivial module to the trivial module. □

Notation 5.7. ¿From now on let 𝔐 be chosen as in 5.6, and let F be the associated functor as described in 5.2.

Lemma 5.8. The following holds:

(i) There is a trivial-source kG-module T in A with associated character χ_{29} + χ_{38} and vertex R, so that T is uniserial with Loewy and socle series

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

(ii) We have

F (T) = \begin{matrix}1a \\ 4b \\ 1a \end{matrix} ⊕ (proj) and F (S_{2}) = 4b.

Proof. (i) ¿From 5.3(ii) we know that there is a self-dual kG-module T in A which is a direct sum of trivial-source modules and has character χ_{29} +χ_{38}. Thus, by 2.7, T = 2·S_{1} +S_{2} as composition factors, and hence from [T, T]^{G} = 2 we conclude that T is indecomposable, and thus of shape as asserted. Finally, R is a vertex of T by making use of 5.4 and [31, II Lemma 12.6(ii)].

(ii) be the projective-free part of F (T). Then, by 5.6(ii), [26, Lemma A.3], and 3.3(iii) we conclude that X is of the shape asserted. Now the splitting-off method, see [26, Lemma A.1], yields F (S_{2}) = 4b. □

Lemma 5.9. The following holds:

(i) There is a trivial-source kG-module U in A with associated character χ_{38} + χ_{55} and vertex P, so that U has Loewy and socle series

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

(ii) We have

F (U) = \begin{matrix}4b \\ 1a 1b \\ 4b \end{matrix} ⊕ (proj) and F (S_{3}) = 1b.

(iii) S_{3} is a trivial-source kG-module in A with associated character χ_{51} and vertex P.

Proof. (i) By 5.3(iv), there is a self-dual kG-module X being the direct sum of trivial-source kG-modules and having character χ_{29} + χ_{38} + χ_{55}. Hence, by 2.7, we have X = 2 · S_{1} + 2 · S_{2} + S_{3} as composition factors.

Assume first that X is indecomposable. Then from self-duality and [U, U]^{G} = 3 we conclude that U has Loewy and socle series

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

But from ⟨χ_{29} + χ_{38}, χ_{29} + χ_{38} + χ_{55}⟩_{G} = 2 and the structure of T in 5.8(i) we conclude that there is an embedding of T into X, and hence X has T ⊕ S_{2} as submodule, implying that S_{3} is an epimorphic image, and thus a direct summand of X, a contradiction.

Hence X is decomposable. Assume now that there is a direct summand with character χ_{38}, or a direct summand with character χ_{55}. Then in either case it follows from self-duality that X has a direct summand isomorphic to S_{2}. But by 2.7 the module S_{2} is not liftable, hence is not a trivial-source module, a contradiction.

Thus we conclude that there is a trivial-source kG-module U with character χ_{38} + χ_{55}, and again by self-duality U has shape asserted. Moreover, by 5.4 and [31, II, Lemma 12.6(ii)], U has P as its vertex.

(ii) Let X be the projective-free part of F (U). Then, by 5.8(ii), [26, A.3 Lemma], [26, A.1 Lemma], and 3.3(i), we get that X is a trivial-source kG′-module in A′ with vertex P such that [4b, X]^{G′} ≠ 0. Hence, 3.3(i) yields the shape of X as asserted. This, by 5.6(ii) and using stripping-off again, implies the statement on F (S_{3}).

(iii) follows from (ii), 3.3(i), [26, A.3 Lemma], and 2.7; the statement on the vertex also follows from Knörr’s result [16, 3.7.Corollary]. □

Lemma 5.10. The following holds:

(i) There is a trivial source kG-module V in A with associated character χ_{38} + χ_{62} and vertex R, so that V has Loewy and socle series

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

(ii) We have

F (V) = \begin{matrix}4b \\ 1a 6 \\ 4b \end{matrix} ⊕ (proj) and F (S_{4}) = 6.

Proof. (i) By 5.3(iii), there is a self-dual kG-module V which is the direct sum of trivial-source kG-modules and has character χ_{38}+χ_{62}. Hence, by 2.7, we have V = S_{1}+2·S_{2}+S_{4} as composition factors. Thus from [V, V]^{G} = 2 we conclude that V is indecomposable, and of shape as asserted. Finally, R is a vertex of T by making use of 5.4 and [31, II Lemma 12.6(ii)].

(ii) Let X be the projective-free part of F (V). Then, by 5.8(ii), [26, A.3 Lemma], [26, A.1 Lemma], and 3.3(iii), we get that X is a trivial-source kG′-module in A′ with vertex R such that [4b, X]^{G′} ≠ 0. Hence, 3.3(iii) yields the shape of X as asserted. This, by 5.6(ii) and using stripping-off again, implies the statement on F (S_{4}). □

Lemma 5.11. We have Ext_{kG}^{1}(S_{4}, S_{3}) = 0 = Ext_{kG}^{1}(S_{3}, S_{4}).

Proof. 5.10(ii),5.9(ii), and 3.3(iv) that Ext_{A}^{1}(S_{4}, S_{3}) ≅ Ext_{A′}^{1}(F (S_{4}), F (S_{3})) ≅ Ext_{A′}^{1}(6, 1b) = 0. The second statement follows similarly. □

Lemma 5.12. The following holds:

(i) There is a trivial source kG-module W in A with associated character χ_{62} +χ_{89} and vertex P, such that W has Loewy and socle series

W = \begin{matrix}S_{4} \\ S_{2} S_{5} \\ S_{4} \end{matrix} .

(ii) We have α = 0.

(iii) We have

F (W) = \begin{matrix}6 \\ 4a 4b \\ 6 \end{matrix} ⊕ (proj) and F (S_{5}) = 4a.

Proof. (i)–(ii) By 5.3(v), there is a self-dual kG-module X being the direct sum of trivial-source kG-modules and having character χ_{29} + χ_{62} + χ_{89}. Hence, by 2.7, we have X = S_{1} + S_{2} + α · S_{3} + 2 · S_{4} + S_{5} as composition factors. Now, from ⟨χ_{29}, χ_{29} + χ_{62} + χ_{89}⟩_{G} = 1, using 5.5(i), we infer that [S_{1}, X]^{G} = 1 = [X, S_{1}]^{G}, thus S_{1} is a direct summand of X, so that there is a trivial-source module W with character χ_{62} + χ_{89}, and thus W = S_{2} + α · S_{3} + 2 · S_{4} + S_{5} as composition factors.

Next, from ⟨χ_{51}, χ_{62} + χ_{89}⟩_{G} = 0 and 5.9(iii) we conclude that [S_{3}, W]^{G} = 0 = [W, S_{3}]^{G}, that is S_{3} does not occur neither in the socle nor the head of W. Moreover, since both S_{2} and S_{5} are not liftable, by 2.7, neither of them is a trivial-source module, hence we infer that [S_{2}, W]^{G} = 0 = [W, S_{2}]^{G} and [S_{5}, W]^{G} = 0 = [W, S_{5}]^{G}. Thus we conclude that W/rad(W) ≅ soc(W) ≅ S_{4}, in particular W is indecomposable. By making use of 5.4 and [31, II Lemma 12.6(ii)] we get that P is a vertex of W.

Thus for the heart Y := rad(W)/soc(W) of W we have Y = S_{2} + S_{5} + α · S_{3} as composition factors. Now assume that α ≠ 0. Then, by self-duality, we infer that [S_{3}, Y]^{G} = 1 = [Y, S_{3}]^{G}. Hence, since W/rad(W) ≅ soc(W) ≅ S_{4}, we infer that there are uniserial modules of shape

\begin{matrix}S_{3} \\ S_{4} \end{matrix} and \begin{matrix}S_{4} \\ S_{3} \end{matrix}

contradicting 5.11. Hence we infer α = 0, and W has shape as asserted.

(iii) Let X be the projective-free part of F (W). Then, by 5.10(ii), [26, A.3 Lemma], [26, A.1 Lemma], and 3.3(i), we get that X is a trivial-source kG′-module in A′ with vertex P such that [6, X]^{G′} ≠ 0. Hence, 3.3(i) yields the shape of X as asserted. This, by 5.8(ii) and using stripping-off again, implies the statement on F (S_{5}). □

Proof of Theorem 1.5. First, A and A′ are splendidly stably equivalent of Morita type via the functor F by 4.8 and 5.2. Moreover, by the choice in 5.7, as well as 5.6(ii), 5.8(ii), 5.9(ii), 5.10(ii), and 5.12(iii), all simple A-modules are sent to simple A′-modules via the functor F. Hence, [33, Theorem 2.1(iii)] yields that A and A′ are Morita equivalent, and hence are Puig equivalent. □

Proof of Theorem 1.3. Just as in [23, (6.13)], we conclude that A and B are splendidly derived equivalent, by using the splendid derived equivalence between A′ and B′ given in [45, Example 4.4] and [46, Corollary 2]. □

Proof of Corollary 1.4. Using the character table of G, GAP [9] shows that G has five 3-blocks, of defect 9, 3, 2, 1, and 0. While the former two, according to [43], have non-abelian defect groups, the latter two have cyclic and trivial defect groups, respectively, for which both Conjectures 1.1 and 1.2 are well-known to hold. Hence in this respect the only block of interest is the one of defect 2 under consideration. □

Proof of 1.6. This follows from 1.5 and [47, Theorem 2.3], see [45, Remark 3.7]. □

Acknowledgements

This work was done while the first author was staying in RWTH Aachen University in 2011 and 2012. He is grateful to Gerhard Hiss for his 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)23540007, 2011–2014.

The second author is grateful for financial support in the framework of the DFG (German Science Foundation) Scientific Priority Programmes SPP-1388 ‘Representation Theory’ and SPP-1489 ‘Computer Algebra’, to which this research is a contribution.

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