THE DERIVED CATEGORIES OF SOME BLOCKS OF SYMMETRIC GROUPS AND A CONJECTURE OF BROUÉ

JOSEPH CHUANG

A well-known conjecture of Broué in the representation theory of finite groups involves equivalences of derived categories of blocks. The aim of this paper is to verify this conjecture for defect two blocks of symmetric groups. Actually we prove for these blocks a refinement of Broué’s conjecture due to Rickard.

Thanks are due to the National Science Foundation for supporting my summer 1997 visit to Imperial College, London, where much of this work was done. I am very grateful to Gordon James for his hospitality and for useful suggestions. I would also like to thank Jeremy Rickard, Raphaël Rouquier, Kai Meng Tan, and my advisor Jonathan Alperin for all their assistance and encouragement.

1. Preliminaries

Let p be a prime number and let O be a complete discrete valuation ring with residue field k of characteristic p and fraction field K of characteristic zero, ‘big enough’ for all the groups considered. We will use R to denote a ring which is either O or k, and if V is an O-module, we will usually write V̄ in place of k ⊗_{O} V.

Let G be a finite group. By a block B of G we mean an indecomposable algebra summand of the group algebra OG. Denote by kB the corresponding summand k ⊗_{O} B of kG, and by KB the corresponding summand K ⊗_{O} B of KG. The principal block of G will be denoted by B_{0}(G).

Throughout this paper, all modules are finitely-generated left modules, unless otherwise stated. Also if B and B′ are blocks of finite groups, we will assume that for any RB′-RB-bimodule M the two R actions coincide. Denote by M^{∗} the R-dual Hom_{R}(M, R), which carries a natural RB-RB′-bimodule structure.

Let K^{b}(B) be the category whose objects are bounded complexes of B-modules and whose morphisms are chain maps modulo homotopy equivalence. The (bounded) derived category of B is the triangulated category D^{b}(B) obtained from K^{b}(B) by inverting quasi-isomorphisms. Two blocks B and B′ are said to be derived equivalent if their derived categories are equivalent as triangulated categories. We note that if two blocks are Morita equivalent then they are certainly derived equivalent.

Derived equivalences have recently begun to play an important role in modular representation theory. For instance:

Conjecture 1.1 (Broué [4]). Let B be a block of a finite group G with abelian defect group D. Then B and its Brauer correspondent in N_{G}(D) are derived equivalent.

This conjecture has been verified in some special cases including the following: blocks of p-solvable groups, blocks with cyclic defect [21, 9, 25, 26], certain blocks of finite reductive groups in non-describing characteristic [17], and in isolated cases such as the principal 2-block of A_{5} [24], the principal 2-block of SL_{2}(8) [25], and the principal 3-blocks of A_{6}, A_{7}, A_{8}, S_{6}, P SL_{3}(4), M_{11}, M_{22}, M_{23}, and HS [16].

Date: July 22, 1998.

In [22] Rickard gives necessary and sufficient conditions for two rings to be derived equivalent. Considering the case of derived equivalent algebras in more detail in [23], he proves that a pair of blocks B and B′ are derived equivalent if and only if there exists a bounded complex X of B′-B-bimodules such that

• each term of X is projective as a left B′-module and as a right B-module,

• X^{∗} ⊗_{B′} X is a split complex of B-B-bimodules with homology concentrated in degree 0, isomorphic to B,

• X ⊗_{B} X^{∗} is a split complex of B′-B′-bimodules with homology concentrated

in degree 0, isomorphic to B′. Such a complex is called a Rickard tilting complex.

For background on the representation theory of symmetric groups we refer the reader to James and Kerber [7]. Denote by B^{τ,w} the block of the symmetric groups corresponding to a p-core τ and weight w. B^{τ,w} is a block of OS_{t+wp} and the irreducible characters and simple modules lying in it are {χ^{λ} | λ is a partition of t + wp with p-core τ} and {D^{λ} | λ is a p-regular partition of t + wp with p-core τ\}. In addition let ϕ^{λ} be the Brauer character of D^{λ} and let \overline{χ^{λ}} be the restriction of χ^{λ} to p-regular elements. B^{τ,w} is a block of defect 0 if and only if w = 0, and is a block of cyclic defect if and only if w = 1, in which case its Brauer tree is an open polygon with exceptional multiplicity one.

The blocks of the symmetric groups to which Broué’s conjecture applies, namely those with abelian defect groups, are the blocks B^{τ,w} with w < p. In this case, the defect group D is an elementary abelian p-group of rank w, the normalizer N(D) of D in S_{t+wp} is isomorphic to ((C_{p} ⋊ C_{p−1}) ≀ S_{w}) × S_{t}, and the Brauer correspondent of B^{τ,w} in N(D) is B_{0}((C_{p} ⋊ C_{p−1}) ≀ S_{w}) ⊗ B^{τ,0}. This block is Morita equivalent to the principal block of (C_{p} ⋊C_{p−1})≀S_{w} because B^{τ,0} is a block of defect 0. Moreover, Marcus [13, Example 5.7] has shown that the principal block of (C_{p} ⋊ C_{p−1}) ≀ S_{w} is derived equivalent to the principal block of S_{p} ≀ S_{w}. Thus the Brauer correspondent of B^{τ,w} is derived equivalent to the principal block of S_{p} ≀ S_{w}. Therefore to prove Broué’s conjecture for B^{τ,w} it suffices to show that B^{τ,w} is derived equivalent to the principal block of S_{p} ≀ S_{w}.

In this paper we will restrict our attention to defect two blocks of symmetric groups. Thus we will be considering the blocks B^{τ,2} and assuming that p > 2. We will begin in section 2 by giving a proof that all of the blocks B^{τ,2} are derived equivalent to each other–this is an unpublished result due to Jeremy Rickard. Then it suffices to show that one such block is derived equivalent to the principal block of S_{p} ≀ S_{2}. Fortunately there is a defect two block of the symmetric groups which is in fact Morita equivalent to the principal block of S_{p} ≀ S_{2}. This equivalence, which was originally conjectured by Rouquier, is stated as Theorem 3.1 and the main line of the argument proving it is given in the remainder of section 3 while the required calculations are contained in sections 4 through 7. Lastly in section 8 we show that the equivalences obtained are splendid, in accordance with Rickard’s refinement of Broué’s conjecture.

2. A proof of a result of Rickard

In this chapter we will give a proof of an unpublished result of Rickard which states that all defect two blocks of symmetric groups are derived equivalent.

In [28, §2] Scopes notes that for any p-core τ there exists a sequence of pcores τ = τ_{1}, τ_{2}, …, τ_{s} = ∅ (where ∅ denotes the empty p-core), such that for each i = 1, 2, …, s − 1 one of the following possibilities occurs:

• B^{τ_{i},2} and B^{τ_{i+1},2} form what she calls a [2:k] pair, where k ≥ 2, and are Morita equivalent by her work in [27].

• B^{τ_{i},2} and B^{τ_{i+1},2} form a [2:1] pair, that is to say τ_{i+1} can be obtained from

τ_{i} by removing a single node. Therefore to prove that all defect two blocks are derived equivalent we need only show that blocks forming a [2:1] pair are derived equivalent.

Throughout this chapter, suppose that τ and τ̃ are partitions of t and t − 1 and that B = B^{τ,2} and B̃ = B^{τ̃,2} form a [2:1] pair. We will construct a complex X of B̃-B-bimodules and initially show that it induces an equivalence at the level of characters. Then a general result of Broué will allow us to deduce that X is a Rickard tilting complex.

The following lemma is taken from Lemma 3.3, Corollary 3.7, and Remark 4.4 of [28].

Lemma 2.1 (Scopes). There exists a bijection Φ from the set of partitions of t+2p with p-core τ to the set of partitions of t − 1 + 2p with p-core τ̃, a bijection Ψ from the set of p-regular partitions of t+2p with p-core τ to the set of p-regular partitions of t − 1 + 2p with p-core τ̃, and partitions α, β, γ of t + 2p with p-core τ, with α p-regular, such that the following hold:

a)

Res_{KB̃}^{KB}(χ^{α}) = χ^{Φ(α)} + χ^{Φ(β)},
Res_{KB̃}^{KB}(χ^{β}) = χ^{Φ(α)} + χ^{Φ(γ)},
Res_{KB̃}^{KB}(χ^{γ}) = χ^{Φ(β)} + χ^{Φ(γ)};

and for all λ other than α, β, and γ,

Res_{KB̃}^{KB}(χ^{λ}) = χ^{Φ(λ)}.

b) For all partitions λ of t + 2p with p-core τ, the head of Res_{kB̃}^{kB}(D^{λ}) is isomorphic to D^{Ψ(λ)}. In fact, if λ ≠ α then we have

Res_{kB̃}^{kB}(D^{λ}) ≅ D^{Ψ(λ)}.

c) Let P (D^{α}) be the projective cover of D^{α} as a B-module and let P (D^{Ψ(α)}) be the projective cover of D^{Ψ(α)} as a B̃-module. Then

ch(K ⊗_{O} P (D^{α})) = χ^{α} + χ^{β} + χ^{γ},

and

ch(K ⊗_{O} P (D^{Ψ(α)})) = χ^{Φ(α)} + χ^{Φ(β)} + χ^{Φ(γ)}.

Let M be a B̃-B-bimodule which induces restriction from B to B̃. M is projective both as a right B-module and as a left B̃-module, and the kB̃-kB-bimodule M̄ induces restriction from kB to kB̃. Let δ′ : P′ ⟶ M be a projective cover of M. [25, Lemma 2] together with (2.1b) implies that

P′ ≅ ⨁_{λ} P (D^{Ψ(λ)}) ⊗_{O} P (D^{λ})^{∗},

where the sum runs over all p-regular partitions of t+2p with p-core τ, and P (D^{Ψ(λ)}) and P (D^{λ}) denote the projective covers of D^{Ψ(λ)} and D^{λ} as B̃- and B-modules respectively.

Take P to be a direct summand of P′ isomorphic to P (D^{Ψ(α)}) ⊗_{O} P (D^{α})^{∗}, and let δ be the restriction of δ′ to P. Put

X = (0 → P ⟶^{δ} M → 0),

a complex of B̃-B-bimodules with M in degree 0.

It is easy to check that X induces an equivalence at the level of characters. Indeed by (2.1a),

ch(K ⊗_{O} M) = ( ∑_{λ≠α,β,γ} χ^{Φ(λ)} ⊗ χ^{λ}^{∗}) + (χ^{Φ(α)} + χ^{Φ(β)}) ⊗ χ^{α}^{∗}
+(χ^{Φ(α)} + χ^{Φ(γ)}) ⊗ χ^{β}^{∗} + (χ^{Φ(β)} + χ^{Φ(γ)})⊗χ^{γ}^{∗}.

and by (2.1c) we have

ch(K ⊗_{O} P) = (χ^{Φ(α)} + χ^{Φ(β)} + χ^{Φ(γ)}) ⊗ (χ^{α} + χ^{β} + χ^{γ})^{∗},

thus the (Lefschetz) character of K ⊗_{O} X is

( ∑_{λ≠α,β,γ} χ^{Φ(λ)} ⊗ χ^{λ}^{∗}) − χ^{Φ(γ)} ⊗ χ^{α}^{∗} − χ^{Φ(β)} ⊗ χ^{β}^{∗} − χ^{Φ(α)} ⊗ χ^{γ}^{∗}.

Next we show that X^{∗}⊗B̃X is split as a complex of B-B-bimodules and X⊗_{B} X^{∗} is split as a complex of B̃-B̃-bimodules, and that both of these complexes have homology concentrated in degree 0. The following argument is modeled on part of the proof of [25, Theorem 6].

Consider P̄^{∗}⊗_{kB̃}M̄. It is a projective kB-kB-bimodule because P̄^{∗} is a projective kB-kB̃-bimodule and M̄ is a projective right kB-module. Also, using (2.1b) we have that for all λ ≠ α,

P̄^{∗} ⊗_{kB̃} M̄ ⊗_{kB} D^{λ} ≅ P̄^{∗} ⊗_{kB̃} D^{Ψ(λ)}
≅ Hom_{kB̃}(P̄, D^{Ψ(λ)}) = 0,

so as a right kB-module, P̄^{∗} ⊗_{kB̃} M̄ is a direct sum of copies of \overline{P (D^{α})^{∗}}. On the other hand the quotient module

(P̄^{∗} ⊗_{kB̃} P̄′)/(P̄^{∗} ⊗_{kB̃} P̄)

as a right kB-module has no summand isomorphic to \overline{P (D^{α})^{∗}} because the same holds for P̄′/P̄.

Evidently, as right kB-modules

P̄^{∗} ⊗_{kB̃} M̄ and (P̄^{∗} ⊗_{kB̃} P̄′)/(P̄^{∗} ⊗_{kB̃} P̄)

are projective and do not have any nonzero isomorphic direct summands. Thus we may apply [25, Lemma 1] to the surjective map

id ⊗δ̄′ : P̄^{∗} ⊗_{kB̃} P̄′ → P̄^{∗} ⊗_{kB̃} M̄

and conclude that the map

id ⊗δ̄: P̄^{∗} ⊗_{kB̃} P̄ → P̄^{∗} ⊗_{kB̃} M̄

is also surjective. Now

X^{∗} ⊗B̃ X =

(0 → M^{∗} ⊗B̃ P ⟶^{(δ^{∗}⊗id, id ⊗δ)} P^{∗} ⊗B̃ P ⊕ M^{∗} ⊗B̃ M

⟶^{id ⊗δ−δ^{∗}⊗id} P^{∗} ⊗B̃ M → 0).

As id ⊗δ̄ is surjective, so is id ⊗δ̄ − δ̄^{∗} ⊗ id. Therefore by Nakayama’s lemma id ⊗δ − δ^{∗} ⊗ id is surjective and hence splits because P^{∗} ⊗B̃ M is projective. By duality (δ^{∗} ⊗ id, id ⊗δ) is injective and splits. We have shown that X^{∗} ⊗B̃ X is a split complex, with homology concentrated in degree 0. The proof for X ⊗_{B} X^{∗} is similar. Together with the preceding character calculation, this implies that X is a Rickard tilting complex, by Theorem 2.3 of [3]. So B and B̃ are derived equivalent.

3. The Main Theorem

Let

ρ = (\frac{(p − 1)p}{2}, (\frac{(p − 2)(p − 1)}{2})^{2}, (\frac{(p − 3)(p − 2)}{2})^{3}, …, 6^{p−3}, 3^{p−2}, 1^{p−1}),

a partition of

r = \frac{(p − 1)p}{2} + 2 · (\frac{(p − 2)(p − 1)}{2}) + … + (p − 2) · 3 + (p − 1) · 1.

It is easy to verify that ρ is a p-core, so B^{ρ,2} is a block of S_{r+2p} of defect two. We can now state our main result:

Theorem 3.1. B^{ρ,2} is Morita equivalent to the principal block of S_{p} ≀ S_{2}.

By the result obtained in the previous section and the discussion at the end of section 1, we have

Corollary 3.2. Broué’s conjecture (1.1) holds for all defect two blocks of symmetric groups.

Raphaël Rouquier conjectured (3.1) based on his observation of examples of matching decomposition matrices (see (3.5)). I am indebted to him for communicating to me his belief that it should be true. Note that Okuyama [16] has verified Broué’s conjecture for the principal 3-block of S_{6} using a different method.

In proving the main theorem we will make extensive use of B_{0}(S_{2p}), the principal block of S_{2p}. We now describe some functors between the module categories of B^{ρ,2}, B_{0}(S_{2p}), and B_{0}(S_{p} ≀ S_{2}).

From now on identify S_{p} ≀ S_{2} with the subgroup of S_{2p} which stabilizes the partition of {1, 2, …, 2p} into {1, 2, …, p} and {p+1, p+2, …, 2p}. Let D be the subgroup of S_{p} ≀ S_{2} generated by the p-cycles (1 2 … p) and (p + 1 p + 2 … 2p). Note that D is a defect group of each of the blocks B^{ρ,2}, B_{0}(S_{2p}), and B_{0}(S_{p} ≀ S_{2}).

The functors

↑: RB_{0}(S_{p} ≀ S_{2}) -mod ⟶ RB_{0}(S_{2p}) -mod

and

↓: RB_{0}(S_{2p}) -mod ⟶ RB_{0}(S_{p} ≀ S_{2}) -mod

are defined to be the induction Ind_{RB_{0}(S_{p}≀S_{2})}^{RB_{0}(S_{2p})} and restriction Res_{RB_{0}(S_{p}≀S_{2})}^{RB_{0}(S_{2p})}, respectively We also define a functor

⇓: RB^{ρ,2} -mod ⟶ RB_{0}(S_{2p}) -mod

which we call ‘restriction’, as follows.

View S_{2p} × S_{r} as a subgroup of S_{r+2p} containing S_{2p}, in the obvious way. RB_{0}(S_{2p}) ⊗_{R} RB^{ρ,0} is a block of R[S_{2p} × S_{r}] and is Morita equivalent to RB_{0}(S_{2p}) because RB^{ρ,0} is a block of defect 0. The equivalence is induced by the functor

Hom_{RB^{ρ,0}}(P_{R}, −) : RB_{0}(S_{2p}) ⊗_{R} RB^{ρ,0} -mod ⟶ RB_{0}(S_{2p}) -mod,

where P_{R} is a projective indecomposable RB^{ρ,0}-module (so \overline{P_{O}} ≅ P_{k} ≅ D^{ρ}). We define ⇓ to be the composition of the functors Res_{RB_{0}(S_{2p})⊗_{R}RB^{ρ,2}}^{RB^{ρ,2}} and Hom_{RB^{ρ,0}}(P_{R}, −).

Note that the functors ↑, ↓, and ⇓ are exact and send projective modules to projective modules. In addition they commute with reduction modulo p and send self-dual modules to self-dual modules.

The functors ↓ and ↑ give a ‘stable equivalence’ between the blocks kB_{0}(S_{2p}) and kB_{0}(S_{p} ≀ S_{2}):

Lemma 3.3. There exists a 1-1 correspondence between non-projective indecomposable kB_{0}(S_{2p})-modules and non-projective indecomposable kB_{0}(S_{p} ≀ S_{2})-modules such that if V and W are corresponding modules, then

V ↓ ≅ W ⊕ (projective kB_{0}(S_{p} ≀ S_{2})−module).

and

W ↑ ≅ V ⊕ (projective kB_{0}(S_{2p})−module).

Proof. If s is an element of S_{2p} not in S_{p}≀S_{2} then D ∩ sDs^{−1} = 1 and (S_{p} ≀ S_{2}) ∩ sDs^{−1} = 1, so by Green Correspondence (see [2]) the statement is true if kB_{0}(S_{2p}) and kB_{0}(S_{p}≀S_{2}) are replaced by kS_{2p} and k[S_{p}≀S_{2}]. Brauer’s Second Main Theorem states that corresponding modules lie in corresponding blocks, and Brauer’s Third Main Theorem tells us that principal blocks correspond, so the proposition follows.

The following lemma is not needed for the arguments in this section. It will be used in section 7 and is included here while the definition of ⇓ is fresh in mind.

Lemma 3.4. Let U be a kB^{ρ,2}-module with vertex D. Then U ⇓ has at most one indecomposable summand which is not relatively S_{2p−1}-projective.

Proof. S_{2p}×S_{r} contains the normalizer of D in S_{2p+r}, so by Green Correspondence, the restriction of U to S_{2p} × S_{r} is the direct sum of an indecomposable k[S_{2p} × S_{r}]-module with vertex D and an ‘error term’ which is relatively projective for subgroups of the form D ∩ sDs^{−1} where s ∉ S_{2p} × S_{r}.

Each of these subgroups is contained in a conjugate of Q, where Q is the subgroup of S_{2p} × S_{r} generated by the p-cycle (1 2 … p). Also, Q is a Sylow p-subgroup of S_{2p−1}, so a k[S_{2p} ×S_{r}]-module is relatively Q-projective if and only if it is relatively S_{2p−1}-projective. Thus Res_{kB_{0}(S_{2p})⊗_{k}kB^{ρ,0}}^{kB^{ρ,2}}(U) has at most one indecomposable summand which is not relatively S_{2p−1}-projective. The same statement holds for U ⇓ because the functor Hom_{kB^{ρ,0}}(D^{ρ}, −) induces a Morita equivalence between kB_{0}(S_{2p}) ⊗_{k} kB^{ρ,0} and kB_{0}(S_{2p}) and moreover preserves vertices.

In the following sections (4 through 7) we will work toward a proof of the following.

Proposition 3.5. There exist 1-1 correspondences (which are given explicitly at the end of section 4) between irreducible characters of KB^{ρ,2} and irreducible characters of KB_{0}(S_{p} ≀ S_{2}) and between isomorphism classes of simple kB^{ρ,2}-modules and isomorphism classes of simple kB_{0}(S_{p} ≀ S_{2})-modules such that

a) with respect to these one-to-one correspondences, the decomposition matrices of B^{ρ,2} and B_{0}(S_{p} ≀ S_{2}) coincide,

b) if χ and χ′ are corresponding irreducible characters, then χ ⇓ = χ′ ↑,

c) if T and T′ are corresponding simple modules, then T ⇓ ≅ T′ ↑.

(3.5a) will follow from the explicit calculation of the decomposition matrices of B^{ρ,2} and B_{0}(S_{p} ≀ S_{2}) in section 4, while (3.5b) and (3.5c) will be proved in sections 6 and 7 respectively.

We conclude this chapter by proving the main theorem (3.1) under the assumption that (3.5) holds:

By composing ⇓ with ↓ and taking R to be either O or k, we obtain functors

F_{O} : B^{ρ,2} -mod ⟶ B_{0}(S_{p} ≀ S_{2}) -mod

and

F_{k} : kB^{ρ,2} -mod ⟶ kB_{0}(S_{p} ≀ S_{2}) -mod .

(3.3) and (3.5c) together imply that if T is a simple kB^{ρ,2}-module then F_{k}(T) is isomorphic to the direct sum of the corresponding simple kB_{0}(S_{p} ≀ S_{2})-module T′ and a projective kB_{0}(S_{p} ≀ S_{2})-module.

F_{O} and F_{k} are exact and take projective modules to projective modules, and if U is a B^{ρ,2}-module, then \overline{F_{O}(U)} ≅ F_{k}(Ū). Hence there exists a B_{0}(S_{p} ≀ S_{2})- B^{ρ,2}-bimodule M which is projective both as a right B^{ρ,2}-module and as a left B_{0}(S_{p} ≀ S_{2})-module and such that the functor M ⊗_{B^{ρ,2}} − is isomorphic to F_{O} and the functor M̄ ⊗_{kB^{ρ,2}} − is isomorphic to F_{k}. Write M = N ⊕ Q, where Q is a projective B_{0}(S_{p} ≀ S_{2})-B^{ρ,2}-bimodule and N has no projective summand.

By [11, Lemma 2.6], the kB_{0}(S_{p} ≀ S_{2})-kB^{ρ,2}-bimodule N̄ has no projective summand. Therefore, if T is a simple kB^{ρ,2}-module then the kB_{0}(S_{p} ≀ S_{2})-module N̄ ⊗_{kB^{ρ,2}} T has no projective summand, by [11, Proposition 2.3]. Now M̄ = N̄ ⊕Q̄, so we must have N̄ ⊗_{kB^{ρ,2}} T ≅ T′.

In addition, the fact that M̄ is projective both as a right kB^{ρ,2}-module and as a left kB_{0}(S_{p} ≀ S_{2})-module implies that the same holds for N̄. Thus we have an exact functor

N̄ ⊗_{kB^{ρ,2}} − : kB^{ρ,2} -mod ⟶ kB_{0}(S_{p} ≀ S_{2}) -mod

which takes projective modules to projective modules and induces a 1-1 correspondence between the isomorphism classes of simple kB^{ρ,2}-modules and the isomorphism classes of simple kB_{0}(S_{p} ≀ S_{2})-modules. Moreover (3.5a) implies that with respect to this 1-1 correspondence, kB^{ρ,2} and kB_{0}(S_{p} ≀ S_{2}) have the same Cartan invariants, so N̄ ⊗_{kB^{ρ,2}} − takes indecomposable projective modules to indecomposable projective modules. It follows that a right adjoint to N̄ ⊗_{kB^{ρ,2}} − takes a simple kB_{0}(S_{p} ≀ S_{2})-module T′ to the corresponding simple module T. Then using the argument in the proof of [27, Theorem 4.2], we see that N̄ ⊗_{kB^{ρ,2}} − induces an equivalence between kB^{ρ,2} -mod and kB_{0}(S_{p} ≀ S_{2}) -mod. This proves the analogue of (3.1) over the residue field k.

To see that (3.1) indeed holds over O, consider the natural homomorphism

η_{N} : B^{ρ,2} → End_{B_{0}(S_{p}≀S_{2})}(N)

induced by the right action of B^{ρ,2} on N. Since N is projective as a left B_{0}(S_{p} ≀ S_{2})- module, we can identify \overline{End_{B_{0}(S_{p}≀S_{2})}(N)} with End_{kB_{0}(S_{p}≀S_{2})}(N̄), and the natural homomorphism

η_{N̄} : kB^{ρ,2} → End_{kB_{0}(S_{p}≀S_{2})}(N̄)

is obtained from η_{N} by tensoring over O with k. We already know that the former map is an isomorphism, because N̄ induces an equivalence. This implies that η_{N} is surjective, by Nakayama’s lemma. But B^{ρ,2} and End_{B_{0}(S_{p}≀S_{2})}(N) are O-free with the same finite rank, as the isomorphism over the residue field k shows; hence η_{N} is an isomorphism.

An analogous argument shows that the natural homomorphism

η′_{N} : B_{0}(S_{p} ≀ S_{2}) → End_{B^{ρ,2}}(N)

induced by the left action of B_{0}(S_{p} ≀ S_{2}) on N is also an isomorphism. We conclude that B^{ρ,2} and B_{0}(S_{p} ≀ S_{2}) are Morita equivalent.

4. The Blocks B^{ρ,2} and B_{0}(S_{p} ≀ S_{2})

In this section we calculate the decomposition numbers for the blocks B^{ρ,2} and B_{0}(S_{p} ≀ S_{2}) and verify that (3.5a) holds. First we describe the partitions of r + 2p with p-core ρ, introducing nonstandard notation.

Lemma-Definition 4.1. Each partition of r + 2p with p-core ρ appears exactly once in the following list:

• For each 1 ≤ i < j ≤ p, there is a unique partition [i, j] of r + 2p such that

[i, j]/ρ = [figure: j; p − j; i; p − i] ,

a disjoint union of two hook partitions.

• For each 1 ≤ i ≤ p, there is a unique partition [i] of r + 2p such that

[i]/ρ = [figure: i; p − i; i; p − i].

• For each 1 ≤ i ≤ p, there is a unique partition [i, i] of r + 2p such that

[i, i]/ρ = [figure: i; p − i; i; p − i].

Note that [i, j] is p-singular if and only if i = 1, [i] is p-singular if and only if i = 1, and [i, i] is p-singular if and only if i = 1.

Remark. For those familiar with abacus displays for partitions, we mention that this is the 𝔘_{2,3,...,p+1} notation defined by Scopes in [28], except that we have replaced angled brackets by square brackets.

To determine the decomposition numbers of the block B^{ρ,2}, we will make use of results of Richards in [20], which we summarize briefly. A partition is said to have p-weight w if one needs to remove w rim p-hooks from it to reach its p-core. If λ is any partition of p-weight two define ∂λ to be the difference between the leg lengths of the two rim p-hooks removed from λ to obtain its p-core (∂λ is independent of the way in which the p-hooks are removed). Each λ has either two rim p-hooks or one rim 2p-hook and one rim p-hook. If ∂λ = 0 we assign λ a color: if λ has two rim p-hooks, say that λ is black if the longer leg length is even and white if it is odd. If λ has a rim 2p-hook, say it is black if the leg length of this hook is congruent to 0 or 3 modulo 4, white if it is congruent to 1 or 2 modulo 4. Given any partition λ of p-weight two, define λ̃ to be the largest partition of p-weight two (in dominance order) such that λ̃ has the same p-core as λ, ∂λ = ∂λ̃, λ and λ̃ have the same color if ∂λ = 0, and λ̃ ⊲ λ.

Proposition 4.2 ([20, Theorem 4.4]). Define integers d_{μλ} for all pairs (μ, λ) of partitions of p-weight two such that μ and λ have the same p-core and λ is p-regular by

\overline{χ^{μ}} = ∑_{λ} d_{μλ}ϕ^{λ}.

Then d_{μλ} = 1 if μ = λ or μ = λ̃ or both λ̃ ⊲ μ ⊲ λ and ∂λ − ∂μ = ±1. Otherwise d_{μλ} = 0.

By consulting Figure 1, it is not hard to check the following lemma.

Lemma 4.3.

a) (Dominance order for partitions of r + 2p with p-core ρ.) For 1 ≤ i ≤ j ≤ p and 1 ≤ i′ ≤ j′ ≤ p,

\begin{aligned}[i, j] ⊴ [i′, j′] & ⇔ & i ≤ i′ and j ≤ j′, \\ [i, j] ⊴ [j′] & ⇔ & j ≤ j′, \\ [i] ⊴ [i′, j′] & ⇔ & i ≤ i′ and i′ ≤ j′, \\ [i] ⊴ [i′] & ⇔ & i ≤ i′. \end{aligned}

b) For 1 ≤ i < j ≤ p,

∂([i, j]) = j − i, ∂([i]) = 0, and ∂([i, i]) = 0.

Also, [i] is black if i is odd and white if i is even, while [i, i] is white if i is odd and black if i is even.

Using (4.2) and (4.3) we get

Proposition 4.4 (Decomposition numbers for B^{ρ,2}).

In the following equations, read ϕ^{[i,j]} as zero if either i < 1 or j > p, and read ϕ^{[i]} as zero if either i < 1 or i > p.

For 1 ≤ i < j ≤ p and j − i ≥ 2,

\overline{χ^{[i,j]}} = ϕ^{[i,j]} + ϕ^{[i+1,j]} + ϕ^{[i,j+1]} + ϕ^{[i+1,j+1]};

for 2 ≤ i ≤ p,

\overline{χ^{[i−1,i]}} = ϕ^{[i−1,i]} + ϕ^{[i]} + ϕ^{[i,i]} + ϕ^{[i−1,i+1]} + ϕ^{[i,i+1]};

for 1 ≤ i ≤ p,

\overline{χ^{[i]}} = ϕ^{[i]} + ϕ^{[i+1,i+1]} + ϕ^{[i,i+1]};

and for 1 ≤ i ≤ p,

\overline{χ^{[i,i]}} = ϕ^{[i,i]} + ϕ^{[i+1]} + ϕ^{[i,i+1]}.

Next we will consider the block B_{0}(S_{p} ≀ S_{2}). We fix some notation for the representations of the wreath product G ≀ S_{2} where G is any finite group. Let σ be the non-identity element of S_{2} and identify S_{2} with its image in G ≀ S_{2} = (G × G) ⋊ S_{2}. Let F be any field of characteristic not equal to two. Given an F G-module V, the tensor product V ⊗V is a F [G×G]-module, The action of G×G may be extended to G≀S_{2} in two ways, letting σ act by σ(v_{1} ⊗v_{2}) = v_{2} ⊗v_{1} or by σ(v_{1} ⊗v_{2}) = −v_{2} ⊗v_{1}. The resulting F [G≀S_{2}]-modules will be denoted (V ⊗V)^{+} and (V ⊗V)^{−} respectively. Their direct sum is isomorphic to Ind_{G×G}^{G≀S_{2}}(V ⊗ V).

By the general theory of representations of wreath products we have

Lemma-Definition 4.5. a) Let

χ^{i,j} = Ind_{S_{p}×S_{p}}^{S_{p}≀S_{2}}(χ^{(i,1^{p−i})} · χ^{(j,1^{p−j})}),  (1 ≤ i < j ≤ p),
χ^{i^{+}} = (χ^{(i,1^{p−i})} · χ^{(i,1^{p−i})})^{+},  (1 ≤ i ≤ p),
χ^{i^{−}} = (χ^{(i,1^{p−i})} · χ^{(i,1^{p−i})})^{−},  (1 ≤ i ≤ p).

These are all the irreducible characters of KB_{0}(S_{p} ≀ S_{2}).

b) Let

D^{i,j} = Ind_{S_{p}×S_{p}}^{S_{p}≀S_{2}}(D^{(i,1^{p−i})} ⊗ D^{(j,1^{p−j})}),  (2 ≤ i < j ≤ p),
D^{i^{+}} = (D^{(i,1^{p−i})} ⊗ D^{(i,1^{p−i})})^{+},  (2 ≤ i ≤ p),
D^{i^{−}} = (D^{(i,1^{p−i})} ⊗ D^{(i,1^{p−i})})^{−},  (2 ≤ i ≤ p).

These modules are self-dual and are up to isomorphism all the simple kB_{0}(S_{p} ≀ S_{2})-modules. Denote by ϕ^{i,j}, ϕ^{i^{+}}, and ϕ^{i^{−}} the corresponding irreducible Brauer characters.

Decomposition numbers for B_{0}(S_{p} ≀ S_{2}) may be calculated using the decomposition numbers in the principal block of S_{p} and by consideration of characters values at σ.

Proposition 4.6 (Decomposition numbers for B_{0}(S_{p} ≀ S_{2})).

In the following formulas, read ϕ^{i,j} as zero if i < 2 or j > p and read ϕ^{i^{+}} and ϕ^{i^{−}} as zero if i < 2 or i > p.

For 1 ≤ i < j ≤ p and j − i ≥ 2,

\overline{χ^{i,j}} = ϕ^{i,j} + ϕ^{i+1,j} + ϕ^{i,j+1} + ϕ^{i+1,j+1};

for 2 ≤ i ≤ p,

\overline{χ^{i−1,i}} = ϕ^{i−1,i} + ϕ^{i^{+}} + ϕ^{i^{−}} + ϕ^{i−1,i+1} + ϕ^{i,i+1};

for 1 ≤ i ≤ p,

\overline{χ^{i^{+}}} = ϕ^{i^{+}} + ϕ^{(i+1)^{+}} + ϕ^{i,i+1};

and for 1 ≤ i ≤ p,

\overline{χ^{i^{−}}} = ϕ^{i^{−}} + ϕ^{(i+1)^{−}} + ϕ^{i,i+1}.

The following notation will be useful:

χ^{i^{±}} = \{ \begin{matrix}χ^{i^{+}} & if i is odd \\ χ^{i^{−}} & if i is even \end{matrix} , χ^{i^{∓}} = \{ \begin{matrix}χ^{i^{−}} & if i is odd \\ χ^{i^{+}} & if i is even \end{matrix} ,

ϕ^{i^{±}} = \{ \begin{matrix}ϕ^{i^{+}} & if i is odd \\ ϕ^{i^{−}} & if i is even \end{matrix} , ϕ^{i^{∓}} = \{ \begin{matrix}ϕ^{i^{−}} & if i is odd \\ ϕ^{i^{+}} & if i is even \end{matrix} ,

D^{i^{±}} = \{ \begin{matrix}D^{i^{+}} & if i is odd \\ D^{i^{−}} & if i is even \end{matrix} , D^{i^{∓}} = \{ \begin{matrix}D^{i^{−}} & if i is odd \\ D^{i^{+}} & if i is even \end{matrix} .

Comparing (4.4) and (4.6), we find that the decomposition matrices of B^{ρ,2} and B_{0}(S_{p} ≀ S_{2}) coincide with respect to the following 1-1 correspondences between irreducible characters and between simple modules:

\begin{matrix}χ^{[i,j]} ⟷ χ^{i,j}, & (1 ≤ i < j ≤ p) \\ χ^{[i]} ⟷ χ^{i^{±}}, & (1 ≤ i ≤ p) \\ χ^{[i,i]} ⟷ χ^{i^{∓}}, & (1 ≤ i ≤ p). \end{matrix} and \begin{matrix}D^{[i,j]} ⟷ D^{i,j}, & (2 ≤ i < j ≤ p) \\ D^{[i]} ⟷ D^{i^{±}}, & (2 ≤ i ≤ p) \\ D^{[i,i]} ⟷ D^{i^{∓}}, & (2 ≤ i ≤ p). \end{matrix}

5. The Block B_{0}(S_{2p})

We begin by listing the partitions of 2p with empty p-core, introducing nonstandard notation. We include the diagrams for these partitions because it helps to be familiar with them in proving 6.6.

Lemma-Definition 5.1. Every partition of 2p with empty p-core is one of the following:

• ⟨i, p + 1⟩ = (p + i, 1^{p−i}), (1 ≤ i ≤ p).

[figure: i; p; p − i]

• ⟨0, i⟩ = (i, 1^{2p−i}), (1 ≤ i ≤ p).

[figure: i; p − i; p]

• ⟨i, j⟩ = (j, i + 1, 2^{p−j}, 1^{j−i−1}), (1 ≤ i < j ≤ p).

[figure: i; j − i; p − j; j − i − 1]

Note that all ⟨i, p+1⟩ are p-regular, all ⟨0, i⟩ are p-singular, and ⟨i, j⟩ is p-singular if and only if i = 1 and j = 2.

Remark. We find it convenient for our purposes to deviate slightly from the 𝔘_{2,2,...,2} notation used in [15] and [5], replacing ⟨i⟩ by ⟨i, p + 1⟩ and ⟨i, i⟩ by ⟨0, i⟩.

Decomposition numbers for B_{0}(S_{2p}) may be calculated using (4.2). They also appear in [15].

Proposition 5.2 (Decomposition numbers for B_{0}(S_{2p})).

In the following equations read ϕ^{⟨i,j⟩} as zero if i < 1 or j > p or both i = 1 and j = 2.

For 0 ≤ i < j ≤ p + 1, j − i ≥ 3, and (i, j) ≠ (0, p + 1) we have

\overline{χ^{⟨i,j⟩}} = ϕ^{⟨i,j⟩} + ϕ^{⟨i+1,j⟩} + ϕ^{⟨i,j+1⟩} + ϕ^{⟨i+1,j+1⟩};

for 0 ≤ i ≤ p − 1,

\overline{χ^{⟨i,i+2⟩}} = ϕ^{⟨i,i+2⟩} + ϕ^{⟨i+1,i+2⟩} + ϕ^{⟨i,i+3⟩} + ϕ^{⟨i+1,i+3⟩} + ϕ^{⟨i+2,i+3⟩};

and for 0 ≤ i ≤ p,

\overline{χ^{⟨i,i+1⟩}} = ϕ^{⟨i,i+1⟩} + ϕ^{⟨i,i+2⟩} + ϕ^{⟨i+2,i+3⟩}.

Next we review the work of Erdmann and Martin on kB_{0}(S_{2p}), collecting those facts we need. In [14] Martin shows that if D^{μ} and D^{λ} are simple kB_{0}(S_{2p})-modules, then the dimension of Ext_{kB_{0}(S_{2p})}^{1}(D^{μ}, D^{λ}) is either 0 or 1 and is 0 if μ = λ. Note that

dim_{k} Ext_{kB_{0}(S_{2p})}^{1}(D^{μ}, D^{λ}) ≅ dim_{k} Ext_{kB_{0}(S_{2p})}^{1}(D^{λ}, D^{μ})

because all simple kS_{2p}-modules are self-dual.

We define a graph Q, which is called the Ext-quiver of kB_{0}(S_{2p}), as follows. The vertices are identified with p-regular partitions of 2p with empty p-core and there is an edge between two vertices λ and μ if and only if the Ext^{1}-space between the corresponding simple kB_{0}(S_{2p})-modules is one-dimensional. The graph Q is determined by Martin in [14].

Proposition 5.3 ([14, Theorem 3.1]). In Q, the vertices ⟨i, j⟩ and ⟨i, j + 1⟩ are connected by an edge for all 1 ≤ i < j ≤ p; the vertices ⟨i, j⟩ and ⟨i − 1, j⟩ are connected by an edge for all 2 ≤ i < j ≤ p + 1; and the vertices ⟨i − 2, i⟩ and ⟨i, i + 1⟩ are connected by an edge for all 3 ≤ i ≤ p. These are all of the edges.

We have included Q as Figure 2, as we will be using it to describe the structure of modules in kB_{0}(S_{2p}). To this end it is also helpful to single out the following subgraphs of Q, which are called induced lines [5]

⟨1, p⟩ — ⟨2, p⟩ — ⋯ — ⟨p − 2, p⟩ — ⟨p, p + 1⟩,

⟨1, p − 1⟩ — ⟨2, p − 1⟩ — ⋯ — ⟨p − 3, p − 1⟩ — ⟨p − 1, p⟩ — ⟨p − 1, p + 1⟩,

⟨1, p − 2⟩ — ⟨2, p − 2⟩ — ⋯

⋯ — ⟨p − 4, p − 2⟩ — ⟨p − 2, p − 1⟩ — ⟨p − 2, p⟩ — ⟨p − 2, p + 1⟩,

⋮

⟨1, p − i⟩ — ⟨2, p − i⟩ — ⋯

⋯ — ⟨p − i − 2, p − i⟩ — ⟨p − i, p − i + 1⟩ — ⋯ — ⟨p − i, p + 1⟩,

[figure: ⟨p, p + 1⟩; ⟨p − 1, p + 1⟩; ⟨p − 2, p + 1⟩; ⟨p − 3, p + 1⟩; ⟨3, p + 1⟩; ⟨2, p + 1⟩; ⟨1, p + 1⟩; ⟨p − 1, p⟩; ⟨p − 2, p⟩; ⟨p − 3, p⟩; ⟨3, p⟩; ⟨2, p⟩; ⟨1, p⟩; ⟨p − 2, p − 1⟩; ⟨p − 3, p − 1⟩; ⟨3, p − 1⟩; ⟨2, p − 1⟩; ⟨1, p − 1⟩; ⟨4, 5⟩; ⟨3, 5⟩; ⟨2, 5⟩; ⟨1, 5⟩; ⟨3, 4⟩; ⟨2, 4⟩; ⟨1, 4⟩; ⟨2, 3⟩; ⟨1, 3⟩; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .]

Figure 1. The graph Q.

⋮
⟨1, 4⟩ — ⟨2, 4⟩ — ⟨4, 5⟩ — ⟨4, 6⟩ — ⋯ — ⟨4, p⟩ — ⟨4, p + 1⟩,
⟨1, 3⟩ — ⟨3, 4⟩ — ⟨3, 5⟩ — ⋯ — ⟨3, p⟩ — ⟨3, p + 1⟩,
⟨2, 3⟩ — ⟨2, 4⟩ — ⋯ — ⟨2, p⟩ — ⟨2, p + 1⟩.

Let kQ be the path algebra of Q. In [5] Erdmann and Martin describe a two-sided ideal I of Q such that Λ = kQ/I is isomorphic to the basic algebra of kB_{0}(S_{2p}). We use this description to deduce the following lemma.

Lemma 5.4.

a) Let λ be a vertex in Q. The projective cover of D^{λ} has Loewy length five: the 1st and 5th layers are each isomorphic to D^{λ}, the 2nd and 4th layers are each isomorphic to ⊕_{μ}D^{μ}, the sum being over all μ adjacent to λ in Q, and the 3rd layer is isomorphic to a direct sum of two copies of D^{λ} (except when λ = ⟨1, p+1⟩, in which case only one copy appears) and ⊕_{ν}D^{ν}, the sum being over all ν in Q such that the distance between λ and ν is two and they do not lie on a common induced line.

b) Let λ—μ—ν be a subgraph of Q and let V be a kB_{0}(S_{2p})-module with head isomorphic to D^{λ}. If D^{ν} is a composition factor of V, then so is D^{μ}.

c) There is up to isomorphism a unique kB_{0}(S_{2p})-module with Loewy structure

\begin{matrix}D^{⟨1,3⟩} \\ D^{⟨3,4⟩} ⊕ D^{⟨2,3⟩} \\ D^{⟨1,3⟩} \end{matrix} ;

for each 4 ≤ i ≤ p, there is up to isomorphism a unique kB_{0}(S_{2p})-module with Loewy structure

\begin{matrix}D^{⟨1,i⟩} \\ D^{⟨2,i⟩} ⊕ D^{⟨1,i−1⟩} \\ D^{⟨1,i⟩} \end{matrix} ;

and for each 1 ≤ i ≤ p − 1, there is up to isomorphism a unique kB_{0}(S_{2p})- module with Loewy structure

\begin{matrix}D^{⟨i,p+1⟩} \\ D^{⟨i+1,p+1⟩} ⊕ D^{⟨i,p+1⟩} \\ D^{⟨i,p⟩} \end{matrix} .

d) Let μ_{1}—λ—μ_{2} be a subgraph of Q which is not contained in any induced line. If V is a submodule of the projective cover of D^{λ} such that both D^{μ_{1}} and D^{μ_{2}} appear in the head of V, then D^{λ} does not appear in the head of V.

e) For any vertex λ in Q and induced line L containing λ we define a Brauer

character

η_{λ,L} = 2ϕ^{λ} + ∑_{μ} ϕ^{μ},

where the sum runs over all μ in Q adjacent to λ and not contained in L.

The Brauer character of any relatively S_{2p−1}-projective kB_{0}(S_{2p})-module is a linear combination with nonnegative integer coefficients of the η_{λ,L} and the Brauer character of the projective cover of D^{⟨1,p+1⟩}.

f) Given a kB_{0}(S_{2p})-module W let Γ(W) be the full subgraph of Q with vertex set {λ | D^{λ} is a composition factor of W }. If W is indecomposable then Γ(W) is connected.

Proof. a) [15, Theorem 2.1].

b) If D^{ν} is a composition factor of V, then by the structure of the projective cover of D^{λ}, given in (a), some path of length two from ν to λ in Q is nonzero when viewed as an element of Λ. Thus, by part 4 of Theorem 3.1 of [5], any path of length two from ν to λ in Q is nonzero in Λ. In particular the path ν → μ → λ is nonzero in Λ, and hence D^{μ} is a composition factor of V.

c) We prove the last statement only as a similar argument proves the others. For i = 1, it follows directly from the structure of the projective cover of D^{⟨1,p+1⟩}. Suppose 2 ≤ i ≤ p − 1. There are three paths in Q of length two from ⟨i, p + 1⟩ to itself. From the structure of the projective cover of D^{⟨i,p+1⟩}, we see that the k-span of these paths in Λ is two-dimensional. Therefore, by part 3 of Theorem 3.1 of [5], the path

D^{⟨i,p+1⟩} → D^{⟨i−1,p+1⟩} → D^{⟨i,p+1⟩}

is nonzero in Λ. The required statement then follows easily.

d) From Theorem 3.1 of [5] we see that the k-subspace of Λ generated by all paths of length two in Q starting and ending at the vertex λ is spanned by the two paths

D^{λ} → D^{μ_{1}} → D^{λ}  and  D^{λ} → D^{μ_{2}} → D^{λ}.

The desired statement then follows from knowledge of the structure of the projective cover of D^{λ}, given in (a). e) Let V be a relatively S_{2p−1}-projective kB_{0}(S_{2p})-module. We may as well

assume that V is nonzero and indecomposable. Then V a direct summand of Ind_{kB}^{kB_{0}(S_{2p})}(U) for some indecomposable kB-module U, where B is a block of S_{2p−1}.

Note that removing a node from a partition of 2p with empty p-core yields a partition of 2p−1 whose p-core is a hook partition (see (5.1)). Thus, by the Branching Rule and Frobenius reciprocity, the p-core associated to the block B must be a hook partition.

Suppose first that U is a simple module. If B = B^{(p,1^{p−1}),0}, then U is isomorphic to the simple projective module D^{(p,1^{p−1})} and a straightforward character calculation shows that inducing this module to kB_{0}(S_{2p}) gives a projective cover of D^{⟨1,p+1⟩}. Otherwise B is a defect one block of S_{2p−1} whose p-core is a hook partition. In this case, by [14, Lemma 4.3], there exists a vertex λ in Q and an induced line L containing λ such that Ind_{kB}^{kB_{0}(S_{2p})}(U) has Loewy structure

\begin{matrix}D^{λ} \\ ⊕_{μ}D^{μ} \\ D^{λ} \end{matrix} ,

where the sum runs over all μ in Q adjacent to λ and not contained in L. Note that in particular Ind_{kB}^{kB_{0}(S_{2p})}(U) is indecomposable.

We’ve shown then that if U is simple then V = Ind_{kB}^{kB_{0}(S_{2p})}(U) and the character of V is either the Brauer character of the projective cover of D^{⟨1,p+1⟩} or η_{λ,L} for some λ and some L containing λ. Before moving on, we mention that the modules obtained by inducing two non-isomorphic simple B-modules to kB_{0}(S_{2p}) do not have a composition factor in common; this is a consequence of [14, Lemma 4.3].

We now turn to the general case: suppose only that U is indecomposable. It will suffice to demonstrate that Ind_{kB}^{kB_{0}(S_{2p})}(U) is indecomposable, for then V = Ind_{kB}^{kB_{0}(S_{2p})}(U) will have a filtration by kB_{0}(S_{2p})-modules induced from simple kS_{2p−1}-modules, and the special case above will apply to each of these kB_{0}(S_{2p})-modules and imply the desired result.

If B = B^{(p,1^{p−1}),0}, then any indecomposable kB-module is simple and the special case above applies. So assume that B is a defect one block of S_{2p−1}. If U is projective, then a character calculation shows that Ind_{kB}^{kB_{0}(S_{2p})}(U) is indecomposable. So assume that U is non-projective. Then, because the Brauer tree of kB is an open polygon with exceptional multiplicity one, it is easy to see that the composition factors of U are distinct; call them

T_{1}, T_{2}, ..., T_{n}.

Hence Ind_{kB}^{kB_{0}(S_{2p})}(U) has a filtration with factors isomorphic to

W_{1}, W_{2}, ..., W_{n},

where W_{i} = Ind_{kB}^{kB_{0}(S_{2p})}(T_{i}). We’ve seen that each W_{i} has no subquotient isomorphic to a direct sum of two copies of a simple module. Thus the same property is enjoyed by Ind_{kB}^{kB_{0}(S_{2p})}(U) because W_{i} and W_{j} do not have a composition factor in common when i ≠ j. Therefore Ind_{kB}^{kB_{0}(S_{2p})}(U) admits a module diagram, in the sense of [1]. It is now clear that this diagram must be connected and we conclude that Ind_{kB}^{kB_{0}(S_{2p})}(U) is indecomposable, as desired.

f) This is a general statement which applies to Ext-quivers in general. We omit the proof.

6. Irreducible Characters

In this section we explicitly calculate ‘restrictions’ of irreducible characters of KB^{ρ,2} to KB_{0}(S_{2p}) and inductions of irreducible characters of KB_{0}(S_{p} ≀ S_{2}) to KB_{0}(S_{2p}). The Littlewood-Richardson rule not unexpectedly plays a prominent role, so we review the relevant combinatorics, referring the reader to [7] for details.

Definition. A sequence a_{1}, a_{2}, … of positive integers is called a lattice permutation if any finite truncation of it contains at least as many m’s as (m+1)’s for any positive integer m.

Let λ and μ be partitions with λ ⊇ μ. A L-R λ/μ-tableau is a tableau of shape λ/μ with weakly increasing rows and strictly increasing columns such that the sequence obtained by reading its entries from right to left and top to bottom is a lattice permutation. It is said to be of content ν = (ν_{1}, ν_{2}, …) if it contains ν_{1} 1’s, ν_{2} 2’s, and so forth.

Theorem 6.1 (Littlewood-Richardson rule). Let s and t be positive integers and let μ, ν, and λ be partitions of s, and t, and s+t, respectively. Then the multiplicity of χ^{μ}·χ^{ν} in Res_{S_{s}×S_{t}}^{S_{s+t}}(χ^{λ}) is zero unless λ ⊇ μ in which case it is equal to the number of L-R λ/μ-tableaux of content ν.

The following well-known combinatorial lemma is crucial. We include a sketch of a proof which we will need to refer to later.

Lemma 6.2. Let s and t be positive integers and let μ and ν be partitions of s and t respectively. Suppose θ is a skew diagram which is the disjoint union of μ and ν. Then for any partition λ of s+t, the number of L-R θ-tableaux of content λ is equal to the number of L-R λ/μ-tableaux of content ν.

Proof. Let T be a L-R θ-tableau of content λ, and let a_{1}, a_{2}, …, a_{s}, b_{1}, b_{2}, …, b_{t} be the sequence obtained by reading the entries of T from right to left and top to bottom. The first s terms a_{1}, a_{2}, …, a_{s} come from the part of T with shape μ and being a lattice permutation there is only one possibility: the first μ_{1} terms of the sequence must be 1’s, the next μ_{2} terms must be 2’s, and so on.

We construct a tableau U inductively using the remaining entries b_{1}, b_{2}, …, b_{t}. Let U_{0} be the empty μ-tableau. Suppose that U_{n} has already been constructed. Then let U_{n+1} be the result of appending an m to the end of the b_{n+1}-st row of U_{n}, where m is the row of ν from which b_{n+1} came. It is not too hard to verify that U = U_{t} is a L-R λ/μ-tableax of content ν and that T ↦ U defines a bijection between the set of L-R θ-tableaux of content λ and the set of L-R λ/μ-tableaux of content ν.

We apply (6.1) and (6.2) to our situation:

Lemma 6.3. Throughout let λ be any partition of 2p with empty p-core. a) For 1 ≤ i < j ≤ p,

χ^{[i,j]} ⇓ = χ^{i,j} ↑,

and the multiplicity of χ^{λ} in this character is equal to the number of L-R λ/(i, 1^{p−i})-tableaux of content (j, 1^{p−j}).

b) For 1 ≤ i ≤ p,

χ^{[i]} ⇓ + χ^{[i,i]} ⇓ = χ^{i^{+}} ↑ + χ^{i^{−}} ↑.

Furthermore, the multiplicity of χ^{λ} in χ^{[i]} ⇓ is equal to the number of L-R λ/(i, 1^{p−i})-tableaux of content (i, 1^{p−i}) in which 1 does not appear in a row greater than p − i + 1, while the multiplicity of χ^{λ} in χ^{[i,i]} ⇓ is equal to the number of L-R λ/(i, 1^{p−i})-tableaux of content (i, 1^{p−i}) in which 1 appears in a row greater than p − i + 1.

Proof. By definition,

χ^{i,j} = Ind_{S_{p}×S_{p}}^{S_{p}≀S_{2}}(χ^{(i,1^{p−i})} · χ^{(j,1^{p−j})}),

so by (6.1) and Frobenius reciprocity the multiplicity of χ^{λ} in χ^{i,j} ↑ is as stated, namely the number of L-R λ/(i, 1^{p−i})-tableaux of content (j, 1^{p−j}).

Next, the multiplicity of χ^{λ} in χ^{[i,j]} ⇓ is equal, by the definition of ⇓, to the multiplicity of χ^{λ} ·χ^{ρ} in Res_{S_{2p}×S_{r}}^{S_{2p+r}}(χ^{i,j}), which is in turn equal to the number of L-R [i, j]/ρ-tableaux of content λ, by (6.1). But [i, j]/ρ is the disjoint union of (j, 1^{p−j}) and (i, 1^{p−i}) (see (5.1)), so we may conclude by (6.2) that the multiplicity of χ^{λ} in χ^{[i,j]} ⇓ is equal to the number of L-R λ/(i, 1^{p−i})-tableaux of content (j, 1^{p−j}).

Since χ^{i,j} ↑ and χ^{[i,j]} ⇓ contain χ^{λ} with the same multiplicity for any λ, they must be equal. This completes the proof of (a).

Let θ be a skew diagram which is the disjoint union of (i, 1^{p−i}) and (i, 1^{p−i}). Let Ω be the set of L-R θ-tableaux of content λ, Ω_{[i]} the set of L-R [i]/ρ-tableaux of content λ, and Ω_{[i,i]} the set of L-R [i, i]/ρ-tableaux of content λ.

Consideration of the shape of [i]/ρ and [i, i]/ρ (see (5.1)) reveals a clear bijection between Ω and the disjoint union of Ω_{[i]} and Ω_{[i,i]}: The tableau in Ω with lattice permutation

\underbrace{1, 1, …, 1}_{i}, 2, 3, …, p − i + 1, b_{1}, b_{2}, …, b_{p}

corresponds to the unique tableau in Ω_{[i]} with the same lattice permutation in the case that b_{1} ≤ p − i + 1 and corresponds to the unique tableau in Ω_{[i,i]} with the same lattice permutation in the case that b_{1} > p − i + 1.

In the bijection constructed in the proof of (6.2), the tableau T is mapped to a L-R λ/(i, 1^{p−i})-tableau of content (i, 1^{p−i}) in which the 1’s occur in rows b_{1}, b_{2}, …, b_{i}. It follows that there is a bijection between Ω_{[i]} and the set of L-R λ/(i, 1^{p−i})- tableaux of content (i, 1^{p−i}) in which 1 does not occur in a row greater than p−i+1 and a bijection between Ω_{[i,i]} and the set of L-R λ/(i, 1^{p−i})-tableaux of content (i, 1^{p−i}) in which a 1 appears in a row greater than p − i + 1. In addition, by (6.1) we have that the multiplicity of χ^{λ} in χ^{[i]} ⇓ is equal to |Ω_{[i]}|, and the multiplicity of χ^{λ} in χ^{[i,i]} ⇓ is equal to |Ω_{[i,i]}|. This proves the second part of (b).

The foregoing argument also shows that the multiplicity of χ^{λ} in the character χ^{[i]} ⇓ + χ^{[i,i]} ⇓ is equal to |Ω|. Now by (6.2) |Ω| is equal to the number of L-R λ/(i, 1^{p−i})-tableaux of content (i, 1^{p−i}), which is in turn equal to the multiplicity of χ^{λ} in Ind_{S_{p}×S_{p}}^{S_{2p}}(χ^{(i,1^{p−i})} · χ^{(i,1^{p−i})}), by (6.1). But

Ind_{S_{p}×S_{p}}^{S_{2p}}(χ^{(i,1^{p−i})} · χ^{(i,1^{p−i})}) = Ind_{S_{p}≀S_{2}}^{S_{2p}}(χ^{i^{+}} + χ^{i^{−}}),

so the first part of (b) holds.

Using (6.3b) we can calculate the sum of χ^{i^{+}} ↑ and χ^{i^{−}} ↑. To calculate each individually, we require a result of Littlewood. I would like to thank Gordon James for drawing my attention to [12].

Definition (Littlewood). Suppose that λ is a partition of 2n. Define the 2-sign ε_{2}(λ) of λ as follows. If the 2-core of λ is not empty then define ε_{2}(λ) to be 0. Otherwise, to reach the empty core one removes n rim 2-hooks, each of which is the partition (2) or (1^{2}). Let k be the number of hooks (1^{2}) removed. Then define ε_{2}(λ) to be (−1)^{k}, which is independent of the way the 2-hooks are removed.

In [12], Littlewood proves some formulas for the plethysm of symmetric functions. The following is an immediate corollary of the theorem appearing in chapter 6 of that paper. We have taken r = 2 in Littlewood’s notation, and translated into the language of representation theory.

Proposition 6.4 (Littlewood). Let λ and μ be partitions of 2n and n respectively. Let m^{+} be the multiplicity of χ^{λ} in Ind_{S_{n}≀S_{2}}^{S_{2n}}((χ^{μ} · χ^{μ})^{+}) and let m^{−} be the multiplicity of χ^{λ} in Ind_{S_{n}≀S_{2}}^{S_{2n}}((χ^{μ} · χ^{μ})^{−}). If ε_{2}(λ) = 0 then m^{+} − m^{−} = 0; if ε_{2}(λ) = 1 then m^{+} − m^{−} ≥ 0; and if ε_{2}(λ) = −1 then m^{+} − m^{−} ≤ 0;

Lemma 6.5 (2-signs for partitions of 2p with empty p-core).

ε_{2}(⟨i, p + 1⟩) = \{ \begin{matrix}−1 & if p − i ≡ 1 or 2 mod 4 \\ 1 & if p − i ≡ 0 or 3 mod 4 \end{matrix} , (1 ≤ i ≤ p).

ε_{2}(⟨0, i⟩) = \{ \begin{matrix}−1 & if i ≡ 0 or 1 mod 4 \\ 1 & if i ≡ 2 or 3 mod 4 \end{matrix} , (1 ≤ i ≤ p).

ε_{2}(⟨i, j⟩) = \{ \begin{matrix}0 & if i + j is even \\ −1 & if i + j ≡ 1 mod 4 \\ 1 & if i + j ≡ 3 mod 4 \end{matrix} , (1 ≤ i < j ≤ p).

Proof. The hooks ⟨i, p + 1⟩ = (p + i, 1^{p−i}) and ⟨0, i⟩ = (i, 1^{2p−i}) have empty 2cores and it is easy to verify that their 2-signs are as stated. Next, the partition ⟨i, j⟩ = (j, i + 1, 2^{p−j−1}, 1^{p−i+1}) has 2-core (3, 2, 1) or (4, 3, 2, 1) if i + j is even, so the 2-sign is 0 in these cases. If i+j is odd then ⟨i, j⟩ may be reduced to the empty 2-core by removing a hook (i, 1^{p−j}) and then a hook (j, 1^{p−i}). The statement follows easily.

Finally, we’re prepared for the main result of this chapter. Note that (4.6a) is a consequence.

Proposition 6.6.

For 1 ≤ i < j ≤ p, χ^{[i,j]} ⇓ = χ^{i,j} ↑ =

\{ \begin{matrix}\begin{matrix}χ^{⟨i,j+1⟩} & +χ^{⟨i−1,j+2⟩} & + ⋯ & & +χ^{⟨i+j−p,p+1⟩} \\ +2χ^{⟨i−1,j+1⟩} & +2χ^{⟨i−2,j+2⟩} & + ⋯ & +2χ^{⟨i+j−p,p⟩} & +χ^{⟨i+j−p−1,p+1⟩} \\ +χ^{⟨i−1,j⟩} & +χ^{⟨i−2,j+1⟩} & + ⋯ & & +χ^{⟨i+j−p−1,p⟩} \end{matrix} , & if i + j ≥ p + 2 \\ \begin{matrix}χ^{⟨i,j+1⟩} & +χ^{⟨i−1,j+2⟩} & + ⋯ & & +χ^{⟨1,p+1⟩} \\ +2χ^{⟨i−1,j+1⟩} & +2χ^{⟨i−2,j+2⟩} & + ⋯ & +2χ^{⟨1,p⟩} \\ +χ^{⟨i−1,j⟩} & +χ^{⟨i−2,j+1⟩} & + ⋯ & & +χ^{⟨0,p⟩} \end{matrix} , & if i + j = p + 1 \\ \begin{matrix}χ^{⟨i,j+1⟩} & +χ^{⟨i−1,j+2⟩} & + ⋯ & & +χ^{⟨1,i+j⟩} \\ +2χ^{⟨i−1,j+1⟩} & +2χ^{⟨i−2,j+2⟩} & + ⋯ & +2χ^{⟨1,i+j−1⟩} & +χ^{⟨0,i+j⟩} \\ +χ^{⟨i−1,j⟩} & +χ^{⟨i−2,j+1⟩} & + ⋯ & & +χ^{⟨0,i+j−1⟩} \end{matrix} , & if i + j ≤ p. \end{matrix}

For 1 ≤ i ≤ p, χ^{[i]} ⇓ = χ^{i^{±}} ↑ =

\{ \begin{matrix}χ^{⟨p,p+1⟩} , & if i = p \\ \begin{matrix}χ^{⟨i,i+1⟩} & +χ^{⟨i−1,i+2⟩} & + ⋯ & & +χ^{⟨2i−p,p+1⟩} \\ +χ^{⟨i−1,i+1⟩} & +χ^{⟨i−2,i+2⟩} & + ⋯ & +χ^{⟨2i−p,p⟩} \end{matrix} , & if \frac{p+1}{2} ≤ i ≤ p − 1 \\ \begin{matrix}χ^{⟨i,i+1⟩} & +χ^{⟨i−1,i+2⟩} & + ⋯ & +χ^{⟨1,2i⟩} \\ +χ^{⟨i−1,i+1⟩} & +χ^{⟨i−2,i+2⟩} & + ⋯ & +χ^{⟨0,2i⟩} \end{matrix} , & if 1 ≤ i ≤ \frac{p−1}{2} . \end{matrix}

For 1 ≤ i ≤ p, χ^{[i,i]} ⇓ = χ^{i^{∓}} ↑ =

\{ \begin{matrix}\begin{matrix}χ^{⟨i−1,i⟩} & +χ^{⟨i−2,i+1⟩} & + ⋯ & +χ^{⟨2i−p−1,p⟩} \\ +χ^{⟨i−1,i+1⟩} & +χ^{⟨i−2,i+2⟩} & + ⋯ & +χ^{⟨2i−p−1,p+1⟩}, \end{matrix} & if \frac{p+1}{2} ≤ i ≤ p \\ \begin{matrix}χ^{⟨i−1,i⟩} & +χ^{⟨i−2,i+1⟩} & + ⋯ & & +χ^{⟨0,2i−1⟩} \\ +χ^{⟨i−1,i+1⟩} & +χ^{⟨i−2,i+2⟩} & + ⋯ & +χ^{⟨1,2i−1⟩} \end{matrix} , & if 2 ≤ i ≤ \frac{p−1}{2} \\ χ^{⟨0,1⟩} , & if i = 1. \end{matrix}

Proof. By (6.3), for each partition λ of 2p with empty p-core, the multiplicity of χ^{λ} in χ^{[i,j]} ⇓ = χ^{i,j} ↑ is equal to the number of L-R λ/(i, 1^{p−i})-tableaux of content (j, 1^{p−j}). We list all such:

[figure: 1; ⋯; 1; 2; 3; ⋮; i; j; p − i; p − j + 1] λ = \{ \begin{matrix}⟨i + j − p, p + 1⟩, & if i + j > p \\ ⟨0, i + j⟩, & if i + j ≤ p \end{matrix} ;

[figure: 1; ⋯; 1; 1; 2; ⋮; j − 1; p − j + 1] λ = \{ \begin{matrix}⟨i + j − p − 1, p + 1⟩, & if i + j − 1 > p \\ ⟨0, i + j − 1⟩, & if i + j − 1 ≤ p \end{matrix} ;

if j < p,  [figure: 1; ⋯; 1; 2; 1; ⋯; 1; 3; ⋮; j + 1 − i; p − j + 1]  λ = ⟨i, j + 1⟩;

for all k such that 1 ≤ k < i and i + j − 1 − k ≤ p,

[figure: 1; ⋯; 1; 1; ⋯; 1; 2; ⋮; 1; ⋮; p − i − j + k + 2; p − i − j + k + 3; p − j + 1; j − 1 − k] λ = ⟨k, i + j − 1 − k⟩;

for all k such that 1 ≤ k < i and i + j − k ≤ p,

[figure: 1; ⋯; 1; 1; ⋯; 1; 2; ⋮; ⋮; p − i − j + k + 1; p − i − j + k + 2; p − j + 1; j − k] λ = ⟨k, i + j − k⟩;

for all k such that 1 ≤ k < i and i + j − k ≤ p,

[figure: 1; ⋯; 1; 2; 1; ⋯; 1; 3; ⋮; 1; ⋮; p − i − j + k + 2; p − i − j + k + 3; p − j + 1; j − k] λ = ⟨k, i + j − k⟩;

for all k such that 1 ≤ k < i and i + j + 1 − k ≤ p,

[figure: 1; ⋯; 1; 2; 1; ⋯; 1; 3; ⋮; ⋮; p − i − j + k + 1; p − i − j + k + 2; p − j + 1; j + 1 − k] λ = ⟨k, i + j + 1 − k⟩.

We hope the reader agrees that these are all the possibilities and concludes that χ^{[i,j]} ⇓ and χ^{i,j} ↑ are as stated.

Next consider χ^{[i]} ⇓. By (6.3), for all partitions λ of 2p with empty p-core, the multiplicity of χ^{λ} in χ^{[i]} ⇓ is the number of L-R λ/(i, 1^{p−i})-tableaux of content (i, 1^{p−i}) in which 1 does not appear in a row greater than p − i + 1. We list all such tableaux:

[figure: 1; ⋯; 1; 2; 3; ⋮; i; i; p − i; p − i + 1] λ = \{ \begin{matrix}⟨2i − p, p + 1⟩, & if 2i > p \\ ⟨0, 2i⟩, & if 2i ≤ p \end{matrix} ;

if i < p,  [figure: 1; 2; 1; ⋯; 1; 3; ⋮; p − i + 1]  λ = ⟨i, i + 1⟩;

for all k such that 1 ≤ k < i and 2i − k ≤ p,

[figure: 1; ⋯; 1; 1; ⋯; 1; 2; ⋮; ⋮; p − 2i + k + 1; p − 2i + k + 2; p − i + 1; i − k] λ = ⟨k, 2i − k⟩;

for all k such that 1 ≤ k < i and 2i + 1 − k ≤ p,

[figure: 1; ⋯; 1; 2; 1; ⋯; 1; 3; ⋮; ⋮; p − 2i + k + 1; p − 2i + k + 2; p − i + 1; i + 1 − k] λ = ⟨k, 2i + 1 − k⟩.

So χ^{[i]} ⇓ is as stated.

The calculation for χ^{[i,i]} ⇓ proceeds similarly: if λ is a partition of 2p with empty p-core, then the multiplicity of χ^{λ} in χ^{[i,i]} ⇓ is the number of L-R λ/(i, 1^{p−i})- tableaux of content (i, 1^{p−i}) in which a 1 appears in a row greater than p − i + 1. We list all such tableaux:

[figure: 1; ⋯; 1; 1; 2; ⋮; i; i − 1; p − i; p − i + 1] λ = \{ \begin{matrix}⟨2i − p − 1, p + 1⟩, & if 2i − 1 > p \\ ⟨0, 2i − 1⟩ & if 2i − 1 ≤ p \end{matrix} ;

if i > 1,  [figure: 1; ⋯; 1; 2; ⋮; 1; p − i + 1]  λ = ⟨i − 1, i⟩;

for all k such that 1 ≤ k < i − 1 and 2i − 1 − k ≤ p,

[figure: 1; ⋯; 1; 1; ⋯; 1; 2; ⋮; 1; ⋮; p − 2i + k + 2; p − 2i + k + 3; p − i + 1; i − 1 − k] λ = ⟨k, 2i − 1 − k⟩;

for all k such that 1 ≤ k < i and 2i − k ≤ p,

[figure: 1; ⋯; 1; 2; 1; ⋯; 1; 3; ⋮; 1; ⋮; p − 2i + k + 2; p − 2i + k + 3; p − i + 1; i − k] λ = ⟨k, 2i − k⟩.

So χ^{[i,i]} ⇓ is as stated.

Finally we verify the expressions given for χ^{i^{±}} ↑ and χ^{i^{∓}} ↑. Given a partition λ of 2p with empty p-core, let m^{+} and m^{−} be the multiplicities of χ^{λ} in χ^{i^{+}} ↑ and χ^{i^{−}} ↑ respectively. By (6.3b), we have

χ^{i^{+}} ↑ + χ^{i^{−}} ↑ = χ^{[i]} ⇓ + χ^{[i,i]} ⇓,

So m^{+} + m^{−} is equal to the multiplicity of χ^{λ} in χ^{[i]} ⇓ + χ^{[i,i]} ⇓. In addition (6.4) tells us that if ε_{2}(λ) = 0 then m^{+} − m^{−} = 0, if ε_{2}(λ) = 1 then m^{+} − m^{−} ≥ 0, and if ε_{2}(λ) = −1 then m^{+} − m^{−} ≤ 0. It follows, using (6.5) as well as the calculation of χ^{[i]} ⇓ and χ^{[i,i]} ⇓ above, that χ^{i^{±}} ↑ and χ^{i^{∓}} ↑ are as stated.

7. Simple Modules

The goal of this section is to prove (3.5c). We begin with some more character calculations. We know from chapter 6 how to ‘restrict’ irreducible characters of KB^{ρ,2} to KB_{0}(S_{2p}) and how to induce irreducible characters of KB_{0}(S_{p} ≀ S_{2}) to KB_{0}(S_{2p}). We also know the decomposition numbers for all three blocks. Thus it is straightforward, albeit tedious, to calculate the ‘restriction’ of irreducible Brauer characters of kB^{ρ,2} to kB_{0}(S_{2p}) and the induction of irreducible Brauer characters of kB_{0}(S_{p} ≀ S_{2}) to kB_{0}(S_{2p}).

Lemma 7.1.

For 2 ≤ i < j ≤ p, ϕ^{[i,j]} ⇓ = ϕ^{i,j} ↑ =

\{ \begin{matrix}\begin{matrix}ϕ^{⟨i,j+1⟩} & +ϕ^{⟨i−1,j+2⟩} & + ⋯ & +ϕ^{⟨i+j−p,p+1⟩} \\ +2ϕ^{⟨i−1,j+1⟩} & +2ϕ^{⟨i−2,j+2⟩} & + ⋯ & +2ϕ^{⟨i+j−p−1,p+1⟩} \\ +ϕ^{⟨i−1,j⟩} & +ϕ^{⟨i−2,j+1⟩} & + ⋯ & +ϕ^{⟨i+j−p−1,p⟩}, \end{matrix} & if i + j ≥ p + 2 \\ \begin{matrix}ϕ^{⟨i,j+1⟩} & +ϕ^{⟨i−1,j+2⟩} & + ⋯ & +ϕ^{⟨2,i+j−1⟩} \\ +2ϕ^{⟨i−1,j+1⟩} & +2ϕ^{⟨i−2,j+2⟩} & + ⋯ & +2ϕ^{⟨1,i+j−1⟩} \\ +ϕ^{⟨i−1,j⟩} & +ϕ^{⟨i−2,j+1⟩} & + ⋯ & +ϕ^{⟨1,i+j−2⟩}, \end{matrix} & if i + j ≤ p + 1. \end{matrix}

For 2 ≤ i ≤ p, ϕ^{[i]} ⇓ = ϕ^{i^{±}} ↑ = ϕ^{⟨i,i+1⟩}.

For 2 ≤ i ≤ p, ϕ^{[i,i]} ⇓ = ϕ^{i^{∓}} ↑ =

\{ \begin{matrix}ϕ^{⟨p,p+1⟩} + 2ϕ^{⟨p−1,p+1⟩} + ϕ^{⟨p−1,p⟩}, & if i = p \\ \begin{aligned}ϕ^{⟨i+1,i+2⟩} & +ϕ^{⟨i,i+1⟩} & +ϕ^{⟨i−1,i+2⟩} & + ⋯ & +ϕ^{⟨2i−p,p+1⟩} \\ & +2ϕ^{⟨i−1,i+1⟩} & +2ϕ^{⟨i−2,i+2⟩} & + ⋯ & +2ϕ^{⟨2i−p−1,p+1⟩} \\ & +ϕ^{⟨i−1,i⟩} & +ϕ^{⟨i−2,i+1⟩} & + ⋯ & +ϕ^{⟨2i−p−1,p⟩}, \end{aligned} & if \frac{p+3}{2} ≤ i ≤ p − 1 \\ \begin{aligned}ϕ^{⟨i+1,i+2⟩} & +ϕ^{⟨i,i+1⟩} & +ϕ^{⟨i−1,i+2⟩} & + ⋯ & +ϕ^{⟨2,2i−1⟩} \\ & +2ϕ^{⟨i−1,i+1⟩} & +2ϕ^{⟨i−2,i+2⟩} & + ⋯ & +2ϕ^{⟨1,2i−1⟩} \\ & +ϕ^{⟨i−1,i⟩} & +ϕ^{⟨i−2,i+1⟩} & + ⋯ & +ϕ^{⟨1,2i−2⟩}, \end{aligned} & if 2 ≤ i ≤ \frac{p+1}{2} . \end{matrix}

Having calculated the Brauer characters of ‘restrictions’ of simple kB^{ρ,2}-modules to kB_{0}(S_{2p}) and inductions of simple kB_{0}(S_{p} ≀ S_{2})-modules to kB_{0}(S_{2p}), we next describe the structure of these kB_{0}(S_{2p})-modules, giving a module diagram for each. For background on module diagrams, see [1].

Proposition 7.2.

For 2 ≤ i ≤ p, both D^{[i]} ⇓ and D^{i^{±}} ↑ are isomorphic to D^{⟨i,i+1⟩}.

For 2 ≤ i < j ≤ p, both D^{[i,j]} ⇓ and D^{i,j} ↑ have diagram

[figure: D^{⟨i−1,j+1⟩}; D^{⟨i−2,j+2⟩}; ⋯; D^{⟨i+j−p,p⟩}; D^{⟨i+j−p−1,p+1⟩}; D^{⟨i,j+1⟩}; D^{⟨i−1,j⟩}; D^{⟨i−1,j+2⟩}; D^{⟨i−2,j+1⟩}; D^{⟨i−2,j+3⟩}; D^{⟨i−3,j+2⟩}; ⋯; D^{⟨i+j−p,p+1⟩}; D^{⟨i+j−p−1,p⟩}; D^{⟨i−1,j+1⟩}; D^{⟨i−2,j+2⟩}; ⋯; D^{⟨i+j−p,p⟩}; D^{⟨i+j−p−1,p+1⟩}]^{(*)}

if i + j ≥ p + 2, and both have diagram

[figure: D^{⟨i−1,j+1⟩}; D^{⟨i−2,j+2⟩}; ⋯; D^{⟨2,i+j−2⟩}; D^{⟨1,i+j−1⟩}; D^{⟨i,j+1⟩}; D^{⟨i−1,j⟩}; D^{⟨i−1,j+2⟩}; D^{⟨i−2,j+1⟩}; D^{⟨i−2,j+3⟩}; D^{⟨i−3,j+2⟩}; ⋯; D^{⟨2,i+j−1⟩}; D^{⟨1,i+j−2⟩}; D^{⟨i−1,j+1⟩}; D^{⟨i−2,j+2⟩}; ⋯; D^{⟨2,i+j−2⟩}; D^{⟨1,i+j−1⟩}]

if i + j < p + 2.

Both D^{[p,p]} ⇓ and D^{p^{∓}} ↑ have diagram

[figure: D^{⟨p−1,p+1⟩}; D^{⟨p−1,p⟩}; D^{⟨p,p+1⟩}; D^{⟨p−1,p+1⟩}.]

For 2 ≤ i ≤ p, both D^{[i,i]} ⇓ and D^{i^{∓}} ↑ have diagram

[figure: D^{⟨i−1,i+1⟩}; D^{⟨i−2,i+2⟩}; ⋯; D^{⟨2i−p,p⟩}; D^{⟨2i−p−1,p+1⟩}; D^{⟨i+1,i+2⟩}; D^{⟨i,i+1⟩}; D^{⟨i−1,i⟩}; D^{⟨i−1,i+2⟩}; D^{⟨i−2,i+1⟩}; D^{⟨i−2,i+3⟩}; D^{⟨i−3,i+2⟩}; ⋯; D^{⟨2i−p,p+1⟩}; D^{⟨2i−p−1,p⟩}; D^{⟨i−1,i+1⟩}; D^{⟨i−2,i+2⟩}; ⋯; D^{⟨2i−p,p⟩}; D^{⟨2i−p−1,p+1⟩}]

if \frac{p+3}{2} ≤ i ≤ p − 1, and both have diagram

[figure: D^{⟨i−1,i+1⟩}; D^{⟨i−2,i+2⟩}; ⋯; D^{⟨2,2i−2⟩}; D^{⟨1,2i−1⟩}; D^{⟨i+1,i+2⟩}; D^{⟨i,i+1⟩}; D^{⟨i−1,i⟩}; D^{⟨i−1,i+2⟩}; D^{⟨i−2,i+1⟩}; D^{⟨i−2,i+3⟩}; D^{⟨i−3,i+2⟩}; ⋯; D^{⟨2,2i−1⟩}; D^{⟨1,2i−2⟩}; D^{⟨i−1,i+1⟩}; D^{⟨i−2,i+2⟩}; ⋯; D^{⟨2,2i−2⟩}; D^{⟨1,2i−1⟩}]

if 2 ≤ i ≤ \frac{p+1}{2}.

Note that in each of the diagrams above, two nodes are joined by an edge if and only if they are labeled by simple modules whose associated partitions are adjacent in Q.

Proof. Let L be the ‘restriction’ of a simple kB^{ρ,2}-module to kB_{0}(S_{2p}) or the induction of a simple kB_{0}(S_{p} ≀ S_{2})-module to kB_{0}(S_{2p}). Since simple modules of kB^{ρ,2} and kB_{0}(S_{p} ≀ S_{2}) are self-dual, so is L.

A special case of a theorem of Knörr [8] says that a simple module lying in a block with abelian defect group D must have D as a vertex. It follows by (3.3) and (3.4) that L = W ⊕ E, where W is indecomposable and E is relatively S_{2p−1}- projective. Suppose E ≠ 0. Then using (7.1) and (5.4e), one can deduce that Γ(W) is not connected, where Γ(W) is the full subgraph of Q with vertex set {λ | D^{λ} is a composition factor of W }. For example, if L = D^{[i,j]} ⇓, where 2 ≤ i < j ≤ p − 1 and i + j ≥ 2, then D^{⟨i+j−p−1,p+1⟩} must be a composition factor of W and either D^{⟨i,j+1⟩} or D^{⟨i+1,j⟩} must be a composition factor too. On the other hand D^{⟨i−k,j+k⟩} is not a composition factor of W for some k with 1 ≤ k ≤ p − j, and it follows that Γ(W) cannot be connected. For other cases, we can argue similarly. Thus, by (5.4f), W is not indecomposable, a contradiction. We have shown that E = 0 and hence that L is indecomposable.

(7.1) tells us what the composition factors of L are. If L has only one composition factor then of course it is simple. For the rest of the proof, assume that L is not simple, so that L is the ‘restriction’ of a simple kB^{ρ,2}-module which is not isomorphic to D^{[i]} for any i, or the induction of a simple kB_{0}(S_{p} ≀S_{2})-module which is not isomorphic to D^{i^{±}} for any i.

L is indecomposable, non-simple and self-dual, so every simple kB_{0}(S_{2p})-module which appears with multiplicity one as a composition factor of L must occur in neither the head nor socle of L. All other composition factors of L appear with multiplicity two, and referring to Figure 2, we see that there are no non-split extensions among these simple modules. Together with the self-duality of L, this forces L to have Loewy length three, with each simple module which appears with multiplicity two occurring once in the first layer and once in the third layer, and all other composition factors occurring once in the second layer. Note that L has no subquotient isomorphic to a direct sum of two copies of a simple module. Thus L has a module diagram, in the sense of [1], and the layers of the diagram are as stated. We now need to show that two nodes in the diagram are joined by an edge if and only if they are labeled by simple modules whose associated partitions are adjacent in Q.

We consider the case L = D^{[i,j]} ⇓, where 2 ≤ i < j ≤ p and i + j ≥ p + 2 (similar reasoning applies in other cases). L is self-dual, so if a node in the first layer labeled by D^{λ} and a node in the second layer labeled by D^{μ} are connected by an edge, then the node in the second layer labeled D^{μ} and the node in the third layer labeled D^{λ} are also connected by an edge, and vice versa. Thus in this case we will simply say that ‘D^{λ} and D^{μ} are connected in the diagram of L’.

If D^{λ} and D^{μ} are connected in the diagram of L then certainly λ and μ are adjacent in Q. We will complete the proof by showing that the converse holds. We handle the case λ = ⟨i−1, j +1⟩, other cases being similar. The nodes of Q adjacent to ⟨i − 1, j + 1⟩ are ⟨i, j + 1⟩, ⟨i − 1, j⟩, ⟨i − 1, j + 2⟩, and ⟨i − 2, j + 1⟩. Certainly D^{⟨i−1,j+1⟩} must be connected with both D^{⟨i,j+1⟩} and D^{⟨i−1,j⟩} in the diagram of L, as the latter two do not have nonzero Ext-spaces with any simple module in the diagram other than D^{⟨i−1,j+1⟩}. Next note that D^{⟨i−1,j+1⟩} must be connected to either D^{⟨i−1,j+2⟩} or D^{⟨i−2,j+1⟩} in the diagram of L because L is indecomposable. Suppose that it is not connected to D^{⟨i−1,j+2⟩}. Then it is connected to D^{⟨i−2,j+1⟩} and this implies that L has a submodule with diagram

[figure: D^{⟨i−1,j+1⟩}; D^{⟨i,j+1⟩}; D^{⟨i−1,j⟩}; D^{⟨i−2,j+1⟩}; D^{⟨i−1,j+1⟩}; D^{⟨i−2,j+2⟩}.]

Since ⟨i − 1, j + 1⟩ — ⟨i − 1, j + 2⟩ — ⟨i − 2, j + 2⟩ is a subgraph of Q, the existence of this submodule contradicts (5.4b). Thus D^{⟨i−1,j+1⟩} is in fact connected to D^{⟨i−1,j+2⟩} in the diagram of L. A similar argument shows that D^{⟨i−1,j+1⟩} is connected to D^{⟨i−2,j+1⟩} in the diagram of L.

The next result together with (7.2) implies (3.5c).

Proposition 7.3. For each of the diagrams in the statement of (7.2), there is up to isomorphism a unique kB_{0}(S_{2p})-module having that diagram.

Proof. We already know this to be true whenever the head of the diagram being considered is simple (5.4c). So induct on the number of factors in the head. We argue for the diagram labeled (*) in the statement of (7.2). The proofs for the other diagrams are similar.

Let L be any kB_{0}(S_{2p})-module with diagram (*). L is an extension of a module with diagram

[figure: D^{⟨i−1,j+1⟩}; D^{⟨i,j+1⟩}; D^{⟨i−1,j⟩}.] (1)

by a module with diagram

[figure: D^{⟨i−2,j+2⟩}; ⋯; D^{⟨i+j−p,p⟩}; D^{⟨i+j−p−1,p+1⟩}; D^{⟨i−1,j+2⟩}; D^{⟨i−2,j+1⟩}; D^{⟨i−2,j+3⟩}; D^{⟨i−3,j+2⟩}; ⋯; D^{⟨i+j−p,p+1⟩}; D^{⟨i+j−p−1,p⟩}; D^{⟨i−1,j+1⟩}; D^{⟨i−2,j+2⟩}; ⋯; D^{⟨i+j−p,p⟩}; D^{⟨i+j−p−1,p+1⟩}]^{(2)}

which is in turn an extension of a module with diagram

[figure: D^{⟨i−2,j+2⟩}; ⋯; D^{⟨i+j−p,p⟩}; D^{⟨i+j−p−1,p+1⟩}; D^{⟨i−1,j+2⟩}; D^{⟨i−2,j+1⟩}; D^{⟨i−2,j+3⟩}; D^{⟨i−3,j+2⟩}; ⋯; D^{⟨i+j−p,p+1⟩}; D^{⟨i+j−p−1,p⟩}; D^{⟨i−2,j+2⟩}; ⋯; D^{⟨i+j−p,p⟩}; D^{⟨i+j−p−1,p+1⟩}] (3)

by the simple module D^{⟨i−1,j+1⟩}.

Since all nonzero Ext-spaces between simple kB_{0}(S_{2p})-modules are one-dimensional, there is up to isomorphism a unique kB_{0}(S_{2p})-module with diagram (1): let U be such a module. By induction there is up to isomorphism a unique kB_{0}(S_{2p})-module with diagram (3). The multiplicity of D^{⟨i−1,j+1⟩} as a composition factor of the projective cover of this module is one, by (5.4a), so it follows that up to isomorphism there is a unique kB_{0}(S_{2p})-module with diagram (2): let V be such a module.

To demonstrate that there is up to isomorphism a unique kB_{0}(S_{2p})-module with diagram (*), then, it suffices to show that the dimension of Ext_{kB_{0}(S_{2p})}^{1}(U, V) is at most one.

We have an exact sequence

0 → ΩU → P → U → 0,

where P is the projective cover of D^{⟨i−1,j+1⟩}. Note that by (5.4a), both D^{⟨i−2,j+1⟩} and D^{⟨i−1,j+2⟩} appear in the head of ΩU. Hence D^{⟨i−1,j+1⟩} does not appear in the head of ΩU, by (5.4c).

We get an exact sequence

0 → Hom_{kB_{0}(S_{2p})}(U, V) → Hom_{kB_{0}(S_{2p})}(P, V)

→ Hom_{kB_{0}(S_{2p})}(ΩU, V) → Ext_{kB_{0}(S_{2p})}^{1}(U, V) → 0.

The multiplicity of D^{⟨i−1,j+1⟩} as a composition factor of V is one, and it appears in the socle of V. It follows easily that every kB_{0}(S_{2p})-homomorphism from P to V factors through the map P → U. Thus the map

Hom_{kB_{0}(S_{2p})}(U, V) → Hom_{kB_{0}(S_{2p})}(P, V)

is surjective and therefore

Hom_{kB_{0}(S_{2p})}(ΩU, V) ≅ Ext_{kB_{0}(S_{2p})}^{1}(U, V).

So we need only to show that the dimension of Hom_{kB_{0}(S_{2p})}(ΩU, V) is at most one.

Let f be any kB_{0}(S_{2p})-homomorphism from ΩU to V and let S be the unique submodule of V which is isomorphic to D^{⟨i−2,j+2⟩}. If the image of f does not contain S, then by the submodule structure of V and the fact that the only composition factors shared by ΩU and V are D^{⟨i−2,j+2⟩}, D^{⟨i−1,j+2⟩}, D^{⟨i−2,j+1⟩}, and D^{⟨i−1,j+1⟩} (see (5.4a)), we have that the image of f must be contained in the unique submodule of V isomorphic to D^{⟨i−1,j+1⟩}. But D^{⟨i−1,j+1⟩} does not appear in the head of ΩU, so we conclude that f = 0.

Now suppose that f_{1} and f_{2} are two nonzero kB_{0}(S_{2p})-homomorphisms from ΩU to V. By the preceding argument, both the image of f_{1} and the image of f_{2} contain S. Hence both f_{1}^{−1}(S) and f_{2}^{−1}(S) have D^{⟨i−2,j+2⟩} as a composition factor. Since the multiplicity of D^{⟨i−2,j+2⟩} as a composition factor of ΩU is one (by (5.4a)), it follows that Z = f_{1}^{−1}(S)∩f_{2}^{−1}(S) has D^{⟨i−2,j+2⟩} as a composition factor. As Hom_{kB_{0}(S_{2p})}(Z, S) is at most one-dimensional there exists a linear combination f of f_{1} and f_{2} such that f(Z) = 0. This implies that the image of f does not have D^{⟨i−2,j+2⟩} as a composition factor and hence does not contain S. So f = 0 by the argument in the previous paragraph and therefore f_{1} and f_{2} are linearly dependent. We have shown that the dimension of Hom_{kB_{0}(S_{2p})}(ΩU, V) is at most one, as desired.

8. Splendid Rickard Equivalences

In [24], Rickard introduces a strengthening of the notion of derived equivalence for principal blocks of finite groups. Harris extends Rickard’s definition for arbitrary blocks in [6] (See also [10]).

Definition (Rickard, Harris). Let G and H be finite groups with a common psubgroup D, and let B and B′ be blocks of G and H with defect group D. Let X be a complex of B-B′-bimodules and suppose that X is a Rickard tilting complex (see chapter 2 for a definition). If each term of X is a relatively ΔD-projective p-permutation O[G × H]-module, then we say that X is splendid and that B and B′ are splendid Rickard equivalent.

Splendid Rickard equivalences are significant because they give rise to derived equivalences between corresponding blocks of centralizers of p-subgroups of D in G and H (see [24], [6], [10], and [19]).

Now we can state Rickard’s refinement of (1.1):

Conjecture 8.1 (Broué-Rickard). Let B be a block of a finite group G with abelian defect group D. Then B and its Brauer correspondent in N_{G}(D) are splendid Rickard equivalent.

After proving a few easy lemmas, we will show that our construction of a derived equivalence between an arbitrary defect two symmetric group block B^{τ,2} and its Brauer correspondent B_{0}((C_{p}⋊C_{p−1})≀S_{2})⊗_{O}B^{τ,0} actually yields a splendid Rickard equivalence and thus affirms (8.1) for these blocks.

Definition. Let G and H be groups with common p-subgroup S. We say that G controls H-fusion of subgroups of S if whenever R is a subgroup of S and h is an element of H such that hRh^{−1} ⊆ S then there exists g ∈ G such that hrh^{−1} = grg^{−1} for all r ∈ R.

Following a suggestion of Rickard (see also [26, Lemma 2.6]), we have

Lemma 8.2. Let G, H, and K be groups with common p-subgroup S. Let M_{1} be a p-permutation O[G × H]-module with vertex ΔP and let M_{2} be a p-permutation O[H ×K]-module with vertex ΔQ, where P and Q are subgroups of S. If G controls H-fusion of subgroups of S, then M = M_{1} ⊗_{OH} M_{2} is a relatively ΔQ-projective p-permutation O[G × K]-module; and if K controls H-fusion of subgroups of S, then M is a relatively ΔP -projective p-permutation O[G × K]-module.

Proof. We may assume that M_{1} = Ind_{ΔP}^{G×H} O and M_{2} = Ind_{ΔQ}^{H×K} O. Define an equivalence relation ∼ on G × H × K by setting

(gp, hq, k) ∼ (g, ph, qk)

for all g ∈ G, p ∈ P, h ∈ H, q ∈ Q , k ∈ K, and let \overline{(g, h, k)} be the equivalence class containing (g, h, k). Then M is isomorphic to a permutation O[G×K]-module with (G × K)-stable basis

Ω = \{\overline{(g, h, k)} | g ∈ G, h ∈ H, k ∈ K\},

where for γ ∈ G and κ ∈ K we define

(γ, κ)\overline{(g, h, k)} = \overline{(γg, h, kκ^{−1})}.

Any element of Ω is in the (G × K)-orbit of \overline{(1, h, 1)} for some h ∈ H, and the stabilizer of \overline{(1, h, 1)} in G × K is

(h, 1)Δ(h^{−1}P h ∩ Q)(h, 1)^{−1} = (1, h)^{−1}Δ(P ∩ hQh^{−1})(1, h).

If G (respectively K) controls H-fusion of subgroups of S, then any subgroup of G × K of this form is conjugate to a subgroup of ΔQ (respectively ΔP).

The next result describes a situation in which the composition of two splendid Rickard equivalences is again a splendid Rickard equivalence.

Lemma 8.3. Let B, B′, and B′′ be blocks of finite groups G, H, and K with common p-subgroup D which is a defect group for all three blocks. If B and B′ are splendid Rickard equivalent and B′ and B′′ are splendid Rickard equivalent, then B and B′′ are splendid Rickard equivalent as long as either G or K controls H-fusion of subgroups of D.

Proof. Let X be a splendid Rickard tilting complex of B′-B-bimodules and let Y be a splendid Rickard tilting complex of B′′-B′-bimodules. Then it is clear that Y ⊗_{B′} X is a Rickard tilting complex of B′′-B-bimodules, and we deduce from (8.2) that it is also splendid.

Lemma 8.4. Let B be a block of a group G and let B′ be a block of a subgroup H with defect group D. Then bimodules inducing the functors Res_{B′}^{B} and Ind_{B′}^{B} are relatively ΔD-projective p-permutation bimodules.

Proof. Let S be a Sylow p-subgroup of H containing D. The O[H × G]-module M inducing restriction from B to B′ is a summand of B′ ⊗_{OH} Ind_{ΔH}^{H×G} O. Now Ind_{ΔH}^{H×G} O is a p-permutation O[H × G]-module with vertex ΔS and B′ is a ppermutation O[H × H]-module with vertex ΔD. Obviously G controls H-fusion of subgroups of S, so using (8.2) we conclude that M is a relatively ΔD-projective p-permutation O[H × G]-module. A similar argument works for induction.

Lemma 8.5. Let G and H be finite groups, B a block of G with defect group D, and C a block of H of defect 0. Then B and B ⊗_{O} C (a block of G×H) are splendid Rickard equivalent.

Proof. Let P be an indecomposable projective C-module. P is a projective O[G×1]- module and we have P^{∨} ⊗_{OG} P ≅ O and P ⊗_{O} P^{∨} ≅ C. It follows easily that B ⊗_{O} P viewed as a complex of (B ⊗_{O} C)-B-bimodules concentrated in degree 0 is a splendid Rickard tilting complex.

For the remainder of this chapter, let D be the permutation group generated by the p-cycles (1 2 … p) and (p + 1 p + 2 … 2p) (as in section 3), and identify (C_{p} ⋊ C_{p−1}) ≀ S_{2} with the normalizer of D in S_{2p}.

Lemma 8.6. Let G and H be finite permutation groups on the set {1, 2, … } which contain (C_{p} ⋊ C_{p−1}) ≀ S_{2}. Then G controls H-fusion of subgroups of D.

Proof. Certainly G controls (C_{p} ⋊ C_{p−1}) ≀ S_{2}-fusion of subgroups of D. Thus it is enough to show that (C_{p} ⋊ C_{p−1}) ≀ S_{2} controls H-fusion of subgroups of D. So suppose R is a p-subgroup of D and h an element of H such that hRh^{−1} ⊆ D. Let Ω be the subset of {1, 2, … } consisting of elements not fixed by R. Then hRh^{−1} ⊆ D implies that h(Ω) ⊆ {1, 2, …, 2p}, so there exists w ∈ S_{2p} such that w(x) = h(x) for all x ∈ Ω, i.e such that hrh^{−1} = wrw^{−1} for all r ∈ R. The desired result is then a consequence of a classic result of Burnside as D is an abelian Sylow p-subgroup of S_{2p} whose normalizer in S_{2p} is (C_{p} ⋊ C_{p−1}) ≀ S_{2}.

Now we are ready to prove that (8.1) holds for any block B^{τ,2}. We showed that B^{τ,2} and its Brauer correspondent B_{0}((C_{p} ⋊ C_{p−1}) ≀ S_{2}) ⊗_{O} B^{τ,0} are derived equivalent by demonstrating a chain of derived equivalences between blocks. Each of the links in this chain is one of the following pairs of blocks:

1. A [2:k] pair of defect two blocks of symmetric groups, where k ≥ 2,

2. A [2:1] pair of defect two blocks of symmetric groups, 3. B^{ρ,2} and B_{0}(S_{p} ≀ S_{2}),

4. B_{0}(S_{p} ≀ S_{2}) and B_{0}((C_{p} ⋊ C_{p−1}) ≀ S_{2}),

5. B_{0}((C_{p} ⋊ C_{p−1}) ≀ S_{2}) and B_{0}((C_{p} ⋊ C_{p−1}) ≀ S_{2}) ⊗_{O} B^{τ,0}. Each block in this list is a block of a permutation group which contains (C_{p} ⋊

C_{p−1}) ≀ S_{2}. Thus in view of (8.6) and (8.3), to prove that the blocks B^{τ,2} and B_{0}((C_{p} ⋊ C_{p−1}) ≀ S_{2}) ⊗_{O} B^{τ,0} are splendid Rickard equivalent it suffices to show that each pair of blocks in the list are splendid Rickard equivalent:

1. Scopes constructs in [27] a Morita equivalence between blocks forming a [2:k] pair when k ≥ 2. Puig reinterprets her work in [18] and a consequence of Theorem 1.7 of this paper is that a bimodule M inducing Scopes’s equivalence is a direct summand of a bimodule inducing restriction from one block to the other. Thus M, viewed as a complex concentrated in degree 0 is a splendid Rickard equivalence, by (8.4).

2. In section 2 we constructed a Rickard tilting complex X for blocks forming a [2:1] pair. X is in fact a splendid Rickard tilting complex: the term in degree zero is a bimodule inducing restriction from one block to the other and is thus a relatively ΔD-projective p-permutation module by (8.4); and the term in degree -1 is projective.

3. Recall that the functor F_{O} defined in section 3 is the composition of the following three functors:

Res_{B_{0}(S_{2p})⊗_{O}B^{ρ,0}}^{B^{ρ,2}} : B^{ρ,2} -mod ⟶ B_{0}(S_{2p}) ⊗_{O} B^{ρ,0} -mod,
Hom_{B^{ρ,0}}(P_{O}, −) : B_{0}(S_{2p}) ⊗_{O} B^{ρ,0} -mod ⟶ B_{0}(S_{2p}) -mod,
Res_{B_{0}(S_{p}≀S_{2})}^{B_{0}(S_{2p})} : B_{0}(S_{2p}) -mod ⟶ B_{0}(S_{p} ≀ S_{2}) -mod .

The first and third are induced by relatively ΔD-projective p-permutation bimodules, by (8.4), and the second is also, by (8.5). It follows by (8.2) that the B_{0}(S_{p} ≀ S_{2})-B^{ρ,2}-bimodule inducing the functor F_{O} is a relatively ΔDprojective p-permutation bimodule. At the end of section 3 we described a summand N of this bimodule which induces a Morita equivalence between B^{ρ,2} and B_{0}(S_{p} ≀ S_{2}). We can conclude that N, viewed as a complex of B_{0}(S_{p} ≀ S_{2})-B^{ρ,2}-bimodules concentrated in degree zero, is a splendid Rickard tilting complex.

4. Marcus ([13], Example 5.7) shows that B_{0}(S_{p} ≀ S_{2}) and B_{0}((C_{p} ⋊ C_{p−1}) ≀ S_{2}) are splendid Rickard equivalent.

5. B_{0}((C_{p} ⋊ C_{p−1}) ≀ S_{2}) and B_{0}((C_{p} ⋊ C_{p−1}) ≀ S_{2}) ⊗_{O} B^{τ,0} are splendid Rickard equivalent, by (8.5).

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Department of Mathematics, University of Chicago, Chicago, IL

E-mail address: jchuang@math.uchicago.edu