BROUÉ’S ABELIAN DEFECT GROUP CONJECTURE FOR ALTERNATING GROUPS

ANDREI MARCUS

Abstract. We establish Broué’s abelian defect group conjecture for the alternating groups, using the Chuang-Rouquier theorem proving this for the symmetric groups and a descent result.

1. Introduction

A conjecture of M. Broué states that a block algebra of a finite group is derived equivalent to its Brauer correspondent if the defect group of the block is abelian. We establish this conjecture for the alternating groups, based on the Chuang-Rouquier theorem proving this for the symmetric groups and on a descent result coming from Clifford theory.

To be more precise, let G = A_{n}, O a complete discrete valuation ring with algebraically closed residue field k of characteristic p > 0, let b be a block of OG with defect group D, and let c be the Brauer corresponding of ON_{G}(D).

We show that D is abelian, then the algebras A = bOG and B = cON_{G}(D) are splendidly derived equivalent, that is, there is a bounded complex X of (A, B)- bimodules such that its components are p-permutation modules whose indecomposable summands have vertices contained in δ(D) = {(u, u) | u ∈ D}, and such that X ⊗_{B} X^{∨} ≃ A in the homotopy category of complexes of (A, A)-bimodules, and X^{∨} ⊗_{A} X ≃ B in the homotopy category of (B, B)-bimodules, where X^{∨} denotes the O-dual of X.

Moreover, we show that there is such an equivalence which is compatible with p′-outer automorphism groups, which means in our case the existence of a tilting complex having an Aut(G)/G-grading. The complete statement of the main result is given in Theorem 3.1 below. This additional condition is especially important in the case of principal blocks, where it is used to reduce the conjecture to the case of simple groups.

In our proof, we use that the conjecture is known to hold for the symmetric group S_{n} by the work of J. Rickard, J. Chuang, R. Kessar and R. Rouquier, and we show how to “go down” to A_{n}. Important inspiration comes from the paper [5] of P. Fong and M. Harris, who verified the weaker “isotypy form” of the conjecture for A_{n}, by using Rouquier’s paper [11] on S_{n}. Note that a similar procedure was devised by E. Dade in [4] leading to the verification of his Invariant Projective Conjecture for A_{n}, and we are able to adapt some techniques from there.

Date: August 19, 2002.

1991 Mathematics Subject Classification. Primary 20C20; Secondary 20C30, 16W50.

Key words and phrases. symmetric groups, alternating groups, blocks, abelian defect groups, Rickard equivalences, splendid tilting complex, group graded algebras.

The author was supported by a Fulbright fellowship.

In Section 2 we consider O-algebras graded by the cyclic group C_{n} of order n not divisible by p. The main technical result is that a bimodule over two such algebras is C_{n}-graded if and only if the group Ĉ_{n} of linear characters of C_{n} acts on it. If a complex X induces a Rickard equivalence between two strongly C_{n}-graded algebras R and S, then we obtain a Rickard equivalence between the 1-components R_{1} and S_{1} provided that X is a complex of C_{n}-graded bimodules.

We discuss blocks of symmetric and alternating groups in Section 3, where we prove Theorem 3.1. The interesting case in when the block b of OS_{n} is self associated, that is, it is invariant under the action of Ĉ_{2}. Then the algebra bOS_{n} is C_{2}-graded, and so is its Brauer correspondent cON_{S_{n}}(D); we only need to show that the terms of the complex inducing a Rickard equivalence between these blocks are also C_{2}-graded.

We refer the reader to [13] for equivalences between blocks, and to [7] for equivalences induced by graded bimodules. All the facts on the representation theory of symmetric groups which we use here are contained in [2] and [5].

2. Algebras graded by a cyclic group

2.1. Let C_{n} = ⟨σ⟩ be the cyclic group of order n, and let (K, O, k) be a p-modular system, where p does not divide n, such that K contains a primitive n-th root ϵ of unity. The group

Ĉ_{n} = Hom(C_{n}, K^{×})

of characters of C_{n} is isomorphic to C_{n}, and we have that Ĉ_{n} = ⟨σ̂⟩, where σ̂(σ) = ϵ. 2.2. Let R = ⨁_{g∈C_{n}} R_{g} be a C_{n}-graded O-algebra, not necessarily strongly graded. Then Ĉ_{n} acts on R as automorphisms of C_{n}-graded algebras by

ρ̂r_{g} = ρ̂(g)r_{g},

for all g ∈ C_{n}, ρ̂ ∈ Ĉ_{n}, and the components of R can be recovered as

R_{σ^{j}} = {r ∈ R | σ̂r = ϵ^{j}r},

for j = 0, …, n − 1. We may form the skew group algebra

R ∗ Ĉ_{n} = {rρ̂ | r ∈ R, ρ̂ ∈ Ĉ_{n}},

with multiplication defined by

(rρ̂)(r′ρ̂′) = r · ρ̂r′ · ρ̂ρ̂′.

Proposition 2.3. The category R-Gr of C_{n}-graded (left) R-modules is isomorphic to the category R ∗ Ĉ_{n}-Mod.

Proof. Let M = ⨁_{g∈C_{n}} M_{g} be a C_{n}-graded R-module. Then M becomes an R∗Ĉ_{n}- module with multiplication defined by

(rρ̂)m_{g} = ρ̂(g)rm_{g},

for all r ∈ R, g ∈ C_{n}, m_{g} ∈ M_{g} and ρ̂ ∈ Ĉ_{n}. If f : M → M′ is a map of C_{n}-graded R-modules, then one easily verifies that f is also R ∗ Ĉ_{n}-linear.

Conversely, let M be an R ∗ Ĉ_{n}-module, and let

M_{σ^{j}} = {m ∈ M | σ̂m = ϵ^{j}m}.

Then every element m ∈ M is written uniquely as m = ∑_{j=0}^{n−1} m_{σ^{j}}, with m_{σ^{j}} ∈ M_{σ^{j}} for j = 0, …, n − 1, where

m_{σ^{j}} = n^{−1} ∑_{i=0}^{n−1}(ϵ^{−j}σ̂)^{i}m.

Again, it is a routine to verify that R_{g}M_{h} ⊆ M_{gh} for all g, h ∈ C_{n}, and if f : M → M′ is a map of R ∗ Ĉ_{n}-modules, then f(M_{g}) ⊆ M′_{g} for all g ∈ C_{n}. □

2.4. Let R and S be two C_{n}-graded O-algebras. Then Ĉ_{n} acts on R ⊗_{O} S^{op} diagonally, by

ρ̂(r ⊗ s) = ρ̂r ⊗^{ρ̂^{−1}}s,

for all ρ̂ ∈ Ĉ_{n}, r ∈ R and s ∈ S, so we may consider the skew group algebra (R ⊗_{O} S^{op}) ∗ Ĉ_{n}. As in Proposition 2.3, one can prove that the category R-Gr-S of C_{n}-graded (R, S)-bimodules is isomorphic to the category (R ⊗_{O} S^{op}) ∗ Ĉ_{n}-Mod.

If M is an (R, S)-bimodule and ρ̂ ∈ Ĉ_{n}, then the ρ̂-th conjugate ρ̂M of M is defined by

ρ̂M = (R ⊗_{O} S^{op})ρ̂ ⊗_{R⊗_{O}S^{op}} M.

Observe that we obtain an isomorphic (R, S)-bimodule if we set ρ̂M = M as Omodules, and multiplication

(r ⊗ s) ·ρ̂ m = ^{ρ̂^{−1}}(r ⊗ s) · m,

for all m ∈ M, r ∈ R, s ∈ S and ρ̂ ∈ Ĉ_{n}. It easily follows that if T is another C_{n}-graded O-algebra and N is an (S, T)-bimodule, then for all ρ̂ ∈ Ĉ_{n} there is an isomorphism

ρ̂(M ⊗ N) ≃ ρ̂N ⊗_{S} ρ̂N

of (R, T)-bimodules. 2.5. We are going to apply the above constructions to obtain a descent theorem for Rickard equivalences, which can also be regarded as an analogue of [4, Theorem 12.2].

Let G^{+} be a normal subgroup of the finite group G, with G/G^{+} ≃ C_{n}. Let b be a block of OG with defect group D ≤ G^{+}, let H = N_{G}(D), H^{+} = N_{G^{+}}(D), and let c ∈ OH be the Brauer correspondent of b. If e is a block of OG^{+} covered by b, then the Brauer correspondent f ∈ OH^{+} of e is covered by c, by the Harris-Knörr correspondence.

The group Ĉ_{n} acts on on the blocks of OG and OH, and for each ρ̂ ∈ Ĉ_{n}, the Brauer correspondent of ρ̂b is ρ̂c. We denote by Ĉ_{n,b} the stabilizer of b under this action. The group C_{n} acts by conjugation of the blocks of OG^{+} and OH^{+}, and for each g ∈ C_{n}, the Brauer correspondent of ^{g}e is ^{g}f. Let C_{n,e} denote the stabilizer of e in C_{n}. Consider the central idempotent

b^{+} = ∑_{ρ̂∈[Ĉ_{n}/Ĉ_{n,b}]} ρ̂b = ∑_{g∈[C_{n}/C_{n,e}]}^{g}e

of OG^{+}, where [C_{n}/C_{n,e}] denotes a full set of representatives for the left cosets of C_{n,e} in C_{n}. The second equality follows by [4, Lemma 9.9]. Let c^{+} be the similarly defined central idempotent of OH^{+}, and consider the strongly C_{n}-graded algebras R = b^{+}OG = OGeOG and S = c^{+}OH = OHeOH. Note that R is Morita equivalent to eOGe and S is Morita equivalent to fOHf.

With these notations, we have the following result, which is more general than we actually use in the case of alternating groups. Theorem 2.6. Let X be a complex of (bOG, cOH)-bimodules inducing a Rickard equivalence between bOG and cOH, and consider the complex

Y = ⨁_{ρ̂∈[Ĉ_{n}/Ĉ_{n,b}]} ρ̂X

of (R, S)-bimodules.

If ρ̂Y ≃ Y as complexes of (R, S)-bimodules for all ρ̂ ∈ Ĉ_{n}, then the block algebras eOG^{+} and fOH^{+} are Rickard equivalent.

Proof. Since Ĉ_{n} is cyclic of order prime to p, the assumption implies, by [6, Proposition 3.11] and [13, Lemma 2.13], that Y extends to a complex of (R ⊗_{O} S^{op}) ∗ Ĉ_{n}- modules, so by 2.4, Y is a complex of of C_{n}-graded (R, S)-bimodules. It follows by [7, Theorem 4.7] that the 1-component Y_{1} of Y is a Rickard tilting complex of (R_{1}, S_{1})-bimodules, hence eY_{1}f is a Rickard tilting complex of (eOG^{+}, fOH^{+})- bimodules. □

3. Blocks of symmetric and alternating groups

Let us state the main result of the paper. Theorem 3.1. Let p > 2, G = S_{n}, G^{+} = A_{n}, G̃ = Aut(G^{+}), b^{+} a block of OG^{+} with nontrivial abelian defect group D, H^{+} = N_{G^{+}}(D), and c^{+} ∈ OH^{+} the Brauer correspondent of b^{+}. Then there exists a splendid tilting complex of G̃/G^{+}-graded (b^{+}OG̃, c^{+}OH̃)-bimodules.

Actually, we only need to consider the case p > 2. Indeed, if p = 2, then by [5, Lemma (7.A)], D ≃ C_{2} × C_{2}. In this case Broué’s conjecture holds (even in the extended form) by [13, Section 6.3].

The proof of the theorem consists of several steps. 3.2. The block b^{+} is C_{2}-invariant. Let b be a block of OG covering b^{+} and let c ∈ OH be the Brauer correspondent of b. We denote σ̂b = b^{∗}, where C_{2} = ⟨σ̂⟩. If b ≠ b^{∗}, then bOG is isomorphic to b^{+}OG^{+} and cOH is isomorphic to c^{+}OH^{+}. Consequently, if X is a splendid tilting complex of (bOG, cOH)-bimodules, then X is also a splendid tilting complex of (b^{+}OG^{+}, c^{+}OH^{+})-bimodules. 3.3. Assume that b = b^{∗}, that is, b is self associated. Then b = b^{+}, c = c^{∗} = c^{+}, and bOG and cOH are strongly C_{2}-graded algebras. We can aply Theorem 2.6 if we show that the splendid equivalence constructed in [2] and [3] is induced by a complex of C_{2}-graded bimodules. As this equivalence is a composition of several equivalences, we shall examine the steps one by one.

Recall that the bloc b corresponds uniquely to a p-core κ and a p-weight w < p, and D ≃ C_{p} × ⋯ × C_{p} (w times). Write n = pw + t. Then, by [2, Section 3], the algebra cOH is isomorphic to ON_{S_{pw}}(D) ⊗_{O} OS_{t}c_{0}, where c_{0} is the block of defect zero of OS_{t} corresponding to the p-core κ. Recall also that since b is self associated, κ is also self associated, that is, its diagram is symmetric with respect to the main diagonal. Moreover, the block ON_{S_{pw}}(D) is isomorphic to O((C_{p} ⋊ C_{p−1}) ≀ S_{w}). 3.4. It was conjectured by R. Rouquier that there are blocks of weight w of symmetric groups which are Morita equivalent to the principal block B_{0}(S_{p} ≀ S_{w}) of O(S_{p} ≀ S_{w}). This conjecture was proved in [2, Section 4], where one of these blocks was defined as follows.

Consider an abacus having w+i(w−1) beads on the i-th runner, i = 0, 1, …, p−1, and let ρ be the p-core having this abacus representation. Note that the core ρ is self-associated.

Let V be a set containing the disjoint union U = U_{1}∪⋯∪U_{w} of sets of cardinality p, and let e be a block of OS(V) with defect group D corresponding to the p-core ρ. Let Ñ be the subgroup of S(U) consisting of permutations sending each U_{i} to some U_{j}, let N = Ñ × S(V \ U), and let f ∈ ON be the Brauer correspondent of e. Then the group Ñ is isomorphic to S_{p} ≀ S_{w}, and the algebra fON is isomorphic to B_{0}(S_{p} ≀ S_{w}) ⊗_{O} OS_{r}f_{0}, where f_{0} is the block of defect zero corresponding to the core ρ, and r = |V \ U|.

By [2, Theorem 2], the Green correspondent M of eOS(V) with respect to (S(V) × S(V), S(V) × N, δ(D)) induces a Morita equivalence between eOS(V) and fON. We claim that M is a C_{2}-graded (eOS(V), fON)-bimodule. Indeed, since e ∈ OA(V), M′ := Res_{S(V)×N}^{S(V)×S(V)}(eOS(V)) is C_{2}-graded, so M′ is an (eOS(V) ⊗_{O}

(fON)^{op}) ∗ Ĉ_{2}-module. Since M is the unique direct summand of M′ with vertex δ(D), it follows that M is Ĉ_{2}-invariant, hence M is an (eOS(V) ⊗_{O} (fON)^{op}) ∗ Ĉ_{2}- module. The claim now follows by 2.4. 3.5. To see that there is a C_{2}-graded Rickard equivalence between O((C_{p} ⋊ C_{p−1}) ≀ S_{w}) ⊗_{O} OS_{t}c_{0} and B_{0}(S_{p} ≀ S_{w}) ⊗_{O} OS_{t}c_{0}, note first that if R = R_{1} ⊕ R_{−1} and S = S_{1} ⊕ S_{−1} are C_{2}-graded algebras, then R ⊗_{O} S is C_{2}-graded, where

\begin{aligned}(R ⊗_{O} S)_{1} & = R_{1} ⊗_{O} S_{1} ⊕ R_{−1} ⊗_{O} S_{−1}, \\ (R ⊗_{O} S)_{−1} & = R_{1} ⊗_{O} S_{−1} ⊕ R_{−1} ⊗_{O} S_{1}. \end{aligned}

Moreover, the wreath product R ≀ S_{w} = R^{⊗w} ∗ S_{w} is C_{2}-graded by

deg(r_{1} ⊗ ⋯ ⊗ r_{w})σ = sgn(σ) deg r_{1} … deg r_{w},

where r_{1}, …, r_{w} ∈ R are homogeneous elements and σ ∈ S_{w}.

By [12] there is a Rickard equivalence between O(C_{p} ⋊C_{\frac{p−1}{2}} ) and B_{0}(A_{p}), which, by [6, Example 5.5], extends to a C_{2}-graded equivalence between O(C_{p} ⋊C_{p−1}) and B_{0}(S_{p}), induced by a complex X. Then by [6, Theorem 4.3], the complex X ≀ S_{w} induces a Rickard equivalence between O((C_{p} ⋊C_{p−1})≀S_{w}) and B_{0}(S_{p} ≀S_{w}). Recall that if x = x_{i_{1}} ⊗ ⋯ ⊗ x_{i_{w}} ∈ X^{⊗w}, x_{i_{j}} ∈ X_{i_{j}} then the action of σ ∈ S_{w} is defined by

σx = (−1)^{ϵ_{σ}(i_{1},...,i_{w})}x_{i_{σ^{−1}(1)}} ⊗ ⋯ ⊗ x_{i_{σ^{−1}(w)}}

for some ϵ_{σ}(i_{1}, …, i_{w}) ∈ {0, 1}. If r = r_{i_{1}} ⊗ ⋯ ⊗ r_{i_{w}} ∈ O(C_{p} ⋊ C_{p−1})^{⊗w} and s = s_{i_{1}} ⊗ ⋯ ⊗ s_{i_{w}} ∈ B_{0}(S_{p})^{⊗w} and ρ, σ, τ ∈ S_{w}, then the multiplication on X ≀ S_{w} is defined by

rσ · xρ · sτ = (r ·^{σ}x ·^{σρ}s)στρ.

Then the above discussion immediately shows that X ≀S_{w} is a complex of C_{2}-graded (O((C_{p} ⋊ C_{p−1}) ≀ S_{w}), B_{0}(S_{p} ≀ S_{w}))-bimodules. 3.6. Next we show that there is a C_{2}-graded Morita equivalence between the block c_{0}OS_{t} and f_{0}OS_{r} of defect zero.

We have that c_{0} ∈ OA_{t} and f_{0} ∈ OA_{r} since the p-cores κ and ρ are selfassociated, but these idempotents decompose as c_{0} = c′ + c′′ and f_{0} = f′ + f′′ in OA_{t} and OA_{r} respectively, where c′, c′′, respectively f′, f′′ are C_{2}-conjugated.

Let V′ be a (c′OA_{t}, f′OA_{r})-bimodule inducing a Morita equivalence. We may take V′ = U′ ⊗_{O} W′, where U′ is the unique simple left c′OA_{t}-module, and W′ is the unique simple right f′OA_{r}-module. Let V′′ = U′′ ⊗_{O} W′′, where U′′ and W′′ are the C_{2}-conjugates of U′ and W′ respectively. Then V := V′⊕V′′ is a (c_{0}OA_{t}⊗_{O} (f_{0}OA_{r})^{op})-module, invariant under the conjugation action of OS_{t} ⊗_{O} (OS_{r})^{op}. It follows that V extends to the diagonal subalgebra Δ = Δ(c_{0}OS_{t} ⊗_{O} (f_{0}OS_{r})^{op}), hence by [6, Theorem 3.4], Ind_{Δ}^{c_{0}OS_{t}⊗_{O}(f_{0}OS_{r})^{op}} V induces the desired C_{2}-graded Morita equivalence. 3.7. It remains to show that there is a C_{2}-graded derived equivalence between bOS_{n} and eOS(V). Rickard [10] has conjectured that any two blocks of the same weight w of symmetric groups are derived equivalent. He proposed a candidate for a tilting complex which is a generalization of Scopes’ Morita equivalence [14]. The conjecture has been recently verified by Chuang and Rouquier [3]. Actually, the derived equivalence between bOS_{n} and eOS(V) is obtained as a composition of equivalences between blocks forming a so called [w : k] pair, defined as follows.

Assume that aOS_{n} is a block of weight w of OS_{n} corresponding to an abacus whose j-th runner has k more beads than the (j − 1)-th runner. Switching the number of beads on these two runners, we obtain a block bOS_{n−k} of weight w of OS_{n−k}.

3.7.1. If k ≥ w, Scopes [14] proved that aOS_{n} and bOS_{n−k} are Morita equivalent. We deal with this situation first, since it already uses the main idea. Observe that M := aOS_{n}b is an (aOS_{n}, bOS_{n−k}⊗_{O}OS_{k})-bimodule. Then the Morita equivalence is induced by M ⊗_{OS_{k}} O. One of the problems is that usually a and b are not both self-associated. If a is not self-associated, the aOS_{n} is isomorphic to a^{+}OA_{n}, where a^{+} is the unique block of OA_{n} covered by a. Let A = aOS_{n} is a is self-associate, and let A = a^{+}OA_{n} if a is not self-associated. Similarly, let B = bOS_{n−k} if b is self-associated, and let B = b^{+}OA_{n−k} if b is not self-associated.

In any case, A and B are C_{2}-graded algebras, where if A = a^{+}OA_{n}, then A_{1} = A and A_{−1} = 0. As in 2.4, we may consider the σ̂-th conjugate

σ̂M = (A ⊗_{O} B^{op})σ̂ ≃ O^{−} ⊗_{O} M ⊗_{O} O^{−}

of M, where O^{−} denotes the sign representation. Then σ̂M can be identified with a^{∗}OS_{n}b^{∗} as (A, B)-bimodules. The above discussion shows that σ̂M is isomorphic to M as (A, B)-bimodules. Since Ĉ_{2} is cyclic, it follows that M extends to an (A ⊗_{O} B^{op}) ∗ Ĉ_{2}-module, and moreover, M is an ((A ⊗_{O} B^{op}) ∗ Ĉ_{2}, OS_{k})-bimodule. Consequently, M ⊗_{OS_{k}} O is an (A ⊗_{O} B^{op}) ∗ Ĉ_{2}-module, hence a C_{2}-graded (A, B)- bimodule by 2.4.

3.7.2. For arbitrary k, Rickard’s complex is a generalization of Scopes’ bimodule. We recall its construction following [10] and [2]. Let

r = max{i ∈ ℕ | i(k + i) ≤ w},

and for 0 ≤ i ≤ r let b_{i} be the block of OS_{n−k−i} having w −i(k +i) and represented by an abacus obtained from the abacus of b by moving i of the beads on the j-th runner onto the (j − 1)-th runner. Consider the (aOS_{n}, bOS_{n−k})-bimodule

Y_{i} = aOS_{n}b_{i} ⊗_{b_{i}OS_{n−k−i}} b_{i}OS_{n−k}b.

Using the map

b_{i−1}OS_{n−k−i+1}b_{i} ⊗_{b_{i}OS_{n−k−i}} b_{i}OS_{n−k−i+1}b_{i−1} → b_{i}OS_{n−k−i+1}

induced by multiplication, and the bimodule isomorphisms

\begin{aligned}aOS_{n}b_{i−1} ⊗_{b_{i−1}OS_{n−k−i+1}} b_{i−1}OS_{n−k−i+1}b_{i} & ≃ aOS_{n}b_{i}, \\ b_{i}OS_{n−k−i+1}b_{i−1} ⊗_{b_{i−1}OS_{n−k−i+1}} b_{i−1}OS_{n−k}b & ≃ b_{i}OS_{n−k}b, \end{aligned}

one obtains a map Y_{i} → Y_{i−1} of (aOS_{n}, bO_{n−k})-bimodules. In order to obtain a complex, the additional structure of these bimodules is needed. Let

X_{i} = (aOS_{n}b_{i} ⊗_{OS_{k+i}} O) ⊗_{b_{i}OS_{n−k−i}} (O^{−} ⊗_{OS_{i}} b_{i}OS_{n−k}b).

The map Y_{i} → Y_{i−1} induces a map X_{i} → X_{i−1} simply because S_{i−1} ≤ S_{i} and S_{k+i−1} ≤ S_{k+i}. By [2],

X := (⋯ → 0 → X_{r} → ⋯ → X_{1} → X_{0} → 0 → …)

is a splendid tilting complex of (aOS_{n}, bOS_{n−k})-bimodules.

Define A as in 3.7.1, and similarly let B_{i} = b_{i}OS_{n−k−i} if b_{i} is self-associated, and B_{i} = b^{+}OA_{n−k−i} if b_{i} is not self-associated. Observe that B_{0} = B. By the procedure described in 3.7.1, aOS_{n}b_{i} is a C_{2}-graded (A, B_{i})-bimodule, b_{i}OS_{n−k}b is a C_{2}-graded (B_{i}, B)-bimodule, so Y_{i} ≃ aOS_{n}b_{i} ⊗_{B_{i}} b_{i}OS_{n−k}b, the map Y_{i} → Y_{i−1} and X_{i} are C_{2}-graded. To se that the map X_{i} → X_{i−1} is also C_{2}-graded, let

\begin{matrix}X′_{i} := ((aOS_{n}b_{i−1} ⊗_{OS_{k+i−1}} O) & ⊗_{B_{i−1}} b_{i−1}OS_{n−k−i−1}b_{i})⊗_{B_{i}} \\ (b_{i}OS_{n−k−i+1}b_{i−1} & ⊗_{B_{i−1}} (O^{−} ⊗_{OS_{i−1}} b_{i−1}OS_{n−k}b)). \end{matrix}

It is clear that the map Y_{i} → Y_{i−1} induces a map X′_{i} → X_{i−1} of (A ⊗_{O} B^{op}) ∗ Ĉ_{2}- modules, and the obvious map X_{i} → X′_{i} is also (A ⊗_{O} B^{op}) ∗ Ĉ_{2}-linear. 3.8. It remains to see that the compatibility with p′-outer automorphism groups also holds. Fortunately, there is almost nothing to do. With the notations of 3.1, assume that b^{+} is the principal block of OG^{+} and b the principal block of OG.

Denoting G̃ = Aut(G^{+}) and H̃ = NG̃(D), we have that G ≤ G̃, and G = G̃ if n ≠ 6 and |G̃/G| = 2 if n = 6.

Let n ≠ 6. If b ≠ b^{∗}, then the algebras bOG and b^{+}OG^{+} are isomorphic, and in this case, the compatibility holds by [6, (5.4)]. If b = b^{∗} = b^{+}, then the required compatibility just means that there is a C_{2}-graded Rickard equivalence between bOG and cOH, and this is what we have proved above.

Finally, let n = 6, so |G^{+}| = 2^{3} · 3^{2} · 5. If p = 5, then there is a G̃/G^{+}-graded Rickard equivalence between b^{+}OG̃ and c^{+}OH̃ by [12] and [6, Example 5.5]. If p = 3, then D ≃ C_{3} × C_{3}. In this case Okuyama constructed in [9] (by using a different method) a Rickard equivalence between b^{+}OG^{+} and c^{+}OG^{+}, and this is compatible with p′-extensions by [8, Example 3.11].

Acknowledgments

This paper was written while the author was visiting the University of Chicago where he was supported by a Fulbright fellowship. He is grateful to J. Alperin, J. Chuang, P. Fong, R. Kessar and J. Rickard for many useful discussions on the subject.

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9. T. Okuyama, Some examples of derived equivalent blocks of finite groups, preprint 1998.

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14. J. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), 441–455.

“Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj-Napoca, Romania

E-mail address: marcus@math.ubbcluj.ro