SYMMETRIC GROUPS, WREATH PRODUCTS, MORITA

EQUIVALENCES, AND BROUÉ’S ABELIAN DEFECT GROUP CONJECTURE

JOSEPH CHUANG AND RADHA KESSAR

Abstract. It is shown that for any prime p, and any non-negative integer w less than p, there exist p-blocks of symmetric groups of defect w, which are Morita equivalent to the principal p-block of the group S_{p} ≀ S_{w}. Combined with work of J. Rickard, this proves that Broué’s abelian defect group conjecture holds for p-blocks of symmetric groups of defect at most 5.

1. Introduction

Broué has made some deep conjectures involving the derived categories of blocks of finite groups. One such is his Abelian Defect Group Conjecture [1]: a p-block of a finite group G with abelian defect group D and its Brauer correspondent in N_{G}(D) should be derived equivalent over a complete discrete valuation ring with residue field of characteristic p.

We show that this conjecture is true for a family of blocks of symmetric groups; this family is given by a certain combinatorial criterion and contains blocks whose defect groups have arbitrarily large rank (as long as we allow p to vary). Our work is the construction of Morita equivalences of blocks in this family and blocks of wreath products of symmetric groups. These equivalences, which were conjectured to exist by Rouquier, appear below in Theorem 2. When composed with certain derived equivalences described by Andrei Marcus in [8] they produce derived equivalences predicted by Broué’s conjecture.

We construct Morita equivalences only for blocks in the aforementioned family, but taken together with unpublished work of Jeremy Rickard they imply that Broué’s conjecture holds for all p-blocks of symmetric groups whose defect groups have order less than or equal to p^{5} (Corollary 3 below). This generalizes the main result of [3], in which the same was proved for blocks whose defect groups have order less than or equal to p^{2}.

2. Preliminaries

Let p a prime number and let O be a complete discrete valuation ring with maximal ideal J such that its residue field k = O/J has characteristic p and is algebraically closed, and its fraction field K has characteristic 0. To indicate reduction modulo J of elements of O or of group algebras over O, we use the conventional bar notation. We will write ⊗ in place of ⊗_{O}. Let G be a finite group. We state the Abelian Defect Group Conjecture of Broué:

1991 Mathematics Subject Classification. 20Cxx.

Conjecture 1. (Broué) Let b be a block idempotent of OG with abelian defect group D, and let c be the Brauer correspondent block idempotent of ON_{G}(D). Then OGb and ON_{G}(D)c are derived equivalent, i.e., there is an equivalence of triangulated categories between their derived module categories.

We will be making use of the Brauer quotient and the Brauer homomorphism, which we briefly describe (see [2] for more details). Let M be an OG-module and P a p-subgroup of G. The Brauer quotient

M(P) = M^{P} / ( ∑_{Q<P} Tr_{Q}^{P}(M^{Q}) + J M^{P} )

is defined by taking the quotient of the P -fixed points of M by relative traces from proper subgroups and then passing to the residue field; M(P) is a kN_{G}(D)-module. If M is a summand of a permutation module, then M(P) ≠ 0 if and only if M has a direct summand which has a vertex containing P. In the case that M = OG and G acts by conjugation, we have OG(P) = kC_{G}(P), and the natural quotient map

Br_{P}^{G} : (OG)^{P} → kC_{G}(P),

which may be given by the rule

Br_{P}^{G} (∑_{g∈G} a_{g}g) = ∑_{g∈C_{G}(P)} \overline{a_{g}}g,

is a homomorphism of O-algebras, called the Brauer homomorphism with respect to P. If H is a subgroup of G containing P, and i ∈ (OG)^{H} is an idempotent, then OGi is a summand of OG as an O(G × H)-module, and writing ΔP for the diagonally embedded subgroup {(x, x) | x ∈ P } ≤ G × H, we have that (OGi)(ΔP) and kC_{G}(P) Br_{P}^{G}(i) are isomorphic as kN_{G×H}(ΔP)-modules.

We introduce some combinatorics relating to partitions. Removing rim phooks from the diagram of a partition λ we obtain the diagram of a partition which is called the p-core of λ; it is independent of the manner in which the hooks are removed. The number of hooks removed is called the p-weight of λ. A partition with p-weight 0 is called a p-core. For our purposes it is useful to represent partitions on an abacus with p runners, following Gordon James [6, pages 77-78 ]. We label the runners of the abacus 0, …, p−1, from left to right, and label its rows 0, 1, …, from the top down. If λ = (λ_{1}, λ_{2}, …) is a partition with less than or equal to m nonzero parts then we may represent λ on the abacus using m beads: for i = 1, …, m, we write λ_{i}+m−i = s+pt, with 0 ≤ s ≤ p − 1, and place a bead on the s-th runner in the t-th row. Sliding a bead up one row on its runner into a previously vacant position corresponds to removing a p-hook. Thus sliding all the beads in an abacus representation of a partition up their runners as far as possible produces an abacus representation of the p-core of that partition.

Fix an abacus representation of λ, and for i = 0, …, p − 1, let λ_{1}^{(i)} be the number of unoccupied positions on the i-th runner which occur above the bottommost bead on that runner, let λ_{2}^{(i)} be the number of unoccupied positions on the i-th runner which occur above the next to bottommost bead on that runner, and so on. Then λ^{(i)} = (λ_{1}^{(i)}, λ_{2}^{(i)}, …) is a partition, and the p-tuple (λ^{(0)}, …, λ^{(p−1)}) is called the p-quotient of λ. We note that it depends on the number of beads used in the abacus representation. The weight of λ is equal to |λ^{(0)}| + … + |λ^{(p−1)}| .

Given a p-core κ and a nonnegative integer w, consider partitions with p-core κ and p-weight w. Choosing m so that any such partition has less than or equal to m nonzero parts and representing these partitions on an abacus with m beads, we have a p-quotient for each one. This gives a bijection between this set of partitions and the set of p-tuples (σ_{0}, …, σ_{p−1}) of partitions satisfying |σ^{(0)}| + … + |σ^{(p−1)}| = w.

3. Blocks of symmetric groups

For any nonnegative integer n denote by S_{n} the symmetric group of degree n, and for any finite set V denote by S(V) the symmetric group on V. The irreducible characters χ^{λ} of KS_{n} (or of KS(V), if |V | = n) are indexed by partitions λ of n. Nakayama’s rule (see, e.g., [6, page 245, Theorem 6.1.20]) states that two characters are in the same p-block of KS(V) if and only if the associated partitions have the same p-core. Hence the p-blocks of characters of KS(V) are labelled by p-cores, and all the partitions associated to characters in a given p-block have the same p-weight. Thus it makes sense to speak of the weight of a p-block.

Let e be the primitive central idempotent of OS(V) corresponding to a p-block which has weight w and is labelled by a p-core κ. Let U be a subset of V of cardinality pw and let D be a Sylow p-subgroup of S(U). Then D is a defect group of OS(V)e (see, e.g., [6, page 263, Theorem 6.2.39]). Thus OS(V)e has abelian defect groups if and only if w < p.

As our main interest is Conjecture 1, we will consider only blocks with abelian defect groups; we will therefore assume that w < p for the remainder of this paper.

We have

C_{S(V)}(D) = D × S(V − U)

and

N_{S(V)}(D) = N_{S(U)}(D) × S(V − U).

A theorem of Puig [9] tells us that

Br_{D}^{S(V)}(e) = 1_{kD} ⊗ \overline{e_{0}}, (1)

where e_{0} is the block idempotent of OS(V − U) corresponding to the pblock labelled by the p-core κ. Thus the Brauer correspondent of OS(V)e in N_{G}(D) is

ON_{S(U)}(D) ⊗ OS(V − U)e_{0}.

This block is Morita equivalent to ON_{S(U)}(D) because OS(V − U)e_{0} is a block of weight 0 and therefore defect 0. The block ON_{S(U)}(D) is in turn derived equivalent to the principal block of O(S_{p} ≀ S_{w}), by a result of Andrei Marcus [8, Example 5.7] based on the existence of derived equivalences for blocks with cyclic defect groups, which was proved by Rickard in [11]. Hence the Brauer correspondent of any p-block of weight w of a symmetric group is derived equivalent to the principal block of O(S_{p} ≀ S_{w}). We now state our result, which was originally conjectured by Rouquier.

Theorem 2. Given any w < p there exist blocks of weight w of symmetric groups which are Morita equivalent to the principal block of O(S_{p} ≀ S_{w}). In particular there exist blocks of weight w for which Broué’s Abelian Defect Group Conjecture holds.

Jeremy Rickard, in an important unpublished theorem (announced in [12]), which built on the work of Scopes in [15], constructed derived equivalences of blocks; he proved in particular that for w ≤ 5 (no restriction on p), all p-blocks of symmetric groups of weight w are derived equivalent. In light of this result and the comments above, our theorem has the following corollary.

Corollary 3. Broué’s Abelian Defect Group Conjecture (Conjecture 1) is true for all blocks of symmetric groups of weight less than or equal to 5.

Remark 3.1. The proof given here is an improvement over that in [3], in which the case w = 2 was handled using a more convoluted argument.

4. Proof of the theorem

Let ρ be a p-core which satifies the following property: ρ has an abacus representation in which each runner other than the leftmost one (the 0-th runner) has at least w − 1 more beads than the runner to its immediate left. Having chosen such a ρ, we may assume that there are at least w beads on each runner; let m be the number of beads in such an abacus representation of ρ. We will be considering partitions with p-core ρ and p-weight v ≤ w. Any such partition can be represented on an abacus with m beads and we do so. We remark that one candidate for ρ is the p-core which has an abacus display with w + i(w − 1) beads on the i-th runner, for i = 0, …, p − 1; this is the smallest ρ possible.

In the second part of the following lemma we give the key property of ρ. If μ and λ are partitions then by μ ⊆ λ we mean that μ_{1} ≤ λ_{1}, μ_{2} ≤ λ_{2}, and so on.

Lemma 4. 1. Let λ be a partition with p-core ρ and weight v ≤ w. Suppose that in the (m-bead) abacus representation of λ there is a bead on the s-th runner in the t-th row. Then on any runner to the right of the s-th runner there are beads in all rows above the t-th row. If v < w, then on any runner to the right of the s-th runner there are beads in all rows on or above the t-th row.

2. Let λ be a partition with p-core ρ and weight v ≤ w, and let μ be a partition with p-core ρ and weight v − 1. If μ ⊆ λ then there exists s with 0 ≤ s ≤ p − 1 such that μ^{(i)} = λ^{(i)} for i ≠ s and μ^{(s)} ⊆ λ^{(s)} with |μ^{(s)}| = |λ^{(s)}|−1. Moreover the complement of the Young diagram of μ in that of λ is the Young diagram of the hook partition (s + 1, 1^{p−s−1}).

Proof. 1. Let 0 ≤ s < s′ ≤ p − 1. Let m_{s} and m_{s′} be the number of beads on the s-th and s′-th runners in the abacus representation of λ. All the beads on the s-th runner lie in the first m_{s} + |λ^{(s)}| rows, while on the s′-th runner there is a bead in each the first m_{s′} − |λ^{(s′)}| rows. We have

(m_{s} + |λ^{(s)}|) − (m_{s′} − |λ^{(s′)}|) = m_{s} − m_{s′} + |λ^{(s)}| + |λ^{(s′)}|
≤ −(w − 1) + v,

which implies the desired result.

2. For i = 1, …, m set α_{i} = λ_{i} + m − i and β_{i} = μ_{i} + m − i. Choose n such that β_{n} < α_{n} and β_{i} = α_{i} for i > n. Write β_{n} = s + pt where 0 ≤ s ≤ p − 1. Then in the abacus representation of μ there is a bead on the s-th runner in row t, while there is no bead in the corresponding spot in the abacus representation of λ. Because the abacus representations of λ and μ have the same number of beads in each runner (as they have the same p-core), there must be a bead on the s-th runner of the abacus representation of λ below the t-th row. By the first part of the lemma, we see that in the abacus representations of μ and λ, there is a bead in the t-th row on each runner to the right of the s-th runner, and that in the abacus representation of λ, there is no bead in the (t + 1)-st row on each runner to the left of the s-th runner. Hence we have

\begin{matrix}β_{n} = s + pt, & α_{n} = (s + 1) + pt; \\ β_{n−1} = (s + 1) + pt, & α_{n−1} = (s + 2) + pt; \\ ⋮ & ⋮ \\ β_{n−(p−s−2)} = (p − 2) + pt, & α_{n−(p−s−2)} = (p − 1) + pt; \\ β_{n−(p−s−1)} = (p − 1) + pt, & α_{n−(p−s−1)} ≥ s + p(t + 1). \end{matrix}

Therefore

μ_{n} = λ_{n} − 1;
μ_{n−1} = λ_{n−1} − 1;
⋮
μ_{n−(p−s−2)} = λ_{n−(p−s−2)} − 1;
μ_{n−(p−s−1)} ≤ λ_{n−(p−s−1)} − (s + 1).

As μ ⊆ λ and |μ| = |λ| − p, the inequality must be equality, and we also must have μ_{i} = λ_{i} and β_{i} = α_{i} for i < n − (p − s − 1). Thus the abacus representation of λ is obtained from that of μ by moving a bead on the s-th runner from the t-th row to the (t + 1)-th row, and all the statements in the lemma hold.

Let V be a set of cardinality pw + r, let U_{1}, …, U_{w} be disjoint subsets of V of cardinality p, and let U be the union of these subsets. In what follows, all groups we consider will be viewed as subgroups of S(V) in an obvious way. For i = 1, …, w, let D_{i} be a Sylow p-subgroup of S(U_{i}), and let a_{i} be the principal block idempotent of OS(U_{i}). For i = 0, …, w let e_{w−i} be the block idempotent of OS(U_{i+1} ∪ … ∪ U_{w} ∪ (V − U)) corresponding to the p-core ρ, let

G_{i} = S(U_{1}) × ⋯ × S(U_{i}) × S(U_{i+1} ∪ … ∪ U_{w} ∪ (V − U)) and let

b_{i} = a_{1} ⊗ ⋯ ⊗ a_{i} ⊗ e_{w−i},

a block idempotent of OG_{i}. We have

G_{i} ≅ \underbrace{S_{p} × ⋯ × S_{p}}_{i} ×S_{(w−i)p+r}.

We set G = G_{0}, b = b_{0}, L = G_{w}, and f = b_{w}. Letting D = D_{1} × ⋯ × D_{w}, we have

C_{G}(D) = D × S(V − U) = D_{1} × ⋯ × D_{w} × S(V − U).

Let Ñ be the subgroup of S(U) consisting of permutations sending each U_{i} into some U_{j}; we note that Ñ is isomorphic to the wreath product S_{p} ≀ S_{w}. Set N = Ñ × S(V − U), a subgroup of G containing N_{G}(D) and L and normalizing L.

Lemma 5. 1. For 0 ≤ i ≤ p − 1, we have that D is a defect group of

OG_{i}b_{i};

2. For 0 ≤ i ≤ p − 1, we have Br_{D}^{G}(b_{i}) = 1_{kD} ⊗ \overline{e_{0}};

3. We have that N stabilizes f, and ONf is a block which is Morita equivalent to the principal block of O (S_{p} ≀ S_{w}) ;

4. As an O(N ×L)-module, ONf is indecomposable with vertex ΔD, and rank_{O}(ONf) = w! · rank_{O}(OLf).

Proof. 1. We have that D_{i+1}×⋯×D_{w} is a defect group of OS(U_{i+1}∪…∪ U_{w} ∪ V)e_{w−i}, as this block has weight w − i < p, while for 0 ≤ j ≤ i, we have that D_{j} is a defect group of OS(U_{j})a_{j}. Hence D is a defect group of OG_{i}b_{i}.

2. First we note that G_{i} ≥ C_{G}(D), so that Br_{D}^{G}(b_{i}) = Br_{D}^{G_{i}}(b_{i}). Next, for 0 ≤ j ≤ i, we have Br_{D_{j}}^{S(U_{i})}(a_{j}) = 1_{kD_{j}}, and by Puig’s result (1), we have

Br_{D_{i+1}×···×D_{w}}^{S(U_{i+1}∪...∪U_{w}∪(V −U))}(e_{w−i}) = 1_{k(D_{i+1}×···×D_{w})} ⊗ \overline{e_{0}}.

Hence

Br_{D}^{G}(b_{i}) = Br_{D}^{G_{i}}(b_{i})
= Br_{D_{1}}^{S(U_{1})}(a_{1}) ⊗ ⋯ ⊗ Br_{D_{i}}^{S(U_{i})}(a_{i}) ⊗ Br_{D_{i+1}×···×D_{w}}^{S(U_{i+1}∪...∪U_{w}∪(V −U))}(e_{w−i})
= 1_{OD} ⊗ \overline{e_{0}}.

3. It is clear that N stabilizes f, and because a_{1} ⊗ ⋯ ⊗ a_{i} is the principal block idempotent of OÑ and e_{0} is a block idempotent of OS(V − U) corresponding to a block of defect 0, the second statement holds.

4. By part 1 of the Lemma, OLf has vertex ΔD. An easy argument using the fact that C_{G}(D) ⊆ L shows that a conjugate of ΔD by an element of N × L not in L × L is not conjugate to ΔD in L × L. Consequently, the stabilizer of OLf in N × L is just L × L, so ONf = Ind_{L×L}^{N×L}(OLf) is indecomposable and has vertex ΔD. As [N : L] = w!, the statement relating ranks is clear.

The lemma shows that the blocks OGb and ONf both have defect group D and are Brauer correspondents. Thus the O(G×G)-module OGb and the O(N × N) -module ONf both have vertex ΔD and are Green correspondents. Let X be the Green correspondent of OGb in G × N, so that X is an indecomposable O(G×N)-module with vertex ΔD, and X is isomorphic to a direct summand of Res_{G×N}^{G×G}(OGb). Because ONf is isomorphic to a direct summand of Res_{N×N}^{G×N}(X), we have Xf ≠ 0, so Xf = X and X is a (OGb, ONf)-bimodule.

Our goal is to show that X induces a Morita equivalence of OGb and ONf. As OGb is a block of a symmetric group of weight w, and ONf is Morita equivalent to the principal block of O (S_{p} ≀ S_{w}) by the lemma, this will imply that Theorem 2 is true.

As an OGb-module X is a progenerator; indeed, OGb is isomorphic to a direct summand of Ind_{G×N}^{G×G}(X), and applying Mackey’s formula we get that OGb is isomorphic to a direct summand of a direct sum of copies of Res_{G×1}^{G×N}(X). Hence by Morita theory, to prove that X induces an Morita equivalence, it suffices to show that the homomorphism

ONf → End_{G}(X)

induced by the action of ONf on X is an isomorphism. Note that it is at least a split monomorphism of (ONf, ONf)-bimodules, because ONf is isomorphic to a direct summand of Res_{N×N}^{G×N}(X). Thus we need only to show that

rank_{O}(End_{G}(X)) ≤ rank_{O}(ONf)

or equivalently, by part 4 of Lemma 5,

(2) rank_{O}(End_{G}(X)) ≤ w! · rank_{O}(OLf).

Now Res_{G×L}^{G×N}(X) is a direct sum of indecomposable modules whose vertices are conjugates of ΔD; indeed, ΔD is a vertex of X and G × L is a normal subgroup of G × N containing ΔD. On the other hand, Res_{G×L}^{G×N}(X) ≅ X ⊗_{N} ON ≅ X ⊗_{N} ONf is a summand of Ind_{N×L}^{G×L}(ONf), and by the Green correspondence the latter is a direct sum of an indecomposable module with vertex ΔD and indecomposable modules with strictly smaller vertices, as ONf has vertex ΔD and N_{G×L}(ΔD) ⊆ N × L. Therefore Res_{G×L}^{G×N}(X) is an indecomposable module with vertex ΔD.

Each of the b_{i} is an idempotent contained in (OG)^{L}; as these idempotents commute with each other, their product is also an idempotent contained in (OG)^{L}. Hence Y = OGb_{0} ⋯ b_{w} is a summand of OGb_{0} as an O(G × L)- module, and we have

Y (ΔD) = OGb_{0} ⋯ b_{w}(ΔD)
≅ kC_{G}(D) Br_{D}^{G}(b_{0} ⋯ b_{w})
= kC_{G}(D) Br_{D}^{G}(b_{0}) ⋯ Br_{D}^{G}(b_{w})
= kC_{G}(D) Br_{D}^{G}(b)
≅ OGb(ΔD),

where the last equality is a consequence of part 2 of Lemma 5. We have OGb ≅ Y ⊕ E, for some O(G × L)-module E. Now OGb(ΔD) ≅ Y (ΔD) ⊕ E(ΔD), and it follows that E(ΔD) = 0. As E is a direct summand of the permutation module OG, this implies that E does not have an indecomposable summand with vertex ΔD. Hence any indecomposable direct summand of OGb as an O(G×L)-module which has vertex ΔD must be isomorphic to a direct summand of Y. Thus Res_{G×L}^{G×N}(X) is isomorphic to a direct summand of Y; consequently there exists an injective ring homomorphism

End_{G}(X) ↪ End_{G}(Y).

Hence in order to show that the inequality (2) holds it suffices to demonstrate that

(3) rank_{O}(End_{G}(Y)) ≤ w! · rank_{O}(OLf).

This will be accomplished by making a calculations with characters.

The endomorphism ring End_{G}(Y) may be identified with b_{w} ⋯ b_{0}OG_{0}b_{0} ⋯ b_{w}. The K-algebra KG_{0}b_{0} is semisimple, so by general theory K ⊗_{O} End_{G}(Y) = b_{w} ⋯ b_{0}KG_{0}b_{0} ⋯ b_{w} is semisimple, and the restriction of any irreducible character of KG_{0}b_{0} to b_{w} ⋯ b_{0}KG_{0}b_{0} ⋯ b_{w} is either an irreducible character or the zero character. Moreover each irreducible character of b_{w} ⋯ b_{0}KG_{0}b_{0} ⋯ b_{w} arises exactly once in this way. Thus

(4)  rank_{O}(End_{G}(Y)) = ∑_{χ∈Irr(KG_{0}b_{0})} χ(b_{w} ⋯ b_{0})^{2}.

We note that

(5) χ(b_{w} ⋯ b_{0}) = b_{w} Res_{G_{w}}^{G_{w−1}}(⋯ b_{2} Res_{G_{2}}^{G_{1}}(b_{1} Res_{G_{1}}^{G_{0}}(χ)) ⋯)(1).

The principal p-block of KS_{p} is the block labelled by the empty p-core. So for i = 1, …, w, we have Irr(KS(U_{i})a_{i}) = {ν_{0}, ⋯, ν_{p−1}}, where for j = 0, …, p − 1, we write

ν_{j} = χ^{(j+1,1^{p−j−1})}

for the sake of convenience. Then for i = 0, …, w, we have

Irr(KG_{i}b_{i}) = {ν_{j_{1}} × ⋯ × ν_{j_{i}} × χ^{λ}} ,

where the j_{γ}’s run from 0 to p − 1 and λ runs over the set of partitions with p-core ρ and p-weight w − i.

Let C be the set of p-tuples c = (c_{0}, …, c_{p−1}) of nonnegative integers satisfying c_{0} + ⋯ + c_{p−1} = w. For i = 0, …, w and c ∈ C, we define Irr(KG_{i}b_{i}, c) to be the set of characters ν_{j_{1}} × ⋯ × ν_{j_{i}} × χ^{λ} ∈ Irr(KG_{i}b_{i}) such that for j = 0, …, p − 1, we have

|λ^{(j)}| + # {γ | j_{γ} = j} = c_{j}.

For i = 0, …, w we have a disjoint union

Irr(KG_{i}b_{i}) = ∐_{c∈C} Irr(KG_{i}b_{i}, c).

Note that for a fixed c ∈ C, there are \frac{w!}{c_{0}!···c_{p−1}!} characters in Irr(KG_{w}b_{w}, c), and all have the same degree.

Suppose that 0 ≤ i ≤ w − 1 and χ = ν_{j_{1}} × ⋯ × ν_{j_{i}} × χ^{λ} ∈ Irr(KG_{i}b_{i}). Then using the Littlewood-Richardson rule (see, e.g., [6, page 93, Theorem 2.8.13]) and part 2 of Lemma 4 we get

e_{i+1} Res_{G_{i+1}}^{G_{i}}(χ) = ∑_{s=0}^{p−1} ∑_{μ} ν_{j_{1}} × ⋯ × ν_{j_{i}} × ν_{s} × χ^{μ}, (6)

where the second sum runs over all partitions μ with p-core ρ and p-weight w−(i+1) such that μ^{(j)} = λ^{(j)} for j ≠ s and μ^{(s)} ⊆ λ^{(s)} with |μ^{(s)}| = |λ^{(s)}|− 1 . Note that if χ ∈ Irr(KG_{i}b_{i}, c) then every constituent of e_{i+1} Res_{G_{i+1}}^{G_{i}}(χ) is in Irr(KG_{i+1}b_{i+1}, c).

Now fix χ^{λ} ∈ Irr(KG_{0}b_{0}, c). For each φ ∈ Irr(KG_{w}b_{w}, c), we may apply (6) repeatedly to deduce that the multiplicity of φ as a constituent of

b_{w} Res_{G_{w}}^{G_{w−1}}(⋯ b_{2} Res_{G_{2}}^{G_{1}}(b_{1} Res_{G_{1}}^{G_{0}}(χ)) ⋯)

is

f(λ^{(0)}) · . . . · f(λ^{(p−1)}),

where for any partition σ of n we denote by f(σ) the number of nested sequences τ_{1} ⊆ ⋯ ⊆ τ_{n} of partitions such that |τ_{i}| = i for i = 1, …, n, and τ_{n} = σ. It follows from the branching rule (see, e.g., [6, page 59, Theorem 2.4.3]) that f(σ) is the dimension of the irreducible character χ^{σ} of S_{n}. It is now immediate that

∑_{σ⊢n} f(σ)^{2} = n!,

and we will use this fact below. Now using equation (5) we have

χ^{λ}(b_{w} ⋯ b_{0}) = f(λ^{(0)}) · . . . · f(λ^{(p−1)}) · ∑_{φ∈Irr(KG_{w}b_{w},c)} φ(1). (7)

Squaring both sides of this equation and summing over all characters in Irr(KG_{0}b_{0}, c) we get

∑_{χ^{λ}∈Irr(KG_{0}b_{0},c)} χ^{λ}(b_{w} ⋯ b_{0})^{2}

= ∑_{χ^{λ}∈Irr(KG_{0}b_{0},c)} f(λ^{(0)})^{2} · . . . · f(λ^{(p−1)})^{2} · (  ∑_{φ∈Irr(KG_{w}b_{w},c)} φ(1))^{2}

= ( ∑_{μ_{0}⊢c_{0}} f(μ_{0})^{2}) · . . . · ( ∑_{μ_{p−1}⊢c_{p−1}} f(μ_{p−1})^{2}) ·

·\frac{w!}{c_{0}! ⋯ c_{p−1}!} · ( ∑_{φ∈Irr(KG_{w}b_{w},c)} φ(1)^{2})

= w! · ∑_{φ∈Irr(KG_{w}b_{w},c)} φ(1)^{2},

and in turn summing this over all c ∈ C and using equation (4) we get

rank_{O}(End_{G}(Y)) = w! · ∑_{φ∈Irr(KG_{w}b_{w})} φ(1)^{2}.

Remembering that L = G_{w} and f = b_{w} we see that the right hand side of this equation is equal to w! · rank_{O}(OLf) and that therefore the inequality (3) holds. This completes the proof of Theorem 2.

5. Splendid Rickard Equivalences

For blocks of group algebras there is an important strengthening of the notion of derived equivalence which is called splendid Rickard equivalence. This notion was introduced by Rickard [13] for principal blocks and later extended to the case of arbitrary blocks by Harris [5] and also by Linckelmann [7]. In the situation of Conjecture 1 it is hoped that there exists this strengthened form of a derived equivalence. In this section we give some pertinent definitions and explain briefly why we have not only derived equivalences but splendid Rickard equivlances in Theorem 2.

Let G and H be a finite groups and let b and c be block idempotents of OG and OH. The corresponding blocks OGb and OHc are called Rickard equivalent if there exists a bounded complex C of (OGb, OHc)-bimodules, each projective as left OGb-module and as right OHc-module, such that End_{OGb}(C) and OHc are homotopy equivalent as complexes of (OHc, OHc)- bimodules, and End_{OHc}(C) and OGb are homotopy equivalent as complexes of (OGb, OGb)-bimodules. In this situation C is called a Rickard complex and the functors C⊗_{OHc}? and Hom_{OGb}(C,?) induce inverse equivalences of the derived bounded categories D^{b}(OGb) and D^{b}(OHc). Of course, if C has only one nonzero term then OGb and OHc are Morita equivalent.

Suppose that OGb and OHc have a common defect group D and that each term of C, considered as an O(G × H)-module, is a direct summand of a permutation module and is relatively ΔD-projective, where ΔD is the diagonal subgroup {(x, x) | x ∈ D} ≤ G×H. Then we say that C is splendid and that OGb and OHc are splendidly Rickard equivalent. If C has only one nonzero term we say that OGb and OHc are splendidly Morita equivalent. Note that in this case, we get that a source algebra of OGb is isomorphic as interior D-algebra to a source algebra of OHc (see, e.g., [10, Remark 7.5]).

Here is a simple situation in which we have a splendid Morita equivalence:

Lemma 6. Let G_{1} and G_{2} be finite groups, and let b_{1} and b_{2} be block idempotents of OG_{1} and OG_{2}. Then if OG_{2}b_{2} has defect group 1, then OG_{1}b_{1} and OG_{1}b_{1} ⊗ OG_{2}b_{2} (a block of G_{1} × G_{2}) are splendidly Morita equivalent.

Proof. Let i be a primitive idempotent in OG_{2}b_{2}. Then the O((G_{1} × G_{2}) × G_{1})-module OG_{1}b_{1} ⊗ OG_{2}i is a summand of a permutation module and has vertex ΔD, where D is a defect group of OG_{1}b_{1}. Furthermore OG_{1}b_{1}⊗OG_{2}i induces a Morita equivalence between OG_{1}b_{1} and OG_{1}b_{1} ⊗ OG_{2}b_{2}.

In order to describe a situation in which a composition of two splendid Rickard equivalences produces again a splendid Rickard equivalence, we consider the control of fusion of p-subgroups. For D any p-subgroup of G, define F(G, D) to be the set of homomorphisms ψ : P → D, where P is any subgroup of D for which there exists g ∈ G such that ψ(x) = gxg^{−1} for all x ∈ P.

Lemma 7. Let G, H, and L be finite groups with a common p-subgroup D, and let b, c, and f be block idempotents of OG, OH, and OL with defect group D. Suppose that OGb and OHc are splendidly Rickard equivalent and that OHc and OLf are splendidly Rickard equivalent. If either F(H, D) ⊆ F(G, D) or F(H, D) ⊆ F(L, D), then OGb and OLf are splendidly Rickard equivalent.

Proof. See, e.g., [3, Lemma 8.3].

We now return to the notation used in section 4 and assume that w ≤ 5. Consider an arbitrary p-block of the symmetric groups which has weight w. We may take this block to be OS(V′)e, where V′ is a finite set containing U and e is a block idempotent of OS(V′) corresponding to a p-core κ. Then D is a defect group of OS(V′)e. We have that

N_{S(V′)}(D) = N_{S(U)}(D) × S(V′ − U),

and the Brauer correspondent of OS(V′)e in N_{S(V′)}(D) is

ON_{S(U)}(D) ⊗ OS(V′ − U)e′_{0},

where e′_{0} is the block idempotent of OS(V′ −U) corresponding to the p-core κ. We consider the following pairs of blocks:

1. OS(V′)e and OGb;

2. OGb and ONf;

3. ONf and the principal block of OÑ;

4. the principal block of OÑ and ON_{S(U)}(D);

5. ON_{S(U)}(D) and ON_{S(U)}(D) ⊗ OS(V′ − U)e′_{0}.

In his unpublished work ([12]) Rickard proved that the blocks in the first pair are splendidly Rickard equivalent. Andrei Marcus [8] showed that the blocks in the fourth pair are splendidly Rickard equivalent based on the existence of splendid Rickard equivalences for blocks with cyclic defect groups, proved by Rouquier in [14]. The blocks in the third pair, and those in the 5th pair are splendidly Morita equivalent by Lemma 6. Finally the blocks in the second pair are splendidly Morita equivalent; indeed, in section 4 we showed that an equivalence is induced by a bimodule X which has vertex ΔD and is a summand of a permutation module, namely OG.

We can use Lemma 7 to combine these splendid equivalences, as long as we can show that F(G_{1}, D) ⊆ F(G_{2}, D) for any groups G_{1} and G_{2} appearing in the list above. We may assume that G_{1} and G_{2} are subgroups of S(W) which contain N_{S(U)}(D), where W is a finite set containing U. Suppose P is a p-subgroup of D and g_{1} an element of G_{1} such that g_{1}P g_{1}^{−1} ⊆ D. Let T be the subset of U consisting of elements not fixed by P. Then because g_{1}P g_{1}^{−1} ⊆ D we have that g_{1}(T) ⊆ U. Thus there exists ω ∈ S(U) such that ω(t) = g_{1}(t) for all t ∈ T. Then we have ωxω^{−1} = g_{1}xg_{1}^{−1} for all x ∈ P. Now as D is a Sylow p-subgroup of S(U) and is abelian, a theorem of Burnside (see, e.g., [4, page 240, Theorem 1.1]) tells us that there exists τ ∈ N_{S(U)}(D) such that τxτ^{−1} = ωxω^{−1} for all x ∈ P. Thus we have τxτ^{−1} = g_{1}xg_{1}^{−1} for all x ∈ P, where τ ∈ G_{2}. So indeed F(G_{1}, D) ⊆ F(G_{2}, D).

We have shown then that OS(V′)e and its Brauer correspondent are splendidly Rickard equivalent. So we indeed get a splendid Rickard equivalence in Theorem 1.

References

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St John’s College, Oxford OX1 3JP, United Kingdom

E-mail address: chuang@maths.ox.ac.uk

University College, Oxford OX1 4BH, United Kingdom

E-mail address: kessar@maths.ox.ac.uk