On the Equivalence of Some Conjectures of Dade and Robinson Charles W. Eaton ^{1} School of Mathematics and Statistics University of Birmingham Birmingham B15 2TT England email: eatonc@for.mat.bham.ac.uk

Abstract

We demonstrate that Conjecture 4.1 of [7] and Dade’s projective conjecture are equivalent in a way which is compatible with the p-local rank. In addition we show that in order to verify Robinson’s conjecture it suffices to consider only those groups with no non-central normal p-subgroup.

1 Introduction

We consider two variants of Alperin’s weight conjecture representing two different approaches to the problem. Both imply the weight conjecture and are refinements of the conjectures given in [4]. Conjecture 4.1 of [7] (which we shall refer to as ‘Robinson’s conjecture’) arises from the application of the results of [5] in order to make predictions about the relations between the local structure and the ordinary characters in a block, whilst keeping the notion of relative projectivity so fundamental to modular representation theory. Dade’s projective conjecture (see [2]) is a more direct refinement of the conjectures in [4], and is part of a series of conjectures formulated with the objective of giving a reduction to finite simple groups for the last (and strongest) of them. The object of this paper is twofold. Firstly to achieve a reduction to the case where all normal p-subgroups are central for Robinson’s conjecture (this was done for Dade’s conjecture in [2]), and secondly to use this to demonstrate the equivalence of the two conjectures (as was predicted in [7]). It is clear that there is much to be gained from the dual consideration of these conjectures, as is evidenced in [9].

Let G be a finite group and p a prime. Given a chain of p-subgroups σ : Q_{0} < Q_{1} < ⋯ < Q_{n} of G, define the length |σ| = n, the final subgroup V^{σ} = Q_{n}, the initial subgroup V_{σ} = Q_{0}, the k-th initial subchain σ_{k} : Q_{0} < Q_{1} < ⋯ < Q_{k}, and the normalizer G_{σ} = N_{G}(σ) = N_{G}(Q_{0}) ∩ N_{G}(Q_{1}) ∩ ⋯ ∩ N_{G}(Q_{n}). Write C(G|Q) for the set of those chains with initial subgroup Q.

A p-subgroup Q of G is radical if O_{p}(N_{G}(Q)) = Q, where O_{p}(N_{G}(Q)) is the maximal normal p-subgroup of the normalizer N_{G}(Q). The p-chain σ is said to be radical if Q_{i} = O_{p}(N_{G}(σ_{i})) for each i. Denote by R = R(G) the set of all radical p-chains of G.

Let R be a complete local discrete valuation ring, with field of fractions K of characteristic zero and residue field k = R/J(R) algebraically closed of characteristic p. Suppose also that R contains a primitive |G|^{3}-root of unity. See [10, 2.2] for a discussion on the nature and existence of such a ring. When we refer to a p-block B of G, this means a block of kG or RG as appropriate to the context.

1Partially supported by EPSRC award no. 95700575

If X ⊆ C = C(G) and B is a p-block of G, then write

X (G, B) = {σ ∈ X (G) : Blk(G_{σ}|B) ≠ ∅},

where Blk(G_{σ}|B) is the set of p-blocks of G_{σ} whose Brauer correspondent in G is B (note that by [4, 3.2] every p-block of G_{σ} does have a Brauer correspondent in G). Following [1] we define the p-local rank, plr(B) of B to be max{|σ| : σ ∈ R(G, B)}, and following [7] the p-local rank, plr(G), of G is plr(G) = max{|σ| : σ ∈ R(G)}.

Let B be a p-block of G (of any defect), d an integer and λ ∈ Irr(O_{p}(Z(G))). If T, H ≤ G, then we say that χ ∈ Irr(T) covers, or lies over μ ∈ Irr(H) if (χ|_{T ∩H}, μ|_{T ∩H}) ≠ 0, and write Irr(T, μ) for the set of such characters. Robinson’s conjecture predicts that

k_{d}(G, B, λ) = ∑_{σ∈R(G)/G}(−1)^{|σ|} ∑_{μ∈Irr_{d}(V_{σ},λ)/G_{σ}} f_{0}^{(B)} (\frac{I_{G_{σ}}(μ)}{V_{σ}}) .

Here if H is a section of G, then f_{0}^{(B)}(H) denotes the number of blocks of defect zero of H corresponding to B under the Brauer correspondence and the natural correspondence with a quotient group. I_{G}(μ) is the inertial subgroup. The subscript d indicates that we count only characters of defect d (following Dade, we say that χ ∈ Irr(H) has defect d if p^{d}χ(1)_{p} = |G|_{p}).

Dade’s projective conjecture states that if B has defect groups not equal to O_{p}(G) and O_{p}(G) ≤ Z(G) ∩ G′ is cyclic, then

∑_{σ∈R(G|O_{p}(G))/G}(−1)^{|σ|}k_{d}(G_{σ}, B, λ) = 0.

Our main results are the following:

Theorem 1.1 (i) If Robinson’s conjecture holds for all p-blocks of p-local rank at most n of finite groups G with O_{p}(G) ≤ Z(G) ∩ G′ cyclic, then the conjecture holds for all p-blocks of p-local rank n.

(ii) If Robinson’s conjecture holds for all finite groups G with plr(G) ≤ n and O_{p}(G) ≤ Z(G) ∩ G′ cyclic, then the conjecture holds for all finite groups with p-local rank n. In particular in proving the conjecture it suffices to check it just for those groups with O_{p}(G) ≤ Z(G) ∩ G′ cyclic.

Theorem 1.2 Dade’s projective conjecture holds for every finite group if and only if Robinson’s does.

The paper is organised as follows. In Section 2 we define and review some properties of relatively projective characters as studied in [5] and gives a brief summary of the Clifford-theoretic tools needed for the rest of the paper. Section 3 consists of a proof of Theorem 1.1, whilst a proof of Theorem 1.2 is given in Section 4.

We observe that, with some work, these results may also be achieved using the results of [2].

2 Characters afforded by relatively projective modules and the Dade correspondence

Definition Let H ≤ G be finite groups. We say that χ ∈ Irr(G) is H-projective with respect to R if there is some H-projective RG-module M affording χ. If R is given then we just say that χ is H-projective. We say that a p-subgroup of G is a vertex for χ if it is a vertex for some RG-module affording χ.

Write w(G, H) for the number of H-projective irreducible characters of G, and Irr(G, H) for the set of such characters.

Suppose that Q ◁ G is a p-group.

Proposition 2.1 ( [5] and [7]) The Q-projective irreducible characters covering μ are in 1-1 correspondence with the p-blocks of defect zero of I_{G}(μ)/Q. Further, the number of Q-projective irreducible characters of defect d lying in a given p-block B of G is

w_{d}(G, B, Q) = ∑_{μ∈Irr_{d}(Q)/G} f_{0}^{(B)} (\frac{I_{G}(μ)}{Q}) .

Proposition 2.2 ( [7]) χ ∈ Irr(G, μ) is Q-projective if and only if

χ(1)_{p} = [G : Q]_{p}μ(1).

A trivial but useful observation following from these results will be needed when dealing with chains of p-subgroups:

Corollary 2.3 Let G be a finite group and Q◁G be a p-subgroup of G. Let μ ∈ Irr(Q). If w(G, μ, Q) ≠ 0, then Q = O_{p}(I_{G}(μ)).

Suppose that N ◁ G and μ ∈ Irr(N) is G-stable. Then following [3], we may construct a central extension Ĝ of G by a central cyclic subgroup Ŵ (whose order has prime divisors which are also divisors of |N| - see [5, 2.1]), so that μ extends to an irreducible character, say μ̂, of Ĝ. Let G̃ = Ĝ/N̂, where N̂ is naturally isomorphic to N and N̂ ∩ Ŵ = 1. Let W̃ be the image of Ŵ under the natural homomorphism, so that G̃/W̃ ≅ G/N. Let θ̂ be the linear character of Ŵ covered by μ̂ and θ̃ ∈ Irr(W̃) be identified with the complex conjugate \overline{θ̂}. For each d there is a 1-1 correspondence Irr_{d}(G, μ) ↔ Irrd̃(G̃, θ̃) given by χ = χ̃μ̂ (regarding χ as a character of Ĝ), where d̃= d+log_{p}(μ(1)_{p}|W̃ |_{p}/|N|_{p}). If two characters χ_{1}, χ_{2} ∈ Irr(G, μ) lie in the same block of G, then the corresponding χ̃_{1}, χ̃_{2} lie in the same block of G̃. We call the blocks of G̃ thus associated to a given block B of G the Dade correspondents of B with respect to μ. The Dade correspondence commutes with the Brauer correspondence. Full details of the above are given in [3], and it should be noted that those results are purely an elementary version of the results of [2]. Note that W̃ ≤ Z(G̃) ∩ [G̃, G̃]. In general we denote objects associated to G̃ with a ‘˜ ’ and objects associated to Ĝ with a ‘ˆ ’.

A trivial computation using Proposition 2.2 tells us that when N is a p-group, χ is N-projective if and only if the corresponding χ̃ is W̃ -projective.

Remark Note that given a block B of G, there is a unique block B̂ of Ĝ corresponding to B (since Ŵ ≤ Z(Ĝ)). Then B̂ dominates the Dade correspondents of B through an RĜ-module affording μ̂ in the sense of [6]. This will be important when we come to use [1, 4.1].

3 Finite groups with normal p-subgroups

In this section we consider what happens when a finite group G has a non-central normal p-subgroup. We use methods developed by Robinson in [7] to ‘prepare’ the formula predicted by Robinson’s conjecture, so that we may apply the results detailed in Section 2 to obtain a proof of Theorem 1.1.

Let λ ∈ Irr(O_{p}(Z(G))) and let d be an integer. Write U = O_{p}(G). By Clifford’s theorem

k_{d}(G, B, λ) = ∑_{μ∈Irr(U,λ)/G} k_{d}(G, B, μ).

We aim to prove the following result, which then allows us to apply the results of the last section:

Proposition 3.1 Let B be a p-block of G. Then

∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}) = ∑_{μ∈Irr(U,λ)/G} (  ∑_{σ∈R(I_{G}(μ))/I_{G}(μ)} (−1)^{|σ|}w_{d}(I_{G}(μ)_{σ}, B, μ, V_{σ})) .

Denote by C_{U}(G) the set of all p-chains in C(G) whose initial subgroup contains U (not necessarily properly). By [8, 1.1] we may replace R in the first alternating sum of Proposition 3.1 with C_{U}, since the initial subgroup of every radical p-chain is radical and so contains U. We obtain

∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}) = ∑_{σ∈C_{U} (G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}).

For each σ ∈ C_{U}(G), w_{d}(G_{σ}, B, λ, V_{σ}) = ∑_{μ∈Irr(U,λ)/G_{σ}} w_{d}(G_{σ}, B, μ, V_{σ}), so

∑_{σ∈C_{U} (G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}) = ∑_{σ∈C_{U} (G)/G}(−1)^{|σ|} ( ∑_{μ∈Irr(U,λ)/G_{σ}} w_{d}(G_{σ}, B, μ, V_{σ})) .

Lemma 3.2

∑_{σ∈C_{U} (G)/G}(−1)^{|σ|} ( ∑_{μ∈Irr(U,λ)/G_{σ}} w_{d}(G_{σ}, B, μ, V_{σ}))

= ∑_{μ∈Irr(U,λ)/G} (  ∑_{σ∈C_{U} (G)/I_{G}(μ)}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ})) .

Proof: Observe that the pair (σ, μ) lies in an orbit of length [G : G_{σ}][G_{σ} : I_{G_{σ}}(μ)] = [G : I_{G_{σ}}(μ)] in the first instance and [G : I_{G}(μ)][I_{G}(μ) : I_{G_{σ}}(μ)] = [G : I_{G_{σ}}(μ)] in the second, and that w_{d}(G_{σ}, B, μ, V_{σ}) is constant under conjugation of (σ, μ) in G. □

We conclude that if

k_{d}(G, B, μ) = ∑_{σ∈C_{U} (G)/I_{G}(μ)}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ})

for each μ ∈ Irr(U, λ), then the conjecture holds for that choice of B, d and λ. We fix μ ∈ Irr(U, λ) and write H = I_{G}(μ). Clifford theory then allows us to move from counting characters of G and G_{σ} to counting characters of H and H_{σ}:

Lemma 3.3 k_{d}(G, B, μ) = k_{d}(H, B, μ) and

∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ}) = ∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ} ∩ H).

Proof: For each σ ∈ C_{U}(G), I_{G_{σ}}(μ) = H_{σ}, and so Clifford’s theorem gives a 1-1 correspondence Irr(H_{σ}, μ) ↔ Irr(G_{σ}, μ) given by induction of characters, and note that G = G_{σ} for the chain σ = U of length 0. This correspondence is clearly defectpreserving. Suppose that χ ∈ Irr(G_{σ}, μ) corresponds to η ∈ Irr(H_{σ}, μ). Now χ is V_{σ}-projective if and only if η is V_{σ} ∩ H-projective (for let φ ∈ Irr(V_{σ} ∩ H, μ) be a character covered by η. Now I_{V_{σ}}(μ) = H ∩ V_{σ}, so by Clifford’s theorem φ^{V_{σ}} = θ for some θ ∈ Irr(V_{σ}, μ). But

(χ|_{V_{σ}}, θ) = (χ|_{V_{σ}}, φ^{V_{σ}}) = (χ|_{V_{σ}∩H}, φ) = (η^{G_{σ}}|_{V_{σ}∩H}, φ) = (η^{G_{σ}}|_{H_{σ}}, φ^{H_{σ}}).

But (η, η^{G_{σ}}|_{H_{σ}}) ≠ 0 and (η, φ^{H_{σ}}) ≠ 0, so χ covers θ. We have χ(1)_{p} = [G_{σ} : H_{σ}]_{p}η(1)_{p} and

[G_{σ} : V_{σ}]_{p}θ(1)_{p} = [G_{σ} : V_{σ}]_{p}[V_{σ} : V_{σ} ∩ H]_{p}φ(1)_{p} = [G_{σ} : H_{σ}]_{p}[H_{σ} : V_{σ} ∩ H]_{p}φ(1)_{p}.

So χ(1)_{p} = [G_{σ} : V_{σ}]_{p}θ(1)_{p} if and only if η(1)_{p} = [H_{σ} : V_{σ} ∩ H]_{p}φ(1)_{p}, and we may apply Proposition 2.2).

Every p-block of H_{σ} containing an irreducible character lying over μ has a Brauer correspondent in G_{σ} and the character correspondence given by induction respects the Brauer correspondence, i.e., if η ∈ Irr(H_{σ}, μ), then η^{G_{σ}} lies in the p-block of G_{σ} which is a Brauer correspondent of the p-block of H_{σ} containing η. Hence w_{d}(G_{σ}, B, μ, V_{σ}) = w_{d}(H_{σ}, B, μ, V_{σ} ∩ H) since the Brauer correspondence is transitive, and we’re done. □

We use the theory of deficient p-chains introduced in [7] in order to overcome the problem of summing over I_{G}(μ)-orbits of p-chains of G (note in particular that a given σ ∈ C_{U}(G) need not lie in the stabilizer of μ). Many of the arguments that follow have their origins in [7]. Given a subgroup T of G, we say that a p-chain σ : Q_{0} < ⋯ < Q_{n} ∈ C(G) is T -deficient if Q_{n} ∩ T ≤ O_{p}(G). Given a p-chain σ ∈ C(G) we call the longest deficient initial subchain the T -deficient part, and denote it by d_{T} (σ). Note that d_{T} (σ) may be empty. Denote the empty chain by ∅.

Returning to our original hypotheses, for brevity we write D(G) = D_{H}(G) for the set of non-empty H-deficient chains in C_{U}(G). Write D^{♯}(G) = D(G) − {U} and d(σ) = d_{H}(σ).

Observe that we may write ∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ} ∩ H) as

∑_{τ∈D(G)/H} ( ∑_{{σ∈C(G|V^{τ})/Hτ \atop d(σ)=σ_{0}}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U))+ ∑_{{σ∈C_{U} (G)/H \atop d(σ)=∅}}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}∩H),

since V_{τ} ∩ H = U when τ ∈ D(G). We show that it suffices to consider only those chains contained in the stabilizer of their deficient part.

Lemma 3.4

∑_{τ∈D(G)/H} (  ∑_{{σ∈C(G|V^{τ})/Hτ \atop d(σ)=σ_{0}}} (−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U))

= ∑_{τ∈D(G)/H} ( ∑_{{σ∈C(Gτ |V^{τ})/Hτ \atop d(σ)=σ_{0}}} (−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U))

Proof: We wish to cancel the contributions to the alternating sum of those chains σ ∈ C_{U}(G) with σ ∉ C_{U}(N_{G}(d(σ))). We do this by pairing such chains with identical stabilizers but with lengths of differing parity. Note that the pairs of chains need not necessarily have the same, or even conjugate deficient part.

Let σ : Q_{0} < ⋯ < Q_{n} < ⋯ < Q_{m} ∈ C_{U}(G) with d(σ) = σ_{n} non-empty. Suppose that σ ∉ C_{U}(N_{G}(d(σ))), i.e., Q_{m} ≰ N_{G}(Q_{i}) for some i ≤ n. Choose i minimal with this property. Given i, choose j maximal such that Q_{i} ◁ Q_{j}. So i ≤ j < m and Q_{j} ≤ N_{Q_{j+1}}(Q_{i}) < Q_{j+1}. There are two cases to consider: (i) Q_{j} = N_{Q_{j+1}}(Q_{i}) and (ii) Q_{j} ≠ N_{Q_{j+1}}(Q_{i}).

(i) If Q_{j} = N_{Q_{j+1}}(Q_{i}), then form the chain \overline{σ} : Q_{0} < ⋯ < Q_{j−1} < Q_{j+1} < ⋯ < Q_{m} from σ by deleting the term Q_{j}. If i = j, then Q_{j} ≠ N_{Q_{j+1}}(Q_{i}) since Q_{j+1} is a p-group, so in particular we must have j ≠ 0 for case (i) to occur, and V\overline{σ} = V_{σ}. Again using the fact that case (i) cannot occur when i = j, N_{H}(\overline{σ}) ≤ N_{H}(Q_{j}) and so H_{σ} = H\overline{σ}. Also Q_{i} is still a term of \overline{σ} and j < m, so \overline{σ} ∉ C_{U}(N_{G}(d(\overline{σ}))).

(ii) If Q_{j} ≠ N_{Q_{j+1}}(Q_{i}), then form the chain \overline{σ} : Q_{0} < ⋯ < Q_{j} < N_{Q_{j+1}}(Q_{i}) < Q_{j+1} < ⋯ < Q_{m} from σ. Of course V\overline{σ} = V_{σ}, and it is easy to see that N_{H}(σ) ≤ N_{Q_{j+1}}(Q_{i}) and so H_{σ} = H\overline{σ}. As above Q_{i} is still a term of the deficient part of \overline{σ} and Q_{m} is still the final subgroup of \overline{σ}, so \overline{σ} ∉ C_{U}(N_{G}(d(\overline{σ}))). In both cases (\overline{σ})^{g} = \overline{(σ^{g})} and \overline{\overline{σ}} = σ, so we may cancel the contributions from orbits of chains containing σ and \overline{σ}, since (−1)^{|\overline{σ}|} = (−1)^{|σ|+1}. □

The next lemma will later allow us to reduce the summation to a consideration of chains of H rather than G, the key to proving Proposition 3.1.

Lemma 3.5 Given τ ∈ D(G), denote by C^{⋆}(G_{τ}|V^{τ}) the set of those chains σ : Q_{0} < ⋯ < Q_{n} in C(G_{τ}|V^{τ}) satisfying Q_{i} = V^{τ}(H ∩ Q_{i}) for each i = 0, …, n. We also allow τ to be the empty chain, in which case we set G_{τ} = G and set C_{U}^{⋆}(G) = C_{U}(H). Then

∑_{{σ∈C_{U} (G)/H \atop d(σ)=∅}} (−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ} ∩ H) = ∑_{{σ∈C^{⋆}_{U} (G)/H \atop d(σ)=∅}} (−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ} ∩ H)

(representing the case τ is empty), and when τ ∈ D(G) we have

∑_{{σ∈C(Gτ |V^{τ})/Hτ \atop d(σ)=σ_{0}}} (−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U) = ∑_{{σ∈C^{⋆}(Gτ |V^{τ})/Hτ \atop d(σ)=σ_{0}}} (−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U).

Proof: Let τ ∈ D(G), and let σ : Q_{0} < ⋯ < Q_{n} ∈ C(G_{τ}|V^{τ}) with d(σ) = σ_{0} and suppose that σ ∉ C^{⋆}(G_{τ}|V^{τ}), i.e., Q_{i} ≠ V^{τ}(H ∩ Q_{i}) for some i. Choose i minimal with this property, noting that i > 0. Now

Q_{i} > V^{τ}(H ∩ Q_{i}) ≥ V^{τ}(H ∩ Q_{i−1}) = Q_{i−1}

by the minimality of i. There are two cases to consider: (i) V^{τ}(H ∩ Q_{i}) = Q_{i−1} and (ii) V^{τ}(H ∩ Q_{i}) ≠ Q_{i−1}.

(i) Suppose that V^{τ}(H ∩ Q_{i}) = Q_{i−1}. Observe that i > 1, since if i = 1 and case (i) occurs, then V^{τ}(H ∩ Q_{1}) = V^{τ}, and so H ∩ Q_{1} = H ∩ V^{τ} = U, contradicting our assumption that d(σ) = σ_{0}. Form the chain \overline{σ} : Q_{0} < ⋯ < Q_{i−2} < Q_{i} < ⋯ < Q_{n} from σ by deleting the term Q_{i−1}. Since i > 1, we must have V\overline{σ} = V_{σ}, and it is easy to see that N_{H}(\overline{σ}) ≤ N_{H}(Q_{i−1}), so N_{H}(\overline{σ}) = N_{H}(σ). Also \overline{σ} ∈ C(G_{τ}|V^{τ}), with d(\overline{σ}) = \overline{σ}_{0} and \overline{σ} ∉ C^{⋆}(G_{τ}|V^{τ}).

(ii) If V^{τ}(H ∩Q_{i}) ≠ Q_{i−1}, then form the chain \overline{σ} : Q_{0} < ⋯ < Q_{i−1} < V^{τ}(H ∩Q_{i}) < Q_{i} < ⋯ < Q_{n} from σ. V\overline{σ} = V_{σ} since i > 0. Clearly N_{H}(σ) ≤ N_{H}(V^{τ}(H ∩ Q_{i})), so that N_{H}(\overline{σ}) = N_{H}(σ). Since V^{τ}(H ∩ Q_{i}) ∩ H = H ∩ Q_{i} > U, we have d(\overline{σ}) = \overline{σ}_{0}, and \overline{σ} ∉ C^{⋆}(G_{τ}|V^{τ}).

In both cases (\overline{σ})^{g} = \overline{(σ^{g})} and \overline{\overline{σ}} = σ, so we may cancel the contributions from orbits of chains containing σ and \overline{σ}, since (−1)^{|\overline{σ}|} = (−1)^{|σ|+1}. Hence the second part of the Lemma holds. To consider those chains with deficient part the empty chain we may use a similar argument to cancel all chains in which Q_{i} ≠ H ∩ Q_{i} for some i. We leave this to the reader. □

Lemma 3.6

∑_{{σ∈C^{⋆}_{U} (G)/H \atop d(σ)=∅}} (−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ} ∩ H) = ∑_{{σ∈C_{U} (H)/H \atop d(σ)=∅}} (−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}),

and if τ ∈ D(G), then

∑_{{σ∈C^{⋆}(Gτ |V^{τ})/Hτ \atop d(σ)=σ_{0}}} (−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U) = ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U).

Proof: Suppose that τ ∈ D(G) and σ ∈ {α ∈ C^{⋆}(G_{τ}|V^{τ}) : d(α) = α_{0}}, where σ : Q_{0} < ⋯ < Q_{n}. Then by definition Q_{i} = V^{τ}(H ∩ Q_{i}) for each i ≥ 0, and so

|Q_{i}| = \frac{|V^{τ}||H ∩ Q_{i}|}{|V^{τ} ∩ H ∩ Q_{i}|} = \frac{|V^{τ}||H ∩ Q_{i}|}{|U|}

(recall that H ∩ V^{τ} = U). Since the Q_{i}’s are distinct, the chain

φ(σ) : H ∩ V^{τ} < H ∩ Q_{1} < ⋯ < H ∩ Q_{n}

has distinct terms. Since H ∩ V^{τ} = U, we have φ(σ) ∈ C(H_{τ}|U).

Conversely, if θ ∈ C(H_{τ}|U), where θ : P_{0} < ⋯ < P_{n}, then define the chain

ψ(θ) : V^{τ}P_{0} < V^{τ}P_{1} < ⋯ < V^{τ}P_{n}.

Observe that V^{τ}P_{i} = V^{τ}(H ∩ V^{τ}P_{i}) for each i and that ψ(θ) ∈ {C^{⋆}(G_{τ}|V^{τ}) : d(α) = α_{0}}. Hence φ and ψ are inverse maps between {α ∈ C^{⋆}(G_{τ}|V^{τ}) : d(α) = α_{0}} and C(H_{τ}|U) which preserve length. It is clear that chains in {α ∈ C^{⋆}(G_{τ}|V^{τ}) : d(α) = α_{0}} are conjugate in H_{τ} if and only if their images in C(H_{τ}|U) are conjugate in H_{τ}. Hence we have a bijection {α ∈ C^{⋆}(G_{τ}|V^{τ})/H_{τ} : d(α) = α_{0}} ↔ C(H_{τ}|U)/H_{τ}. It remains to check that N_{H_{τ}} (φ(σ)) = N_{H_{τ}} (σ). But this is clear since N_{H_{τ}} (σ) ≤ N_{H_{τ}} (φ(σ)) and (since τ is stabilized by H_{τ}) N_{H_{τ}} (φ(σ)) ≤ N_{H_{τ}} (ψ(φ(σ))) = N_{H_{τ}} (σ).

The first equality of the lemma is immediate since C_{U}^{⋆}(G) is by definition C_{U}(H). □

Proof of Proposition 3.1 We have seen that

∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}) = ∑_{μ∈Irr(U,λ)/G} (  ∑_{σ∈C_{U} (G)/I_{G}(μ)}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ}))

(see Lemma 3.2 and the discussion preceding it). We fix μ ∈ Irr(U, λ) and write H = I_{G}(μ). By Lemmas 3.3, 3.4, 3.5 and 3.6, we may write ∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ}) as

∑_{τ∈D(G)/H} ( ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U)) + ∑_{{σ∈C_{U} (H)/H \atop d(σ)=∅}}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}),

where the final alternating sum represents those chains in C_{U}(G) with empty deficient part.

Now if we set α ∈ D(G) to be the chain α = U (of length zero), then

∑_{σ∈C(H_{α}|U)/H_{α}}(−1)^{|α|+|σ|}w_{d}((H_{α})_{σ}, B, μ, U) + ∑_{σ∈C_{U} (H)/H,d(σ)=∅}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ})

= ∑_{σ∈C_{U} (H)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}),

as we are considering on the one hand chains whose initial subgroup is U and on the other chains whose initial subgroup strictly contains U. It suffices to show that

∑_{τ∈D^{♯}(G)/H} ( ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U)) = 0,

since then we have

∑_{σ∈C_{U} (G)/H}(−1)^{|σ|}w_{d}(G_{σ}, B, μ, V_{σ}) = ∑_{σ∈C_{U} (H)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ})

+ ∑_{τ∈D^{♯}(G)/H} ( ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U))

= ∑_{σ∈C_{U} (H)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ}) = ∑_{σ∈R(H)/H}(−1)^{|σ|}w_{d}(H_{σ}, B, μ, V_{σ})

as required.

Suppose that τ ∈ D^{♯}(G), and consider a chain σ ∈ C(H_{τ}|U), σ : Q_{0} < ⋯ < Q_{n}, of length |σ| > 0. Then U < Q_{1} ◁ N_{H_{τ}} (σ) = I_{N_{Hτ} (σ)}(μ), i.e., Q_{1} is a normal p-subgroup of (H_{τ})_{σ} strictly containing U and stabilizing μ, and so w_{d}((H_{τ})_{σ}, B, μ, U) = 0 by Corollary 2.3. Hence

∑_{τ∈D^{♯}(G)/H} ( ∑_{σ∈C(H_{τ} |U)/H_{τ}}(−1)^{|τ|+|σ|}w_{d}((H_{τ})_{σ}, B, μ, U)) = ∑_{τ∈D^{♯}(G)/H}(−1)^{|τ|}w_{d}(H_{τ}, B, μ, U).

But notice that we may pair each deficient chain τ ∈ D^{♯}(G) satisfying V_{τ} = U with another chain in D^{♯}(G) with initial term strictly containing U, since we have defined D^{♯}(G) to exclude the chain τ = U. Clearly paired chains lie in H-orbits of the same size, and the lengths of the chains in each pair differ in parity, so we may cancel their contributions to this last alternating sum. This gives ∑_{τ∈D^{♯}(G)/H}(−1)^{|τ|}w_{d}(H_{τ}, B, μ, U) = 0, and Proposition 3.1.

Proof of Theorem 1.1

(i) Let B be a p-block of G with plr(B) = n. Write U = O_{p}(G). We have seen (by Proposition 3.1, its proof and the accompanying discussion) that in order to verify Robinson’s Conjecture for B (for a given λ ∈ Irr(O_{p}(Z(G))) and integer d) it suffices to show that

k_{d}(I_{G}(μ), μ) = ∑_{σ∈R(I_{G}(μ),B′)/I_{G}(μ)}(−1)^{|σ|}w_{d}(I_{G}(μ)_{σ}, B′, μ, V_{σ})  (1)

for each μ ∈ Irr(U, λ) and each block B′ of I_{G}(μ) which is a Brauer correspondent of B. Fix μ ∈ Irr(U, λ) and write H = I_{G}(μ). We now use the results described in Section 2 and the notation introduced there freely, with H in place of G.

Let B̃ be the sum of Dade correspondents for B′ with respect to μ. It is clear that R(H̃/W̃)/H̃ and R(H/U)/H may be identified, and since every radical p-subgroup contains O_{p}(H) and U ≤ O_{p}(H), we may further identify R(H)/H and R(H̃)/H̃. For each σ ∈ R(H), define σ̃ ∈ R(H̃) in the obvious way. We may make the identification Irr_{d}(H_{σ}, B′, μ, V_{σ}) ↔ Irrd̃(H̃σ̃, B̃, θ̃, Vσ̃), where d̃ = d + log_{p}(μ(1)|W̃ |/|U|), since the Dade correspondence commutes with the Brauer correspondence.

Hence (1) if and only if kd̃(H̃, B̃, θ̃) = ∑_{σ̃∈R(H̃)/H̃}(−1)^{|σ̃|}wd̃(H̃σ̃, B̃, θ̃, Vσ̃). Now W̃ ≤ O_{p}(H̃), and if we do not have W̃ = O_{p}(H̃), then we may repeat the whole process until we do get equality (this must eventually happen, since G is finite), and we have left- and right-hand-sides of the statement of Robinson’s conjecture for the component blocks of B̃ in place of B. Hence if the conjecture holds for every component of B̃, then it holds for B. Observe that each component of B̃ is related to B either by the Brauer correspondence or by domination, and so by [1, 3.2,4.1] each has p-local rank at most n, completing the proof of part (i).

The proof of (ii) is almost identical to that of part (i), using the p-local rank of a group rather than that of a block.

4 The equivalence of Robinson’s conjecture and Dade’s projective conjecture

Lemma 4.1 Let B be a p-block of G. Suppose that plr(B) = n and that Robinson’s conjecture holds for finite groups of p-local rank strictly less than n. Write U = O_{p}(G). Then

∑_{σ∈R(G|U)/G}(−1)^{|σ|}k_{d}(G_{σ}, B, λ) = ∑_{σ∈R(G|U)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, U)

if and only if

k_{d}(G, B, λ) = ∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}).

The analogous result holds considering radical p-chains for G rather than B, and the p-local rank of a group instead of that for a block.

Proof: Note that by the inductive hypotheses both equalities hold for all blocks of p-local rank less than n. Suppose that the former holds.

Let R_{0}(G, B) be the set of all radical p-subgroups of B (i.e., terms of radical p-chains of length zero). We have

∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ})
= ∑_{Q∈R_{0}(G)/G} (∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}w_{d}(N_{G}(Q)_{σ}, B, λ, Q))
= ∑_{Q∈R_{0}(G)/G} (∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}k_{d}(N_{G}(Q)_{σ}, B, λ))
=  ∑_{σ∈R(G)/G}(−1)^{|σ|}k_{d}(G_{σ}, B, λ)

We may pair every chain Q_{0} < ⋯ < Q_{r} in R(G) in which Q_{0} ≠ U with the chain U < Q_{0} ⋯ < Q_{r} in R(G) of length r + 1. These two chains have the same stabilizer, so their contributions cancel. We may cancel all chains in R(G) in this way except for the chain U. Hence

∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ}) = k_{d}(G, B, λ)

and Robinson’s conjecture is satisfied in that case.

Now suppose that B satisfies the second equality. Then

k_{d}(G, B, λ)

=  ∑_{σ∈R(G)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, V_{σ})
= ∑_{Q∈R_{0}(G)/G} (∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}w_{d}(N_{G}(Q)_{σ}, B, λ, Q))
=  ∑_{σ∈R(G|U)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, U)
+ ∑_{U≠Q∈R_{0}(G)/G} (∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}k_{d}(N_{G}(Q)_{σ}, B, λ))

Replacing each chain Q_{0} < ⋯ < Q_{r} in R(G) in which U ≠ Q_{0} by U < Q_{0} ⋯ < Q_{r}, we have

∑_{U≠Q∈R_{0}(G)/G} (  ∑_{σ∈R(N_{G}(Q)|Q)/N_{G}(Q)}(−1)^{|σ|}k_{d}(N_{G}(Q)_{σ}, B, λ))

= ∑_{σ∈R(G|U)/G}(−1)^{|σ|+1}k_{d}(G_{σ}, B, λ) − k_{d}(G, B, λ),

and so

∑_{σ∈R(G|U)/G}(−1)^{|σ|}k_{d}(G_{σ}, B, λ) = ∑_{σ∈R(G|U)/G}(−1)^{|σ|}w_{d}(G_{σ}, B, λ, U)

as required. □

Proof of Theorem 1.2 Suppose that Robinson’s conjecture holds. Then it follows immediately from Lemma 4.1 that Dade’s projective conjecture also holds.

Suppose that Dade’s projective conjecture holds. Let plr(B) = n and suppose that Robinson’s conjecture holds for every block with p-local rank strictly less than n. Then by Lemma 4.1 Robinson’s conjecture holds for every p-block with p-local rank n of every finite group H with O_{p}(H) ≤ Z(H) ∩ H′ cyclic. Then by Theorem 1.1 Robinson’s conjecture holds for B. Since Robinson’s conjecture is identical to Dade’s for blocks of p-local rank one of finite groups with O_{p}(H) ≤ Z(H) ∩ H′, the result follows by induction on the p-local rank.

ACKNOWLEDGEMENTS

The material for this paper consists of part of my doctoral research, and I thank my supervisor Geoffrey Robinson for all his help. I also thank Burkhard Külshammer and Wayne Wheeler for their useful comments and suggestions.

References

[1] J.An and C.W.Eaton, The p-local rank of a block, J. Group Theory 3(2000), 369-380.

[2] E.Dade, Counting characters in blocks II, J. Reine Angew. Math. 448(1994), 97-190.

[3] C.W.Eaton and G.R.Robinson, On a minimal counterexample to Dade’s projective conjecture, to appear, J. Algebra.

[4] R.Knörr and G.R.Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39(1989), 48-60.

[5] B.Külshammer and G.R.Robinson, Characters of relatively projective modules, II, J. London Math. Soc. (2) 36(1987), 59-67.

[6] M.Murai, Normal subgroups and heights of character, J. Math. Kyoto Univ. 36(1996), 31-43.

[7] G.R.Robinson, Local structure, vertices and Alperin’s conjecture, Proc. London Math. Soc. (3) 72(1996), 312-330.

[8] G.R.Robinson, Further consequences of conjectures like Alperin’s, J. Group Theory 1(1998), 131-141.

[9] G.R.Robinson, Cancellation theorems related to conjectures of Alperin and Dade, to appear, J. Algebra.

[10] J.Thévenaz, “G-Algebras and Modular Representation Theory,” Clarendon Press, Oxford, 1995.