Weight conjectures for Ordinary Characters Geoffrey R. Robinson School of Mathematics and Statistics University of Birmingham Edgbaston Birmingham B15 2TT England e-mail : g.r.robinson@bham.ac.uk, Fax UK 121 4143389

INTRODUCTION: When p is a prime, Dade’s Projective Conjecture (DPC) for a p-block B whose defect group is non-central gives an alternating sum which (conjecturally) expresses the number of irreducible characters of defect d in B lying over a linear character λ of a central p-subgroup Z ( denoted k_{d}(B, λ)) in terms of an alternating sum whose terms can all be calculated p-locally. This conjecture is a refinement of the Knörr-Robinson formulation ( see [8]) of Alperin’s Weight Conjecture (AWC) ( see [1] ). DPC is only formulated for finite groups G such that O_{p}(G) ≤ Z(G), and for p-blocks of G whose defect group(s) strictly contain O_{p}(G). We will give a precise statement of this conjecture later.

In the paper [14], we stated the Ordinary Weight Conjecture ( OWC) which can be stated for any finite group G and any p-block of G, and in the paper [15], we explored different ways to formulate this, exploring a connection with fusion in the case of principal blocks. This also expresses the number of irreducible characters of defect d in a block B in terms of an alternating sum whose terms can all be calculated p-locally. However, the local computations can be made inside p-local subgroups N_{G}(U) and only involve U-projective irreducible characters ( ie characters which are afforded by lattices with vertex inside U). In [14], we proved that if we sum over OWC for all non-negative integers d, then the result is equivalent to Alperin’s Weight Conjecture. In [6], ( containing some results from his 1999 PhD thesis), C.W. Eaton proves that OWC is equivalent to DPC. We hope to illustrate that once the equivalence of OWC and DPC is accepted, then OWC is often easier to work with, and can reveal theoretical insights not immediately obvious from the original form of DPC. In particular, using the subpair ( see Alperin-Broué [2] for the definition of subpairs) formulation of OWC, we show directly that only chains of B-subpairs which start with Alperin-Goldschmidt B-subpairs (see, for example Kessar-Linckelmann-Robinson [7] for the definition of such subpairs ) contribute to the alternating sum. This has previously been shown ( with some difficulty) in the papers [7],[14], and [17] to be the case for a minimal counterexample to DPC, but the proof is (we believe) more conceptual and illuminating if we work directly with OWC. Furthermore, this is a general fact about OWC which holds for any p-block and does not need any assumption about minimality of a putative counterexample. In particular, the fact that only self-centralizing B-subpairs (often called Brauer subpairs) contribute to the alternating sums of OWC means that in our local computations, we can immediately deal with blocks of the normalizer of a B-subpair which cover nilpotent blocks of a certain normal subgroup. This allows us to exploit work of Külshammer-Puig [10], which is applicable in such a situation.

The heart of this paper is to show that the Morita equivalences exhibited by Külshammer-Puig in [10] respect ordinary weights. In this work, the relationship between ordinary weights and characters of relatively projective modules as in Külshammer-Robinson [11] is crucial. This compatibility means that when considering the contribution to the alternating sum of OWC from chains of B-subpairs beginning with an Alperin-Goldschmidt B-subpair (U, b_{U}), we can perform the computations in a group L̃ = L̃(B, U), which is a certain p′ central extension

1 → Z̃ → L̃ → L → 1

of the Külshammer-Puig extension

1 → Z(U) → L → N_{G}(U, b_{U})/C_{G}(U) → 1.

The 2-cocycle which occurs is determined in a transparent fashion by the block b_{U} of C_{G}(U).

When working with L̃ as above, the contribution to the alternating sum of OWC from chains of B-subpairs beginning with (U, b_{U}) is determined by knowledge of:

i) The 2-cocyle associated to the p-block b_{U} of C_{G}(U). ii) The fusion within N_{G}(U, b_{U}) of B-subpairs containing (U, b_{U}), which is faithfully reflected in the fusion of p-subgroups of L̃ containing (a natural isomorphic copy of ) U. iii) The action of N_{G}(U, b_{U}) on ordinary irreducible characters of U, which may be identified with an action of L̃ on irreducible characters of U.

All of these, except the 2-cocycle, are determined by knowledge of N_{G}(U, b_{U})/UC_{G}(U), that is, the group of outer automorphisms of (U, b_{U}) induced by morphisms within the Brauer category of B. Since L̃ encodes both knowledge of the 2-cocycle and the structure of N_{G}(U, b_{U})/UC_{G}(U), this explains why the necessary computations may be performed using L̃.

Let λ̃ be a faithful linear character of Z̃. Then since C_{L}(O_{p}(L)) ≤ O_{p}(L), we see that the irreducible characters of L̃ which lie over λ̃ all lie in the same p-block of L̃. Thus the necessary computations within L̃ are with the whole twisted group algebra of a p-constrained group.

We will see how, for each G-conjugacy class of Alperin-Goldschmidt Bsubpairs of G with representative (U, b_{U}), and for each non-negative integer d, there is an alternating sum which gives contribution to the alternating sum expression of OWC for k_{d}(B), and may be calculated within N_{G}(U, b_{U})- in fact within the group L̃ as above.

SECTION 1: NOTATION

Let p be a rational prime number, and let (𝕂, R, F) be a “sufficiently large” p-modular system. This includes the requirement that R is a complete discrete valuation ring of characteristic 0, with field of fractions 𝕂, such that F = R/J(R) is algebraically closed of characteristic p. We also require that 𝕂 contains a primitive |H|_{p}-th root of unity for any finite group H under consideration. As usual, for a non-zero integer n and a prime number q, the highest power of q dividing n is denoted by n_{q}. The hypotheses on F and R imply that 𝕂 contains p′-roots of unity of all orders. Hence we may, and do, suppose that ℚ ⊆ 𝕂 and every root of unity in 𝕂 is contained in the complex field ℂ.

Let X be a finite group. We may identify complex characters of X with 𝕂-valued characters of X. Whenever we speak of characters from now on, we will be referring to complex characters. As is now customary, we say that an irreducible character, χ, of X with p^{d}χ(1)_{p} = |X|_{p} has defect d.

We will sometimes refer to blocks of the group algebra RX simply as “pblocks of X”. Let b be a block of RX. We denote the set of irreducible characters in b by Irr(b). We denote the multiplicative identity of b by 1_{b}.

As is now customary, we employ the notation L < M for finite groups L and M to indicate that L is a proper subgroup of M.

In contrast to some notation employed in earlier papers, we denote by N (X) the set of all (strictly increasing) normal chains of (possibly trivial) p-subgroups of X of the form

σ = (Q_{1} < … < Q_{n}),

where Q_{i} ⊲ Q_{n} for i = 1, …, n, and we omit the empty chain. For such a chain σ, we let |σ| = n, the number of subgroups appearing in σ. We let V^{σ} denote the maximal p-subgroup appearing in σ and let X_{σ} denote the stabilizer of σ under the natural conjugation action of X on N (X). We let V_{σ} denote the first subgroup of σ. We define the sum of blocks b_{σ} of RX_{σ} as in [8] ( as we are dealing with normal chains, this is just the sum of the usual Brauer correspondents of b for X_{σ}). When Y is a p-subgroup of X, we denote by N (X, Y) the set of all chains from N (X) with initial subgroup Y. We will somtimes have occasion to work with related p-subgroup complexes, in particular P(X), which is analogous to N (X), except that the p-subgroups in the chains are not necessarily normal in the maximal p-subgroup of the chain. Another useful collection of chains, though not in general a simplicial complex, is the collection of radical p-chains R(X), which were exhibited in [8], but so named later by Dade. These are the chains σ ∈ N (X) with the property that for any initial segment τ of σ, the largest subgroup V^{τ} is equal to O_{p}(X_{τ}). We will use variants of the notation established for N with P and R as necessary. It follows from Lemma 3.2 and Proposition 3.3 of [8] ( after identifying N (X, U) with N (X/U, 1) in the case that U is a normal p-subgroup of X ) that alternating sums of the type which appear in this paper taken over N (X, U)/X can be freely interchanged with the corresponding sums over P(X, U)/X or over R(X, U)/X. Such interchangeability is made more explicit in the case that U > 1 in Dade [4]. In any case, we will freely engage in it throughout this paper without further explanation.

For b′ a sum of blocks of RX and d a non-negative integer, we let k_{d}(b′) denote the number of irreducible characters of defect d in b′. More generally, for γ a linear character of Y, we denote the number of irreducible characters of defect d in b′ which lie over γ by k_{d}(b′, γ). We let Irr(b′) denote the set of irreducible characters in b′, Irr_{d}(b′) denote the set of irreducible characters of defect d in b′ and Irr_{d}(b′, γ) denote the the set of irreducible characters of defect d in b′ which lie over γ.

For a p-subgroup, Q, of X, we let Br_{Q} denote the composition of the natural epimorphism from RX to F X followed by the vector space projection from F X onto F C_{X}(Q) with kernel F [X\C_{X}(Q)]. Then Br_{Q} restricts to a homomorphism of R-algebras from RX^{Q} ( the subalgebra of RX consisting of the Q-fixed points under conjugation action) to F C_{X}(Q).

If B is a block of X, and Y is a section of X such that Brauer correspondence is defined, we use a superscript (B) to denote “in Brauer correspondents of B” for invariants of p-blocks of Y. It is occasionally convenient to extend this notation to p-blocks of a p-power finite central extension, say Ỹ, of Y, since there is a canonical bijection between p-blocks of Y and p-blocks of Ỹ . We use the notation f_{0}(A) to denote the number of projective simple modules associated to the reduction ( mod J(R)) of A, when A is a sum of p-blocks of Y ( or Ỹ ) in that situation.

Let us now consider a finite group G such that O_{p}(G), the unique largest normal p-subgroup of G, is contained in the centre, Z(G), of G. We set Z = O_{p}(G) and we fix a block, B, of RG whose defect groups strictly contain Z.

Dade’s Projective Conjecture ( henceforth abbreviated to DPC) asserts that whenever B has defect group greater than Z, we should have

∑_{σ∈N (G,Z)/G}(−1)^{|σ|+1}k_{d}(B_{σ}, λ) = 0

for each linear character, λ, of Z and every positive integer d. The notation /G indicates that we are taking a full set of representatives for the G-orbits under the given action.

REMARKS: We remind the reader that, as we are dealing with a “blockwise” conjecture, it follows that for a linear character, μ, of a central p′-subgroup, W, of G, either all irreducible characters in B (and, in fact, in every B_{σ}) lie over μ, or none do.

We also remind the reader that ( as remarked in [16]), if the formula

∑_{σ̄∈N (Ḡ,Z̄)/Ḡ}(−1)^{|σ|}k_{d′}(B̄σ̄) = 0

( which we henceforth denote by WDPC (for Ḡ in this case)) holds whenever d′ is a positive integer, Ḡ is a homomorphic image of G of the form G/Z_{1} for a subgroup, Z_{1}, of Z, and B̄ is the unique block of RḠ which is the image of B under the natural epimorphism from RG to RḠ, then we have

∑_{σ∈N (G,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}, λ) = 0

for each linear character, λ, of Z, and each positive defect d.

Dade’s Projective Conjecture is equivalent to the following assertion ( in the usual sense that a minimal counterexample to one is a minimal counterexample to the other)

ORDINARY WEIGHT CONJECTURE: Whenever H is a finite group ( not necessarily with O_{p}(H) central), and B is a block of RH, then for each non-negative integer d we have

k_{d}(B) = ∑_{σ∈N (H)/H}(−1)^{1+|σ|} ∑_{μ∈Irr_{d}(V_{σ})/H_{σ}} f_{0}^{(B)}(I_{H_{σ}}(μ)/V_{σ}).

REMARK: We warn the reader, as remarked in several papers before, that since the terms in the alternating sum depend not only on the chain stabilizer H_{σ}, but also on the initial subgroup V_{σ}, many of the usual contractibility results used for p-subgroup complexes do not apply. However, those which do not affect the first subgroup of the chain can still be used.

We note that if we consider the contribution to the above alternating sum of OWC from all chains beginning with (conjugates of) a fixed p-subgroup U, then this will be zero unless U = O_{p}(N_{H}(U)), by now standard arguments. This is because the contractions used to cancel this contribution in the case U < O_{p}(N_{H}(U)) do not disturb the first subgroup of the chain.

We also point out a dichotomy which occurs between the case when B has defect 0, and when B has positive defect. For when B has defect 0, it is clear that only the singleton chain σ = 1 makes a non-zero contribution to the right hand side of OWC, and then only for d = 0, in which case that contribution is f_{0}(B) = k(B). By contrast, when B has positive defect, then ( for every d,) each chain σ ∈ N (H, 1) makes zero contribution to the right hand side of OWC.

We remind the reader that Alperin’s Weight Conjecture is hidden within OWC. If we consider the contribution to the right hand side of OWC from the trivial character of each V_{σ} ( and sum over all d), then the result should be ℓ(B) ( for notice that if a normal chain σ starts U < V … and μ is the trivial character of U, then V/U ≤ O_{p}(I_{H_{σ}}(μ)/U), so no p-block of the latter group has defect 0. Hence when μ is the trivial character, only chains of length one can contribute.

We will usually find it more convenient to work with the B-subpair formulation of the Ordinary Weight Conjecture ( OWC). Let N_{B} denote the collection of normal chains of B-subpairs of the form (σ, b_{n}) = ((U_{1}, b_{1}) < … < (U_{n}, b_{n})). This means that the chain σ = U_{1} < … < U_{n} is in N (H), and that each subpair (U_{i}, b_{i}) is a B-subpair, with the usual ( strict) inclusion. We note that the stabilizer, H_{(σ,b_{n})}, of this chain is just I_{H_{σ}}(b_{n}). We let β_{(σ,b_{n})} denote the block 1_{b_{n}}RH_{(σ,b_{n})} of RH_{(σ,b_{n})}. For μ an irreducible character of V_{σ}, we let H_{(σ,b_{n},μ)} be the inertial subgroup of μ in H_{(σ,b_{n})}, and we define the block β_{(σ,b_{n},μ)} accordingly. We let f_{1}(β_{(σ,b_{n},μ)}) denote the number of irreducible characters in β_{(σ,b_{n},μ)} which contain V_{σ} in their kernels and lie in p-blocks of defect 0 of H_{(σ,b_{n},μ)}/V_{σ}.

Then, in its subpair version, when B has positive defect, OWC may be stated as:

ORDINARY WEIGHT CONJECTURE ( SUBPAIR FORMULA-TION): For each non-negative integer d, we have

k_{d}(B) = ∑_{σ∈N_{B}/H}(−1)^{1+|σ|} ∑_{μ∈Irr_{d}(V_{σ})/H_{σ}} f_{1}(β_{(σ,b_{n},μ)}).

REMARK: We may assume that all the orbit representatives of chains of Bsubpairs are chosen to be contained in a fixed maximal B-subpair (D, b_{D}), and that they are always chosen so that N_{D}(V_{1}) is a defect group for the block 1_{b_{1}}RN_{H}(V_{1}, b_{1}). In that case, it is shown in [14] that for the chain (σ, b_{n}) as above, if we have C_{D}(V_{1}) ≠ Z(V_{1}), then we have f_{1}(β_{(σ,b_{n},μ)}) = 0 for every μ ∈ Irr(V_{σ}). Hence we may further assume in OWC that the initial subgroup V_{σ} satisfies C_{D}(V_{σ}) = Z(V_{σ}). It also follows from the results of [17],[18] that there is zero contribution to the alternating sum from all those chains (σ, b_{n}) starting with a fixed B-subpair (V, b_{V}) for which O_{p}(N_{H}(V, b_{V})/V C_{H}(V)) ≠ 1. We will see an alternative derivation of this fact in the next section.

REMARK: It may be helpful to note for future reference that ( as is wellknown) for any B-subpair (U, b_{U}), we have V = O_{p}(N_{H}(U)) ≤ N_{H}(U, b_{U}). One explanation for this is that, given any simple F C_{H}(U)-module S in b_{U}, we have I_{N_{H}(U)}(S) ≤ N_{H}(U, b_{U}). Hence every simple F N_{H}(U)-module T which lies over S is induced from N_{H}(U, b_{U}). But V acts trivially on T, so that V must be contained in N_{H}(U, b_{U}).

SECTION 2: WEIGHTS AND KÜLSHAMMER-PUIG CONSTRUCTIONS

For the convenience of the reader, we will review some of the results of the paper [11] of Külshammer-Robinson in the form we will need them. Let N be a normal subgroup of a finite group G. We remind the reader that an irreducible character χ of G is said to be N-projective if it may be afforded by an Nprojective (R-free) RG-module. For a given irreducible character μ of N, [11] gives us a bijection between N-projective irreducible characters of G which lie over (G-conjugates of) μ and N-projective irreducible characters of the inertial subgroup I_{G}(μ) which lie over μ ( as usual in Clifford Theory, the bijection arises via induction of appropriate characters from the inertial subgroup).

In view of this, we continue the discussion in the case that μ is G-stable. Then [11] goes on to prove that there is a finite p′-central extension

1 → Z̃ → G̃ → G/N → 1,

and a faithful linear character λ̃ of Z̃ such that the number of N-projective irreducible characters of G which lie over μ is the number of blocks of defect 0 of RG̃ which lie over λ̃. It is not stated explicitly in [11], but inspection of the constructions reveals that the 2-cocycle α of G/N used to construct G̃ is the p′-part ( in the Schur Multiplier of G/N) of the 2-cocycle of G/N afforded by μ. The order of Z̃ is the order of α. If N is a p-group, then α is trivial.

We note that it follows in particular from these results that in the statement of OWC, the expression

∑_{μ∈Irr_{d}(V_{σ})/H_{σ}} f_{0}^{(B)}(I_{H_{σ}}(μ)/V_{σ})

is precisely the number of V_{σ}-projective irreducible characters of H_{σ} in Brauer correspondents of B.

The discussion above gives us particular reason to consider the p′-part of 2-cocycles.

LEMMA 2.1 : Let X be a finite group, Y be a normal subgroup of X. Let μ be an X-stable irreducible character of Y. Then there is an RY -module whose isomorphism type is X-stable, affording character μ. Furthermore, the associated factor set consists of eμ(1)-th roots of unity, where e = e(Y) is the exponent of Y/Y′.

PROOF: We may replace X by a central extension

1 → Ẑ → X̂ → X → 1,

with Ẑ cyclic, having a normal subgroup Ŷ canonically isomorphic to Y such that the character μ̃ of Ŷ naturally identified with μ extends to an irreducible character μ̂ of X̂. Furthermore, this central extension arises by considering a 𝕂Y -module affording μ with associated representation τ, and considering the associated projective (in Schur’s sense) representation σ, chosen so that for a fixed transversal T to Y in X, we have ytτ = yσtτ and det(tτ) = 1 for all t ∈ T, y ∈ Y. Then det(xτ) is an e-th root of unity for all x ∈ X, and the associated factor set consists of eμ(1)-th roots of unity.

Now μ̂ may be afforded by an RX̂-module, as R is a principal ideal domain, and the result follows.

LEMMA 2.2 : Let N be a normal subgroup of the finite group G. Let M be an RN-module whose reduction (mod J(R)) is irreducible, and let μ be the character afforded by M. Then:

i) M is ( up to isomorphism) the unique RN-module affording character μ. ii) If μ is G-stable, then the p′-part of the 2-cocycle of G/N associated to μ is the same 2-cocycle as that associated to M/J(R)M.

PROOF: i) This is well-known, but we include a proof for completeness. Suppose that L_{1} : N → GL(h, R) is the matrix representation afforded by M ( after a choice of R-basis) and that L_{2} : N → GL(h, R) is another representation of N affording character μ. Then we can certainly find an invertible matrix X ∈ GL(h, 𝕂), which we may assume has entries in R, but not all in J(R), such that X^{−1}L_{1}(n)X = L_{2}(n) for all n ∈ N. Now X has the form UDV, where U, V ∈ GL(n, R) and D ∈ M_{n}(R) is diagonal with each entry on its main diagonal dividing the next ( in R). We may assume that the first diagonal entry of D is 1.

Replacing L_{1}, L_{2} by R-equivalent representations if necessary, we may assume that U = V = I. Now assume that r is the maximal integer such that d_{rr} ∉ J(R). If r = h, then D ∈ GL(n, R), so that L_{1}, L_{2} are equivalent over R. Suppose that r < h. Then since X^{−1}L_{1}(n)X ∈ GL(h, R) for all n ∈ N, we see that the (i, j)-entry of L_{1}(n) is in J(R) whenever both i > r and j ≤ r. This contradicts the fact that the reduction modJ(R) of N is irreducible. Hence L_{1} and L_{2} are indeed equivalent over R. ii) By part i) and Lemma 2.1, the RN-isomorphism type of M is G-stable. Furthermore, the associated factor set consists of eμ(1)-th roots of unity, where e is the exponent of N/N′. This induces a cocycle ( necessarily of p′-order) associated to the reduction (mod J(R)) of M. Since there is a canonical bijection between p′-roots of unity in R and p′-roots of unity in F, we conclude that the induced cocycle is just the p′-part of the 2-cocycle associated to μ.

This result is particularly relevant in the case when a covered block of a normal subgroup is nilpotent. It is well-known ( eg from results of Broué and Puig [3]) that all characters of height 0 in a nilpotent block remain irreducible ( mod p).

Now let X be a finite group, B be a block of RX with defect group D. Let (D, b_{D}) be a maximal B-subpair, and (U, b_{U}) be a subpair contained in (D, b_{D}) such that U is non-trivial, C_{D}(U) = Z(U), and N_{D}(U) is a defect group for the block 1_{b_{U}} RN_{X}(U, b_{U}).

Let G be a subgroup of N_{X}(U, b_{U}) containing UC_{X}(U), and let N = UC_{X}(U). Then b′_{U} = 1_{b_{U}} RN is a G-stable nilpotent block of RN.

Let b_{U}^{∗} be the unique block of RG covering the block b′_{U} of RN. Now the block b_{U} has |Z(U)| irreducible characters, all of the same degree. One of these, say θ_{1}, has Z(U) in its kernel. For each linear character, λ, of Z(U), there is a unique character θ_{λ} in b_{U} ( viewed as a block of C_{X}(U)), such that θ_{λ}(zc) = λ(z)θ_{1}(c) whenever z ∈ Z(U) and c ∈ C_{X}(U) is p-regular ( θ_{λ}(t) = 0 for elements t not expressible as such a product). We note that an irreducible character of G lies in the block b_{U}^{∗} if and only if it lies over (a G-conjugate of) some θ_{λ}.

The irreducible characters in b′_{U} are well-known to all have the form μ ∗ θ_{λ}, where μ is an irreducible character of U lying over the linear character λ of Z(U). Such a character is afforded by an RN-module of the form V_{1} ⊗V_{2}, where V_{1} is an RU-module affording μ and V_{2} is an RC_{X}(U)-module affording θ_{λ} ( the latter module being unique up to isomorphism). We have μ∗θ_{λ}(uc) = μ(u)θ_{λ}(c) whenever u ∈ U and c ∈ C_{X}(U) is p-regular ( μ ∗ θ_{λ}(t) = 0 for elements t not expressible as such a product).

Now, using Lemma 2.1 again, we may assume that the isomorphism type of V_{1} (as RN-module) is I_{G}(μ)-stable, and that the associated factor set consists of p-power roots of unity. We know from Lemma 2.2 that the isomorphism type of V_{2} is I_{G}(λ)-stable, and that the p′-part of the associated 2-cocycle is the restriction to I_{G}(λ)/N of the 2-cocycle of G/N afforded by θ_{1} ( the 2-cocycle afforded by θ_{1} has order prime to p because ( by a Theorem of P. Fong), θ_{1} can be realized over an extension of ℚ obtained by adjoining p′-roots of unity). Hence the p′-part of the 2-cocycle of I_{G}(μ)/N afforded by V_{1} ⊗ V_{2} is the restriction to I_{G}(μ)/N of the 2-cocycle of G/N afforded by θ_{1}.

Let us choose a fixed irreducible character χ = μ ∗ θ_{λ} of N, and let us note before proceeding further that I_{G}(χ) = I_{G}(μ) ≤ I_{G}(λ). Now an irreducible character ψ of G lying in the block b_{U}^{∗} is U-projective if and only if it is Nprojective, since U ⊲ G and U is a defect group for the unique block b′_{U} of RN covered by b_{U}^{∗}.

We may set H = G/N, and construct a p′-central extension

1 → Z̃ → H̃ → H → 1,

with Z̃ cyclic, constructed from H and the 2-cocycle of H afforded by θ_{1}. We may choose a faithful linear character λ̃ of Z̃ ( independent of the choice of μ ) so that if we set H(μ) = I_{G}(μ)/N, and consider the corresponding subgroup H̃(μ) of H̃, then there is a bijection between N-projective irreducible characters of G lying over χ and Z̃-projective irreducible characters of H̃(μ) lying over λ̃.

Since Z̃ is a p′-group, we notice that an irreducible character of H̃(μ) is Z̃-projective if and only if it lies in a block of defect zero of RH̃(μ).

We conclude that the number of U-projective irreducible characters of G which lie in the block b_{U}^{∗} and lie over μ is the number of blocks of defect 0 of RH̃(μ) which lie over the linear character λ̃ of Z̃. We also note that this quantity is determined by the subgroup of the outer automorphism group of U induced by the action of I_{G}(μ), together with the restriction to H(μ) of the 2-cocycle of RN afforded by θ_{1}.

We recall some results of Külshammer and Puig [10] which are helpful in this situation above. In this situation, they construct an extension

1 → Z(U) → L → G/C_{G}(U) → 1,

which has a normal subgroup isomorphic to U and whose Sylow p-subgroup is isomorphic to a defect group of the block b_{U}^{∗} of G.

They also prove that the block b_{U}^{∗} is Morita equivalent to a block β_{U} of the group algebra RL̃, where

1 → W̃ → L̃ → L → 1

is the p′-central extension obtained from L using the 2-cocycle of G/C_{G}(U) afforded by θ_{1}. Now W̃ ≅ Z̃. Let γ̃ be the linear character of W̃ corresponding to λ̃. Let Ũ be the (normal) subgroup of L̃ which is naturally identified with U, and for μ as above, let μ̃ be the irreducible character of Ũ corrresponding to μ.

Since CL̃(Ũ) = W̃ × Ũ, we note that Irr(β_{U}) consists of all of Irr(L̃, γ̃). For μ ∈ Irr(U), we set μ̃ ∗ γ̃ to be the obvious irreducible character of Ũ × W̃ .

What we have demonstrated above is:

THEOREM 2.3: Let G be as above. Then :

i) The actions of G on Irr(U) and the action of L̃ on Irr(U) can be canonically identified. ii) For any irreducible character χ = μ ∗ θ_{λ} in b_{U}, (where μ ∈ Irr(U) and μ lies over the linear character λ of Z(U)), there is a (defect preserving) bijection between U-projective irreducible characters of G in the block b_{U}^{∗} which lie over χ and Ũ-projective irreducible characters of L̃ which lie over μ̃ ∗ γ̃.

One way of interpreting this theorem is that, in the situation above, the Külshammer-Puig Morita equivalence respects ordinary weights. We now proceed to show that this respect of weights can be extended to stabilizers of chains of B-subpairs. We remark also that Külshammer and Puig prove that the fusion of b_{U}^{∗}-subpairs containing (U, b_{U}) is the fusion of the p-subgroups of L̃ containing Ũ.

Now consider a normal chain

(σ, b_{n}) = ((U, b_{U}) < … < (V_{n}, b_{n}))

of B-subpairs with stabilizer X_{(σ,b_{n})}.

This time, we set G = C_{X}(U)X_{(σ,b_{n})}, N = C_{X}(U)V_{n}. Now, given x ∈ N_{G}(V_{n}, b_{n}) ∩ C_{X}(U), we have [V_{n}, U, x] = [U, x, V_{n}] = 1, so that [x, V_{n}, U] = 1. Hence [V_{n}, x] ≤ Z(U) and x ∈ X_{(σ,b_{n})}. Hence we have X_{(σ,b_{n})} ≥ N_{G}(V_{n}, b_{n}), the opposite inclusion being trivially true. Hence we have X_{(σ,b_{n})} = N_{G}(V_{n}, b_{n}) and also V_{n}C_{X_{(σ,bn)}}(U) = N_{N}(V_{n}, b_{n}), Notice also that if x above is p-regular, we obtain [V_{n}, x, x] = 1 and then [V_{n}, x] = 1, so that

O^{p}(C_{X_{(σ,bn)}}(U)) = O^{p}(C_{X}(V_{n})).

Now the block b_{n}^{∗} = 1_{b_{U}} RG is the unique block of RG covering the block b_{U} of RC_{X}(U). Then b_{n}^{∗} covers a unique nilpotent block β_{n}^{∗} of RN with defect group V_{n}, which is the unique block of RN lying over the block b_{U} of RC_{X}(U) and is G-stable. Notice that (V_{n}, b_{n}) is a maximal β_{n}^{∗}-subpair.

Now we have G = NN_{G}(V_{n}, b_{n}), and in this situation, the results of Külshammer-Puig Theorem tell us that the block b′_{n} = 1_{b_{n}}RX_{(σ,b_{n})} is Morita equivalent to b_{n}^{∗}. Notice that

G/N ≅ N_{G}(V_{n}, b_{n})/N_{N}(V_{n}, b_{n}) = X_{(σ,b_{n})}/V_{n}C_{X_{(σ,bn)}}(U).

When these two isomorphic groups are naturally identified, this Morita equivalence amounts to the fact that the 2-cocycle of RG/N afforded by the unique extension of θ_{1} to N which contains V_{n} in its kernel is the same as the 2-cocycle of

X_{(σ,b_{n})}/V_{n}C_{X_{(σ,bn)}}(U)

afforded by the unique irreducible character in the block 1_{b_{n}}RN_{N}(V_{n}, b_{n}) which has V_{n} in its kernel. For future use, we let ψ_{1} denote the unique irreducible character of C_{X_{(σ,bn)}}(U) which contains Z(U) in its kernel in the block 1_{b_{n}}RC_{X_{(σ,bn)}}(U), and we define ψ_{λ} similarly for each λ ∈ Irr(Z(U)). Notice that since 1_{b_{n}} is primitive as a central idempotent of RC_{X}(V_{n}) and O^{p}(C_{X_{(σ,bn)}}(U)) = O^{p}(C_{X}(V_{n})), it is also primitive as a central idempotent of RC_{X_{(σ,bn)}}(U).

Suppose now that we wish to count U-projective irreducible characters in these Morita equivalent blocks. We first need to compare the actions of G and X_{(σ,b_{n})} on Irr(U). Since G = C_{X}(U)X_{(σ,b_{n})}, these are clearly the same. Then for a given representative μ of a G-orbit on Irr(U), lying over a linear character λ of Z(U), we need to compare the number of U-projective irreducible characters of I_{G}(μ) which lie over the irreducible character θ_{λ} of C_{X}(U) with the number of U-projective irreducible characters of I_{X_{(σ,bn)}}(μ) which lie over the irreducible character ψ_{λ} of C_{X_{(σ,bn)}}(U).

In the discussion above, we can replace G by I_{G}(μ) and X_{(σ,b_{n})} by I_{X_{(σ,bn)}}(μ), and we still have Külshammer-Puig Morita equivalences as above. Theorem 2.3 and its pre-amble tell us that the number of U-projective irreducible characters of I_{G}(μ) which lie over the irreducible character θ_{λ} of C_{X}(U) can be computed by working within the central extension L̃ of the Külshammer-Puig extension

1 → Z(U) → L → G/C_{X}(U) → 1.

A similar conclusion applies for the group X_{(σ,b_{n})} with an analogous group L̃_{(σ,b_{n})}. But the Külshammer-Puig Morita equivalences above arise because L̃ ≅ L̃_{(σ,b_{n})}. Hence we have:

THEOREM 2.4 : For G, (σ, b_{n}) as above, the actions of G on Irr(U) and the action of X_{(σ,b_{n})} on Irr(U) are the same. Furthermore, for any irreducible character μ of U, there is a (defect preserving) bijection between U-projective irreducible characters of G in the block b_{n}^{∗} = 1_{b_{U}} RG which lie over μ and Uprojective irreducible characters of the block 1_{b_{n}}RX_{(σ,b_{n})} which lie over μ .

REMARK: This result tells us that the ordinary weights associated to X_{(σ,b_{n})} can equally well be calculated with respect to the group C_{G}(U)X_{(σ,b_{n})}. The advantage of this is that ( the restriction of) the 2-cocycle of N_{X}(U, b_{U})/UC_{X}(U) afforded by the irreducible character θ_{1} is always the 2-cocycle we have to deal with when we consider normal chains of B-subpairs beginning with (U, b_{U}). Furthermore, in calculating the contribution to the alternating sum of the generalized weight conjecture from chains of B-subpairs beginning with (U, b_{U}), we can instead replace N_{X}(U, b_{U}) by a central extension L̃ ( obtained by using the above 2-cocycle) of the Külshammer-Puig extension

1 → Z(U) → L → N_{X}(U, b_{U})/C_{X}(U) → 1

and the block 1_{b_{U}} RN_{X}(U, b_{U}) by the block of RL̃ consisting of all irreducible characters lying over the faithful linear character γ̃ of W̃ .

In other words, the ordinary weight conjecture may be rephrased as:

ORDINARY WEIGHT CONJECTURE, KÜLSHAMMER-PUIG FORM: Let B be a block of RX with defect group D, and let (D, b_{D}) be a maximal B-subpair. Let A(B) be a set of representatives for the X-conjugacy classes of B-subpairs (U, b_{U}) contained in (D, b_{D}) such that:

i) C_{D}(U) = Z(U). ii) N_{D}(U) is a defect group for 1_{b_{U}} RN_{X}(U, b_{U}).

For each (U, b_{U}) ∈ A(B), let α(U, B) be the 2-cocycle of N_{X}(U, b_{U})/UC_{X}(U) afforded by the unique irreducible character in b_{U} with Z(U) in its kernel. Let L = L(B, U) be the Külshammer-Puig extension

1 → Z(U) → L → N_{X}(U, b_{U})/C_{X}(U) → 1

and let L̃ = L̃(B, U) be the central extension

1 → W̃ → L̃ → L → 1,

obtained using the Külshammer-Puig 2-cocycle α = α(B, U). Let γ̃ = γ̃(B, U) denote the canonically corresponding faithful linear character of W̃ .

Let N_{p}(L, U) denote the set of normal chains of p-subgroups of L whose initial term is U. This may be canonically identified with N_{p}(L̃, Ũ). Let

w(U, B, d) = ∑_{σ∈N_{p}(L̃,Ũ)/L̃}(−1)^{|σ|+1} ∑_{μ∈Irr_{d}(Ũ)/L̃_{σ}} f_{0}(IL̃_{σ}(μ̃)/Ũ, γ̃),

where Irr_{d}(Ũ) is the set of irreducible characters of defect d of Ũ, and

f_{0}(IL̃_{σ}(μ̃)/Ũ, γ̃)

is the number of blocks of defect 0 of RIL̃_{σ}(μ̃)/Ũ which lie over the linear character γ̃ of (the natural image of ) W̃ Then for each non-negative integer d we have

k_{d}(B) = ∑_{(U,b_{U})∈A(B)} w(U, B, d).

REMARK: This form of the ordinary weight conjecture makes it clear that w(U, B, d) = 0 for each d if we have O_{p}(N_{X}(U, b_{U})/UC_{X}(U)) ≠ 1, for then O_{p}(L̃) > Ũ, and the relevant alternating sum vanishes for the usual trivial reasons.

This means that we may restrict attention to Alperin-Goldschmidt B-subpairs, which satisfy:

i) N_{D}(U) is a defect group for 1_{b_{U}} RN_{X}(U, b_{U}). ii) C_{D}(U) = Z(U). iii)O_{p}(N_{X}(U, b_{U}))/UC_{X}(U)) = 1.

Letting AG(B) denote a set of representatives for the X-conjugacy classes of Alperin-Goldschmidt B-subpairs, we may replace the previous version of the Ordinary Weight Conjecture by:

REDUCED ORDINARY WEIGHT CONJECTURE, KÜLSHAMMER-PUIG FORM: For each non-negative integer d we have

k_{d}(B) = ∑_{(U,b_{U})∈AG(B)} w(U, B, d).

REMARKS: It is worth singling out how to evaluate the quantity w(D, B, d). The unique chain of B-subpairs beginning with (D, b_{D}) is the singleton chain (D, b_{D}). All irreducible characters in the block β_{D} = 1_{b_{D}}RN_{X}(D, b_{D}) are Dprojective. In this case, the Külshammer-Puig Morita equivalence reduces to a Morita equivalence arising in earlier work of W.F.Reynolds [13], and Külshammer [9]. Using the results of Külshammer-Robinson [11], it follows that w(D, B, d) = k_{d}(β_{D}) for all non-negative integers d. This amounts to saying that the equality of OWC is satisfied ( in a somewhat degenerate fashion) by the block β_{D}.

This also illustrates that it is not generally the case that the quantity w(U, B, d) is non-negative. For if this were so, it would imply that

k_{d}(B) ≥ k_{d}(β_{D})

for each non-negative integer d, which is not always the case.

It also illustrates that OWC reduces in a particularly straightforward way when N_{X}(D, b_{D}) controls fusion of B-subpairs. For, in that case, (D, b_{D}) is ( up to conjugacy) the only Alperin-Goldschmidt B-subpair ( see Kessar-Linckelmann-Robinson[7]), and to verify OWC, it suffices to prove that k_{d}(B) = k_{D}(β_{D}) for each non-negative integer d.

More generally, if there is a normal subgroup V of D such that N_{X}(V, b_{V}) controls strong fusion of B-subpairs, then it is shown in Kessar-Linckelmann-Robinson [7] that (V, b_{V}) ≤ (U, b_{U}) and N_{X}(U, b_{U}) ≤ N_{X}(V, b_{V}) for each Alperin-Goldschmidt B-subpair (U, b_{U}) ≤ (D, b_{D}). It follows that w(U, B, d) = w(U, β_{V} , d) whenever (U, b_{U}) ∈ AG(B), where β_{V} = 1_{b_{V}} RN_{X}(V, b_{V}).

Hence we have:

THEOREM 2.5: Suppose that there is a normal subgroup V of D such that N_{X}(V, b_{V}) controls strong fusion of B subpairs, and such that β_{V} = 1_{b_{V}} RN_{X}(V, b_{V}) satisfies OWC. Then B satisfies OWC if and only if k_{d}(B) = k_{d}(β_{V}) for each non-negative integer d.

We also remark that it is easy to adapt the method of [16], together with the above constructions, to prove directly that OWC is true for p-solvable groups. In fact, this is somewhat more straightforward for OWC, given the work above, than it is for DPC. For, let G be a minimal ( among p-solvable groups) counterexample to OWC, in the sense that first [G : Z(G)], then |G| are minimized among p-solvable counterexamples, and let B be a block for which it fails. Then the results of Eaton [6] ( which do not require p-solvability of G) show that O_{p}(G) ≤ Z(G). Let N = O_{p′,p}(G) > Z(G)O_{p′}(G). Let β be a block of RN covered by B, D be a defect group for B, chosen so that D_{0} = D ∩ N is a defect group for β. Then, as β is a nilpotent block of RN, a maximal β-subpair (D_{0}, β_{D_{0}}) controls strong fusion of β-subpairs. Then by the results of Kessar-Linckelmann-Robinson, [7], N_{G}(D_{0}, b_{D_{0}}) controls strong fusion of B-subpairs. Let β^{∗} = 1_{b_{D_{0}}} RN_{G}(D_{0}, b_{D_{0}}). The Külshammer-Puig theorem shows that B and β^{∗} are Morita equivalent, so that k_{d}(B) = k_{d}(β^{∗}) for all non-negative integers d. On the other hand, we know by minimality of G that OWC is valid for the block β^{∗}. Hence the remarks above imply that OWC holds for B.

It is easy to check that if B is a tame block ( p = 2), then N_{X}(U, b_{U})/UC_{X}(U) has trivial Schur-multiplier whenever (U, b_{U}) is an Alperin-Goldschmidt Bsubpair, so that the Külshammer-Puig 2-cocycles are all trivial in this case. This makes it easy to verify directly,(using results of Brauer and of Olsson) that OWC is true for all blocks B of tame type. We remark that DPC is known to be true for blocks of tame type by work of K. Uno [19]. We illustrate the proofs of OWC for tame blocks with one:

EXAMPLE: Suppose that D is semi-dihedral of order 2^{e} ≥ 16. Suppose that (U, b_{U}) is a proper Alperin-Goldschmidt B-subpair. Then it is easy to check that U is either quaternion of order 8 or a Klein four group, and that there are at most two proper elements in AG(B). Furthermore, we always have N_{G}(U, b_{U})/UC_{G}(U) ≅ S_{3}. Let us consider the case where there is one element (U, b_{U}) of AG(B) with U quaternion of order 8, and another element (V, b_{V}) of AG(B) with V a Klein four group.

Now S_{3} has trivial Schur multiplier, so w(U, B, d) can be calculated using L = GL(2, 3), and all normal chains of 2-subgroups of L which start with O_{2}(L) ≅ U. Now U has one irreducible character of defect 2, which is L-stable, the trivial character of defect 3, and one other L-orbit of irreducible characters of defect 3, for which the inertial subgroup is a Sylow 2-subgroup S of L.

The calculation of w(U, B, 2) reduces to f_{0}(RL/O_{2}(L)) − f_{0}(RS/O_{2}(L)). Hence w(U, B, 2) = 1.

The contribution to w(U, B, 3) from the trivial character of U is also [f_{0}(RL/O_{2}(L)) − f_{0}(RS/O_{2}(L))] = 1. The contribution to w(U, B, 3) from the other orbit of linear characters of U is more complicated to evaluate. The chain O_{2}(L) < S has stabilizer S. According to reduced OWC, we first need to compute the orbits of S on Irr(U). We have accounted for the orbit of the trivial character and the orbit of the irreducible character of degree 2. However, the non-trivial linear characters break up into two orbits under the action of Sone with stabilizer O_{2}(L), and one with stabilizer S. The orbit with stabilizer S makes a contribution −1 to w(U, B, 3). Hence we find that w(U, B, 3) = 0.

Similar arguments work for (V, b_{V}), except that this time we must work with L = S_{4}, and all irreducible characters of V have defect 2, the nontrivial characters falling into a single L-orbit of length 3 whose stabilizer is a 2-group. As above, the orbit of non-trivial linear characters makes contribution −1 to w(V, B, 2) and the orbit of the trivial character makes contribution 1 to w(V, B, 2), so that w(V, B, 2) = 0.

What of w(D, B, d)? In this case, N_{G}(D, b_{D})/DC_{G}(D) = 1, as this index is odd, while Aut(D) is a 2-group. In this case, we find that L = L̃ ≅ D, and hence w(D, B, e) = k_{e}(D) = 4, w(D, B, e − 1) = k_{e−1}(D) = 2^{e−2} − 1, while w(D, B, d) = k_{d}(D) = 0 for 0 ≤ d < e − 1.

Hence we find that OWC predicts that: k_{e}(B) = 4 k_{e−1}(B) = 2^{e−2} − 1 k_{2}(B) = 1. This is in accordance with the results of Olsson [12]. We note that if we restrict attention to the contribution from the trivial character, we get a contribution of 1 from each of the subpairs (U, b_{U}), (V, b_{V}) and (D, b_{D}), so predicting 3 modular simple modules according to Alperin’s weight conjecture.

The pattern for other tame blocks is similar. There are at most two nonconjugate proper Alperin-Goldschmidt B-subpairs to consider. For any such subpair, (U, b_{U}), we have that U is quaternion of order 8 or a Klein 4 group. In the case that U is a Klein 4-group, we get that w(U, B, d) = 0 for all d ( but 1_{b_{U}} RN_{G}(U, b_{U}) contributes one modular weight). In the case that U is a quaternion group of order 8, we get that w(U, B, d) = 0 for all d ≠ 2, w(U, B, 2) = 1 ( and 1_{b_{U}} N_{G}(U, b_{U}) contributes one modular weight). In all cases, we obtain w(U, B, d) = k_{d}(1_{b_{D}}RN_{G}(D, b_{D})) for all integers d ( and 1_{b_{D}}N_{G}(D, b_{D}) contributes one modular weight when |D| > 8.). In the case that defect group D is a Klein 4-group, or quaternion of order 8, then 1_{b_{D}}RN_{G}(D, b_{D}) contributes three modular weights in the case that [N_{G}(D, b_{D}) : DC_{G}(D)] = 3 one modular weight in the case that [N_{G}(D, b_{D}) : DC_{G}(D)] = 1. In this case, there are no proper Alperin-Goldschmidt B-subpairs to consider. These results all accord with the results of Olsson [12].

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