Further consequences of conjectures like Alperin’s

by

GEOFFREY R. ROBINSON

ISAAC NEWTON INSTITUTE FOR MATHEMATICAL SCIENCES CAMBRIDGE March 26 1997

INTRODUCTION AND STATEMENT OF MAIN RESULTS

In this paper, we are concerned with consequences of the conjectures of [7] (which are all related to, or descended from, Alperin’s weight conjecture [2]). While these conjectures are difficult to check directly, any ”theoretical” predictions which they make can be tested, and we make some further such predictions here, which in some cases have independent interest in their own right. We assume familiarity with the notation and results of [7] and the earlier papers [4],[8]. Our main results are

THEOREM : Let G be a finite group, p be a prime, λ be a linear character of a central p-subgroup,Z, of G. Suppose that the subgroup, H, of G controls strong p-fusion in G, and that the principal p-blocks of G,H, respectively, both satisfy Conjecture 4.1 of [7] ( for any choice of d,Z and λ). Then there is a height-preserving ( equivalently, defect preserving) bijection between irreducible characters in the principal p-block of G which lie over λ and irreducible characters in the principal p-block of H which lie over λ.

THEOREM : Let G be a finite group, p be a prime, B be a p-block of G with non-trivial defect group D, and let H = N_{G}(D, b), where (D, b) is a maximal B-subpair. Then if H controls strong fusion of B-subpairs in G, and B satisfies the formulae of [7] for each linear character λ of each central p-subgroup of G, there is ( for each linear character λ of each central p-subgroup of G), a height-preserving bijection between irreducible characters in B which lie over λ and irreducible characters in b^{∗} which lie over λ, where b^{∗} is the unique block of H which covers b.

REMARK : These two theorems extend results of Alperin [2], and show that a stronger form of a conjecture of Alperin [1] would follow from Conjecture 4.1 of [7]. In section 3 of the paper, we also show that many chains can be omitted when checking conjecture 4.1 of [7] for principal blocks. In section 5, we remark that Alperin’s conjecture makes some strong predictions which may be related to other current conjectures. In section 6, we use the solvable subgroup complex to show that if G has no non-trivial solvable normal subgroup, then k(G) is locally determined in a rather precise way.

SECTION 1 : PERMISSIBLE OPERATIONS ON COMPLEXES

In [7], several formulae were given which were related to ( or refinements of ) Alperin’s weight conjecture, and which involved alternating sums over radical chains of p-subgroups ( up to conjugacy). In this paper, we will have occasion to work with various of the other ( now familiar ) p-subgroup complexes, as well as radical chains. Some care must be taken, however, since the functions involved in the alternating sums depend not just on the chain stabilizer, but also on the initial subgroup of the chain. This means that Proposition 3.3 of [4] can not always be applied. In particular, ( in the notation of [4]), the alternating sum over P need not be the same as the alternating sum over E for the above type of function , as the reductions used may alter the first subgroup of the chain. The argument of ([4], Proposition 3.3 ) showing that chains in P\U make no contribution to the relevant alternating sums still remains valid, since those reductions do not affect the first subgroup of a chain.

We indicate here how to eliminate chains in P\N in a way which does not disturb the initial subgroup of any chain. Given a chain σ = Q_{o} < Q_{1} < … < Q_{n} of P\N, we construct σ′ as follows : choose i minimal subject to Q_{i} < Q_{n}, and choose j maximal subject to Q_{i} < Q_{j}. If N_{Q_{j}+1}(Q_{i}) = Q_{j}, delete Q_{j} from σ; otherwise, insert N_{Q_{j+1}}(Q_{i}) between Q_{j} and Q_{j+1}. Then it is easy to check that σ′′ = σ, |σ′| = |σ|, and G_{σ} = G_{σ′}. Furthermore, σ and σ′ have the same initial subgroup.

We leave it as an exercise for the reader to verify that chains in P\R may be eliminated in a similar fashion. Hence we have : LEMMA 1.1 : Let A be an Abelian group, and f : P → A be a G-stable function such that f(σ) depends only on the chain stabilizer G_{σ} and the initial subgroup of the chain σ. Then :

∑_{σ∈P/G}(−1)^{|σ|}f(σ) = ∑_{σ∈U/G}(−1)^{|σ|}f(σ) = ∑_{σ∈N /G}(−1)^{|σ|}f(σ) = ∑_{σ∈R/G}(−1)^{|σ|}f(σ).

We may also record that the above arguments include as a special case :

COROLLARY 1.2 : Let A,f be as in Lemma 1.1 and let U be a non-trivial p-subgroup of G which is not radical.Then chains in P which begin with ( conjugates of ) U make zero contribution to the first alternating sum appearing in Lemma 1.1 .

SECTION 2 : RE-CASTING CONJECTURES

Theorem 2.3 of [7] states that ( for a fixed prime p), Alperin’s weight conjecture is equivalent to the assertion that whenever B is a p-block of positive defect of a finite group G, then

k(B) = ∑_{σ∈R/G}(−1)^{|σ|+1} ∑_{μ∈Irr(V_{σ})/G_{σ}} f_{o}^{(B)}(\frac{I_{G_{σ}}(μ)}{V_{σ}}).

In view of the results of section 1, we may replace R by P or N or U. Similarly,the later refinements ( in [7]) of this conjecture may be stated using any of the complexes above. However, if G has a non-trivial central p-subgroup , Z, say, it need not be the case that all chains in P have an initial subgroup which contains Z, so we need to modify the definition of Irr_{d}(V_{σ}, λ) used in [7] ( for λ a linear character of Z). So for σ a chain in P with initial subgroup V, Irr_{d}(V, λ) will henceforth denote {μ ∈ Irr(V) : p^{d}μ(1) = |V |, ⟨Res_{V ∩Z}^{V}(μ), Res_{V ∩Z}^{V}(λ)⟩ ≠ 0}.

To some extent, this is a technical inconvenience, as chains whose initial subgroup does not contain Z make no contribution to the alternating sums considered, but it allows a uniformity of statement.

Now suppose that there is a non-trivial central p-subgroup,Z say, of G, and let λ be a linear character of Z. Since there is a bijection between p-blocks of G and of G/Z, an easy induction argument shows that Alperin’s weight conjecture ( in its equivalent form above), implies that whenever the p-block B has positive defect, we should have:

k(B, λ) = ∑_{σ∈R/G}(−1)^{|σ|+1} ∑_{μ∈Irr(V_{σ},λ)/G_{σ}} f_{o}^{(B)}(\frac{I_{G_{σ}}(μ)}{V_{σ}}),

where Irr(V_{σ},λ) is defined as in [7] ( subject to the above modification of the definition of Irr_{d}(V_{σ}, λ) given above).

The first step is to note that we may pass to G/ker(λ) and assume that Z is cyclic. Then we compare G/Ω_{1}(Z) with G, using a suitable inductive hypothesis. Finally, we need only note that, given Galois conjugate linear characters of Z, the same number of irreducible characters of B lie over each of them. Hence refining the previous formula to take account of linear characters of Z still results in a formula equivalent to Alperin’s weight conjecture. However, if we now refine the formulae according to character defects, we obtain the following conjecture which appears to be a genuine strengthening of Alperin’s weight conjecture :

CONJECTURE 2.1 : Let G be a finite group, B be a p-block of positive defect of G, d be a positive integer, λ be a linear character of a central p-subgroup,Z, of G. Then

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

( and the formula remains valid if P is replaced by any of N,U or R ).

REMARK : Notice that Z is contained in I_{G_{σ}}(μ) for each σ,μ, so that if Z is not contained in V_{σ}, then

f_{o}^{(B)}(\frac{I_{G_{σ}}(μ)}{V_{σ}}) = 0.

The other conjectures in [7] may be recast in a similar fashion, but we do not state them explicitly.

SECTION 3 : PRINCIPAL BLOCKS AND CONTROL OF FUSION

Suppose now that B is the principal p-block of the finite group G. Let D be a Sylow p-subgroup of G. Let S be a set of representatives for the G-conjugacy classes of subgroups U of D such that N_{D}(U) ∈ Syl_{p}(N_{G}(U)) and C_{D}(U) = Z(U).

LEMMA 3.1 : For any positive integer d, and any linear character λ of any central psubgroup,Z, of G, we have :

∑_{σ∈N /G}(−1)^{|σ|+1} ∑_{μ∈Irr_{d}(V_{σ},λ)/G_{σ}} f_{o}^{(B)}(\frac{I_{G_{σ}}(μ)}{V_{σ}})

=

∑_{U∈S} ∑_{\{σ∈N:V_{σ}=U\}/N_{G}(U)}(−1)^{|σ|+1} ∑_{μ∈Irr_{d}(U,λ)/G_{σ}} f_{o}^{(B)}(\frac{I_{G_{σ}}(μ)}{V_{σ}}).

( Note that, by earlier results, if U is in S, but U is not radical, then chains beginning with U make no contribution to the right-hand alternating sum).

PROOF : Choose a subgroup, U, of D, so that ( without loss of generality ), N_{D}(U) is a Sylow p-subgroup of N_{G}(U). It suffices to prove that, in the left hand alternating sum above, chains starting with ( conjugates of ) U make no contribution unless C_{D}(U) = Z(U). If U is not radical, this is clear. Furthermore, if U is radical, the contribution from nonradical chains starting with ( conjgates of ) U vanishes. Hence we consider a radical chain σ = U < V_{1}.... < V_{n} such that N_{D}(U) ∈ Syl_{p}(N_{G}(U), and

f_{o}^{(B)}(I_{G_{σ}}(μ)/U) ≠ 0

for some μ in Irr(U). Let E_{σ} be a Sylow p-subgroup of G_{σ}. Then E_{σ}^{x} ⊆ N_{D}(U) for some x ∈ N_{G}(U), so we may suppose that D contains a Sylow p-subgroup of G_{σ}. Now Brauer’s Third Main theorem and the argument used in the proof of Theorem 5.1 of [7] show that C_{D}(U) = Z(U).

COROLLARY 3.2 : If U ∈ S, but U ∉ Syl_{p}(O_{p′,p}(N_{G}(U)), then there is no contribution to the right-hand side of the equation appearing in Lemma 3.1 from chains starting with U.

PROOF : Let V = D ∩ O_{p′,p}(N_{G}(U), which is a Sylow p-subgroup of O_{p′,p}(N_{G}(U)). Then V > U. Set H = N_{G}(U). By a Frattini argument, we have H = O_{p′}(H)N_{H}(V). Set L = N_{H}(V), and assume ( as we may ) that N_{D}(U) ⊆ L. Then L controls strong p-fusion in H. Each chain σ ∈ N whose initial subgroup is U is conjugate within H to a chain whose subgroups are all contained in N_{D}(U), and two chains of this form which are H-conjugate are already L-conjugate. Now if σ = U < V_{1} < … < V_{n} is a chain in N with V_{n} ⊆ N_{D}(U), then a standard argument shows that N_{H}(V_{n}) = O_{p′}(N_{H}(V_{n}))N_{L}(V_{n}) = O_{p′}(C_{G}(V_{n}))N_{L}(V_{n}) ( note that C_{G}(V_{n}) ⊆ H). Now O_{p′}(C_{G}(V_{n})) acts trivially on Irr(U), and Dedekind’s modular law yields:

G_{σ}(= H_{σ})) = O_{p′}(C_{G}(V_{n}))L_{σ} ⊆ O_{p′}(H_{σ})L_{σ}.

Hence we have H_{σ} = O_{p′}(H_{σ})L_{σ} and H_{σ}/O_{p′}(H_{σ}) ≅ L_{σ}/O_{p′}(L_{σ}). Notice that H_{σ} and L_{σ} have the same action on Irr(U). Furthermore, for each μ in Irr(U), we have I_{G_{σ}}(μ) = O_{p′}(H_{σ})I_{L_{σ}}(μ). Since B is the principal block, we have

f_{o}^{(B)}(I_{G_{σ}}(μ)/U) = f_{o}^{(B_{1})}(I_{H_{σ}}(μ)/U) = f_{o}^{(B_{2})}(I_{L_{σ}}(μ)/U),

where B_{1} is the principal p-block of H, B_{2} is the principal p-block of L. It follows readily that the contribution from chains starting with U is

∑_{\{σ∈N (L^{∗}):V_{σ}^{∗}=U^{∗}\}/L^{∗}} ∑_{μ∈Irr_{d}(U,λ)/L_{σ}^{∗}} f_{o}^{(B_{2})}(\frac{I_{L_{σ}^{∗}}(μ)}{U^{∗}}),

where L^{∗} = L/O_{p′}(L). Since V^{∗} = O_{p}(L^{∗}) > U^{∗}, this alternating sum vanishes by the results of section 1. Now we are in a position to prove :

THEOREM 3.3 : Let G be a finite group such that (for all relevant d,Z,λ), the formula of conjecture 2.1 holds for the principal p-block of G, and p divides |G|. Let D ∈ Syl_{p}(G). Let F denote {Q ⊆ D : N_{D}(Q) ∈ Syl_{p}(N_{G}(Q)), C_{D}(Q) = Z(Q)andQ ∈ Syl_{p}(O_{p′,p}(N_{G}(Q))}.

Let F_{o} be a set of representatives for the distinct conjugacy classes of p-subgroups of G meeting F. Then for each d,λ , we have

k_{d}(B, λ) = ∑_{\{U∈F_{o}\}} ∑_{\{σ∈N:V_{σ}=U\}/N_{G}(U)} ∑_{μ∈Irr_{d}(U,λ)/G_{σ}} f_{o}^{(B)}(\frac{I_{G_{σ}}(μ)}{V_{σ}}).

Furthermore, if G has a subgroup H ≥ D which controls strong p-fusion in G, and B̃ ( the principal p-block of H ), also satisfies the equality of conjecture 2.1 ( for all relevant d,λ), then for each positive integer d, each linear character λ of any central p-subgroup,Z, of G, we have: k_{d}(B, λ) = k_{d}(B̃, λ).

PROOF: The first statement is an immediate consequence of Corollary 3.2. To prove the second, we choose a subgroup ,U, in S(G) ( in the notation of 3.1). Then, as H controls strong p-fusion in G, we have N_{G}(U) = C_{G}(U)N_{H}(U). Since N_{D}(U) is a Sylow p-subgroup of N_{G}(U), C_{D}(U) is also a Sylow p-subgroup of C_{G}(U), so C_{G}(U) has a normal p-complement, as C_{D}(U) = Z(U). Thus N_{G}(U) = O_{p′}(C_{G}(U))N_{H}(U).

It follows readily that U is a Sylow p-subgroup of O_{p′,p}(N_{G}(U)) if and only if U is a Sylow p-subgroup of O_{p′,p}(N_{H}(U). Since H controls the fusion of elements of S(G), we may now proceed by arguments similar to those of Corollary 3.2 to deduce that

k_{d}(B, λ) = ∑_{U∈F_{o}(H)} ∑_{\{σ∈N:V_{σ}=U\}/N_{H}(U)}(−1)^{|σ|+1} ∑_{μ∈Irr_{d}(U,λ)/H_{σ}} f_{o}^{(B̃)}(\frac{I_{H_{σ}}(μ)}{V_{σ}}) = k_{d}(B̃, λ).

For principal blocks, we can also strengthen Theorem 5.1 of [7]:

COROLLARY 3.4 : Let G be a finite group of order divisible by the prime p and such that the equality of Conjecture 2.1 holds for the principal p-block, B, of G ( for each d, λ ). Suppose that for some central p-subgroup, Z, of G, and some linear character, λ, of Z, there is an irreducible character of defect d in B which lies over λ.

Then there is a p-subgroup,U, of G such that :

(i) Z(U) ∈ Syl_{p}(C_{G}(U)) .

(ii) U ∈ Syl_{p}(O_{p′,p}(N_{G}(U))).

(iii) U has an irreducible character,μ, of defect d, lying over λ.

REMARKS: It should be noted that J.L. Alperin proved in [2] that it would follow from his weight conjecture that if B, the principal p-block of G, is a controlled block ( in the case of the principal p-block, this means that a Sylow normalizer controls strong p-fusion ), and we set H = N_{G}(D), ( for D a Sylow p-subgroup of G ), then k(B) = k(B̃) and ℓ(B) = ℓ(B̃), where B is the principal p-block of H.

It might be tempting to wonder whether, in view of Corollary 3.4, it would suffice to restrict attention to subgroups of D in Goldschmidt’s conjugation family when calculating k_{d}(B, λ) for the principal p-block in a group where the equalities predicted by Conjecture 2.1 are known to hold. We give an example to show that this need not be the case:

EXAMPLE : Let H be the semi-direct product of C_{3}×C_{3} with SL(2, 3) in its natural action, and let G = H × H. Let U = O_{3}(H), S ∈ Syl_{3}(H), V = O_{3}(G), T = S × S ( and p = 3, B be the principal 3-block of G, which in this case is the whole group algebra RG, where R is an appropriate complete discrete valuation ring of characteristic 0 whose residue field has characteristic 3). In this case, Goldschmidt’s conjugation family (F_{1},say) is {U × S, S × U, S × S}. The reason that V is not in F_{1} is that N_{G}(V)/V does not posses a strongly 3-embedded proper subgroup. We will show, however, that chains in R which start with V make a non-zero contribution to one of the alternating sums appearing in the statement of Conjecture 2.1. The family F_{o} of Theorem 3.3 is ( for our G) : {U × S, S × U, S × S, V }, and we note that S and T are self-normalizing in H,G, respectively. The chains in R which start with V are ( up to conjugacy ): σ : V,τ : V < U × S,α : V < S × U, β : V < T, γ : V < U × S < T, δ : V < S × U < T. We note that G_{σ} = G , G_{τ} = H × S , G_{α} = S × H , G_{β} = G_{γ} = G_{δ} = T. The contribution to the alternating sum of Conjecture 2.1 ( with Z trivial) to k_{4}(B, 1) from chains beginning with V is ( using some of the results of [6] and [7]), easily seen to be : #(V-projective irreducible characters of RG) - 2#(V-projective irreducible characters of R(H × S)) + #(V-projective irreducible characters of RT), which ( using [6] again, for example ) is 1 − 4 + 4 = 1.

SECTION 4 : SUBPAIRS AND CONTROLLED BLOCKS

Let B be a p-block of the finite group G with non-trivial defect group D, and let (D, b) be a maximal B-subpair. Given a chain σ = V_{o} < V_{1} < ..... < V_{n} in P, σ will make zero contribution to any alternating sum of the form appearing in Conjecture 2.1 unless, perhaps, V_{n} is contained up to conjugacy in D. Suppose, then that V_{n} ⊆ D. Then there is a unique chain of B-subpairs, σ′ = (V_{o}, B_{o}) < (V_{1}, b_{1}) < … < (V_{n}, b_{n}) ≤ (D, b), and we note that G_{σ′} = I_{G_{σ}}(b_{n}) ( using the uniqueness of (V_{i}, b_{i}) ≤ (D, b) ).

Now T r_{G_{σ′}}^{G_{σ}} (1_{b_{n}}) is the block idempotent of a block of G_{σ} which is Morita equivalent to the unique block of G_{σ′} covering b_{n}. By the uniqueness ( up to conjugacy ) of maximal Bsubpairs,( and making use of some of the results of [4] and [7]), we may re-state Conjecture 2.1 in an equivalent way, using the language of subpairs:

CONJECTURE 2.1′ : Let G,B be as above. Let P_{s}(B), ( similarly N_{s}(B), etc. ) be a full set of representatives for the distinct G-conjugacy classes of chains,σ, of B-subpairs (V_{o}, b_{o}) < (V_{1}, b_{1}) < … < (V_{n}, b_{n}) with (V_{n}, b_{n}) ⊆ (D, b), C_{D}(V_{o}) = Z(V_{o}), and with (D, b) containing a Sylow subpair of G_{σ}. Then whenever d is a positive integer, and λ is a linear character of a central p-subgroup of G, then we have :

k_{d}(B, λ) = ∑_{σ∈calN_{s}(B)}(−1)^{|σ|+1} ∑_{μ∈Irr_{d}(V_{σ},λ)/G_{σ}} f_{o}(I_{G_{σ}}(μ)/V_{σ}, b_{n})

( and N_{s}(B) can be replaced by any of P_{s}(B),U_{s}(B) or R_{s}(B) ), where σ is the chain (V_{o}, b_{o}) < … < (V_{n}, b_{n}) and f_{o}(I_{G_{σ}}(μ)/V_{σ}, b_{n}) denotes the number of blocks of defect 0 of I_{G_{σ}}(μ)/V_{σ} which lie in the unique block of I_{G_{σ}}(μ) covering the block b_{n} of C_{G}(V_{n}). Now we set H = N_{G}(D, b). Our next result shows that Conjecture 2.1′ implies a conjecture of Alperin [1] in a somewhat stronger form:

THEOREM 4.1: Let G, B, D, b and H be as above, and assume that H controls strong fusion of B-subpairs. Let d be a positive integer,λ be a linear character of a central p-subgroup of G. Then if the formula of conjecture 2.1 holds for k_{d}(B, λ), we have k_{d}(B, λ) = k_{d}(B̃, λ), where B̃ is the Brauer correspondent of B for N_{G}(D).

PROOF : Let σ = (V_{o}, b_{o}) < … < (V_{n}, b_{n}) be a chain in N_{s}(B) with V_{n} < D. By the hypothesis on H, we have N_{G}(V_{n}, b_{n}) = C_{G}(V_{n})N_{H}(V_{n}, b_{n}). Hence G_{σ} = C_{G}(V_{n})H_{σ}. Thus H_{σ} and G_{σ} act in the same way on Irr(V_{o}). For any μ ∈ Irr(V_{o}), we have I_{G_{σ}}(μ) = C_{G}(V_{n})I_{H_{σ}}(μ). Let D_{σ} = D ∩ G_{σ}(= D ∩ H_{σ}). We also note that C_{G}(V_{n})I_{D_{σ}}(μ) is normal in I_{G_{σ}}(μ). Suppose that I_{G_{σ}}(μ)/V_{o} has a block of defect zero whose unique irreducible character covers b_{n} ( when viewed as a character of I_{G_{σ}}(μ)). Since C_{D}(V_{n}) = Z(V_{n}), we see that b_{n} has defect group V_{n} as a block of C_{G}(V_{n}), so contains a unique simple module ( using the well-known structure of blocks with central defect group). Since N_{D}(V_{n}) stabilizes b_{n}, I_{D_{σ}}(μ) certainly stabilizes the unique simple module of b_{n}. Hence the unique block of C_{G}(V_{n})I_{D_{σ}}(μ) which covers b_{n} has defect group I_{D_{σ}}(μ). Since the unique block of I_{G_{σ}}(μ) which covers b_{n} also covers a block of defect zero of I_{G_{σ}}(μ)/V_{o}, we must have V_{o} = I_{D_{σ}}(μ).

Now let (D_{σ}, b_{n+1}) be a Sylow subpair of G_{σ} contained in (D, b). Then the unique block of I_{G_{σ}}(μ) which covers b_{n} has defect group I_{D_{σ}}(μ) ( which is V_{o}). We conclude that f_{o}(I_{G_{σ}}(μ)/V_{o}, b_{n}) is the number of blocks of I_{G_{σ}}(μ) which cover b_{n} and have defect group V_{o}. This is 1 if I_{D_{σ}}(μ) = V_{o}, 0 otherwise. If this last integer is 1, then we also have f_{o}(I_{G_{τ}} (μ)/V_{o}, b) = 1, where τ = (V_{o}, b_{o}) < … < (V_{n}, b_{n}) < (D, b). Conversely, if we have f_{o}(I_{G_{τ}} (μ)/V_{o}, b) ≠ 0, then we conclude that I_{D_{τ}} (μ)(= I_{D_{σ}}(μ)) = V_{o}, so that f_{o}(I_{G_{τ}} (μ)/V_{o}, b) = 1. But also, as I_{D_{σ}}(μ) = V_{o}, V_{o} is a defect group for the unique block of I_{G_{σ}}(μ) covering b_{n}. Hence we also find that f_{o}(I_{G_{σ}}(μ)/V_{o}, b_{n}) = 1. We deduce that the contribution to the alternating sum we are considering from the chains σ and τ cancel each other. Since H controls strong fusion of B-subpairs in (D, b), we may conclude that the formula of Conjecture 2.1′ reduces in this case to :

k_{d}(B, λ) = ∑_{μ∈Irr_{d}(D,λ)} f_{o}(I_{H}(μ)/D, b),

which ( using the results of [6] ) is the number of D-projective irreducible characters of defect d of H which lie over the block b of C_{G}(D) and over λ. Thus k_{d}(B, λ) = k_{d}(b, λ) where b is the unique block of H covering b, and the result follows.

SECTION 5: CONNECTIONS WITH OTHER EQUIVALENCES

In [8], a connection is mentioned between the verification of a special case of one of the equivalent forms of Alperin’s weight conjecture presented there, and a theorem of Külshammer and Puig [5] ( which was then in preprint form. The result of [5] is in fact much more general than is required for this particular application. In this section, we point out how the above-mentioned formula from [8] makes predictions ( in some cases, as yet unverified ) in situations where other equivalences are known ( or conjectured) to be present. The reformulation of Alperin’s conjecture in question is that if N is a normal subgroup of the finite group G, and B is a p-block of G, then the number of simple B-modules should the sum over p-subgroups,Q, of N ( up to G-conjugacy ) of the number of simple N_{G}(Q)/Qmodules in Brauer correspondents of B which lie over blocks of defect zero of N_{N}(Q)/Q.

A case of special interest, to which other cases can usually be reduced, is when B covers a G-stable block,b say, of N, which has defect group D, ( say). Let b^{∗} be the corresponding block of N_{N}(D). A theorem of Harris and Knörr [3] states that there is a unique Brauer correspondent of B for N_{G}(D) ( which covers b^{∗}), and we denote this by B^{∗}. Now suppose that we have ℓ(b) = ℓ(b^{∗}). This occurs, for example, if b is nilpotent. Both Alperin’s weight conjecture and a conjecture of Brouŕme imply that this should also occur when D is Abelian. A long-standing conjecture predicts that this should also be the case when b and b^{∗} are stably equivalent. Given the above equality, if we assume that Alperin’s weight conjecture holds for b, we must conclude that for Q < D, no projective simple N_{N}(Q)/Q module can lie in a Brauer correspondent of b. In that case, no simple modules of N_{G}(Q)/Q lie in Brauer correspondents of B and over projective simple modules of N_{N}(Q)/Q). If follows easily that the above formula reduces under the current hypothesis to ℓ(B) = ℓ(B^{∗}). Hence we see that Alperin’s weight conjecture implies :

CONJECTURE 5.1 : Let B be a p-block of G which covers a G-stable block b of the normal subgroup ,N, of G. Let b^{∗} be the Brauer correspondent of b for N_{N}(D), and let B^{∗} be the unique Brauer correspondent of B for N_{G}(D). Then if ℓ(b) = ℓ(b^{∗}), we also have ℓ(B) = ℓ(B^{∗}).

REMARK : If b as above is nilpotent, then the main theorem of [5] proves that B and B^{∗} are in fact Morita equivalent. It would be of interest to know under what conditions, assuming that b and b^{∗} above are derived equivalent, it necessarily follows that B and B^{∗} are also derived equivalent

SECTION 6: SOLVABLE CONTROL OF CONJUGACY CLASSES

Let S denote the simplicial complex associated to the poset of non-trivial solvable subgroups of G. We consider the generalized character

Γ = ∑_{σ∈S/G}(−1)^{|σ|}Ind_{G_{σ}}^{G}(γ_{σ})

where γ_{σ} is the complex character of G_{σ} afforded by its conjugation action on itself. For each prime p, the methods of [4] easily adapt to show that

∑_{σ∈S/G}(−1)^{|σ|}Ind_{G_{σ}}^{G}(RG_{σ})

is a virtual projective RG-module ( R an appropriate local ring whose residue field has characteristic p). The key observation is that for each non-trivial p-subgroup,Q, of G, S^{Q} is N_{G}(Q)-contractible via X → XQ → Q. As p was arbitrary, the generalized character Γ vanishes on all of G^{#}. Hence Γ is an integer multiple, m say, of the regular character. Since Γ(1) = m|G|, we deduce that

m = ∑_{σ∈S/G}(−1)^{|σ|}

as Ind_{G_{σ}}^{G}(γ_{σ}) has dimension |G| for each σ. On the other hand, counting the multiplicity of the trivial character of G in Γ using Frobenius reciprocity yields:

m = ∑_{σ∈S/G}(−1)^{|σ|}k(G_{σ}).

Since the G-orbit of σ contains [G : G_{σ}] simplices, we deduce that

k(G) − 1 = ∑_{σ∈S^{#}}(−1)^{|σ|+1}(k(G_{σ}) − 1)/[G : G_{σ}],

where S^{#} denotes the collection of non-empty simplices of S.

References

[1] J.L. Alperin, Local Representation Theory in Proc. Symp. Pure Math. 37, (AMS, Providence, 1980), 369-376.

[2] J.L. Alperin,Weights for finite groups in Proc. Symp. Pure Math 47, (AMS, Providence, 1987), 369-379.

[3] M.E. Harris and R. Knörr, Brauer correspondence for covering blocks of finite groups, Comm. Alg., 13, (5), (1985), 1213-1218.

[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 L.Puig, Extensions of nilpotent blocks, Invent. Math., 102, (1990), 17-71.

[6] B. Külshammer and G.R. Robinson, Characters of Relatively Projective Modules, II, J. London Math. Soc. (2), 36, (1987), 59-67.

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

[8] G.R. Robinson and R. Staszewski, More on Alperin’s Conjecture, Astérisque, 181-82, (1990), 237-255.