ARITHMETICAL PROPERTIES OF BLOCKS Geoffrey R. Robinson Department of Mathematics and Computer Science University of Leicester Leicester LE1 7RH England

SECTION 1 : BACKGROUND

Richard Brauer began the development of the theory of modular representations of finite groups, and one of his main motivations was to obtain arithmetical information about complex irreducible characters of finite groups. Given a finite group,G, and a prime p, we may consider the group algebra 𝕂G, where 𝕂 is the field of fractions of a complete discrete valuation ring R of characteristic 0. We assume that F = R/π is algebraically closed of characteristic p, where π is the unique maximal ideal of R, and we also assume that R contains “sufficiently many” ( though a finite number of ) p-power roots of unity ( in particular, enough to ensure that 𝕂 contains a splitting field for any finite group we consider in this article).

Under these assumptions, 𝕂 contains roots of unity of all orders prime to p. Also, the structure of the group algebra 𝕂G is very much like the structure of the complex group algebra ℂG. 𝕂G is the direct sum of k(G) full matrix algebras over 𝕂 , where k(G) is the number of conjugacy classes of G, and by identifying a cyclotomic subfield of ℂ with an isomorphic copy in 𝕂, we may deduce information about complex irreducible characters from knowledge of characters of representations over 𝕂.

On the other hand, the group algebra 𝕂G contains the group ring RG, and we have the natural epimorphism from RG to F G (reduction ( mod π), or , more informally, reduction (mod p)). Now RG inherits a natural metric from the valuation ring structure on R, and it is an easy exercise to verify that , given any idempotent, e, of F G, and a pre-image, E say, of e in RG, the limit lim_{n→∞} E^{p^{n}} exists and is an idempotent of RG. From this, it is easy to deduce that any decomposition of 1_{G} as a sum of mutually orthogonal primitive idempotents of F G “lifts” to a decomposition of 1_{G} as a sum of the same number of mutually orthogonal primitive idempotents of RG. Furthermore, central idempotents of F G lift to central idempotents of RG .

We may certainly write F G as a direct sum of mutually annihilating indecomposable algebras, say F G = B_{1} ⊕ B_{2} … ⊕ B_{m}, and it follows from the discussion above that

RG = B̃_{1} ⊕ B̃_{2} … ⊕ B̃_{m},

where the B̃_{i}’s are mutually annihilating indecomposable two-sided ideals of RG ( and the number of summands is the same as for F G). The former decomposition corresponds to the unique expression of 1_{G} as a sum of mutually orthogonal primitive idempotents of Z(F G) ( the centre of the group algebra F G ). The summands of these decompositions are known as blocks of F G ( respectively RG). Each block of F G is a finite-dimensional algebra in its own right, and by considering the module categories xdvi mod − B_{i} we can simplify the study of the module category mod − F G.

It is easy to see that, given an indecomposable F G-module, M say, there is exactly one value of i for which 1_{B_{i}} does not annihilate M, so we say that the module M “belongs” to the block B_{i}. More generally, we say that a finite-dimensional F G-module N belongs to the block B_{j} if each of its indecomposable summands is in the block B_{j} ( by the way, under our assumptions, the Krull-Schmidt theorem is available in both mod−F G and mod−RG). In a similar fashion, we can assign certain RG-modules ( including all indecomposable modules ) to blocks of RG. There are no non-zero morphisms in mod − F G between modules which lie in distinct blocks ( and similarly for mod − RG).

Since R is a principal ideal domain with field of fractions 𝕂, each finite-dimensional representation of G over 𝕂 is equivalent to a representation of G over R ( in other words, each character afforded by a finite-dimensional 𝕂G-module may be afforded by an R-free RG-module ). Given an irreducible character χ, say of 𝕂G, any RG-module which affords χ is certainly indecomposable, so is assigned to a unique block of RG ; we assign the character χ to this block ( the block is the unique B̃_{i} such that χ(1B̃_{i}) ≠ 0, so does not depend on the particular choice of RG-module affording χ ).

Brauer proved that the number of irreducible characters assigned to the block B̃_{i} is the Rrank of its centre Z(B̃_{i}), which is also the F -dimension of Z(B_{i}). We let Irr(B̃_{i}) denote the set of irreducible characters assigned in this way to B̃_{i}, and we let k(B̃_{i}) denote the cardinality of Irr(B̃_{i}). If k(B̃_{i}) = 1, then it readily follows that B̃_{i} is isomorphic to a full matrix algebra over R, hence B_{i} is isomorphic to a full matrix algebra over F, and there is a unique simple F G-module assigned to B_{i}, which is projective. Conversely, given a projective simple F G-module, its endomorphism algebra is a full matrix algebra over F which occurs ( up to isomorphism ) as one of the blocks of the group algebra F G. Hence we see Brauer’s result: the blocks of RG which contain a unique irreducible character are in bijection with the isomorphism types of projective simple F G-modules. Irreducible characters which belong to such blocks can be characterized by a purely arithmetical condition. If χ is such a character, lying in a block B̃_{i}, then χ(1)χ is the character afforded by the projective right RG-module B̃_{i}, so that χ vanishes on all p-singular elements of G ( that is to say, elements of order divisible by p ). It follows easily from this that χ(1_{G}) is divisible by |G|_{p}. Conversely, if χ is any irreducible character of RG whose degree is divisible by |G|_{p}, then the associated central idempotent of 𝕂G defined by ∑_{g∈G}(χ(1)χ(g^{−1})/|G|)g is an idempotent of Z(RG) , and is primitive. This idempotent yields a block of RG which contains no irreducible character of 𝕂G other than χ. Hence we see the second part of Brauer’s result: the number of isomorphism types of projective simple F G-modules is the number of irreducible characters of 𝕂G of degree divisible by |G|_{p}.

Blocks of RG which contain a unique irreducible character are known as blocks of defect 0 of RG. Speaking informally, such blocks “ignore the presence of the prime p”. Their module categories have a very transparent structure. Indeed, so far, we have not made any assumptions about the prime p, and if p does not divide |G|, we see that all blocks of RG are blocks of defect 0. In the more interesting case that p divides |G|, there is always at least one block which is not of defect 0, the principal block, that is, the block of RG to which the trivial character is assigned. We remarks also that if O_{p}(G) ≠ 1, then O_{p}(G) acts trivially on each simple F G-module, which means that no such simple module can be projective, and there are no blocks of defect 0 of RG.

One of the main themes of this article is that the computation of several familiar block-theoretic invariants of RG can ( conjecturally, at least) be precisely described in terms of numbers of blocks of defect 0 of sections of G ( that is to say, groups of the form M/N where M is a subgroup of G and N ◁ M ). Though proofs of these conjectures do not appear to be imminent, there is growing evidence that the conjectures are valid for ( at least ) large classes of finite groups.

The group G acts by conjugation on the group algebra RG ( similarly on F G). Given a subgroup, H, of G, we have the relative trace map T r_{H}^{G} : RG^{H} → Z(RG) defined via αT r_{H}^{G} = ∑_{t∈H/G} α^{t}. We note that ImT r_{H}^{G} is an ideal of Z(RG), which we denote by RG_{H}^{G}. J.A. Green ( [24]) proved that for each block B̃_{i} of RG, there is a subgroup,D_{i}, of G ( unique up to conjugacy) which is minimal subject to 1B̃_{i} ∈ RG_{D_{i}}^{G}, and that D_{i} is a p-group. The p-group D_{i} is called the defect group of the block B̃_{i}. This definition agrees with an earlier definition of defect group given by Brauer. An arithmetical characterization of the order of D_{i} was given by Brauer : [G : D_{i}]_{p} is the largest power of p which divides χ(1)_{p} for each χ ∈ Irr(B̃_{i}). There are very real senses in which the complexity of the group-theoretic structure of the defect group D_{i} measures the compexity of the structure of the group algebra B_{i}. We note in particular that D_{i} is trivial if and only B̃_{i} has defect 0. More generally, we say that B̃_{i} is a block of defect d_{i} if D_{i} has order p^{d_{i}}. Another arithmetical characterization of d_{i} has been given by Brauer ( with a different proof by J.A. Green)

: p^{2a−d_{i}} is the exact power of p dividing dim_{F} (B_{i}), where the Sylow p-subgroups of G have order p^{a}.

SECTION 2 : p-LOCAL STRUCTURE

Up to now, we have explained how questions about arithmetical properties of irreducible characters can be “localized” in a number-theoretic sense by concentrating on the particular prime p. The existence of defect groups has also hinted that group-theoretic structure is likely to play a role in deeper investigations. This is indeed the case. A p-local subgroup of the group G is a subgroup of the form N_{G}(Q), where Q is a non-trivial p-subgroup of G. It is very much in the spirit of Brauer’s investigations to express the belief that block theoretic invariants of blocks of positive defect of RG should be determined “p-locally”. In [1], J.L. Alperin described the “main problem of block theory” as the determination of the values of the irreducible characters of G on p-singular elements in terms of p-local information.

A key tool in relating structure of blocks to p-local subgroups is the Brauer morphism. For each non-trivial p-subgroup , Q, of G, the Brauer morphism, denoted Br_{Q}, is the vector space projection from F G^{Q} to F C_{G}(Q) whose kernel is spanned by the Q-conjugacy class sums of those Q-conjugacy classes of cardinality greater than one. The Brauer morphism is indeed an algebra homomorphism, and it restricts to an algebra homomorphism from Z(F G) to Z(F N_{G}(Q)). For each block B_{i} of F G, Br_{Q}(1_{B_{i}}) is either 0, or may be decomposed as a sum of primitive idempotents of Z(F N_{G}(Q)). Each of these idempotents corresponds to a block of F N_{G}(Q) , and we call the blocks which occur Brauer correspondents of B_{i}. One characterization of the defect group D_{i} of the block B_{i} is that it is a p-subgroup which is maximal subject to the property that Br_{D_{i}}(1_{B_{i}}) ≠ 0.

An indication of the sort of result we would like to prove is provided by Brauer’s First Main Theorem: for each p-subgroup ,D say, of G, there is a bijection between the blocks of F G with defect group D and the blocks of F N_{G}(D) with defect group D. In particular, if D is non-trivial, this number is p-locally determined. Evidence that this programme has some chance of success is provided by the following result, which is a re-casting of an old result of Brauer and Feit [6] :

THEOREM 2.1 (Brauer and Feit [6]) : The number of irreducible characters of 𝕂G which lie in blocks of positive defect of RG is bounded in terms of p-local information.

The key point in the proof of the above result ( proved by Brauer and Feit in a somewhat sharper form ), is that a block of RG of defect d > 0 contains fewer than p^{2d} irreducible characters . Hence, given S ∈ Syl_{p}(G) ( whose order, p^{a} say, and isomorphism type are p-locally determined,by the way ), we see that the number of irreducible characters in blocks of positive defect of RG is less than rsp^{2a}, where s is the number of non-trivial subgroups of S, and r is the maximum over all non-trivial subgroups, D, of S of the number of blocks of F N_{G}(D) with defect group D. Perhaps worth mentioning is that, in particular, if RG has no blocks of defect zero ( or, more generally, an explicit bound for the number of blocks of defect zero of F G is known),then k(G) is bounded in terms of p-local information, so ( by a result of E. Landau), |G| is bounded in terms of p-local information.

In making more precise statements ( or, at least, predictions), an important tool has been the p-subgroup complex S_{p}(G) ( and some of its variants ), associated to the poset of non-trivial p-subgroups of G. This has been studied by several authors ( e.g. K.S. Brown, [12] , D. Quillen, [42], P.J. Webb [56] ). Its n-simplices are strictly increasing chains of non-trivial p-subgroups of the form Q_{0} < Q_{1} < … < Q_{n}. We also include the empty simplex. There is an obvious boundary map, but that need not concern us here. S_{p}(G) encodes much information about the p-local structure of G. There is an obvious action of G by conjugation on S_{p}(G). Given a simplex, σ = Q_{0} < Q_{1} < … < Q_{n}, of S_{p}(G), the stabilizer of σ is ⋂_{i=0}^{n} N_{G}(Q_{i}). Hence the chain stabilizers not only include the p-local subgroups of G ( when n = 0), but also record how various p-local subgroups of G intersect with each other. We will often have occasion to consider G-stable ( that is, constant on conjugacy classes of subgroups) functions f : { subgroups of G } → A for some Abelian group A. We may form the alternating sum

∑_{σ∈S_{p}(G)/G}(−1)^{|σ|}f(G_{σ}),

where |σ| denotes the number of subgroups appearing in σ. The vanishing or otherwise of this alternating sum is a precise way of measuring whether f is “p-locally determined” or not. For further discussion of a possible precise definition of p-local control, see Thévenaz [54].

SECTION 3 : ALPERIN’S CONJECTURE

For a finite-dimensional algebra A, we will denote the number of isomorphism types of simple A-modules by ℓ(A), the dimension of Z(A) by k(A), and the number of isomorphism types of projective simple A-modules by f_{o}(A). When B is a block and we employ a superscript (B), it is to be read “in Brauer correspondents of B ”.

Alperin’s weight conjecture ( see [3]) predicts that for each block B_{i} of F G, we should have

ℓ(B_{i}) = ∑_{Q} f_{o}^{(B_{i})}(F N_{G}(Q)/Q),

where Q runs through p-subgroups of G ( up to conjugacy). In other words, this is a precise assertion that , as long as B_{i} has positive defect, ℓ(B_{i}) is p-locally determined. Furthermore, in that case, it expresses a fundamental invariant of the non-semi-simple algebra in terms of simple algebras involved in group algebras of smaller groups.

EXAMPLE : M. Cabanes [13] has proved that the equality predicted by Alperin’s weight conjecture is satisfied whenever G is a finite group of Lie type of characteristic p ( see also [3] and [34] ). The interested reader might care to check directly the case of GL(n, 2) ( for p = 2) , which is illuminating, and in many ways typical.

Other cases where the weight conjecture has been verified include [4],[38] ( see also [26] ).

Brauer gave the fundamental formula

k(B_{i}) = ∑_{x} ℓ^{B_{i}}(F C_{G}(x)),

where x runs through p-elements of G ( up to conjugacy) . This formula allows us to compute the number of irreducible characters of 𝕂G assigned to B_{i}. This may be combined with Alperin’s Conjecture above to give a precise prediction of the way that k(B_{i}) is p-locally determined when B_{i} is a block of positive defect of F G. In [27], Knörr and the author gave a different ( but equivalent ) formulation of Alperin’s weight conjecture in terms of irreducible characters of 𝕂G, using the p-subgroup complex S_{p}(G). It is important to understand that the conjectures are equivalent in the sense that one is true ( for a particular prime p ) for all p-blocks of all finite groups if and only if the other is. In general, the fact that the formula predicted by one of the conjectures holds for a particular block does not,a priori, imply that the formula predicted by the other conjecture holds.

Knörr and the author showed in [27] that Alperin’s conjecture is equivalent in the above sense to the assertion that whenever B_{i} is a block of positive defect of F G, we should have

∑_{σ∈S_{p}(G)/G}(−1)^{|σ|}ℓ^{(B_{i})}(F G_{σ}) = 0.

We also showed that we always have

∑_{σ∈S_{p}(G)/G}(−1)^{|σ|}k^{(B_{i})}(F G_{σ}) = ∑_{σ∈S_{p}(G)/G}(−1)^{|σ|}ℓ^{(B_{i})}(F G_{σ}).

This last equality may be viewed as saying that k(B_{i}) − ℓ(B_{i}) is p-locally determined in a precise fashion. Brauer also developed a theory of lower defect groups which would be too much of a diversion for us to explain fully here ( but see, e.g., Olsson [41]). It is a means by which every p-subgroup of G which occurs as a Sylow p-subgroup of the centralizer of some p-regular element of G may be uniquely assigned ( as a so-called lower defect group) to a block of F G. All lower defect groups which are assigned to a block are contained up to conjugacy in the defect group ( in the sense used earlier ) of that block. The multiplicity of 1_{G} as a lower defect group for the block B_{i} is dim_{F} ((B_{i})_{1}^{G}), and we denote it by ℓ_{0}(B_{i}). It can be shown that k(B_{i}) − ℓ_{0}(B_{i}) is p-locally determined in a similar fashion to that in which k(B_{i}) − ℓ(B_{i}) is.

In [46], R. Stazsewski and the author proposed an “equivariant” version of Alperin’s conjecture which was compatible with the action of a group of automorphisms of G, and with Clifford Theory ( and gave various p-subgroup complex formulations of this conjecture). We also proved that a minimal counterexample to Alperin’s conjecture ( for the prime p) has no normal subgroup of index p. For other reformulations of the Alperin weight conjecture, the reader is referred ( for example ) to Thévenaz [55] and Thévenaz and Webb [56].

SECTION 4:THE ALPERIN-McKAY AND DADE CONJECTURES

Brauer proved several results concerning the number of irreducible characters of 𝕂G assigned to a block B_{i} whose degree is divisible by a particular power of p. Many of these were expressed in terms of an invariant called the “height” of a character, which is the amount by which the power of p dividing the character degree exceeds a−d_{i}. In recent times, a more convenient notion, introduced by E.C. Dade, has been that of the defect of a character. This is simply the amount by which the power of p dividing the character degree is less than a. So for χ ∈ Irr(B_{i}), d(χ), the defect of χ satisfies p^{d(χ)}χ(1)_{p} = p^{a}. Each element of Irr(B_{i}) has defect at most d_{i}. For each non-negative integer d, we denote by k_{d}(B̃_{i}) the number of irreducible characters of defect d in Irr( B̃_{i} ). With this notation, the number of irreducible characters of height 0 in B_{i} ( in Brauer’s terminology ) is k_{d_{i}}(B_{i}). The conjecture which has become known as the Alperin-McKay conjecture asserts that if b_{i} is the unique Brauer correspondent of B_{i} for N_{G}(D_{i}), then k_{d_{i}}(b_{i}) = k_{d_{i}}(B_{i}). This conjecture was proved for blocks of p-solvable groups by Okuyama and Wajima [39], but is still open in general. The history of the conjecture is that McKay conjectured that we should have k_{a}(F N_{G}(S)) = k_{a}(F G) where S is a Sylow p-subgroup of G, and then Alperin made the more general block-theoretic version of the conjecture.

Some links between the p-subgroup complex version of Alperin’s conjecture and the Alperin-McKay conjecture were mentioned in [27]. A significant advance in predictions about p-local control of character defects was made when E. C. Dade announced ( in 1990, at MSRI, Berkeley) his ordinary conjecture which asserts that if O_{p}(G) = 1_{G}, then we should have

∑_{σ∈S_{p}(G)/G}(−1)^{|σ|}k_{d}^{(B_{i})}(F G_{σ}) = 0

for each non-negative integer d, if B_{i} has positive defect ( see also [15]). This is a refinement of the Knörr-Robinson formulation of Alperin’s conjecture, since when the formula appearing in the conjecture is summed over all non-negative d, we recover the Knörr-Robinson version of Alperin’s conjecture. Furthermore, when d = d_{i}, Dade’s conjecture reduces to k_{d_{i}}(B_{i}) = k_{d_{i}}^{(B_{i})}(F N_{G}(D_{i})), which is the Alperin-McKay Conjecture. Dade also gave a “projective” version of the conjecture ( in [16])( when O_{p}(G) is central, but not necessarily trivial ), and announced the existence of stronger “invariant” forms of the conjecture, which were compatible with Clifford-Theory, the strongest of which could be reduced to a question about finite simple groups. At present, the statement of some of these conjectures has not been published. In any case, the “ordinary” form of the conjecture may be viewed as a simultaneous generalization of the Alperin weight conjecture and the Alperin-McKay Conjecture. While special cases of these conjectures have been checked by several authors for particular groups ( e.g. [40]) , all the conjectures remain open, and no counterexample has been found to any of them to date.

SECTION 5 : FURTHER REFINEMENTS AND CONSEQUENCES In the physical sciences, the validity of of a scientific theory may be tested by examining

predictions made by the theory which were not made by earlier theories. While a different standard of proof is required in Mathematics, it is nevertheless often the case that the interest or worth of a Mathematical question can be judged by the new avenues of investigation it suggests, and the previously unseen consequences it would have if it were answered in the manner anticipated by its proposer. In this section, we will give formulations and refinements of the previous conjectures which ( if correct ) reduce the determination of the fundamental block-invariants to questions about blocks of defect zero of proper sections of the group G, in accordance with our goal stated at the beginning. We will then examine some more easily stated predicted consequences which are of interest in their own right.

For a chain σ in S_{p}(G), we let V_{σ} denote its initial subgroup ( which is assumed to be trivial if σ is empty). For λ a linear character of a central p-subgroup Z, and μ an irreducible character of a p-subgroup U of G, we let Irr(U, λ) denote the set of irreducible characters of U which lie over Res_{U∩Z}^{Z}(λ), and for each non-negative integer d, we let Irr_{d}(U, λ) denote the set of irreducible characters of defect d in Irr(U, λ). Whenever B_{i} is a block of RG,

In [47] and [50], it is proved ( making some use of results from [30]) that Alperin’s weight conjecture is equivalent to: BASIC CONJECTURE 1 : Whenever λ is a linear character of a central p-subgroup, Z, of G, and B_{i} is a block of positive defect of F G, then we have :

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

This may be refined to : BASIC CONJECTURE 2: Whenever λ is a linear character of a central p-subgroup, Z, of G, and B_{i} is a block of positive defect of F G, then for each non-negative integer d we have we have :

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

A number of remarks are in order here :

a) If B_{i} has defect group D_{i}, we can restrict attention to chains σ for which V_{σ} is conjugate to a subgroup of D_{i}.

b) If V_{σ} ≠ O_{p}(N_{G}(V_{σ})) ( that is, V_{σ} is not radical ), then chains beginning with ( conjugates of) V_{σ} make zero contribution to the above sum.

c) Basic conjecture 2 is probably equivalent to the “projective form” of Dade’s conjecture, but notice that the formulae of the above conjectures do not require us to assume that O_{p}(G) is trivial, or central. We will see presently that this has some advantages.

d) Further refinements of these conjectures have been made, but we omit the details here. It is a somewhat surprising fact that the special case of basic conjecture 1 in which λ is the trivial character implies the general case with arbitrary λ ( in the sense that one holds for all finite groups if and only if the other does ). In fact, it can be shown ( using the results of [51]) that a minimal counterexample to basic conjecture 1, say G, satisfies O_{p}(G) = 1 . It is not clear at present ( at least to the author) that this is the case with basic conjecture 2.

e) If the conjectures are correct, then the calculation of numbers of irreducible characters of given defect in a block can be reduced to questions about blocks of defect zero of proper sections of G , and numbers of characters of given defect of p-subgroups of G.

f) In [47], it is shown that, if the block B_{i} has defect group D_{i}, then chains beginning with V_{σ} will make no contribution to the alternating sum unless there is a conjugate U, of V_{σ} for which C_{D_{i}}(U) = Z(U).

g) If B_{i} is the principal block, then it is shown in [50] that V_{σ} makes no contribution to the alternating sums above unless V_{σ} is a Sylow p-subgroup of O_{p′,p}(N_{G}(V_{σ})).

h) The reader will observe that if basic conjecture 1, is correct, then when λ is trivial, the contribution to the alternating sum from the trivial characters of the various V_{σ}’s should be ℓ(B_{i}).

Let us examine some consequences of the above remarks. Brauer proved that the defect group D_{i} of B_{i} must be radical. If no proper subgroup of D_{i} is radical and contains its centralizer in D_{i}, then remarks a) and b) and f) together yield that if basic conjecture 2 is correct, and the defect group D_{i} of B_{i} has no proper radical subgroup which contains its centralizer in D_{i}, then there is a defect preserving bijection between Irr(B_{i}) and Irr(b_{i}), where b_{i} is the unique Brauer correspondent of B_{i} for N_{G}(D_{i}). Furthermore, this may be refined according to characters lying over any linear character of a central p-subgroup. Note, in particular, that this covers the situation that D_{i} is Abelian.

An even more straightforward consequence of the conjectures above is that if B_{i} has non-Abelian defect group D_{i}, then each irreducible character χ in Irr(B_{i}) satisfies χ(1)_{p} < [G : Z(D_{i})]_{p} ( see [47]). This is a kind of “dual” statement to Brauer’s height zero conjecture, to be mentioned later. This is a new and simple prediction which had not previously been made ( although Brauer had proved an analogous result with |Z(D_{i})| replaced by the exponent of Z(D_{i})). A more precise version of this statement is that ( if the basic conjectures above are true, then the defect of any irreducible character in B_{i} is the defect of an irreducible character of a radical p-subgroup U, of G, which is contained in D_{i} and satifies C_{D_{i}}(U) = Z(U). An interesting special case of this is when G is a finite group of Lie type in characteristic p, where the prediction may be stated the p-defect of any irreducible character of G is the defect of an irreducible character of the unipotent radical of some parabolic subgroup of G. After I mentioned this special case of the prediction in my Newton Institute lectures, M. Geck [21] verified it directly for all “good” primes and all types.

We recall that a subgroup H (of order divisible by p) of the group G is said to control strong fusion for p if [G : H] is not divisible by p, and we have N_{G}(Q)/C_{G}(Q) ≅ N_{H}(Q)/C_{H}(Q) for each p-subgroup, Q, of H. Although we do not have space to reproduce the arguments here, it is proved in [50] that if basic conjecture 2 above holds, and G possesses a subgroup H which controls strong p-fusion, then there is a defect preserving bijection between the irreducible characters in the respective principal blocks of RH and RG ( and this may be refined as above to characters lying over a fixed linear character of a central p-subgroup of G ) ( see also [2] and [3]).

REMARK : A typical example of control of strong fusion is provided when G is a symmetric group S_{ap+b}(b < p) and H is the subgroup S_{ap}. In this particular situation ( and similar ones for linear and unitary groups), M. Enguehard ( [17],[18]) verified that the predicted defect preserving bijection between the two principal blocks does exist ( before the conjectures as stated above were even made!) by constructing a “perfect isometry” between the two blocks ( see the next section) .

Another intriguing prediction can be seen by pursuing an example in Alperin’s article ( [3]). Suppose first that p = 2, and let H = N_{G}(S), where S is a Sylow 2-subgroup subgroup of the group G. Let B_{1} denote the principal 2-block of G, b_{1} denote the principal 2-block of H. Then if ℓ(B_{1}) = ℓ(b_{1}) and Alperin’s conjecture is correct, we may conclude that H controls strong 2-fusion in G, so that the should be a defect preserving bijection between Irr(B_{1}) and Irr(b_{1}). The same argument may be applied with odd p, but the classification of finite simple groups is required- it is necessary to know that if a finite group has a proper strongly p-embedded subgroup, then it has a p-block of defect zero- the most difficult case to check when p is odd is when the Sylow p-subgroups of G are cyclic, but fortunately, this has been done by J.P. Zhang in [59]. The fact that a finite group with a strongly (2)-embedded proper subgroup has a 2-block of defect zero can be proved directly by representation-theoretic means.

Another situation, to which others can often be reduced by Clifford Theory, is when a block, B, of F G, lies over a G-stable block b with defect group D of a normal subgroup N. Let b′ denote the Brauer correspondent for of b for N_{N}(D). Then Harris and Knörr ( 25])have shown that there is a unique Brauer correspondent of B, say B′, for N_{G}(D). Using the results of [46], the author showed in [50], that it would follow from Alperin’s weight conjecture that if ℓ(b) = ℓ(b′), then ℓ(B) = ℓ(B′). In particular, we should have ℓ(B) = ℓ(B′) if D is Abelian, or TI in N_{N}(D) with respect to N, or if b is a nilpotent block. Notice that no assumption was made about the structure of a defect group of B. In fact, when b is a nilpotent block, the predicted equality is known to hold, by a result of Külshammer and Puig ( [32]) ( their result is much more general, and was proved before this consequence was predicted). It would seem that a direct verification of the other predicted equalities would be difficult by currently available methods.

SECTION 6: STRUCTURAL METHODS

We have seen that Alperin’s Conjecture and its various descendants make some sweeping predictions which, it must be admitted, appear at the moment to be far from a proof. It is difficult to foresee what methods will be required to solve the problems raised. There are reasons to believe that it will not be possible to describe the structure of the general block in the kind of detail that has been achieved in the case of ( for example) blocks with cyclic defect group ( [14]), nilpotent blocks,( see [11]) and blocks of tame type ( that is blocks whose defect groups are dihedral, semidihedral or generalized quaternion- e.g. [19]). Nevertheless, determination of the kind of invariants that we have discussed so far may be accessible from coarser information. These invariants are unchanged, for example, under Morita equivalence of blocks. This has been exploited extensively in the block theory of p-solvable groups. A more general type of equivalence of module categories which preserves the block theoretic invariants we have been discussing is that of derived equivalence. For example, while the structure of a block of F G with cyclic defect group D is very strongly controlled by the structure of its Brauer correspondent for F N_{G}(D), the two blocks are not usually Morita equivalent; they are, however, always derived equivalent ( see Linkelmann [35], Rickard [44], and Rouquier [52] ).

Fundamental work on the role of derived equivalences in the representation theory of finitedimensional algebras and in the representation theory of finite groups has been done in recent years by J. Rickard (e.g. [43], [44], [45] ).

The preservation of the above block-theoretic invariants under derived equivalence is explained by the fact that if blocks B and B′ are derived equivalent, then there is a “perfect isometry” from Irr(B) to Irr(B′) . The definition of perfect isometry is due to M. Broué ( see [8]). In particular, the existence of a perfect isometry induces a bijection ( with signs) from Irr(B) to Irr(B′), and it sends generalized characters in B which vanish on all p-singular elements to similar generalized characters in B′. In particular, if such a perfect isometry exists, then the blocks B and B′ satisfy ℓ(B) = ℓ(B′).

A possible direction to look for a “structural” explanation of coincidence of block-theoretic invariants is to find derived equivalences between blocks, or, failing that, to at least find perfect isometries. For example, we have :

BROUÉ’S CONJECTURE: If B̃ is a block of RG with Abelian defect group D, and B̃′ is its Brauer correspondent for RN_{G}(D), then B̃ and B̃′ are derived equivalent.

This conjecture has been the source of intense activity in recent years. It would, for example, provide a “structural” explanation for Alperin’s conjecture ( and the Alperin-McKay conjecture) in the case of blocks with Abelian defect group. In the strong form above, a solution still seems distant. However, much progress has been made, especially in the case of principal blocks, towards directly establishing the existence of a perfect isometry compatible with the Brauer morphism ( called an isotypie) predicted if a particularly favourable type of derived equivalence were present. Fong and Harris have proved this conjecture for principal blocks for the primes 2 and 3 ( using the classification of finite simple groups) - see [20] for the case p = 2.

In the case of finite groups of Lie type, remarkable connections have been found between the existence of perfect isometries and Deligne-Lusztig varieties, and the quest for perfect isometries has led to deep new questions about the Deligne-Lusztig theory. In some ways, the Abelian defect case is the “generic” one for simple groups of Lie type. If G is such a group, and p is neither the characteristic nor a divisor of the order of the Weyl group of G, then G has Abelian Sylow p-subgroups, so all blocks of F G have Abelian defect groups. The reader is referred to work of Broué, Malle and Michel [10], ( and Broué and Malle [9] for some related matters).

We wish to emphasize, however, that there are many instances where there is known ( or conjectured) to be a coincidence of block theoretic invariants, yet there is no possibility of a derived equivalence ( or even perfect isometry) between the blocks in question. For example, when p = 2 and G is a Suzuki group, Broué and Serre have shown that the quadratic forms represented by the Cartan matrices of the principal 2-blocks of G and its Sylow 2-normalizer are inequivalent over the integers, and it follows from this that there can be no perfect isometry between the two principal blocks ( see [8]). However, there is a defect-preserving bijection between irreducible characters in these blocks, and the two blocks ( over F ) have the same number of isomorphism types of simple modules. In this case, the Sylow 2-subgroups of G have class 2. Another interesting case ( yielding infinitely many examples for each prime, where the defect group has class 2 ) is that of G = GL(3, q), where q is a power of p. In this case, for each block , B̃, of positive defect, of RG, there is a defect-preserving bijection between Irr(B̃), and Irr(B̃^{∗}), where B̃^{∗} the Brauer correspondent for the Sylow p-normalizer in G ( i.e. the Borel subgroup). Furthermore, ℓ(B) ≠ ℓ(B^{∗}), so the two blocks are not derived equivalent.

SECTION 7 : COHOMOLOGICAL CONNECTIONS

The results of this section are at the moment somewhat speculative and incomplete, but we believe that the partial results are more than suggestive, and that the speculation is far from idle.

We have seen in section 4 that the various conjectures we have discussed imply that if a subgroup H controls strong p-fusion in G, then there should be a defect preserving bijection between irreducible characters in the principal block of RH and irreducible characters in the principal block of RG. It has long been known that if H controls strong p-fusion in G, then we have the isomorphism H^{∗}(G, F) ≅ H^{∗}(H, F). Using topological methods, G. Mislin [37] proved the opposite implication : if there is an isomorphism in cohomology, H^{∗}(G, F) ≅ H^{∗}(H, F), then the subgroup H must control strong p-fusion in G. So we deduce that if the conjectures we are discussing are correct, then control of cohomology in the above sense should imply control of the block-theoretic invariants we have considered for the principal block of RG. Perhaps, more generally, better understanding of the cohomology ring H^{∗}(G, F) will lead to better understanding of the block-theoretic invariants for the principal block of RG.

Let us examine a possible algebraic approach to Mislin’s result which may give us some insight into trying to understand control of block-theoretic invariants, at least for the principal block. Let S be a Sylow p-subgroup of G, contained in a subgroup H. We suppose that H does not control strong p-fusion in G, and we wish to prove that H^{∗}(G, F) ≠ H^{∗}(H, F).

There must be a non-trivial subgroup, Q say, of S such that N_{G}(Q) ≠ C_{G}(Q)N_{H}(Q), so we may assume that we have chosen Q maximal with respect to this property. Then we easily check that Q has the following properties :

a) N_{S}(Q) ∈ Syl_{p}(N_{G}(Q)).

b) Z(Q) ∈ Syl_{p}(C_{G}(Q).

c) Q ∈ Syl_{p}(O_{p′,p}(N_{G}(Q))). d) C_{G}(Q)N_{H}(Q)/Q is strongly p-embedded in N_{G}(Q)/Q ( or Q = S). Condition d) implies that N_{G}(Q)/O_{p′,p}(N_{G}(Q)) has a proper strongly p-embedded subgroup if Q ≠ S. We recall that a subgroup H of order divisible by p of the group G is said to be strongly p-embedded if |H ∩ H^{g}| is prime to p for each g ∈ G\H.

Let us consider the following “local” situation arrived at in the above case ( Y corresponds to N_{G}(Q) - since we are working with the principal block, we can factor out normal subgroups of index prime to p): Y is a finite group, p is a prime, P = O_{p}(Y) satisfies C_{Y} (P) = Z(P) ≠ 1 ( so that O_{p′}(Y) = 1.) There is a proper subgroup, X of Y, such that P ∈ Syl_{p}(X ∩ X^{y}) for all y ∈ Y \X. Then we have

Ind_{X}^{Y}(F) = F ⊕ T,

where T is a projective FY/P-module ( and is non-zero, as X is proper). Let U be any indecomposable summand of T ( viewed as an F Y/P -module). Then a result of Linnell and Stammbach [36] yields that H^{∗}(Y, U) ≠ 0, as Y is p-constrained.

But by Shapiro’s Lemma, we have H^{∗}(Y, Ind_{X}^{Y}(F)) = H^{∗}(X, F), while also

H^{∗}(Y, Ind_{X}^{Y}(F)) = H^{∗}(Y, F) ⊕ H^{∗}(Y, T).

Thus we have

H^{∗}(Y, F) ≠ H^{∗}(X, F).

Hence, at the “local” level, we have the result we wanted. To complete this algebraic proof of Mislin’s theorem, we need to pass from local to global information. More precisely, we need to find an indecomposable summand of T whose Green correspondent has non-vanishing cohomology. By standard arguments, this is equivalent to showing that

H^{∗}(Y, T)/ ∑_{P <Q} T r_{P}^{Y} (H^{∗}(P, T)) ≠ 0,

a purely “local” question. If follows from the proof of 7.1 of Benson,Carlson, and Robinson [5], that this expression is non-zero if Q is Abelian, or , more generally, if N_{G}(Q)/QC_{G}(Q) acts faithfully on Z(Q). Readers experienced in “local analysis” may note parallels with results of Glauberman and Thompson on control of transfer and fusion ( e.g. [21]).

At the time of writing, the general required non-vanishing result remains unproved, (D.J Benson, J.F. Carlson and I are currently investigating this question). We would like, though, to draw attention to the following analogy : Q is a member of Goldschmidt’s conjugation family, and if Alperin’s conjecture is correct, then N_{G}(Q) should contribute at least one weight, while the above line of enquiry indicates that N_{G}(Q) should contribute to H^{∗}(H, F)\H^{∗}(G, F).

Another ( not entirely unrelated ) line of enquiry is to consider the family F of p-subgroups, Q, of G such that Z(Q) is not a Sylow p-subgroup of C_{G}(Q). We may then consider the category (mod − B_{1})_{F}, which has the same objects as mod − B_{1}, but where morphisms are taken modulo morphisms which factor through an F-projective module. It is easy to modify an argument of Broué to show that if H controls strong p-fusion in G, then there is an equivalence of categories between (mod − B_{1})_{F} and the corresponding category for H ( see [5] ). Furthermore, if the earlier conjectures are correct, then control of the category (mod−B_{1})_{F} ( or possibly its integral analogue) should yield control of the block-theoretic invariants we have been considering for B_{1}

SECTION 8 : UNANSWERED QUESTIONS

We have seen that the conjectures we have studied would imply better understanding of some of the problems raised by Brauer, and others since. In this section, we give some examples of some problems which could be solved by somewhat indirect means if the basic conjectures are true, and we also mention some problems whose status would appear likely to be unaffected, even if affirmative answers were found.

One of the questions which would appear likely to be unaffected is the “k(B) − problem” of Brauer: if the block B_{i} has defect group D_{i}, is it true that k(B_{i}) ≤ |D_{i}|? The methods above have been geared towards reducing problems to “p-local” questions, but this particular problem appears to be difficult even when the defect group is normal This problem is still open even when G is p-solvable, although it is now known to be true in that case for p > 211 ( see [22], [48], [49], and [23],[28] for earlier results).

B. Külshammer has shown [29] that an affirmative answer to the above question for p-solvable groups as well as the Alperin-McKay conjecture for arbitrary groups, would imply the following conjecture of Olsson: k_{d_{i}}(B_{i}) ≤ [D_{i} : D′_{i}]. In particular, Dade’s conjecture ( or basic conjecture 2 above ) together with a positive answer to Brauer’s k(B)-problem for p-solvable groups, imply a positive answer to Olsson’s conjecture ( so for larger primes, basic conjecture 2 would suffice now). Similarly, a positive answer to the Alperin-McKay conjecture ( or Dade’s conjecture) would imply a positive answer to Brauer’s Problem 21 : if a block , B, of FG satisfies k(B) = k, and has defect group D, is it true that |D| is bounded as a function of k?. This was proved by B. Külshammer and the author [33], but made use both of the proof in the special case of p-solvable groups in [31] and of Zelmanov’s solution of the restricted Burnside problem ( [57 and [58]).

Another problem of Brauer’s whose resolution seems unclear, even given a positive answer to the conjectures given earlier is the “other half” of Brauer’s “height zero” conjecture the defect group D_{i} of the block B_{i} of F G is Abelian if and only if each irreducible character in Irr(B_{i}) has defect d_{i}. One half follows easily from the conjectures above ( if the defect group is Abelian, then all characters have maximal defect). It does not seem clear at present that the implication in the other direction would follow, even if basic conjecture 2 were proved.

SECTION 9 : CONCLUDING REMARKS

We have seen that many of the “classical” questions of Brauer ( e.g. [7]) about p-local control of block-theoretic invariants have recently been given a previously unimagined degree of precision, thanks to Alperin’s weight conjecture and its descendants, and Broué’s conjecture in the case of Abelian defect groups. In some cases, a structural explanation has been proposed (such as the existence of a derived equivalence); in others, a structural explanation is yet to be forthcoming.

In Mathematics, formulating the right questions is often a crucial step in the development of an area. There is strong evidence that the questions and conjectures mentioned earlier are at least the beginnings of the “right” questions for a general understanding of block-theory. It requires people of imagination and ingenuity to take up the challenge of answering these questions, giving structural explanations, and formulating the correct new questions to further the development of the subject. Indications are that while this will not be an easy task, it will be richly rewarding.

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