On a question of Brauer in modular representation theory of finite groups

Gerhard Hiss

1 Introduction

This note is based on talks I gave in Paris, Osaka, and Kyoto in 1998. Its aim is to discuss a question of Brauer and present a reduction theorem obtained by my student Olav Düvel in his PhD thesis [4]. In the final section, I shall relate Brauer’s question to Donovan’s conjecture.

Throughout this article, k denotes an algebraically closed field of characteristic p > 0. Moreover, G is a finite group, kG its group algebra over k and mod-kG the category of finitely generated right kG-modules.

Brauer’s question belongs to a whole class of problems, namely to relate numerical invariants of G with cohomological invariants of mod-kG. An example for a numerical invariant of G is |G|_{p}, the p-part of the order of G. A cohomological invariant of mod-kG is c(kG), the largest Cartan invariant of kG, i.e.,

c(kG) = max{dim_{k} Hom_{kG}(P, Q) | P, Q ∈ mod-kG PIMs},

where a PIM is a projective indecomposable kG-module. A relation between the two invariants is implied by Maschke’s theorem: |G|_{p} = 1 if and only if c(kG) = 1. Next consider the case that |G|_{p} = p. This implies that a Sylow p-subgroup of G is cyclic. Then, by results of Brauer and others, kG is representation finite and c(kG) ≤ p = |G|_{p}.

These results and further examples might have motivated the following question of Brauer.

Question 1.1 (Brauer [2], Problem 22) Is it true that c(kG) ≤ |G|_{p} for all finite groups G?

In [11], Landrock showed that the Suzuki group Sz(8) for p = 2 provides a negative answer to Brauer’s question.

Brauer’s original question is asking for a particular bound for the Cartan invariants of kG in terms of |G|_{p}. It is natural to modify Brauer’s original question by just asking for some bound.

Question 1.2 Is there a function f_{p} : ℕ → ℕ such that c(kG) ≤ f_{p}(log_{p} |G|_{p}) for all finite groups G?

2 Blocks and some of their invariants

Brauer’s question can be refined by looking at blocks. We have a direct sum decomposition

kG = ⨁_{B} B,

into indecomposable two-sided ideals B of kG, the blocks of kG. Correspondingly, we have a direct sum decomposition of the module category

mod-kG = ⨁_{B} mod-B.

The problem now is to compare numerical invariants of a block B with cohomological invariants of mod-B. First one has to define the “right” numerical invariants of blocks. To a block B one associates a conjugacy class of p-subgroups of G, the defect groups of B. If D is such a defect group for B,

d(B) := log_{p}(|D|)

is called the defect of B (replacing log_{p}(|G|_{p}) in the previous example).

In the following, a block is a finite dimensional k-algebra which is isomorphic to a block of kG for some finite group G. We consider the following invariants of a block B.

• C(B) := Cartan matrix of B (= matrix of Cartan invariants).

• c(B) := largest entry of C(B).

• ℓ(B) := number of simple B-modules (up to isomorphism).

• LL(B) := min{0 ≠ n ∈ ℕ | J(B)^{n} = 0} (the Loewy length of B).

• e(B) := max{dim_{k} Ext_{B}^{1}(S, T) | S, T simple B-modules}.

Note that all these invariants are invariants of the Morita equivalence class of B. It is well known that C(B) is a symmetric matrix whose determinant is a power of p. Moreover, p^{d(B)} is the unique largest invariant factor of C(B) [12, Corollary III.8.13], such that d(B) can be recovered from mod-B.

3 More questions and Düvel’s reduction

In view of Brauer’s question one may ask whether d(B) bounds the invariants of B introduced in the previous section. More precisely, do there exist functions f_{i,p} : ℕ → ℕ, i = 1, …, 4, such that the following hold for all blocks B?

(1) ℓ(B) ≤ f_{1,p}(d(B)),

(2) LL(B) ≤ f_{2,p}(d(B)),

(3) e(B) ≤ f_{3,p}(d(B)),

(4) c(B) ≤ f_{4,p}(d(B))?

Of course, Question (4) is just the block version of Question 1.2.

Remarks 3.1 (a) By a result of Brauer and Feit (see [5, Theorem IV.4.18]) Question (1) has a positive answer, namely ℓ(B) ≤ \frac{1}{4}p^{2d(B)} + 1.

(b) In view of this, a positive answer to Question (4) is equivalent to a simultaneous positive answer to Questions (2) and (3).

(c) If B is replaced by kG (and d(B) by log_{p}(|G|_{p})) in Question (1), then this question does not have a positive answer. To reduce Question (4) to the two Questions (2) and (3), it is therefore essential to turn to blocks.

(d) It has been conjectured by Kiyota and Wada [9], that LL(B) ≤ ρ(C(B)), where ρ(C(B)) is the largest eigenvalue of C(B).

Donovan has formulated a conjecture which appears to go far beyond Brauer’s question.

Conjecture 3.2 (Donovan, see [1], Conjecture M) The number of Morita equivalence classes of blocks of a given defect is finite.

It is obvious that the truth of Donovan’s conjecture would imply a positive answer to Questions (1)–(4) above. To weaken Donovan’s conjecture, let C be a class of finite groups. Say that Donovan’s conjecture holds for C, if the number of Morita equivalence classes among the blocks of a given defect of the group algebras kG for G ∈ C, is finite. Scopes has proved in [15] that Donovan’s conjecture holds for the symmetric groups, the results of Kessar [8] imply that Donovan’s conjecture holds for the alternating groups. In [10] Külshammer has proved that in order to verify Donovan’s conjecture, it suffices to consider blocks of groups generated by the defect groups of the blocks.

In the final section of this note a condition will be introduced allowing to derive the truth of Donovan’s conjecture for a class of groups from a positive answer to Brauer’s question for this class.

We are now going to state Düvel’s reduction theorem, but need to introduce some more notation. First of all, put

Ê_{p′} := {Ĝ | Ĝ perfect, finite group, Z := Z(Ĝ) cyclic , p |̸ |Z|, Ĝ/Z simple}.

The following definitions are due to Düvel.

Definition 3.3 (a) For d ∈ ℕ let

\overline{LL}_{p}(d) := sup\{LL(B) | B block of kĜ, Ĝ ∈ Ê_{p′}, d(B) ≤ d\} ∈ ℕ ∪ \{∞\},

and

ē_{p}(d) := sup{e(B) | B block of kĜ, Ĝ ∈ Ê_{p′}, d(B) ≤ d} ∈ ℕ ∪ {∞}.

(b) Define the maps LL_{p}, e_{p} : ℕ → ℕ recursively by

LL_{p}(0) := 1, LL_{p}(n) := max\{\overline{LL}_{p}(n), LL_{p}(i)LL_{p}(n − i) | 0 < i < n\}, n > 0,

and

e_{p}(0) := 0, e_{p}(n) := max{ē_{p}(n), e_{p}(i) + e_{p}(n − i) | 0 < i < n}, n > 0.

With these notations we can now formulate Düvel’s reduction theorem.

Theorem 3.4 ([4, Theorem 3.2]) Let G be a finite group, B a block of kG of defect d. Then LL(B) ≤ LL_{p}(d) and e(B) ≤ e_{p}(d)ℓ(B).

Remark 3.5 Let d ∈ ℕ. It follows from results of many authors that

sup{LL(B), e(B) | B block of kG, d(B) ≤ d, G ∈ C}

is finite for a large class C of finite Chevalley groups containing, for example, all general linear groups [3], the symmetric and alternating groups [15, 8], the 4-dimensional symplectic groups [14], the Suzuki groups (cyclic defect) and the “small” Ree groups [13]. We hope to be able to extend the class C in the future. The aim is, of course, to show that if we replace C by Ê_{p′} then the above number is still finite.

4 Idea of Düvel’s reduction

It is perhaps not surprising to the experts that Düvel uses a generalized version of Fong reduction—with control of defects of blocks—based on Clifford theory à la Dade, to obtain his theorem. Without going into details, some of the main points of Düvel’s reduction will now be sketched. This section is based on Düvel’s thesis [4].

Let G be a finite group and let N be a normal subgroup of G.

(1) Define a full subcategory S_{G}(N) of mod-kG consisting of those finite dimensional kG-modules whose restriction to N is semisimple. Then every kG-module M has a filtration, of length at most LL(kN), with factors in S_{G}(N).

(2) Let V_{1}, …, V_{t} represent the G-orbits of the isomorphism classes of the simple kN-modules. For 1 ≤ i ≤ t let G_{i} denote the stabilizer of V_{i} in G. We then have a direct sum decomposition

S_{G}(N) = ⨁_{i=1}^{t} S_{G}(N, V_{i}),

where S_{G}(N, V_{i}) is the full subcategory of S_{G}(N) consisting of those kGmodules whose restriction to N is a direct sum of modules G-conjugate to V_{i}.

(3) Fix an integer i with 1 ≤ i ≤ t. Then there is a cyclic central p′-

extension

1 → Z_{i} → { \atop G_{i}/N} → G_{i}/N → 1

and a simple kZ_{i}-module X_{i} such that S_{G}(N, V_{i}) is Morita equivalent to S{ \atop G_{i}/N}(Z_{i}, X_{i}).

(4) Let M be an indecomposable kG-module and B the block of kG containing M. Next, let M′ be an indecomposable subquotient (a factor module of a submodule) of M contained in S_{G}(N).

Since M′ is indecomposable, it is contained in S_{G}(N, V_{i}) for a unique i. By (3), M′ determines a block B_{i} of { \atop G_{i}/N}. Finally, let b_{i} denote the kN-block containing V_{i}. Then

d(B_{i}) + d(b_{i}) ≤ d(B).

(5) The proof bounding the Loewy length proceeds as follows. Let B be a block of kG and let b_{1}, …, b_{s} be the blocks of kN covered by B. Then, since these are conjugate in G, all of the b_{i}s have the same Loewy length. Now let M be a PIM of B of maximal Loewy length. Among all indecomposable subquotients of M′ of M which lie in S_{G}(N), choose one such that LL(B_{i}) is maximal, where B_{i} is the block of { \atop G_{i}/N} determined by M′ as in (4). Then, by (4), d(B_{i}) + d(b_{i}) ≤ d(B).

By induction on the order of G/O_{p′}(G) one may assume that either G is in the class Ê_{p′}, or else that b_{i} and B_{i} belong to groups H with |H/O_{p′}(H)| < |G/O_{p′}(G)|. Then

LL(B) ≤ LL(b_{i})LL(B_{i})
≤ LL_{p}(d(b_{i}))LL_{p}(d(B_{i}))
≤ LL_{p}(d(b_{i}))LL_{p}(d(B) − d(b_{i}))
≤ LL_{p}(d(B)).

Here, the first inequality follows by filtering M with at most LL(b_{i}) factors of S_{G}(N) and the choice of M′, the second by induction, and the third and fourth by properties of the function LL_{p}.

5 Towards Donovan’s conjecture

In the final section a few ideas will be sketched which allow to approach Donovan’s conjecture for certain classes of groups, provided Brauer’s question has a positive answer for these groups.

Proposition 5.1 Let B be a set of blocks. Suppose there exist positive integers d and c and a finite field k_{0} ⊂ k such that for all blocks B ∈ B the following conditions are satisfied.

(a) d(B) ≤ d,

(b) c(B) ≤ c,

(c) B ≅ B_{0} ⊗_{k_{0}} k for some split k_{0}-algebra B_{0}.

Then there are only finitely many Morita equivalence classes among the blocks in B.

Proof. Let B ∈ B. By the results of Brauer and Feit, ℓ(B) ≤ p^{2d}/4 + 1. By Assumption (b), the sum of the entries of C(B) is at most c(p^{2d}/4 + 1)^{2}.

Let P_{1}, …, P_{n} denote a set of representatives for the isomorphism classes of the PIMs of B_{0} and let M := P_{1} ⊕ ⋯ ⊕ P_{n}. Put A_{0} := End_{B_{0}}(M). Since k_{0} is a splitting field for B_{0} by assumption, the projective B-module P_{i} ⊗_{k_{0}} k is indecomposable. Thus P_{1} ⊗_{k_{0}} k, …, P_{n} ⊗_{k_{0}} k is a complete set of representatives for the PIMs of B. It follows that A := A_{0} ⊗_{k_{0}} k ≅ End_{B}(M ⊗_{k_{0}} k) is a basic algebra for B. By the first paragraph of the proof, dim_{k_{0}} A_{0} = dim_{k} A ≤ c(p^{2d}/4 + 1)^{2}.

Since there are only finitely many isomorphism classes among the k_{0}- algebras of fixed, bounded dimension, the above implies that there are only finitely many isomorphism classes among the basic algebras of the blocks in B. Since two k-algebras are Morita equivalent if and only if their basic algebras are isomorphic, the result follows.

Proposition 5.2 Let G be a finite group and B a block of kG. Let φ_{1}, …, φ_{n} denote the k-characters of the simple B-modules. If k_{0} is the finite field containing the ϕ_{i}(g), i = 1, …, n, g ∈ G, then there is a k_{0}-algebra B_{0} with B ≅ B_{0} ⊗_{k_{0}} k. Moreover, k_{0} is a splitting field for B_{0}.

Proof. Let e ∈ Z(kG) be the central idempotent with B = kGe. By Osima’s theorem and Brauer reciprocity it follows that e ∈ k_{0}G (see [12, Theorem III.2.9]). Hence B = B_{0} ⊗_{k_{0}} k with B_{0} := k_{0}Ge. Also, by [6, Corollary 9.23], every B-module is realizable over k_{0} and hence k_{0} is a splitting field for B_{0}.

We conclude this section with an example.

Example 5.3 (Unipotent blocks of GL_{n}(q)). Let d ∈ ℕ.

(1) For 0 ≠ n ∈ ℕ let B_{n,d} denote the set of unipotent blocks of kGL_{n}(q) of defect d, where q runs through the prime powers not divisible by p. Then Assumption (a) of Proposition 5.1 is trivially satisfied. By results of Dipper and James [3] on the decomposition numbers of the general linear groups, (b) is also satisfied. All unipotent characters of GL_{n}(q) are rational valued. The same is true for the irreducible Brauer characters of the unipotent blocks. By Proposition 5.2, Assumption (c) is satisfied with k_{0} = 𝔽_{p}.

(2) Next, fix a prime power q not divisible by p. The results of Jost [7, Theorem 6.2] imply that there is a bound N, depending only on d and p, but not on q, such that every unipotent block of defect d of kGL_{m}(q) for some 0 ≠ m ∈ ℕ is Morita equivalent to a unipotent block of the same defect of kGL_{n}(q) for some n ≤ N.

(3) By first using (2) and then applying (1) for 1 ≤ n ≤ N, it follows that there are only finitely many Morita equivalence classes of unipotent blocks of defect d among the unipotent blocks of kGL_{n}(q), where n runs through the positive integers and q through the prime powers not divisible by p.

References

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Gerhard Hiss Lehrstuhl D für Mathematik RWTH Aachen Templergraben 64 52062 Aachen Germany

E-Mail: Gerhard.Hiss@Math.RWTH-Aachen.DE