A REMARK ON DONOVAN’S CONJECTURE

RADHA KESSAR

1. Introduction

Let p be a prime number and k an algebraically closed field of characteristic p. Let P be a finite p-group. By a P -block of a finite group we will mean a block of a group algebra kG, where G is a finite group such that the corresponding defect groups are isomorphic to P. One of the main outstanding problems in modular representation theory is the following conjecture of Donovan.

Conjecture 1.1. (Donovan, see [1, Conjecture M]) Up to Morita equivalence there are only finitely many k algebras that occur as P -blocks.

The truth of the above conjecture would of course imply the truth of the following conjecture, motivated by Brauer’s Problem 22 (see [2]) and conjectured by Donovan.

Conjecture 1.2. There are only finitely many possibilities for the Cartan entries of P -blocks.

Since by a result of Brauer and Feit ([4]) the number of simple modules in a block of a group algebra is bounded by the order of the defect groups of the block, the truth of 1.2 would imply that there are only finitely many possibilities for the Cartan matrices of P -blocks.

In [6], Olaf Düvel has shown that in order to prove Conjecture 1.2 it suffices to prove that there are only finitely many possibilities for Cartan entries of P -blocks of quasi-simple groups. Conjecture 1.1 has been checked for several families of quasisimple groups. The gap between Conjecture 1.2 and Conjecture 1.1 lies in the fact that the coefficients of block idempotents of P -blocks as linear combinations of group elements lie in arbitrarily large finite subfields of k. This note gives a precise formulation of the gap between the two conjectures; a consequence of Theorem 1.4 below is that if we restrict ourselves to principal blocks, the two conjectures are in fact equivalent.

Let G be a finite group and let σ : kG → kG be the map induced by the Frobenius automorphism of k, that is σ(∑_{g∈G} α_{g}g) = ∑_{g∈G} α_{g}^{p}g. Clearly, σ is an isomorphism of rings, and since the blocks of kG are in one to one correspondence with primitive idempotents of the center of kG, < σ > permutes the blocks of kG. A block B of kG and σ(B) are isomorphic as rings but they may or may not be Morita equivalent as k-algebras. Denote by m(G, B) the number of Morita equivalence classes of algebras in the set {σ^{d}(B) : d ∈ ℕ ∪ {0}. The blocks B and σ(B) have isomorphic defect groups, hence Conjecture 1.1 implies the following.

Conjecture 1.3. There is an integer m, depending only on P such that m(G, B) ≤ m for all finite groups G and all P -blocks B of kG.

In this note we show that Conjecture 1.3 is exactly the gap between Conjectures 1.1 and 1.2. In other words, the following holds.

Theorem 1.4. Conjectures 1.2 and 1.3 are together equivalent to Conjecture 1.1.

We point out that Theorem 1.4 implies that for principal blocks Conjecture 1.2 is equivalent to Conjecture 1.1 since if B is the principal block of kG, σ(B) = B.

2. Proof of Theorem 1.4

Our proof is an application of the following lemma, which under the additional hypothesis that for each x ∈ V, there is a natural number n such that ϕ^{n}(x) = x is a corollary of [3, Chapter 5, 10.4, Proposition 7] (see [5, Corollary 3.5]). Lang’s theorem may be used to dispense with the extra hypothesis.

Lemma 2.1. Let q be a power of p and let V be a finite dimensional k-vector space. Let ϕ : V → V be an automorphism of additive groups and denote by V^{ϕ} the subgroup of V of fixed points under ϕ. If ϕ(λx) = λ^{q}ϕ(x) for all x ∈ V and all λ ∈ k, then V^{ϕ} is an 𝔽_{q}- subspace of V such that V = k ⊗_{𝔽_{q}} V^{ϕ} as k-vector spaces.

Proof. For a ∈ V^{ϕ} and λ ∈ 𝔽_{q}, ϕ(λa) = λ^{q}ϕ(a) = λa, hence V^{ϕ} is 𝔽_{q} invariant. Let I be a k-basis of V, and denote by S = (s_{x,y})_{x,y∈I} the matrix of ϕ with respect to I, so that for any x ∈ I, ϕ(x) = ∑_{j∈I} s_{x,y}x. Since ϕ is an isomorphism of additive groups and since ϕ(λz) = λ^{q}ϕ(z) for all λ ∈ k and all z ∈ V, S is an invertible matrix, that is S ∈ GL(V). By Lang’s theorem therefore, there is a matrix T in GL(V) such that F (T)^{−1}T = S, where F is the automorphism (as abstract group) of GL(V) given by F ((λ_{x,y})) = (λ_{x,y}^{q}). For x ∈ I, let x_{0} = T (x) and let J = {x_{0} : x ∈ I}. Since T is invertible, J is a k-basis of V. By definition of ϕ and T, ϕ(x_{0}) = F (T)ϕ(x) = F (T)S(x) = T (x) = x_{0}. Hence, J ∈ V^{ϕ}. Let a = ∑_{x∈I} λ_{x}x_{0} be an element of V. Then ϕ(a) = ∑_{x∈I} λ_{x}^{q}x_{0}.Hence a ∈ V^{ϕ} if and only if λ_{x} ∈ 𝔽_{q} for each x in I. In other words, J spans V^{ϕ} as 𝔽_{q}-space. Since the elements of J are k-independent, they are 𝔽_{q}-independent, thus completing the proof. □

Proof of 1.4. Suppose that the hypothesis of 1.4 holds. Let m be such that m(G, D) ≤ m for all finite groups G and all P -blocks B of kG. Put M = m!.

Let G be a finite group and let B be a P -block of G. Let t ≤ m be a natural number such that B and σ^{t}(B) are Morita equivalent as k-algebras. Write B = kGb where b is the corresponding block idempotent. Let i be an idempotent of kGb such that ikGi is a basic algebra for the k-algebra kGb; then σ^{t}(i)kGσ^{t}(i) is a basic algebra for the k-algebra kGσ^{t}(b). Since B and σ^{t}(B) are Morita equivalent k-algebras, there is a k-algebra isomorphism ψ : ikGi → σ^{t}(i)kGσ^{t}(i). Put ϕ := ψ^{−1} ∘ σ^{t} : ikGi → ikGi and q := p^{t}. Then ϕ is a ring isomorphism satisfying ϕ(λx) = λ^{q}ϕ(x) for all λ in k and all x ∈ ikGi. By the Lemma, we get that (ikGi)^{ϕ} is an 𝔽_{q}-subspace of ikGi such that ikGi = k ⊗_{𝔽_{q}} (ikGi)^{ϕ} as k-vector spaces. Since ϕ is a homomorphism of rings, (ikGi)^{ϕ} is an 𝔽_{q}-algebra. Thus, since t is a factor of M, it follows that 𝔽_{p^{M}} ⊗_{𝔽_{q}} (ikGi)^{ϕ} is a 𝔽_{p^{M}} -subalgebra of ikGi, with ikGi ≅ k ⊗_{𝔽_{p^{M}}} (ikGi)^{ϕ} as k-algebras.

Thus, we have shown that if A is a basic algebra of a P -block, then there is an 𝔽_{p^{M}} -algebra A_{0} such that dim_{𝔽_{p^{M}}} (A_{0}) = dim_{k}(A) and such that A ≅ k ⊗_{𝔽_{p^{M}}} A_{0}.

Let c be the largest number occurring as a Cartan number of a P -block and let l be the maximum number of simple modules occurring in a P -block. By a Theorem of Brauer and Feit ([4]), l ≤ \frac{1}{4}|P |^{2} + 1. Then, clearly the k-dimension of a basic algebra of a P -block is bounded by c^{2l}. Thus, the basic algebra of any P -block is of the form k ⊗_{𝔽_{q}} A_{0}, where A_{0} is an 𝔽_{p^{M}} -algebra of 𝔽_{p^{M}} dimension at most c^{2l}. Clearly, the possible number of such algebras is bounded by a function of M , c and l, hence by a function of P. □

References

[1] J. L. Alperin, Local representation theory, Proc. Sympos. Pure Math. 37 (1980), 369–375.

[2] R. Brauer, Representations of finite groups, in: Lectures on modern mathematics, Vol. 1, Wiley, 1963.

[3] N. Bourbaki, Algèbre, Masson, 1981.

[4] R. Brauer, W. Feit, On the number of irreducible characters of finite groups in a given block, Proc. Nat. Acad. Sci. U.S.A. 45 (1959),361-365.

[5] F. Digne and J. Michel, Representations of finite groups of Lie type, London Math. Soc. Students Texts 21, Cambridge University Press, 1991.

[6] O. Düvel, On Donovan’s conjecture, preprint 2002

Department of Mathematics, The Ohio State University, 231, W. 18th Avenue, Columbus, OH 43210, U.S.A.

 E-mail address: kessar@math.ohio-state.edu