SOME TOPICS ON DERIVED EQUIVALENT BLOCKS OF FINITE GROUPS

Naoko Kunugi

1. Introduction

Let G be a finite group. Let k be an algebraically closed field of characteristic ℓ > 0. We denote the principal block of kG by B_{0}(G).

We say that two finite groups G and H have the same ℓ-local structure if G and H have a common Sylow ℓ-subgroup P such that whenever Q_{1} and Q_{2} are subgroups of P and f : Q_{1} → Q_{2} is an isomorphism, then there is an element g ∈ G such that f(x) = x^{g} for all x ∈ Q_{1} if and only if there is an element h ∈ H such that f(x) = x^{h} for all x ∈ Q_{1}.

There is a well known conjecture due to Broué.

Conjecture 1.1(Broué [1, 2]). Let G and H be finite groups having the same ℓ-local structure with common Sylow ℓ-subgroup P. If P is abelian then the principal blocks of G and H would be derived equivalent.

If P is not abelian, then there is a counterexample to this conjecture. However , there are some examples that P is not abelian and there is a derived equivalence between the principal blocks of G and H. We will give such examples in §3.

2. General theory

In this section, let G and H be finite groups having the same ℓ-local structure with common Sylow ℓ-subgroup P. We say that a complex of (B_{0}(G), B_{0}(H))-bimodules is splendid if each indecomposable summand of each term of the complex is a direct summand of a module of the form kG ⊗_{kQ} kH for a subgroup Q of P.

Definition 2.1. Let X^{•} be a splendid complex of (B_{0}(G), B_{0}(H))-bimodules. We say that X^{•} induces a splendid stable equivalence if we have isomorphisms

X^{•} ⊗_{B_{0}(H)} X^{•}^{∗} ≅ B_{0}(G) ⊕ Z_{1}, X^{•}^{∗} ⊗_{B_{0}(G)} X^{•} ≅ B_{0}(H) ⊕ Z_{2}

where Z_{1} and Z_{2} are homotopy equivalent to complexes of projective bimodules.

Definition 2.2. Let X^{•} be a splendid complex of (B_{0}(G), B_{0}(H))-bimodules. We say that X^{•} induces a splendid equivalence if we have isomorphisms

X^{•} ⊗_{B_{0}(H)} X^{•}^{∗} ≅ B_{0}(G) ⊕ Z_{1}, X^{•}^{∗} ⊗_{B_{0}(G)} X^{•} ≅ B_{0}(H) ⊕ Z_{2}

where Z_{1} and Z_{2} are homotopy equivalent to 0. The complex X^{•} is called a splendid tilting complex.

The detailed version of this paper will be submitted for publication elsewhere.

By the definition, splendid equivalences induce derived equivalences and homotopy equivalences.

Theorem 2.1 (Rouquier [12]). Let X^{•} be a splendid complex of (B_{0}(G), B_{0}(H))-bimodules. Then the following are equivalent.

(1) The complex X^{•} induces a splendid stable equivalence between B_{0}(G) and B_{0}(H).

(2) For every non-trivial subgroup Q of P, the complex X^{•}(Δ(Q)) induces a splendid equivalence between B_{0}(C_{G}(Q)) and B_{0}(C_{H}(Q)), where Δ(Q) is a diagnal subgroup and

X(Q) = X^{Δ(Q)}/ ∑_{R<Q} Tr_{R}^{Q}X^{Δ(R)}.

In our example in §3 we will use the following method when we prove splendid equivalences. (Step 1) Construct a splendid tilting complex between B_{0}(C_{G}(Q)) and B_{0}(C_{H}(Q)) for every non-trivial subgroup Q of P. (Step 2) Construct a splendid stable equivalence F from B_{0}(G) to B_{0}(H) by gluing the splendid tilting complexes obtained in Step 1 (by using the above theorem). (Step 3) Calculate F (S) for the simple B_{0}(G)-modules. (Step 4) Lift the stable equivalence in Step 2 to a splendid equivalence by looking at the modules calcutated in Step 3.

3. General linear groups and unitary groups

Let q be a power of a prime. Assume that ℓ is odd and ℓ^{e} divides q + 1 but ℓ^{e+1} does not divide q + 1 for some e > 0. Under this condition, we consider representations of the general linear group GL(n, q^{2}) and the unitary group GU(n, q^{2}) for small n. Note that if ℓ > n then the principal ℓ-block of GL(n, q^{2}) is Morita equivalent to its Brauer correspondent by Puig’s result(see [8]).

3.1. GL(2, q^{2}) and GU(2, q^{2}). We have isomorphisms

B_{0}(GL(2, q^{2})) ≅ kZ_{ℓ^{e}} ⊗ B_{0}(SL(2, q^{2})), B_{0}(GU(2, q^{2})) ≅ kZ_{ℓ^{e}} ⊗ B_{0}(SU(2, q^{2})).

The blocks B_{0}(SL(2, q^{2})) and B_{0}(SU(2, q^{2})) have cyclic defect groups, and they are splendid equivalent by Rouquier’s result in [11]. Therefor the principal blocks B_{0}(GL(2, q^{2})) and B_{0}(GU(2, q^{2})) are splendid equivalent.

3.2. GL(3, q^{2}) and GU(3, q^{2}) in characteristic ℓ > 3. In this case, Sylow ℓ-subgroups of GL(3, q^{2}) and GU(3, q^{2}) are abelian. As in case n = 2, we have isomorphisms

B_{0}(GL(3, q^{2})) ≅ kZ_{ℓ^{e}} ⊗ B_{0}(SL(3, q^{2})), B_{0}(GU(3, q^{2})) ≅ kZ_{ℓ^{e}} ⊗ B_{0}(SU(3, q^{2})).

In [5], Waki and the author showed that B_{0}(SU(3, q^{2})) and its Brauer correspondent, which is isomorphic to the Brauer correspondent of B_{0}(SL(3, q^{2})), are splendid equivalent. Therefore B_{0}(SL(3, q^{2})) and B_{0}(SU(3, q^{2})) are splendid equivalent since as we mentioned above B_{0}(SL(3, q^{2})) and its Brauer correspondent are Morita(Puig) equivalent by Puig’s result. Hence we also have B_{0}(GL(3, q^{2})) and B_{0}(GU(3, q^{2})) are splendid equivalent.

3.3. GL(3, q^{2}) and GU(3, q^{2}) in characteristic 3. In this case Sylow 3-subgroups of GL(3, q^{2}) and GU(3, q^{2}) are not abelian. Our main result in this paper is the following theorem.

Theorem 3.1 (with T. Okuyama). Assume that 3^{e} divides q + 1 but 3^{e+1} does not divide q + 1 for e > 0. Then

(1) The principal 3-blocks B_{0}(P SL(3, q^{2})) and B_{0}(P SU(3, q^{2})) are splendid equivalent.

(2) The principal 3-blocks B_{0}(SL(3, q^{2})) and B_{0}(SU(3, q^{2})) are splendid equivalent.

(3) The principal 3-blocks B_{0}(P GL(3, q^{2})) and B_{0}(P GU(3, q^{2})) are splendid equivalent.

(4) The principal 3-blocks B_{0}(GL(3, q^{2})) and B_{0}(GU(3, q^{2})) are splendid equivalent.

Remark 3.1. If e = 1, then the result for (1) has been obtained by [6, 4, 3] and the result for (3) has been obtained by Usami and the author.

4. Outline of proof of theorem

In this section, we give an outline of a proof of Theorem 3.1 (1) and (2). Let G = SL(3, q^{2}), H = SU(3, q^{2}), \overline{G} = P SL(3, q^{2}) and \overline{H} = P SU(3, q^{2}). Let P be a common Sylow 3-subgroup of G and H. We denote the image of a subgroup L of G (or H) in \overline{G} (or \overline{H}) by \overline{L}. For each subgroup \overline{R} of \overline{P}, let \overline{G}_{R} := C\overline{G}(\overline{R}), \overline{H}_{R} := C\overline{H}(\overline{R}), and let \overline{M}_{R} be the Scott module of \overline{G}_{R} × \overline{H}_{R} with vertex Δ(\overline{R′}), where \overline{R′} is a Sylow 3-subgroup of \overline{G}_{R} and \overline{H}_{R}.

(Step 1). There is essentially one subgroup of P (up to conjugate), which we denote by Q, containing Z(P) such that B_{0}(C_{G}(Q)) and B_{0}(C_{H}(Q)) are not Morita equivalent. Then C_{G}(Q) ≅ GL(2, q^{2}) and C_{H}(Q) ≅ GU(2, q^{2}). Let \overline{M}_{Q} → k_{\overline{G}_{Q}×\overline{H}_{Q}} be a Δ(\overline{Q′})- projective cover of k_{\overline{G}_{Q}×\overline{H}_{Q}} and \overline{N}_{Q} → Ω_{Δ(\overline{Q′})}(k_{\overline{G}_{Q}×\overline{H}_{Q}}) be a Δ(\overline{Q})-projective cover of Ω_{Δ(\overline{Q′})}(k_{\overline{G}_{Q}×\overline{H}_{Q}}). Then we have a splendid tilting complex for B_{0}(\overline{G}_{Q}) and B_{0}(\overline{H}_{Q}) of the form

0 ⟶ \overline{N}_{Q} ⟶ \overline{M}_{Q} ⟶ 0.

For a subgroup \overline{R} of \overline{P} not contained in \overline{Q}, the blocks B_{0}(\overline{G}_{R}) and B_{0}(\overline{H}_{R}) are Morita equivalent and the Scott module \overline{M}_{R} gives a splendid tilting complex for these two blocks.

(Step 2). Let M be the Scott module of G × H with vertex Δ(P). Let M → k_{G×H} be a Δ(P)-projective cover of k_{G×H} and N → Ω_{Δ(P)}(k_{G×H}) be a Δ(Q)-projective cover of Ω_{Δ(P)}(k_{G×H}). Consider the following complex

M^{•} : 0 ⟶ N ⟶ M ⟶ 0.

and set \overline{M}^{•} = Inv_{Z(P)×1}(M^{•}). Then the complex \overline{M}^{•} is a splendid complex, and for each non-trivial subgroup \overline{R} of \overline{P}, the complex \overline{M}^{•}(Δ(\overline{R})) coincides with the complex in (Step 1). Therefore by Rouquier’s theorem (Theorem 2.1) we can see that the complex \overline{M}^{•} induces a splendid stable equivalence between B_{0}(\overline{G}) and B_{0}(\overline{H}).

(Step 3). Let F = − ⊗_{B_{0}(G)} \overline{M}^{•}. The principal block of B_{0}(\overline{G}) has 5 simple modules k, S, T_{1}, T_{2} and T_{3} and the principal block of B_{0}(\overline{H}) has 5 simple modules k, φ, θ_{1}, θ_{2} and θ_{3}. Then we have the following lemma.

Lemma 4.1. There exist exact sequences

0 ⟶ Ω^{−1}(U(k, φ)) ⟶ Ω(F (S)) ⟶ k ⊕ k ⟶ 0

and

0 ⟶ Ω^{−1}(U(k, φ, θ_{i})) ⟶ Ω^{2}(F (T_{i})) ⟶ k ⟶ 0

for i = 1, 2 and 3, where U(k, φ) is a uniserial module of length 2 with top k, and U(k, φ, θ_{i}) is a uniserial module of length 3 with top k and socle θ_{i}.

(Step 4). It follows from Lemma 4.1 that the tilting complex defined by a sequence {θ_{1}, θ_{2}, θ_{3}}, {φ, θ_{1}, θ_{2}, θ_{3}} and {φ, θ_{1}, θ_{2}, θ_{3}} of subsets of the set of simple modules (see [6]) gives a derived equivalence between B_{0}(\overline{G}) and B_{0}(\overline{H}). The equivalence is a lift of the stable equivalence given by F (see [7]), and therefore B_{0}(G) and B_{0}(H) are splendid equivalent.

Now we have the splendid tilting complex for B_{0}(\overline{G}) and B_{0}(\overline{H}) of the form

\overline{X}^{•} : 0 ⟶ \overline{Q}_{3} ⟶ \overline{Q}_{2} ⟶ \overline{Q}_{1} ⊕ \overline{N} ⟶ \overline{M} ⟶ 0

where \overline{M} = Inv_{Z(P)×1}(M) and \overline{N} = Inv_{Z(P)×1}(N) and \overline{Q}_{1}, \overline{Q}_{2}, and \overline{Q}_{3} are projective bimodules. Since Inv_{Z(P)×1}(−) induces a one to one correspondence between the set of trivial source k[G × H]-modules with vertex Δ(Z(P)) and the set of projective k[\overline{G} × \overline{H}]- modules, we have a tilting complex of the form

X^{•} : 0 ⟶ Q_{3} ⟶ Q_{2} ⟶ Q_{1} ⊕ N ⟶ M ⟶ 0

for B_{0}(G) and B_{0}(H), where Q_{1}, Q_{2} and Q_{3} are direct sums of trivial source k[G × H]- modules with vertex Δ(Z(P)) and Inv_{Z(P)×1}(X^{•}) = \overline{X}^{•} (see [12, A.4]).

References

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[12] R. Rouquier, Block Theory via stable and Rickard equivalences, in “Modular representation theory of finite groups”, Proceedings of a Symposium held at the University of Virginia, Charlottersville, 1998, pp.101–147, de Gruyter (2001)

Department of Mathematics Aichi University of Education Igaya-cho, Kariya, Aichi 448-8542 Japan

E-mail address: nkunugi@auecc.aichi-edu.ac.jp