Endo-permutation modules arising from the action of a p-group on a defect zero block

Adam Salminen Department of Mathematics University of Evansville 1800 Lincoln Avenue Evansville, IN 47722

as341@evansville.edu

February 10, 2008

Abstract

Let p be an odd prime and let k be an algebraically closed field of characteristic p. Also, let G be a p′-group. Then by Maschke’s theorem we know kG ≅ ∏_{i=1}^{t} End_{k}(V_{i}) as k-algebras. Suppose that a p-group P ≤ Aut(G) stabilizes End_{k}(V_{i_{0}}) for some i_{0}. Such a V_{i_{0}} will be an endopermutation kP -module. Puig showed that the only modules that occur in this way are those whose image is torsion in the Dade group D(P).

If we let G be any finite group and let b be a defect zero block of kG, then kGb ≅ End_{k}(L) for some L. If kGb is P -stable for some p-group P ≤ Aut(G) and Br_{P} (b) ≠ 0, then L will again be an endo-permutation kP -module. We show that if p ≥ 5, then L is torsion in D(P). This result depends on the classification of the finite simple groups.

1 Introduction

Let p be an odd prime, and let k be an algebraically closed field of characteristic p. Suppose that G is a finite p′-group. Then Maschke’s theorem implies that kG ≅ ∏_{i=1}^{t} M_{n_{i}}(k) ≅ ∏_{i=1}^{t} End_{k}(V_{i}). Now suppose that P ≤ Aut(G) is a p-group that stabilizes End_{k}(V_{i_{0}}) for some i_{0}. It is known that V_{i_{0}} is an endopermutation kP -module. Which endo-permutation kP -modules arise in this way?

Theorem 1.1 (Puig, [9]). With the above setup the V_{i} are torsion in the Dade group D(P).

The proof of the above result uses the fact that for every simple p′-group G, Aut(G) has p-rank 1. The proof of this result depends on the classification of finite simple groups. If we drop the assumption that G is a p′-group, then Maschke’s theorem no longer applies. Decompose kG ≅ ∏_{i=1}^{t} B_{i} into a product of indecomposable algebras. Each B_{i} has the form kGb_{i} for some primitive central idempotent b_{i} of kG, and both the B_{i} and b_{i} are called blocks of kG. A block kGb of kG is said to be a defect zero block if kGb ≅ End_{k}(V) for some k-module V. As above, suppose that kGb ≅ End_{k}(V) is a defect zero block of kG which is P -stable for some p-subgroup of Aut(G). Also assume that Br_{P} (b) ≠ 0, then V is an endo-permutation kP -module. In this paper, we investigate which endo-permutation kP -modules appear in this way. It is expected that V is always torsion in D(P). In this paper we show this is true for p ≥ 5. Our first result is a consequence of a result of Carlson and Thévenaz [2].

Theorem 1.2. Let p be odd. Assume the setup and notation of the previous paragraph. If a non-torsion V appears for some G, P and b, then we can find some G, P, V and b with P ≅ C_{p} × C_{p} and V non-torsion in D(P).

Now that we have reduced to P ≅ C_{p} × C_{p}, we can apply the reduction results of [10], which depend on the classification of the finite simple groups. These results allow to reduce the cases: G is a central extension of P SL_{n+1}(q), P SU_{n+1}(q), or D_{4}(q) with p = 3. A recent result of Kessar [6] takes care of the first two cases. So we are left with a single open case for p = 3. In particular we have the following.

Theorem 1.3. Suppose that G is a finite group and kGb ≅ End_{k}(V) is a defect zero block of kG which is P -stable for some p-group P of Aut(G). Also assume that Br_{P} (b) ≠ 0. If p ≥ 5, then V is torsion in D(P). In particular, V is self-dual.

The above result is a special case of the conjecture on the finiteness of the number of source algebra equivalence classes of nilpotent blocks, with defect group P, of finite groups for a fixed p-group P. A proof of this conjecture has been announced by Puig. This paper is organized as follows. Section 2 will recall definitions and basic results on blocks and endo-permutation modules. Section 3 will provide a proof of Theorem 1.2. Section 4 will briefly recall the results of [10], and will contain a proof of Theorem 1.3.

2 Notation and preliminaries

Fix a prime p and an algebraically closed field k of characteristic p. Let G be a finite group, and let b be a block of kG. For a p-subgroup P of G, the Brauer Homomorphism Br_{P} : (kG)^{P} → kC_{G}(P) is defined by ∑_{x∈G} λ_{x}x ↦ ∑_{x∈C_{G}(P)} λ_{x}x,

and it is easy to check that this map is a homomorphism. A defect group of a block b is defined to be a maximal p-subgroup P of G such that Br_{P} (b) ≠ 0. It is well known that any two defect groups are conjugate. A block is said to be a defect zero block if 1 is a defect group. It is also well known that b is a defect zero block if and only if kGb ≅ End_{k}(L) for some k-module L.

We now recall the definitions and some facts about endo-permutation modules and the Dade group. For a more detailed discussion see [3] and [4] or [11]. Recall that if V is a kG-module, then the dual of V^{∗} = Hom_{k}(V, k) is also a kG-module via the action g · α(m) = α(g^{−1}m) for α ∈ V^{∗}, g ∈ G and m ∈ V. A kG-module V is a permutation module if it has a G-stable k-basis.

Definition 2.1 (Dade). Let P be a p-group. A kP -module V is said to be endo-permutation if End_{k}(V) ≅ V ⊗ V^{∗} is a permutation kP -module.

The endo-permutation kP -modules which usually show up in the representation theory of finite groups are those which have a summand of vertex P. Such endo-permutation modules are said to be capped. Dade showed that V is capped if and only if k is a summand of End_{k}(V). We will follow the suggestion of Thévenaz in [11] and use endo-permutation to mean capped endo-permutation unless stated otherwise.

If V is an endo-permutation kP -module, then any two summands of vertex P are isomorphic. Such a summand is called a cap of V. Two endo-permutation kP -modules V and W are said to be equivalent if they have isomorphic caps. It is easy to see that this is an equivalence relation. We denote the class of an endo-permutation module V by [V]. The Dade group of a finite p-group P, denoted D(P), is the set of these equivalence relations with the group operation induced by the tensor product. That is [V] + [W] = [V ⊗ W]. The fact that this operation is well defined is the following.

Theorem 2.2 (Dade). Let p be odd, and let P be a finite p-group. If V and W are endo-permutation kP -modules, then V ⊗ W is an endo-permutation kP - module. Moreover, the cap of V ⊗ W is isomorphic to a summand of V_{0} ⊗ W_{0} where V_{0} and W_{0} are caps of V and W respectively.

If V is a kP -module the Brauer quotient is defined by

V (P) = V^{P} / ( ∑_{Q<P} Tr_{Q}^{P}(V^{Q})) .

Suppose that kGb ≅ End_{k}(V) is a defect zero block and that P is a p-group that acts on kGb. By Corollary 21.4 of [12], V will inherit an action of P, and V (P) ≠ 0 if and only if Br_{P} (b) ≠ 0. Lemma 2.2 of [11] contains the following result which follows from Higmann’s criterion.

Lemma 2.3. Let V be an endo-permutation kP -module(not necessarily capped). The following are equivalent:

(i) V is capped.

(ii) The Brauer quotient (End_{k}(V))(P) is nonzero.

Theorem 1.2 will be a consequence of the following.

Theorem 2.4 (Carlson-Thévenaz, 13.1, [2]). Let p be odd, and let P be a finite p-group. The map

∏_{R/Q} Defres_{R/Q}^{P} : D(P) → ∏_{R/Q} D(R/Q)

is injective where R/Q runs over the set of all sections of P which are cyclic of order p or elementary abelian of rank 2.

The Defres maps are the composition of the ordinary restriction map with the deflation map, which we now describe. Let V be an endo-permutation kP - module. If Q is a normal subgroup of P, then it is easy to see that (End_{k}(V))^{Q} ≅ End_{kQ}(V) and that (End_{k}(V))^{Q} and (End_{k}(V))(Q) are acted on naturally by P/Q. Theorem 4.15 of [3] shows that End_{k}(V)(Q) ≅ End_{k}(V_{Q}) for a unique endo-permutation k(P/Q)-module V_{Q}. In [3], Dade also showed that D(C_{p}) = C_{2} for odd p. Combining this fact with Theorem 2.4, we have the following. If an endo-permutation kP -module V has a non-torsion image [V] in D(P), then there must be a section R/Q of P such that the image of Defres_{R/Q}^{P}(V) is non-torsion in D(R/Q) and R/Q ≅ C_{p} × C_{p}. Another consequence of Theorem 2.4 is that if p is odd then any torsion element of D(P) has order 2. It is also known that −[V] = [V^{∗}] for any [V] ∈ D(P). Therefore [V] is torsion in D(P) if and only if V is self-dual.

3 Proof of Theorem 1.2

In this section, we prove Theorem 1.2. Let G be a finite group. Suppose that b is a defect zero block. So kGb ≅ End_{k}(V) for some k-module V. Further suppose that there is a p-group P ≤ Aut(G) such that b is P -stable. Then Lemma 28.1 of [12] shows that V is a P -module. The group algebra kG is a P permutation module under the action of P. So the summand kGb of kG is also a permutation module by (1.5) of [3]. Therefore V is a (not necessarily capped) endo-permutation kP -module. If we also assume that (End_{k}(V))(P) ≠ 0, then V will be a capped endo-permutation module by Lemma 2.3.

We need the following.

Lemma 3.1. Let p be odd. Suppose that there is a finite group H, a defect zero block kHc ≅ End_{k}(V) of kH which is stable under an action of Q ≅ C_{p} ×C_{p} on H such that the V is non-torsion in D(Q) under this action. Then the action of Q on H is faithful.

Proof. Suppose that Q does not act faithfully on H. Let R ≤ Q be a subgroup which acts trivially on H. Then R clearly centralizes c and will therefore act trivially on V and on the dual, V^{∗}, of V.

Q/R is a cyclic group. This means that V is isomorphic to V^{∗} as kQ/Rmodules, since Q/R is cyclic. But this implies that V and V^{∗} are isomorphic as kQ-modules. But self-dual implies torsion in D(Q) [2]. Therefore, the action of Q on H must be faithful.

We can now prove Theorem 1.2 which we restate for convenience.

Theorem 3.2. Let p be odd. Suppose that there exists a finite group G and a defect zero block kGb ≅ End_{k}(V) which is P stable for some p-subgroup P of Aut(G). Also suppose that (End_{k}(V))(P) ≠ 0 and the image of V in D(P) is non-torsion. Then there exists G, b, P and V as above with P ≅ C_{p} × C_{p}.

Proof. Assume we have G, b, P and V as above with [V] non-torsion in D(P) and (End_{k}(V))(P) ≅ kGb(P) ≠ 0. As we mentioned above, there must be Q ⊴ R ≤ P such that the image of Defres_{R/Q}^{P}(V) is non-torsion in D(R/Q) and R/Q ≅ C_{p} × C_{p}. In the paragraph after Theorem 2.4, we noted that Defres_{R/Q}^{P}(V) ≅ V_{Q} for some endo-permutation R/Q-module V_{Q}.

Let Ĝ = C_{G}(Q) and b̂ = Br_{Q}(b). Then kGb(Q) = kĜb̂ ≅ End_{k}(V_{Q}). So b̂ is a defect zero block of Ĝ. The conjugation action of R/Q on Ĝ induces the same action on kĜb̂ as the one induced by R/Q on V_{Q}. Since V_{Q} is non-torsion in D(R/Q) this map must be injective by Lemma 3.1. This completes the proof of the theorem.

4 Proof of Theorem 1.3

Assume that we can find a finite group G and a defect zero block kGb ≅ End_{k}(V) of kG which is P -stable for a p-subgroup of Aut(G). Also suppose that Br_{P} (b) ≠ 0. If V is not torsion in D(P), then the results of the previous section allow us to assume that P ≅ C_{p} × C_{p}. This situation was considered in [10]. We recall two results from this paper.

Theorem 4.1 ([10]). Suppose that G is a finite group such that P ≅ C_{p} ×C_{p} ≤ Aut(G), and that b is a P -stable defect 0 block of kG such that Br_{P} (b) ≠ 0. Also, suppose that the source V of a simple k(G ⋊ P)b-module M is a finitely generated non-torsion endo-permutation kP -module. Then we can find G, b, V with the above properties where G is a central p′-extension of a simple group.

A detailed proof of this result can be found in [10]. The main idea is to let N be a normal subgroup of G which is maximal with respect to being P -stable. We can find a P -stable block d of kN such that bd ≠ 0. Applying Puig’s algebra theoretic version of Fong reduction reduces the problem to a central p′-extension of G/N. Our choice of N implies that G/N is a minimal normal subgroup of G/N ⋊P. Therefore, G/N is a direct product of isomorphic simple groups. The direct product can be eliminated using the fact, which can be found in [1], that Ten_{Q}^{P}(End_{k}(M)) ≅ End_{k}(Ten_{Q}^{P}(M)).

Now assume that G is a central p′-extension of a simple group. If b is a defect zero block of G, then he assumption that Br_{P} (b) ≠ 0 implies that P ∩Inn(G) = 1. Which means P can be detected in Out(G). Since | Out(G)| ≤ 2 for all sporadic groups, they can’t provide any examples in the above theorem. | Out(G)| = 2 for all alternating groups except A_{6} and | Out(A_{6})| = 4, so these groups will also fail to provide any examples. This leaves the finite simple groups of Lie type. Looking at the structure of Out(G) reduces the problem to P SL_{n}(q), P SU_{n}(q) (with the restrictions listed below) or p = 3 and D_{4}(p) (with the restrictions listed below), E_{6}(q) or ^{2}E_{6}(q). This only involves looking at the structure of Out(G). We actually have the more restrictive condition that P ∩ Inn(G) = 1 with P ⊆ Aut(G), and from this the groups E_{6} and ^{2}E_{6} can also be eliminated, and we have the following result whose detailed proof can be found in [10].

Theorem 4.2 ([10]). Assume that p is odd. Suppose that P = C_{p} × C_{p} ≤ Aut(G) where G is a finite group. Suppose that b is a P -stable defect 0 block of kG such that Br_{P} (b) ≠ 0. Finally, suppose that the source V of a simple k(G⋊P)b-module M is a finitely generated endo-permutation kP -module whose image is non-torsion in D(P). Then we can find G, b, V where one of the following holds.

(i) (a) G is a central p′-extension of A_{n}(q) = P SL_{n+1}(q) with p|(n + 1, q −

1, f) where q = r^{f}; or

(b) G is a central p′-extension of ^{2}A_{n}(q) = P SU_{n+1}(q) with p|(n + 1, q + 1, f) where q = r^{f}; or

(ii) p = 3, q = r^{f} and G is a central extension of D_{4}(q) with 3|f.

The following result of Kessar takes care of the A and ^{2}A cases above.

Theorem 4.3 (Theorem 1.2 [6]). Let p be odd, and let H be a finite group with a normal subgroup N such that H/N is elementary abelian of order p^{2}. Suppose that N is a quasi-simple group, with Z(N) a p′-group and such that N/Z(N) is isomorphic to P SL_{n}(q) or to P SU_{n}(q) where q is a prime power that is not divisible by p. Suppose that b is an H-stable block of kN which is of defect 0. Let U be a simple kHb-module and let (P, W) be a vertex source pair for U. Then [W] has order at most 2 in D(P).

We can now prove the following which was Theorem 1.3.

Theorem 4.4. Let p > 3 be a prime. Suppose that G is a finite group and kGb ≅ End_{k}(V) is a defect zero block of kG which is P -stable for some p-group P of Aut(G). Also assume that Br_{P} (b) ≠ 0. Then V is torsion in D(P).

Proof. Let G, b, V and P be as above. Theorem 1.2 allows us to assume that P ≅ C_{p} ×C_{p}. Then applying Theorem 4.2 we may assume that G is a central p′- extension of P SL_{n}(q) or P SU_{n}(q) for some prime power q which is not divisible by p.

Letting N = G and H = G ⋊ P we can apply Theorem 4.3 and conclude that V must be torsion.

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