THE BROUE CONJECTURE FOR THE FAITHFUL 3-BLOCKS OF 4.M_{22}

by

Jürgen Müller

RWTH-Aachen

and

Mary Schaps

Bar-Ilan

1. Introduction

We consider the Broué conjecture [B] for the faithful 3-blocks of defect 2 in the central extension of the Mathieu group M_{22} by C_{4}. This is a very interesting example, different from those 3-blocks which have been considered to date, because the stable equivalence which is needed between the global and local blocks is not simply restriction, but requires multiplication by an endo-permutation module, using a construction of Puig [P]. There are two other special features to this example. The tilting complex is a ”mixed” elementary tilting complex[S], a generalization of the elementary tilting complexes defined by Okuyama, [O]. Finally, the method of getting the derived equivalences uses Rickard’s Lemma 5.2 directly, rather than his theorem 5.1.

2. The 3-blocks of 4.M_{22}

The Mathieu group M_{22} has Schur multiplier isomorphic to C_{12}. We will be considering the central extension of M_{22} by C_{4}, of order 2^{9} · 3^{2} · 5 · 7 · 11, which we will designate by 4.M_{22} or simply by G. Let z be a generator of the center, an element of order 4.

If K is the field of 9 elements, let us designate by i one of the primitive 4th roots of unity in K. The group algebra K[G] has four blocks of defect 2. One is isomorphic to the principal block of M_{22} and one to the unique non-principal block of 2.M_{22}. There remain two conjugate blocks B_{+} and B_{−} consisting of characters on which z takes the values i and −i respectively. These blocks are interchanged by the unique outer automophism of G of order 2, and thus the group 4.M_{22}.2 contains a block Morita equivalent to B_{+} and B_{−}. According to the second author’s catalogue of 3-blocks of abelian defect group in the ATLAS groups, there are no other 3-blocks Morita equivalent to these block among the ATLAS groups and their subgroups.

Date: Apr. 18, ’04.

2 THE BROUE CONJECTURE FOR 4.M_{22}

Let B_{+} be first of the two conjugate blocks, consisting of characters for which the value at z is a positive multiple of i, and let D →^{∼} C_{3} ×C_{3} be its defect group, which is a 3-Sylow subgroup of G. The normalizer N = N_{G}(D) is a solvable group of order 288. The centralizer C_{G}(D) →^{∼} D × C_{4} is just the product of D and the central cyclic subgroup Z generated by z, and each of the four blocks of the centralizer is stable under the action of N.

The 2-Sylow subgroup H of N_{G}(D) is isomorphic to H →^{∼} C_{4} ⋊ C_{8}, and the relations of N_{G}(D) are

⟨a, b, c, d|a^{8} = b^{4} = c^{3} = d^{3} = e,
ba = ab^{−1}, ca = cda, da = acd^{−1}, cb = bd, db = bc^{−1}, cd = dc⟩.

The element z can be taken to be a^{2}b^{2}. Since D is abelian, the atomizer E is

N_{G}(D)/C_{G}(D) →^{∼} Q_{8},

the quaternion group of order 8.

The group E acts freely on the non-identity elements of D, and thus the group N_{G}(D)/Z, which is the normalizer of the defect group of the principal block of M_{22}, is a Frobenius group, i.e., it is a semidirect product of d by Q_{8} in which the action of Q_{8} on D − e is free. Furthermore, the four blocks of the normalizer are all isomorphic as algebras, so in fact the Brauer correspondent b_{+} of our faithful block B_{+} is isomorphic to the group algebra of a Frobenius group.

The nontrivial cyclic subgroups of D will be denoted by P_{j} for j ∈ I = {1, 2, 3, 4}. In the presentation of the normalizer given above, these will be generated by c, d, cd, c^{−1}d. In order to create a stable equivalence between modules of B_{+} and b_{+}, we must first analyze the centralizers C_{G}(P_{j}). On calculation, these turn out to be isomorphic to P_{i} × SL(2, 3)Y C_{4}, where the ”Y ” notation indicates identification of a common central subgroup, and the C_{4} is generated by the central element z. The 2-Sylow subgroup R is normal, isomorphic to Q_{8}Y C_{4}. Since D is abelian,

D ⊆ C_{G}(P_{j}),

we can view SL(2, 3) as a semi-direct product Q_{8} : P_{k}, for any k ≠ j in I.

The group algebra KC_{G}(P_{j}) has six three blocks, all nilpotent. Two of these blocks, corresponding to the 3-block of defect zero in SL(2, 3), have defect one. The other four have defect 2. Two of them, corresponding to the non-faithful blocks, are isomorphic to KD. The other two, corresponding to the two faithful blocks, are isomorphic to M_{2}(KD). We chose that 3-block of defect 2 in C_{G}(P_{j}) corresponding to the chosen block B_{+} of G and denote it by b_{j}^{+}.

At this point, in order to proceed, we must keep track of the action of D on the matrix algebra. We decompose b_{j}^{+} as a tensor product S_{j} ⊗ KD, where S_{j} is isomorphic as an algebra to M_{2}(K), but has an additional structure as an interior KD-algebra, that is to say, there is a group homomorphism

ϕ : D → S_{j}^{∗}.

The kernal of this homomorphism is P_{j}, and thus S_{j} is actually an interior D/P_{j} algebra. S_{j} is an interior D-module of a particular kind; the conjugation action by ϕ(D) leaves the identity element of the algebra fixed, and permutes the remaining three elements of a vector space basis. If V_{j} is the natural module for S_{j}, then V_{j} is called a KD-endo-permutation module to indicate that its endomorphism module S_{j} is a permutation module.

Let us work this out explicitly, for P_{1}. Let ζ be a primitive eighthroot of unity in the field K of nine elements, let i = ζ^{2} as before. Let f and g be two generators of Q_{8} of order 4, and let us take the standard two-dimensional representation ρ:

ρ(g) = [\begin{matrix}i & 0 \\ 0 & −i \end{matrix}] , ρ(f) = [\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}] .

Then the generator d of P_{2} can be embedded in S_{1} as

Y = [\begin{matrix}ζ^{7} & ζ^{3} \\ ζ^{5} & ζ^{5} \end{matrix}] ,

and we will have Y ρ(g)Y^{−1} = ρ(f). This representation comes from a representation of SL(2, 3) over the field GF (9) of nine elements. This representation is in fact similar to the natural representation of SL(2, 3) over the base field GF (3). In order to define the representation on all of D we set

ρ(c) = [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}]

, and we send the central elements z to a scalar matrix whose diagonal elements are the fourth root of unity corresponding to the chosen block.

Let E be the automizer, a different copy of Q_{8}. We represent the generators by a′ and b, both of order 4, with a′ba^{′−1} = b^{−1}.

We want to find a set of endopermutation modules which is stable under the action of the automizer E. The action of h = a^{′2} = b^{2} sends each element of D to its inverse and thus stabilizes all the cyclic subgroups. Thus we must first show that the endopermutation module which sends d to Y^{−1} is similar to the given endopermutation module in the Dade group. However, since the cyclic subgroups are fixed, both S_{1} and its image are in the relative Dade group for D/P_{1}. Since the Dade group of a cyclic group of prime order is isomorphic to C_{2}, the two modules are not only similar but also self-dual.

Since SL(2, 3) is normal of index 2 in GL(2, 3) and the natural representation is fixed under conjugation, it lifts to two distinct representations of GL(2, 3) which differ outside of SL(2, 3) by a factor of −1. Choosing one of these representations, we can extend ρ to

N_{G}(P_{1}) →^{∼} (P_{1} × SL(2, 3)Y C_{4}) ⋊ C_{2} by defining

ρ(h) = [\begin{matrix}0 & ζ^{5} \\ ζ^{3} & 0 \end{matrix}] .

Note that ρ(h) is the linear transformation mapping the endopermutation module V_{1} to its isomorphic dual. Conjugation by ρ(h) sends ρ(d) to its inverse, and trivally sends ρ(c) to its inverse, which is itself.

The quotient of E by its center h = b^{2} then acts transitively on the cyclic subgroups, so we can define V_{1} = V, V_{2} = V b, V_{3} = V a′, andV_{4} = V ba′ for j = 1, 2, 3, 4 by transfer of structure. What we will do, in fact, is to first construct the induced module of dimension 8.

V ⊕ V b ⊕ V a′ ⊕ V ba′.

The actions of a′ and b consist of permutation of the basis vectors and multiplication by a power of ζ.

Let us now glue these endo-permutation modules together, producing a 16-dimensional endopermutation module

W = V ⊗ V b ⊗ V a′ ⊗ V ba′.

For any element x of D, and in particular for c and d, we get

σ(x) = ρ(x) ⊗ ρ(bxb^{−1}) ⊗ ρ(a′xa^{′−1}) ⊗ ρ(ba′x(ba′)^{−1}).
σ(h) = ρ(h) ⊗ ρ(h) ⊗ ρ(h) ⊗ ρ(h).

By our choices of the modules, W is stable under the action of E. In fact, it will again be true that the actions of a′ and b consist of permuting the basis and multiplying by powers of ζ. Let W^{∗} be the dual of W, and let σ^{∗} be the corresponding representation, obtained by taking the transpose inverse of each matrix

The block algebra b_{+} is a twisted group algebra K[D]^{α}[E] by a cocycle α which is actually a coboundary of a linear character of the 2-Sylow subgroup H, and can be conveniently given by multiplying a′ by the eighth root of unity ζ, whose square is i. Thus the version of the endopermutation W^{∗} module adapted to b_{+} is given by

σ^{∗}(a) = ζσ^{∗}(a′).

Let f_{+} be a block idempotent of the Brauer correspondent b_{+} of B_{+} in N_{G}(D). By Remark 6.8 and Cor. 6.7 of Prop. 6.5 in [P], there is a stable equivalence between the modules of B_{+} and the modules of b_{+} given by:

F (M) = f_{+} · Ind_{D}^{N_{G}(D)}(V^{∗} ⊗ M|_{D}).

This formula, as it is written, is a bit misleading, since that induction is not ordinary module induction but the more complicated process we have described above, and the tensor product is over the center. Furthermore, since we are tring to define a stable equivalence, we can throw away all projective components.

Finally, we are interested in applying this formula when M is a simple module for G. Since the bimodule which gives our stable equivalence has no projective direct summands, the image of M under the equivalence should be indecomposable with maximal vertex. Thus our procedure will be as follows:

(1) Restrict M to N_{G}(D) and chose the unique indecomposable component M′ of maximal vertex. Let τ be the representation.

(2) Take the tensor product of W^{∗} with M′, where for a we take

σ^{∗}(a′) ⊗ τ(a).

In this way we avoid multiplying twice by ζ.

(3) Decompose the result and choose the unique indecomposable component of maximal vertex.

In the next section, we will actually carry this out for the simples of B_{+}.

3. The representation theory of the faithful blocks.

We must now apply this functor to the five simple modules of B, of dimensions 56a, 56b, 64, 160a, and 160b. Since we are only calculating up to stable equivalence, we can eliminate all projective factors of M|_{D}, since these will still be projective after tensoring with V^{∗}, and will thus induce projective N_{G}(D) modules. This means that it suffices to look at the Green correspondents of the simples, and apply the functor to their restriction to D. The Green correspondents will be denoted by 11a, 11b, 10, 16a, and 16b.

We carried out the calculation in the C-Meataxe written by M. Ringe. In each case we tensored by the 16 dimensional module and the resulting module over N_{G}(D) broke up into many indecomposable submodules, most of which were the four projectives of dimension 9 and the unique projective of dimension 18. In each case there was exactly one indecomposable summand of non-maximal but non-trivial vertex, always of dimension divisible by 12. Finally there was one indecomposable summand of maximal vertex.

In order to describe the structure of these indecomposables, we recall that the block has four simples S_{1}, S_{2}, S_{4}, S_{5} of dimension 1, and one, S_{3}, of dimension 2. We will denote the indecomposables corresponding to 56a, 56b, 64, 160a and 160b by M′_{i} for i = 1, 2, ..., 5, where

M′_{1} →^{∼} Ω^{−1}(S_{1}),

M′_{2} →^{∼} Ω(S_{2}),

both of dimension 8, M_{3} has ascending composition series S_{1} −S_{3} −S_{2}.

We had originally chosen to investigate the group 4.M_{22} because the second author’s database of blocks of abelian defect group in the ATLAS groups [S2] had indicated that there should be a very simple tilting complex, which appeared to be one of the elementary tilting complexes defined by Okuyama [O]. With the appearance of both the Heller operator and its transpose in the results of the stable equivalence, we revised this estimate and decided that the tilting complex was still elementary, but of the ”mixed” type defined in [S], with I′ = {1}, I” = {2}, and I_{0} = {3, 4, 5}. If we renumber the simples as 1, 2, 3, 4, 5 to simplify the notations, this would correspond to a tilting complex T:

P_{1} → P_{3} → 0,
0 → P_{3} → P_{2},
0 → P_{3} → 0,
0 → P_{4} → 0,
0 → P_{5} → 0.

We want to apply Rickard’s Lemma 5.2 in [R2]. This lemma requires some preliminary work to define complexes X_{i}, for i = 1, 2, ..., 5, stably equivalent in the derived category to the modules M′_{i}. In this case we can take X_{i} = M′_{i} for i = 3, 4, 5. The module M′_{2} is isomorphic in the bounded derived category to the complex

P_{2} → S_{2}.

This is the mapping cone of the canonical projection, and thus fits into a triangle with S_{2}[1−], where the third complex in the triangle is a complex of projectives. This sufices to show that X_{2} = S_{2}[−1] is stably equivalent to M′_{2}. By the dual argument, X_{1} = S_{1}[1] is stably equivalent to M′_{1}.

The next step is to demonstrate that the X_{i} satisfy the following conditions, where the Hom groups are calculated in the derived category:

(1) Hom(X_{i}, X_{j}[m]) = 0 for m < 0,

(2) Hom(X_{i}, X_{j}) = 0 if i ≠ j,

(3) Hom(X_{i}, X_{j}) = k is i = j,

(4) The X_{i} generates D^{b}(mod(b_{+})) as a derived cetegory.

Since most of the modules are simple, the only possible violations would involve X_{3} with one of first two simples. However, there is no homomorphism of X_{2} into X_{3}[−1], and no homomorphism of X_{3} into X_{1}[−1], so the conditions hold.

That T is a tilting complex follows from the Conditions 1 − 3 in [S], though it might be simpler just to check all the possible shifts. It is clear that it is possible to generate all the projectives from T. Therefore, it remains to check the conditions of Lemma 5.2 for the maps from T to the X_{i}. These conditions are as follows:

(1) Hom(X_{i}, X_{i}) = k,

(2) Hom(T_{i}, X_{j}[m]) = 0 for i ≠ j or m ≠ 0,

The only possible map from T_{2} into X_{3} is homotopic to zero, and there is no well- defined chain map from T_{1} into X_{3}. We conclude from lemma 5.2 that the tilting complex T sends the X_{i} to simple modules. Thus, composing the original stable equivalence which sent the simples of B_{+} to the X_{i} with the stable equivalence induced by T (using the fact that our blocks are all symmetric), we have a stable equivalence sending simples to simples, which is a Morita equivalence by [L] Hence we conclude that T is a tilting complex from b_{+} to a block Morita equivalent to B_{+}, thus proving the Broué conjecture for the faithful blocks of 4.M_{22}.

We wish to express our appreciation to Markus Linckelmann and Raphaël Rouquier for their help with finding the stable equivalence.

References

[B] M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181-182 (1990), 61–92.

[O] T. Okuyama, Some examples of derived equivalent blocks of finite groups, preprint, Hokkaido, 1998.

[L] M. Linckelmann, Stable equivalences of Morita type for self-injective algebras and −-groups. Math. Z., 223(1):87-100, 1996.

[P] L. Puig Une correspondance de modules pour les blocs a groupes de défaut abéliens, Geom. Dedicata, 37, (1991), pp. 9-43.

[R1] J. Rickard, Morita theory for derived equivalence, J. London Math. Soc. 39 (1989), 436–456.

[R2] J. Rickard, Equivalences of derived categories for symmetric algebras, preprint, Bristol.

[S] M. Schaps,Deformations, Tilting, and Decomposition Matrices, Proceedings of ICRAX, Fields Institute Publications, 2005.

[S2] M. Schaps, http://www.cs.biu.ac/ mschaps/BLOCKS/SortAt 3.html.