Algorithmic Methods in Modular Representation Theory

Der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinisch–Westfälischen Technischen Hochschule Aachen vorgelegte

Habilitationsschrift zur Erlangung der venia legendi

von Dr. rer. nat. Klaus Martin Lux aus Bogen/Niederbayern

Contents

Introduction 3

1 Notation and Basic Definitions 9

2 Direct Methods for Studying Representations 15 2.1 Basic Data Structures and Algorithms . . . . . . . . . . . . . 15 2.2 The Norton Criterion . . . . . . . . . . . . . . . . . . . . . . . 27

3 Condensation and Module Categories 39 3.1 Morita Equivalence . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Group Algebras and Permutation Modules . . . . . . . . . . . 48 3.4 Peakword Condensation . . . . . . . . . . . . . . . . . . . . . 54 3.5 An Extension of the MeatAxe . . . . . . . . . . . . . . . . . . 60 3.6 Implementations of the Condensation Method . . . . . . . . . 62

4 Examples for Explicit Morita Equivalences 65 4.1 The Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 The P.I.M.s of HS in Char. 5 . . . . . . . . . . . . . . . . . . 71 4.3 The P.I.M.s of M_{24} in Char. 3 . . . . . . . . . . . . . . . . . . 77 4.4 The P.I.M.s of J_{2} in Char. 2 . . . . . . . . . . . . . . . . . . . 82 4.5 The P.I.M.s of J_{2} in Char. 3 . . . . . . . . . . . . . . . . . . . 92 4.6 The P.I.M.s of J_{2} in Char. 5 . . . . . . . . . . . . . . . . . . . 95 4.7 The P.I.M.s of J_{3} in Char. 3 . . . . . . . . . . . . . . . . . . . 97 4.8 The P.I.M.s of W (E_{6}) in Char. 3 . . . . . . . . . . . . . . . . 105

5 An Algorithm for H^{1}(G, V) 109 5.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6 Faithful Condensation for Sporadic Groups 117

6.1 Faithful Condensation for M_{11} in Char. 2 . . . . . . . . . . . . 118 6.2 Faithful Condensation for M_{11} in Char. 3 . . . . . . . . . . . . 119 6.3 Faithful Condensation for M_{12} in Char. 2 . . . . . . . . . . . . 120 6.4 Faithful Condensation for M_{12} in Char. 3 . . . . . . . . . . . . 121 6.5 Faithful Condensation for J_{1} in Char. 2 . . . . . . . . . . . . . 122 6.6 Faithful Condensation for M_{22} in Char. 2 . . . . . . . . . . . . 123 6.7 Faithful Condensation for M_{22} in Char. 3 . . . . . . . . . . . . 124 6.8 Faithful Condensation for M_{23} in Char. 2 . . . . . . . . . . . . 125 6.9 Faithful Condensation for M_{23} in Char. 3 . . . . . . . . . . . . 126 6.10 Faithful Condensation for HS in Char. 2 . . . . . . . . . . . . 127 6.11 Faithful Condensation for HS in Char. 3 . . . . . . . . . . . . 128 6.12 Faithful Condensation for J_{3} in Char. 2 . . . . . . . . . . . . . 129 6.13 Faithful Condensation for M_{24} in Char. 2 . . . . . . . . . . . . 130 6.14 Faithful Condensation for McL in Char. 2 . . . . . . . . . . . 132 6.15 Faithful Condensation for McL in Char. 3 . . . . . . . . . . . 133 6.16 Faithful Condensation for McL in Char. 5 . . . . . . . . . . . 134

Appendix 135 The GAP-Procedure Faithfulkondburn . . . . . . . . . . . . . . . . . 135 The GAP-Procedure Imprimperm . . . . . . . . . . . . . . . . . . . 138 The GAP-Procedure Imprimpermcan . . . . . . . . . . . . . . . . . . 139 The Definition of the Words z1, …, z10. . . . . . . . . . . . . . . . 141

List of Important Algorithms 143

Frequently Used Symbols 145

List of Notation 146

Index 147

Bibliography 149

Introduction

The main issue of representation theory is the study of the realizations of an abstract algebra as a subalgebra of the endomorphisms of a vector space over some field F. In other words one studies matrix representations of the algebra over F. Originally the starting point of representation theory was the theory of representations of finite groups over the complex numbers studied by G. Frobenius in the end of the last century. As a main tool Frobenius introduced the traces of the matrices of group elements, which he called characters. He could show that the character of a representation gives a lot of useful information about the matrix representation itself. Nevertheless from the beginning of representation theory there always has been a need to construct explicit matrix representations. I. Schur for example constructed representations of the Schur covers of the symmetric groups, see [Sch11], based on the presentations for the Schur covers he had determined beforehand.

In her lecture course on group theory and hypercomplex numbers in 1927, E. Noether used the notion of the group algebra to unify the structure theory of algebras and the representation theory of finite groups. This approach had an important impact on the further development of the representation theory of finite groups. It implied that one could apply the results of the general representation theory of algebras and module theory to the case of finite groups. A typical example is Wedderburn’s theorem on the structure of semisimple algebras. It took Frobenius several years to complete the proof of Wedderburn’s theorem for groups over the complex numbers without knowing about the general proof given by T. Molien for semisimple algebras over the complex numbers, see [vdW85].

It was R. Brauer, a student of I. Schur, who extended the representation theory of finite groups from fields of characteristic zero to fields of finite characteristic in the 1930’s. Unlike in characteristic zero, where the Frobenius characters are a very important tool in studying representations, it turns out that they are much less useful in finite characteristic. However, Brauer solved this problem in the following way. For a given representation over a field of finite characteristic p, he defined a map from the conjugacy classes of elements whose orders are coprime to p, into the complex numbers, which he called the modular character of the representation. Nowadays this class function is called a Brauer character and Brauer showed that it is the right generalization of a character as defined by Frobenius. The Brauer characters have turned out to be one of the central tools for studying the representations of a finite group in finite characteristic. But again, some of the most important results obtained by Brauer are most naturally phrased in the framework of the representation theory of finite-dimensional algebras.

In his work on representation theory of algebras, K. Morita observed in the late 1950’s that from a categorical point of view the representations of a finite-dimensional algebra can be linked to the representations of a usually much smaller algebra called the basic algebra. Morita’s theorems can be interpreted as a generalization of Wedderburn’s theorem on the structure of a simple algebra. The relationship Morita defined between the representations of two algebras is now called a Morita equivalence and it establishes a functorial equivalence between the module categories of the two algebras. One of the topics of this manuscript is the application of Morita’s theory to computational aspects in the representation theory of finite groups.

Although computations always played a certain role in representation theory, not much progress was made in this direction when compared to other branches of computational group theory until the 1970’s. At this time one became interested — due to the classification of finite simple groups project — in the realization of finite simple groups, especially sporadic simple groups as matrix groups over finite fields. There were several reasons to take this approach: first of all, constructing a matrix representation was an efficient way of showing the existence of a previously unknown simple group and secondly it enabled one to study in detail group theoretic properties of the group in question, like for example the classification of the maximal subgroups. The first sporadic simple group found after the Mathieu groups, the so-called Janko group J_{1} was constructed by Z. Janko, see [Jan66], as a matrix group in dimension 7 over the field with 11 elements. For larger sporadic simple groups the use of a computer was inevitable and one of the highlights of the computational approach was the construction of the sporadic simple Janko group J_{4} of order 86.775.571.046.077.562.880. It was shown to exist by J.H. Conway, S.P. Norton and R.A. Parker by constructing it as a group of 112 by 112 matrices over the field with two elements, see [Nor79]. The proof was achieved using a collection of programs developed by Parker. These programs, called the MeatAxe by Parker, contained most of the basic algorithms of linear algebra over finite fields. In addition to these the MeatAxe also included a program, implementing a constructive method for proving the irreducibility of a given matrix representation based on a criterion by Norton.

In the case of J_{4} the matrix representation was predicted by J. Thompson, using information about the subgroup structure of J_{4}. The representation itself is by far the most efficient way of realizing J_{4} if one compares it to realizations as a permutation group, where the smallest degree is 173.067.389 or as a matrix group in characteristic different from 2, where the smallest degree is 1332, see [Jan96].

In [Par84], an outline of the basic MeatAxe programs is given. Moreover it contains a description of the Norton criterion which is used to decide whether a given matrix representation is irreducible. The efficiency of the Norton criterion depends on the probability that the matrix of a randomly chosen element of the group algebra is singular. As pointed out in [Par84], this probability decreases linearly with the size of the underlying field and therefore the resulting algorithm is most efficient over the field with two elements. The first success of the MeatAxe after the existence proof of J_{4} was the determination of the last unknown degree of an irreducible representation of the Mathieu group M_{24} in characteristic 2. This problem had been left unsolved by G. James in his doctoral thesis on the irreducible modular representations of the Mathieu groups. In 1994 a major progress was achieved by D. Holt and S. Rees, who succeeded in generalizing Norton’s criterion in [HR94]. The applicability of their criterion, which is based on irreducible factors of the characteristic polynomial of a randomly chosen matrix instead of its being singular, turns out to be independent of the size of the underlying field. It implies therefore that one can prove the irreducibility of a given matrix representation regardless of the size of the field. However, Holt and Rees also pointed out that proving the reducibility of a given representation constructively can be much harder. This problem includes the task of finding an explicit basis for a subspace invariant under the given matrices. In [HR94] they construct an example where finding an invariant subspace using the generalized Norton criterion becomes very time consuming.

A more sophisticated use of the kernels of singular matrices was made in [LMR94]. Let us suppose we are given a matrix representation of an algebra A. Using basic facts about the representation theory of algebras in general one can show that the kernel of a well chosen element in A carries structural information about the representation itself. More precisely, a suitable element of the given algebra A, whose matrix in the given matrix representation is singular, leads to a primitive idempotent in A. The images of such idempotents can then be used, as was proven in [LMR94], to recover the whole lattice of A-invariant subspaces in the underlying vector space.

In the sequel, the same idea was applied to show that related objects like the socle and the radical series of a given A-module can be determined algorithmically along the same lines, see [LW96a].

Falling back on the character theoretic approach Brauer had taken when studying representations over finite fields, R.A. Parker, G. Hiss and the author developed the program system MOC for calculating with the Brauer characters of a given finite group, see [HJLP97]. MOC was quite successfully applied to the problem of determining the irreducible Brauer characters of sporadic simple groups in blocks of cyclic defect, see [HL89]. Moreover, during the development of MOC it was emphasized that the proofs first produced by MOC could be turned easily into human-readable form. This was a major progress when compared to the situation for the ordinary character tables of sporadic simple groups. Only the character tables themselves are usually accessible in the literature. Note, however, that the techniques underlying the determination of the ordinary character tables can be found in [LP91]. MOC and the MeatAxe were applied jointly in order to determine the irreducible Brauer characters for various simple groups, see [HLP91] and [HL88], but it soon became clear that even these two methods taken together usually do not suffice to determine the irreducible Brauer characters of a sporadic simple group.

In this situation the newest approach of studying matrix representations based on algorithmic versions of Morita equivalence can help, and the main topic of this manuscript will be the description of this techniques. The approach was described for the first time by J. Thackray in his doctoral thesis, see [Tha81], where he called it condensation. As mentioned above, one of the driving forces for the development of new algorithms in representation theory was the classification of finite simple groups. D. Gorenstein, one of the organizers of the classification of finite simple groups project, emphasized the task of studying properties of the finite simple groups after the classification had been established. Among the important properties he lists the irreducible modular representations, see [Gor82]. The use of Morita equivalence became necessary since the irreducible representations of finite simple groups tend to be too large to be studied directly, see for example [JLPW95]. The condensation method suggests that one studies instead the representations of a subalgebra of the group algebra in question, which usually is chosen to be Morita equivalent to the given group algebra. Condensation has turned out to be a very useful tool, since it reduces the dimension of the representations one has to study quite considerably. Ideally, one would like to work with a so-called basic algebra, the smallest algebra that is Morita equivalent to a given group algebra. It is however a hard problem in general to find a basic algebra of a given algebra. If one is interested in just finding a reasonably sized algebra that is Morita equivalent to the given algebra then for the case of group algebras one can successfully make use of the subgroup structure of the given group.

The first chapter of this manuscript introduces the basic terminology and notation needed in the following. The second chapter will give a summary of the methods that deal directly with a given matrix representation. We will describe in detail the implementation of the basic algorithms as part of the C-MeatAxe, a collection of programs, developed by M. Ringe and J. Müller in joint work with the author, which represents an up-to-date version of the original ideas of Parker. We will then discuss Norton’s irreducibility criterion and its generalization due to D. Holt and S. Rees. Again we will give an illustration of the implementation of this criterion in form of the C-MeatAxe program chop, one of the most frequently used programs of the C-MeatAxe.

The third chapter begins with a description of the theory of Morita equivalence and condensation functors. We will then analyze the important case of group algebras. Moreover, a constructive method is presented that finds a smaller algebra that is Morita equivalent to a block of a given group algebra. We will proceed by showing how one determines primitive idempotents algorithmically. Primitive idempotents will then be used to solve the problem encountered by Rees and Holt when generalizing Norton’s criterion. Moreover, we will use primitive idempotents to develop an algorithm for calculating the socle series of a representation. The last part of the chapter will give an account on the various implementations of the condensation method.

In Chapter 4 we will study examples for Morita equivalences by applying the constructive method from Chapter 3. We will start with describing a general approach to find a Morita equivalent subalgebra for a finite group whose table of marks is known. The power of this approach is demonstrated in the case of several finite simple groups. We will use the explicit Morita equivalence and the algorithm for calculating socle series to determine the layers of the socle series for the projective indecomposable modules of the group algebras in question. This will be done by first determining the projective indecomposable modules and their socle series for the smaller algebra and then using the Morita equivalence to get the layers of the socle series for the projective indecomposable modules of the group algebra.

The fifth chapter will begin with a description of an algorithm that gives an upper bound on the first cohomology group of a given module. It was shown to be an important ingredient for determining the maximal subgroups of an arbitrary finite group by Aschbacher and Scott in [AS85]. Reducing the problem to the case of simple groups, it was furthermore shown in [AG84] that the dimension of the first cohomology group H^{1}(G, V) for any finite group G and any faithful simple F_{p}G-module V is less than the dimension of V. Moreover, Guralnick conjectures, see [Gur86], that H^{1}(G, V) is at most two dimensional if V is an absolutely irreducible F_{p}G-module. For large simple groups the standard algorithm for determining the first cohomology group, see [Cel92], assumes that the group is given as a finitely presented group. This can cause the algorithm to be very time consuming for finite simple groups, since presentations for them usually consist of quite a number of generators and relations. The algorithm we present is a modification of the standard algorithm, but uses the fact that a simple group is generated by two elements. Furthermore, it generates random relations in these two generators, which not necessarily are a defining set of relations for the group. This implies that the algorithm can only give an upper bound for the first cohomology group of a given module. Nevertheless, it has been successfully applied to various sporadic simple groups in characteristic 2, where the results show that H^{1}(G, V) is at most two dimensional for all these cases. The biggest calculation done so far showed that the first cohomology group of the 4370-dimensional representation of the Baby Monster over the field with two elements, which was constructed by R.A. Wilson, see [Wil93], is one dimensional, a result which definitely could not be have been achieved using the standard algorithm.

In the last chapter we will determine a subgroup H suitable for condensation in the case that G is a sporadic simple group of order up to 10^{9}. Furthermore, for most of these cases, we will explicitly determine generators for the so-called condensation algebra derived from H. Since the principal block b_{0} of this algebra will be Morita equivalent to the principal block B_{0} of G, representation theoretic questions about B_{0} can be reduced to questions about b_{0}.

First of all, I want to express my gratitude to Prof. Dr. Herbert Pahlings, who taught me representation theory and whose advice and encouragement were of great help. I also want to thank all the members of Lehrstuhl D for the fruitful discussions during the last few years. My special thanks go to the following people. To Alexander Hulpke, the local TEX guru, for answering my numerous questions about TEXand L^{A}TEX, to Frank Celler, the system and hardware guru, for supporting me in the fight with the computer, to Thomas Merkwitz, who supplied me at the right time with his extensions to the tables of marks in GAP, and to Thomas Breuer, who is just there when you need him. Finally, I want to thank Prof. Dr. Joachim Neubüser, whose energy and endurance created this unique environment concerning people and computer equipment, which made it a pleasure for me to work at Lehrstuhl D für Mathematik.

Chapter 1

Notation and Basic Definitions

The main object of study in this manuscript are finite-dimensional algebras, their representations and modules. This preliminary chapter will therefore give an introduction to the basic terminology, which can also be found in the standard textbooks on representation theory.

Definition 1.1.1. Let R be a commutative ring with 1. A ring A is called an R-algebra, if there exists a ring homomorphism ψ from R into the center of A, such that ψ(1_{R}) = 1_{A}.

We will mainly deal with F -algebras, where F is a field. It follows that an F -algebra A has the structure of an F -vector space using the homomorphism ψ. We will restrict ourselves to the study of finite-dimensional F -algebras over some finite field F.

An important example is the group algebra of a finite group over F.

Example 1.1.2. Let G be a finite group and let F be a field. Let F G be an F -vector space with basis b_{g} indexed by the elements g ∈ G. We define a multiplication on F G by defining the product of two basis elements b_{g} and b_{h} to be b_{gh} and extending it linearly to F G. Then F G is a finite-dimensional F -algebra called the group algebra of G over F. To simplify our notation we will identify b_{g} with g in the following.

Definition 1.1.3. A (right) A-module is an F -vector space V supplied with a binary operation from the cartesian product V × A to V satisfying

• v · (a + b) = v · a + v · b for all a, b ∈ A and v ∈ V,

• (v + w) · a = v · a + w · a for all a ∈ A and v, w ∈ V,

• v · (ab) = (v · a) · b for all a, b ∈ A and v ∈ V,

• v · 1_{A} = v for all v ∈ V.

A left A-module is defined in an analogous way. We also say that the F - algebra A is acting linearly on V. Moreover, if A and B are F -algebras and V is a right A-module and a left B-module such that b · (v · a) = (b · v) · a for all a ∈ A, b ∈ B and v ∈ V, then V is called a B-A bimodule.

Example 1.1.4. Let A be an F -algebra. We define the (right) regular Amodule A_{A} to be the F -vector space A with the right multiplication derived from the multiplication in the F -algebra A.

Definition 1.1.5. An F -linear map ϕ from an A-module V to an A-module W is called an A-homomorphism if it commutes with multiplication by elements of A, i.e., if ϕ(v · a) = ϕ(v) · a for all a ∈ A and v ∈ V. We use the notation Hom_{A}(V, W) for the F -vector space of A-homomorphisms from V to W. The A-modules V and W are said to be isomorphic if there is an A-isomorphism between V and W. If we take the F -vector space of A-endomorphisms End_{A}(V) of a given A-module, then End_{A}(V) is an F - algebra, when we define the product of two elements ϕ, ψ ∈ End_{A}(V) to be the composition of maps, i.e., (ϕψ)(v) = ϕ(ψ(v)) for all v ∈ V. Observe that the A-module V has the structure of an (End_{A}(V) − A) bimodule, i.e., ϕ(v · a) = ϕ(v) · a for all ϕ ∈ End_{A}(V), a ∈ A, and v ∈ V.

Definition 1.1.6. Let V be an A-module. An F -subspace of V is called an A-submodule, if it is an A-module supplied with the restricted multiplication. V ≠ {0} is called a simple A-module or an irreducible A-module, if the only A-submodules are {0} and V. Otherwise it will be called reducible. A composition series of V is a strictly increasing sequence of A-submodules such that the quotients of successive members are simple A-modules. The A-module V is called uniserial if it has a unique composition series. It will be called semisimple or completely reducible if it can be written as an internal direct sum of simple submodules. There always exists a largest semisimple submodule of an A-module V, which is called the socle Soc(V) of V. Dually, there is also a smallest A-submodule Rad(V) of V such that V/ Rad(V) is semisimple, which is called the radical of V. If V cannot be written as the internal direct sum of nontrivial A-submodules then V is called an indecomposable A-module. Furthermore, an indecomposable direct summand of the regular A-module A_{A} is called a projective indecomposable module.

Remark 1.1.7. There is a correspondence between the simple A-modules and the projective indecomposable A-modules which is given as follows. If P is a projective indecomposable A-module then P/ Rad(P) is simple. Furthermore, if S is a simple A-module then there exists a unique (up to isomorphism) projective indecomposable module P such that P/ Rad(P) = S.

We call P the projective cover of S and we use the notation P (S) for P in this case. If P is a projective A-module which for all (up to isomorphism) simple A-modules S contains P (S) as a direct summand, then P is called a projective generator for the algebra A.

Definition 1.1.8. Let V be an A-module. Define the sequence

[Soc_{1}(V), …, Soc_{l}(V) = V]

inductively by Soc_{1}(V) = Soc(V) and Soc_{i}(V)/ Soc_{i−1}(V) = Soc(V/ Soc_{i−1}(V)) for i > 1. Define l to be the smallest number with Soc_{l}(V) = V. Then [Soc_{1}(V), …, Soc_{l}(V)] is called the socle series of V, the quotients Soc_{i}(V)/ Soc_{i−1}(V) are called the socle layers, and l is called the socle length of V.

Definition 1.1.9. Let V be an A-module. Define the sequence

[Rad_{1}(V), …, Rad_{m}(V) = {0}]

inductively by Rad_{1}(V) = Rad(V) and Rad_{i}(V) = Rad(Rad_{i−1}(V)) for i > 1. Moreover, define m to be the smallest number such that Rad_{m}(V) = {0}. Then [Rad_{1}(V), …, Rad_{m}(V)] is called the radical series of V. It is well known, see for example [Lan83], that m is equal to the socle length of V.

Remark 1.1.10. Let S be a simple A-module. The smallest extension field K of the finite field F with the property that

End_{A⊗_{F} K}(S ⊗_{F} K) ≅ K

is called the splitting field of S. We will use the fact, see for example [CR81], that K ≅ End_{A}(S). In the case of F being the splitting field of S we call S an absolutely irreducible A-module.

The category whose objects are the finite-dimensional right A-modules and whose morphisms are the A-homomorphisms will be abbreviated by RMod(A) and the category of finite-dimensional left A-modules will be denoted by LMod(A).

The structure of an F -algebra A can be studied by looking at its category of A-modules. Important objects in this category are the simple A-modules, the projective indecomposable A-modules, and more generally the indecomposable A-modules. For a finite-dimensional algebra there are only finitely many simple and projective indecomposable modules but there might be infinitely many indecomposable modules.

Definition 1.1.11. Let V be an F G-module. We define the trace map Tr from End_{F} (V) to End_{F G}(V) in the following way

Tr(ϕ)(v) = ∑_{g∈G} ϕ(vg)g^{−1} for all ϕ ∈ End_{F} (V), v ∈ V.

An F G-endomorphism ψ of V is called projective if it is in the image of the trace map, i.e., if there exists ϕ ∈ End_{F} (V) with Tr(ϕ) = ψ.

The trace map can be used to give a different characterization of projective F G-modules.

Lemma 1.1.12. Let V be an F G-module. Then the following statements hold.

i) The projective F G-endomorphisms form a two-sided ideal of End_{F G}(V).

ii) V is a projective F G-module if and only if Tr(End_{F} (V)) = End_{F G}(V).

Proof: See [Lan83]. ■

Definition 1.1.13. A homomorphism δ_{V} from A to the F -endomorphism ring End_{F} (V) of an F -vector space V is called a representation of A. The concept of an A-representation is equivalent to that of an A-module: a representation gives obviously rise to an A-module structure on V and on the other hand given an A-module V we can naturally define a homomorphism from A to End_{F} (V). The concept of an irreducible A-representation corresponds to that of a simple A-module. Moreover, two representations δ_{V} and δ_{W} are defined to be equivalent if and only if the corresponding Amodules V and W are isomorphic. Phrased in terms of category theory we can state that RMod(A) is equivalent to the category of finite-dimensional A-representations, see for example [Pie82].

For computational purposes, the representations one works with are the matrix representations of A. These are defined as homomorphisms δ from A to the ring of n by n matrices over F for some n ∈ ℕ. Again there is a correspondence between A-representations and matrix representations of A, which can be stated as an equivalence between the category of A-modules and the category of A-matrix representations. A representation δ_{V} gives rise to a matrix representation δ_{V,B} once we have fixed a basis B for the underlying vector space V, where δ_{V,B}(a) denotes the matrix of δ_{V} (a) for a ∈ A with respect to B. Conversely, given an A-matrix representation δ of degree n we can define a representation δ_{V} of A on the vector space V = F^{1×n} by mapping an element a ∈ A to the endomorphism whose matrix with respect to the canonical basis B of V is given by δ(a). Matrices are a fundamental tool for computations in linear algebra and more generally in representation theory. The representation theoretic algorithms we will present will make heavy use of matrices and algorithms dealing with matrices. However, the theory of modules is better suited to describe the underlying theory. Since we can change from the module point of view to the one studying matrices quite easily by the procedure described above, we will use both concepts interchangeably without further comment.

The basic algorithmic problems we have to study are the following:

• Decide whether an A-module is simple.

• Find a nontrivial A-submodule of a reducible A-module.

• Find the socle and the socle series of an A-module.

• Investigate the A-submodule structure of an A-module.

• Give a good description of the representations of A in general.

Before we will further investigate these problems, we will first describe the underlying basic algorithms in linear algebra, which we will use to solve the problems mentioned above.

14 CHAPTER 1. NOTATION AND BASIC DEFINITIONS

Direct Methods for Studying Representations

2.1 Basic Data Structures and Algorithms

We begin our study of algorithms for analyzing and constructing matrix representations over finite fields by first giving an account on the basic data structures and basic matrix algorithms. These are fundamental and will be used in the more sophisticated algorithms we describe later on. We closely follow the actual implementation in the C-MeatAxe, see [Rin94]. The reader should bear in mind that we want to apply the algorithms to the study of the structure of the representations of large finite simple groups. This will involve calculations in large dimensional vector spaces over finite fields of small characteristic. We therefore strive for good performance of the basic matrix algorithms for fields of size up to 256. At the same time we require an efficient use of main memory. This leads us to the following approach based on basic data structures for

• elements of finite fields,

• vectors over finite fields,

• matrices over finite fields.

We first describe the representation of elements of finite fields. For a finite field F we start from the Kronecker construction but we will use a different approach for calculating in F. More precisely, if we have to deal with elements of a finite field F of degree n over its prime field F_{p}, we choose an irreducible polynomial c_{n} ∈ F_{p}[X] of degree n. Then F is isomorphic to F_{p}[X]/(c_{n}) and we calculate once and for all tables for the basic arithmetic in F using F_{p}[X]/(c_{n}). This approach enables us to determine the result of an addition or multiplication when needed by just looking up the corresponding table. In addition to these two tables we also store a table which contains the inverse for all nonzero field elements. An important task we face quite frequently is the explicit embedding of a field into a larger field. Since we want to keep the embedding particularly simple we have to make a careful choice of the polynomials c_{n}. R.A. Parker suggested the following way of selecting c_{n}, which leads us to the nice embeddings we require. We order the monic polynomials in F_{p}[X] following a suggestion of Conway: we map the polynomial X^{n} − a_{n−1}X^{n−1} + … + (−1)^{n}a_{0} ∈ F_{p}[X] to the sequence [a_{n−1}, a_{n−2}, a_{n−3}, …, a_{0}] and take the lexicographic ordering on the sequences.

Definition 2.1.1. Let F be a finite field of degree n over the prime field F_{p}. The Conway polynomial c_{n} of F with respect to F_{p} is defined inductively as the smallest primitive polynomial of degree n according to the ordering defined above such that (X + (c_{n}))^{(p^{n}−1)/(p^{d}−1)} is a root of c_{d} for all divisors d of n.

Remark 2.1.2. A proof that shows the existence of Conway polynomials can be found in [Nic88]. Currently, Conway polynomials are used in several Computer algebra systems, for example GAP and the C-MeatAxe.

Example 2.1.3. The following table gives the list of Conway polynomials for the nonprime fields of size up to 256.

size  c_{n}  size  c_{n}
4 X^{2} + X + 1 64 X^{6} + X^{4} + X^{3} + X + 1
8 X^{3} + X + 1 81  X^{4} + 2X^{3} + 2
9 X^{2} + 2X + 2 121  X^{2} + 7X + 2
16 X^{4} + X + 1 125  X^{3} + 3X + 3
25 X^{2} + 4X + 2 128  X^{7} + X + 1
27 X^{3} + 2X + 1 169  X^{2} + 12X + 2
32 X^{5} + X^{2} + 1 243  X^{5} + 2X + 1
49 X^{2} + 6X + 3 256 X^{8} + X^{4} + X^{3} + X^{2} + 1

Remark 2.1.4. Programs that determine the Conway polynomial for a given finite field F have been written by various people. There is a GAP-procedure written by J. Müller and a C-program written by R.A. Parker. Furthermore, a list of Conway polynomials covering a lot of interesting cases can be found in [JLPW95].

We next describe the data structure that we found suitable for representing the elements in a finite field.

Remark 2.1.5. Let c_{n} be the Conway polynomial for the finite field F of degree n over F_{p}. We use the isomorphism F ≅ F_{p}/(c_{n}) to represent elements in F: if f ∈ F corresponds to the coset s_{f} + (c_{n}) with deg(s_{f}) < n then we lift s_{f} to the polynomial ŝ_{f} = ∑_{i=0}^{n−1}(ŝ_{f})_{i}X^{i} in ℤ[X], whose coefficients are nonnegative and less than p. We represent f by the nonnegative integer (ŝ_{f})_{0} + (ŝ_{f})_{1}p + … (ŝ_{f})_{n−1}p^{n−1} = ŝ_{f}(p).

Based on this representation the data structure for vectors is given as follows.

Remark 2.1.6. Let F have size q = p^{n} ≤ 256 and let e be the largest integer such that q^{e} ≤ 256. Let the field elements f_{1}, f_{2}, …, f_{e} be represented by f̂_{1}, f̂_{2}, …, f̂_{e} according to the previous remark. Then we can store the e elements f_{1}, f_{2}, …, f_{e} in a byte by taking the number f̂_{e}q^{0} + f̂_{e−1}q^{1} + … + f̂_{1}q^{e−1}. This gives us a very compact way of storing vectors. We divide a vector into subvectors of length e and store the corresponding bytes instead. We avoid the problem of packing and unpacking vectors by using an analogous approach to the one for field elements. We determine two lookup tables, one that stores the number corresponding to the addition of two vectors of length e, i.e., of two bytes, and one table that stores the number corresponding to the multiplication of a vector of length e by an element of F. Both tables require 2^{16} bytes in the C-MeatAxe implementation. This approach allows us for example to add two vectors of length e using only one table lookup. We also use further tables for the following purposes

• finding the position and the the value of the first nonzero entry in a byte (2^{16} bytes each),

• extracting the i-th entry of a byte (2^{16} bytes),

• inserting an entry at the i-th position in a byte (2^{16} bytes).

Example 2.1.7. Take the field F with 9 elements and with Conway polynomial c_{2} = X^{2} + 2X + 2, let z = (X + (c_{2})) and take the vector [z, z^{2}, z, 1] of length 4. Since 9^{2} is less than 256 we can store e = 2 entries per byte. This gives us a 2 byte packed vector.

entries  z  z^{2}  z  1
cosets (X + (c_{2})) (X + 1 + (c_{2})) (X + (c_{2})) (1 + (c_{2}))
number  3  4  3  1
byte value  31  28

Hence we save a factor of two for storing a vector, and an addition of two packed vectors using the table lookup will be roughly twice as fast compared to the standard method.

Remark 2.1.8. An alternative way of storing vectors over small prime fields has been suggested by R.A. Parker. It has been implemented by Michler et al., see [FMR^{+}95]. In this approach, vectors are packed less dense but an addition of two vectors will be faster.

We now give a selection of the algorithms that will be fundamental tools on which more sophisticated algorithms are based. This is also reflected in the actual C-MeatAxe implementation, where these algorithms have been incorporated into a library of routines dealing with matrices. We will investigate the following algorithms.

1. Addition of two matrices.

2. Multiplication of two matrices.

3. Calculating the nullspace (or kernel) of a matrix.

4. Determining the inverse of an invertible matrix.

5. Putting a matrix into semi-echelonized form.

6. Determining the closure of a subspace under the action of an F -algebra.

7. Determining a matrix representation of an F -algebra on an invariant subspace and the corresponding quotient space.

Most of the algorithms mentioned above are based on variants of the Gaussian elimination algorithm.

Remark 2.1.9. We describe algorithms in different ways. The basic algorithms will be formulated in a PASCAL like language. The more complex algorithms usually consist of a sequence of calls to subroutines implementing the basic algorithms. We will therefore use a different kind of description indicating only in which order the basic algorithms are called. Note that we usually express that a variable is a packed matrix over a finite field by using capital letters and we will treat a matrix M = (m_{ij}) as a two dimensional array, i.e., M[i] is the i-th (packed) row and M[i][j] denotes the entry m_{ij}. Note that in the description of the algorithms we do not distinguish between a row vector of length n and a 1 by n matrix. Sometimes we interpret a matrix as a sequence of vectors and we use the following basic functions without further comment: the function length, which returns the length of a sequence, the function append, which appends a vector to a given sequence and the function merge, which merges two sequences. Moreover, all the basic arithmetic on vectors and matrices is done using the tables we have mentioned before and should therefore be interpreted like this in the program descriptions. There are usually two variants of the same algorithm implemented in the C-MeatAxe: a stand-alone program and a subroutine as part of the C-MeatAxe subroutine-library.

Algorithm 2.1.10 (add).

INPUT: Two m by n matrices M and N over F in packed form. CALCULATION:

\underline{for} i := 1 \underline{to} m \underline{do}

SUM[i] := M[i] + N[i]; # addition is done using table lookup \underline{od};

OUTPUT: The sum SUM of M and N in packed form.

Algorithm 2.1.11 (multiply).

INPUT: An m by n matrix M and an n by k matrix N over F in packed form. CALCULATION:

\underline{for} i := 1 \underline{to} m \underline{do} PROD[i] := 0; # this is a packed zero vector of length k \underline{for} j := 1 \underline{to} n \underline{do}

PROD[i] := PROD[i] + M[i][j] ∗ N[j]; # this is done using table lookups

\underline{od}; \underline{od};

OUTPUT: The matrix product PROD of M and N in packed form.

Definition 2.1.12. Let M be an m by n matrix over F. We define a sequence pivotindex of length m in the following way: if the row M[i] is zero, then pivotindex[i] = 0 else pivotindex[i] gives the first nonzero position in M[i]. The numbers from 1 to n that do not appear in pivotindex are called the insignificant positions. The matrix M is said to be semi-echelonized or in semi-echelonized form if all rows M[i] are nonzero and for all j greater than i it holds M[j][pivotindex[i]] = 0. If in addition for all j less than i we have M[j][pivotindex[i]] = 0 then M is said to be in (full) echelonized form. If M is in semi-echelonized form then we define pivotfactor as the sequence [M[1][pivotindex[1]], …, M[m][pivotindex[m]]].

Checking whether a given row vector v ∈ F^{1×n} lies in an r-dimensional subspace S ≤ F^{1×n} is an important task. Using a semi-echelonized basis B = [B[1], …, B[r]] of S and the two sequences pivotindex and pivotfactor as defined above this can be solved in the following way. Set v_{0} = v and define v_{i} = v_{i−1} − \frac{v_{i−1}[pivotindex[i]]}{pivotfactor[i]}B[i] for i = 1, …, r. Then v lies in S if and only if v_{r} is the zero row vector. The recursive procedure was called “cleaning the vector” v by R.A. Parker and the vector v_{r} is called the cleaned vector. The next algorithm implements this idea. More precisely, it reads in a semiechelonized basis B of S and a sequence of vectors [v_{1}, …, v_{k}] of V. It cleans the vectors v_{1}, …, v_{k} using B and returns the cleaned vectors.

Algorithm 2.1.13 (clean).

INPUT: An m by n matrix M in semi-echelonized form representing a basis B for a subspace S, the sequences pivotindex, pivotfactor for M, and a k by n matrix N. CALCULATION:

\underline{for} i := 1 \underline{to} k \underline{do} RES[i] := N[i]; \underline{for} j := 1 \underline{to} m \underline{do}

RES[i] := RES[i] − RES[i][pivotindex[j]]/pivotfactor[j] ∗ M[j]; \underline{od};

\underline{od};

OUTPUT: The cleaned matrix RES.

The subroutine version of Algorithm 2.1.13 takes as first argument the matrix M representing the basis B and as second argument the matrix N, whose rows are the vectors to be cleaned. The sequences pivotindex and pivotfactor defined by M are treated as global sequences. Algorithm 2.1.13 is used in the following algorithm, which transforms a matrix into semiechelonized form.

Algorithm 2.1.14 (semi-echelonize).

INPUT: An m by n matrix M over F. CALCULATION:

B := []; pivotindex := []; pivotfactor := []; card := 0; # the cardinality of B \underline{for} i := 1 \underline{to} m \underline{do}

ROW := clean(B, M[i]);

# clean is a subroutine version of the clean algorithm # M[i] is interpreted as a 1 by n matrix \underline{if} ROW ≠ 0 \underline{then}

card := card + 1; B[card] := ROW; index := firstnonzero(ROW); # a procedure returning the index of the first nonzero entry # of a vector pivotindex[card] := index; pivotfactor[card] := B[card][index];

\underline{fi}; \underline{od};

OUTPUT: The matrix B in semi-echelonized form and pivotindex and pivotfactor for B.

The algorithm for calculating the nullspace of a matrix combines the algorithm for cleaning and the algorithm for transforming a matrix into semiechelonized form.

Algorithm 2.1.15 (nullspace).

INPUT: An m by n matrix M over F. CALCULATION:

NSP := ID(m); # ID(m) is the m by m identity matrix pivotindex := []; pivotfactor := []; \underline{for} i := 1 \underline{to} m \underline{do}

\underline{for} j := 1 \underline{to} i − 1 \underline{do} \underline{if} pivotindex[j] ≠ 0

\underline{then}

NSP [i] := NSP [i] − M[i][pivotindex[j]]/pivotfactor[j] ∗ NSP [j];
M[i] := M[i] − M[i][pivotindex[j]]/pivotfactor[j] ∗ M[j];

\underline{fi}; \underline{od}; index := firstnonzero(M[i]); # firstnonzero returns 0 if the row is the zero row pivotindex[i] := index; \underline{if} index > 0 \underline{then}

pivotfactor[i] := M[i][index];

\underline{fi}; \underline{od}; # put nullspace into semi-echelonized form card := 0; B := []; \underline{for} i := 1 \underline{to} m \underline{do}

\underline{if} pivotindex[i] = 0 # row vector is a basis vector for the nullspace \underline{then}

card := card + 1;

B[card] := clean(B, NSP [i]);

bpivotindex[card] := firstnonzero(B[card]); bpivotfactor[card] := B[card][bpivotindex[card]];

\underline{fi}; \underline{od};

OUTPUT: A semi-echelonized basis B for the nullspace of M.

Remark 2.1.16. Alternative algorithms can be found in [FMR^{+}95] and [Coh93].

Algorithm 2.1.17 (inverse).

INPUT: An n by n matrix M. CALCULATION:

INV := ID(n); \underline{for} i := 1 \underline{to} n \underline{do}

k := i;

\underline{while} M[k][i] = 0 \underline{do} k := k + 1; \underline{od}; \underline{if} k > n \underline{then} error;

# the matrix is singular

\underline{fi}; \underline{if} k > i

\underline{then} # swap the rows
T MP := M[i];
M[i] := M[k];
M[k] := T MP;

T MP := INV [i];
INV [i] := INV [k];
INV [k] := T MP;

\underline{fi};

M[i] := 1/M[i][i] ∗ M[i]; # normalize the row INV [i] := 1/M[i][i] ∗ INV [i]; \underline{for} k := 1 \underline{to} n \underline{do} \underline{if} k ≠ i \underline{then}

M[k] := M[k] − M[k][i] ∗ M[i];
INV [k] := INV [k] − M[k][i] ∗ INV [i];

\underline{fi}; \underline{od};

\underline{od};

OUTPUT: The inverse INV of M.

One of the basic tasks we face when analyzing a given matrix representation of an F -algebra on an A-module V is the construction of the smallest A-invariant submodule containing a given sequence of vectors. If a group G is acting as a permutation group on a set S, then there is a well known algorithm that determines the orbit of an element s ∈ S under the action of G. The calculation of the smallest A-invariant submodule is done in an analogous fashion.

Algorithm 2.1.18 (closure).

INPUT: A semi-echelonized sequence of vectors C = [v_{1}, …, v_{r}] and a sequence E of matrices for a generating set of A. CALCULATION:

\underline{for} x := 1 \underline{to} length(C) \underline{do} # note that length is evaluated during the loop! \underline{for} a := 1 \underline{to} length(E) \underline{do}

RES := clean(C, C[x] ∗ E[a]); \underline{if} RES ≠ 0 \underline{then}

C := append(C, RES);
index := firstnonzero(RES);
pivotindex[length(C)] := index;
pivotfactor[length(C)] := RES[index];

\underline{fi};

\underline{od}; \underline{od};

OUTPUT: A semi-echelonized basis C for the smallest A-invariant subspace generated by [v_{1}, …, v_{r}], pivotindex and pivotfactor for C.

We get an important variant of the closure algorithm if we do not add a cleaned nonzero vector to the sequence of basis vectors but the vector itself. In addition we also return the words in the generators that were used to define the basis vectors.

Algorithm 2.1.19 (spinning).

INPUT: A sequence of linearly independent vectors [v_{1}, …, v_{r}] and a sequence of matrices E for a generating set of A. CALCULATION:

preimage := [1, …, r]; gens := [0, …, 0]; B := [v_{1}, …, v_{r}]; C := semiechelonize([v_{1}, …, v_{r}]); \underline{for} x := 1 \underline{to} length(B) \underline{do}

\underline{for} a := 1 \underline{to} length(E) \underline{do} ROW := clean(C, B[x] ∗ E[a]); \underline{if} ROW ≠ 0 \underline{then}

C := append(C, ROW);
B := append(B, B[x] ∗ E[a]);
gens[length(C)] := a;
preimage[length(C)] := x;
index := firstnonzero(ROW);
pivotindex[length(C)] := index;
pivotfactor[length(C)] := ROW [index];

\underline{fi}; \underline{od};

\underline{od};

OUTPUT:

•A standard F -Basis B = [v_{1}, v_{2}, …, v_{s}], where s is the dimension of

⟨v_{1}, …, v_{r}⟩_{F} A ≤_{A} V.

•Two sequences of natural numbers gens = [w_{1}, …, w_{s}] and preimage = [f_{1}, …, f_{s}], such that v_{f_{i}}a_{w_{i}} = v_{i} for 1 ≤ i ≤ s. Furthermore, f_{i} < i for i > r and w_{i} = 0, f_{i} = i for i ≤ r.

Although the spinning algorithm requires a copy of the cleaned basis in order to test for membership and hence requires more storage space, it is a very useful algorithm for the following reason.

Remark 2.1.20. If V is an A-module generated by [v_{1}, …, v_{r}] and ϕ is an A-homomorphism from V to the A-module W, then the knowledge of the sequence [ϕ(v_{1}), …, ϕ(v_{r})] is sufficient to determine the image of an arbitrary element of the standard basis B constructed by the spinning algorithm. This fact has been exploited by R.A. Parker to give a constructive isomorphism test for simple A-modules. We will describe this test in Section 2. Actually, the same idea can be extended to give an isomorphism test for arbitrary A-modules, as was shown by M. Szőke and the author.

The next algorithm is an extension of the algorithm for cleaning vectors. It is used for determining a matrix representation on a quotient module with respect to an A-submodule, which we might have constructed using Algorithm 2.1.18.

Algorithm 2.1.21 (clean and quotient space).

INPUT: An m by n matrix M in semi-echelonized form and a k by n matrix N. CALCULATION:

\underline{for} i := 1 \underline{to} k \underline{do}

\underline{for} j := 1 \underline{to} m \underline{do}

COEF F [i][j] := N[i][pivotindex[j]]/pivotfactor[j];
N[i] := N[i] − N[i][pivotindex[j]]/pivotfactor[j] ∗ M[j];

\underline{od}; pos := 0; # restrict N[i] to insignificant positions \underline{for} s := 1 \underline{to} n \underline{do}

\underline{if} s ∉ pivotindex

\underline{then}

pos := pos + 1;
V EC[i][pos] := N[i][s];

\underline{fi}; \underline{od};

\underline{od}; OUTPUT: Two matrices: COEF F, containing the coefficients in the (partial) decomposition of the rows of N into the rows of M, and the matrix V EC, containing the restriction of the rows of N to the insignificant positions after cleaning.

Remark 2.1.22. The clean and quotient algorithm can be interpreted in the following way. It calculates the decomposition of vectors with respect to the semi-echelonized basis B of the subspace spanned by the rows of M extended by the canonical row vectors at the insignificant positions of B. The C-MeatAxe program zsp incorporates Algorithm 2.1.18 and Algorithm 2.1.21. Given matrix generators for A on an A-module V and a sequence of vectors in V, it first determines the closure of the vectors under A and then writes out a matrix representation for A on the submodule and the quotient module. It proceeds as follows.

Algorithm 2.1.23 (split).

INPUT: A semi-echelonized sequence of vectors [v_{1}, …, v_{r}] and a sequence E of n by n matrices for a generating set of A. CALCULATION:

B := closure(E, [v_{1}, …, v_{r}]); \underline{for} a := 1 \underline{to} length(E) \underline{do}

\underline{call} clean and quotient(B, B ∗ E[a]);

# call the previous algorithm returning COEF F and V EC ESUB[a] := COEF F; \underline{for} i := 1 \underline{to} n \underline{do}

\underline{if} i ∉ pivotindex

\underline{then}

\underline{call} clean and quotient(B, E[a][i]);
EQUOT [a][i] := V EC;

\underline{fi}; \underline{od};

\underline{od};

OUTPUT: A sequence of matrices ESUB for the generating set of A on the subspace ⟨v_{1}, …, v_{r}⟩A, and, if this is a proper subspace, a sequence of matrices EQUOT for the generating set of A on the quotient space.

We still have to find answers to the following two questions:

• How do we prove that a given A-module V is simple, i.e., that a given matrix representation of A is irreducible?

• How do we find vectors which lie in an A-submodule if V is reducible?

We will see in the next section that there are algorithmic answers to both questions, which at the same time are interesting from the representation theoretic point of view. They will lead to new algorithms that can be used to analyze a given module in further detail.

2.2 The Norton Criterion

In the previous section we have described the basic closure Algorithm 2.1.18, which determines the smallest A-invariant subspace containing a given sequence of vectors in an A-module V. In this section we will consider an algorithm for proving the irreducibility of a given A-module. We will introduce the Norton criterion, which is the essential tool for proving the irreducibility of a given A-representation. In the following we give a proof of the criterion based on [Pah89] that avoids the use of a matrix representation on the A-module V. An alternative proof, which uses an explicit matrix representation of A on V, can be found in [HR94]. Furthermore, we will also have a look at the related, but computationally more difficult, problem of finding an A-submodule in a reducible A-module.

If A is an F -algebra over a finite field F there is an obvious way of proving the irreducibility of a given A-module: Run through all nonzero vectors v of V, determine a basis for vA using Algorithm 2.1.18 and check whether vA is a proper submodule of V. The drawback of this “obvious” method is that it depends heavily on the dimension of V and also on the size of the field F. We will see that this is essentially unnecessary. The Norton criterion will give us a much better method, which will satisfy both demands, i.e., it depends no longer on the field size and on the dimension of V.

Before stating the criterion, let us first recall some basic definitions.

Definition 2.2.1. Let V be a finite-dimensional F -vector space. Let V^{∗} be the dual vector space Hom_{F} (V, F). If V is an A-module for the F -algebra A, then V^{∗} is a left A-module in the following way. For λ ∈ V^{∗} and a ∈ A define a·λ(v) := λ(v·a) for all v ∈ V. If W is an F -subspace of V, then we define the annihilator W^{0} of W in V^{∗} to be W^{0} = {λ ∈ V^{∗} | λ(w) = 0 for all w ∈ W }. Note that if W is an A-submodule of V, then W^{0} is a left A-submodule of V^{∗}.

We will use the following facts about dual vector spaces.

Theorem 2.2.2. Let V be an A-module and let W be an A-submodule of V. Then V^{∗}/W^{0} ≅ W^{∗} as left A-modules and (V^{∗})^{∗} ≅ V as (right) Amodules. Let M(V) be the lattice of A-submodules of V and let M(V^{∗}) be the lattice of left A-submodules of V^{∗}. Then the map from M(V) to M(V^{∗}) mapping W to W^{0} is a lattice anti-isomorphism. Moreover, the kernel of the restriction map from V^{∗} to W^{∗} is given by W^{0}. If M is an F -vector space and ϕ ∈ Hom_{F} (V, M), then we denote by ϕ^{∗} ∈ Hom_{F} (M^{∗}, V^{∗}) the linear map defined by ϕ^{∗}(λ)(v) = λ(ϕ(v)) for all λ ∈ M^{∗}, v ∈ V. The kernel of ϕ^{∗} is given by the annihilator of the image of ϕ. Expressed in the language of categories and functors, taking duals of A-modules and A-homomorphisms realizes a categorical anti-equivalence between the category of right A-modules RMod(A) and the category LMod(A) of left A-modules.

Proof: See for example [Kas77]. ■

We now can formulate the Norton criterion for testing irreducibility of a given representation.

Theorem 2.2.3. Let A be an F -algebra, let V be an A-module and let δ: A → End_{F} (V) be the corresponding representation of A. Moreover, let a ∈ A with V > Kernel(δ(a)) > {0}. Then the following statements are equivalent.

i) δ is an irreducible representation of A.

ii) For all nonzero elements v ∈ Kernel(δ(a)) it holds that vA = V and there exists a nonzero element v^{∗} ∈ Kernel(δ^{∗}(a)) with Av^{∗} = V^{∗}.

Proof: If δ is an irreducible representation, then the only A-submodules are {0} and V. Therefore i) implies ii) trivially.

Now assume that there exists a nontrivial A-submodule W. Then 0 < W^{0} < V^{∗} in the dual module V^{∗}. Moreover, the A-submodule W is surely δ(a)-invariant. We distinguish the following cases.

Case 1: W ∩ Kernel(δ(a)) ≠ 0. Then there exists a nonzero vector v ∈ W ∩ Kernel(δ(a)) and hence vA ≤ W.

Case 2: The intersection W ∩Kernel(δ(a)) is trivial. It follows that the restriction δ(a)|_{W}: W → W is injective and since W is finite-dimensional, δ(a)_{W} is bijective, i.e., δ(a)(W) = W. Now, since Kernel(δ(a)^{∗}) = (Image(δ(a))^{0} and W = δ(a)(W) it follows that Kernel(δ(a)^{∗}) < W^{0}. So for all vectors v^{∗} in Kernel(δ(a)^{∗}) it follows that Av^{∗} ≤ W^{0}. ■

Remark 2.2.4. The Norton criterion immediately leads to an algorithm that enables us to verify that a given representation of an F -algebra A is irreducible. It requires that we have some way of running through the representing matrices for the elements of A, given the matrices for a generating system of A. There are different approaches to this problem, see [Rin94] and [HR94]. The approach taken in the C-MeatAxe roughly is the following: given the matrices for the system of generators, we first determine products up to a certain length in the matrix generators and then we investigate the F -linear combinations of these which will be called words in the subsequent algorithms.

Algorithm 2.2.5.

INPUT: A matrix representation δ given on a generating system E of A. CALCULATION:

•Run through the words a in the generators of A

•if Kernel(δ(a)) is nonzero then

–for all vectors v ∈ Kernel(δ(a)) representing the 1-dimensional subspaces of Kernel(δ(a))

∗Determine vA using Algorithm 2.1.23

∗if vA < V ·return matrices for E on vA and V/vA. ·break;

∗fi;

–od;

–Determine a vector v^{∗} such that δ(a)v^{∗} = 0

–Determine Av^{∗} using Algorithm 2.1.23 with transposed input

–if Av^{∗} < V^{∗} then

∗return transposed matrices for E on Av^{∗} and V^{∗}/Av^{∗}.
∗break;

–fi;

–we haven’t found a nontrivial invariant subspace, so return the information that the representation is irreducible.

–break;

•fi;

OUTPUT: Either the information that V is irreducible, or a matrix representation of A on an A-invariant subspace and the corresponding quotient space.

Remark 2.2.6. In an implementation of the algorithm above, one assumes that the run through the words a in A produces random elements δ(a) in δ(A). Moreover, this approach also relies heavily upon the assumption that the probability of finding an element in A whose kernel is nonzero is sufficiently large. If S is a simple A-module, then δ(A) is isomorphic to End_{A}(S)^{k×k} by Wedderburn’s theorem and the number of elements in δ(A) with a nonzero kernel is therefore roughly \frac{1}{| End_{A}(S)|}. Hence the probability of finding singular elements in an irreducible representation decreases linearly with increasing field size. This implies that the algorithm works best for the field with 2 elements but that the random search becomes inefficient when the field size increases. Moreover, to enhance the efficiency in the algorithm, it is convenient to look for “good” elements in the sense that they should have a kernel of the smallest possible dimension, preferably a kernel that is one dimensional. We will show below that the smallest possible dimension in the case of a simple A-module S is exactly the dimension of a splitting field of S over F. Note that as long as we can guarantee that we will meet every matrix in δ(A) at least once, the resulting irreducibility test is deterministic. However in practice, we do not check whether we generate the full image, and the algorithm is therefore only a probabilistic algorithm.

In contrast to what the remark seems to indicate, there is actually no problem of finding singular elements, if we organize our search through δ(A) in a more systematic way. This method has been developed by Holt and Rees in [HR94]. Before we describe their approach let us first investigate the question, what the dimension of the kernel of a singular element can be.

Lemma 2.2.7. Let F be a finite field and let S be a simple A-module for the F -algebra A. Let δ_{S} be the corresponding representation of A and denote the endomorphism ring End_{A}(S) by D. Then for all a in A it follows that dim_{F} (Kernel(δ_{S}(a))) is a multiple of [D : F]. Furthermore, there exists an element a_{0} ∈ A such that dim_{F} (Kernel(δ_{S}(a_{0}))) = [D : F].

Proof: Wedderburn’s theorem tells us that δ_{S}(A) is isomorphic as an F - algebra to D^{k×k}, where k dim_{F} (D) = dim_{F} (S). Moreover, δ_{S}(a) is singular if and only if the corresponding element of D^{k×k} is a singular matrix. The dimension of the kernel of δ_{S}(a) is [D : F] times the D-dimension of the kernel of the corresponding element in D^{k×k}. Hence, if we take a matrix in D^{k×k} whose kernel is one dimensional over D, then the corresponding δ_{S}(a) has a [D: F]-dimensional kernel.

A second proof can be given as follows. Let e be a primitive idempotent of A such that Se is nonzero. Then Se ≅ Hom_{A}(eA, S) ≅ End_{A}(S) = D since eA is a projective A-module. Moreover, S = Se ⊕ S(1 − e) and hence Se = Kernel(δ_{S}(1 − e)). Hence 1 − e is an element whose kernel has the right dimension. Furthermore, if a is an arbitrary element of A, then Kernel(δ_{S}(a)) is a D-module and hence isomorphic to a direct sum of copies of the regular, simple D-module D_{D}. ■

An alternative way of interpreting the Norton criterion is the following. Restrict the representation δ_{S} to the cyclic subalgebra ⟨a⟩ generated by an element a. The cyclic algebra ⟨a⟩ has an irreducible one dimensional representation that maps a to zero and a^{0} = 1_{A} to the identity. We call this representation the trivial representation of ⟨a⟩. The kernel of δ_{V} (a) is then the sum of all submodules on which the algebra ⟨a⟩ acts trivially. For a given A-submodule W of V the intersection with this space is either nontrivial, in which case we find a vector in the intersection generating an A-submodule contained in W. Otherwise it follows that W as an ⟨a⟩-module does not contain the trivial composition factor at all. This implies that W^{0} as a left ⟨a⟩-module contains all (left) trivial ⟨a⟩-submodules. Hence the A-submodule generated by any vector generating a trivial ⟨a⟩-submodule is contained in W^{0}. In this context it becomes clear that we can apply the Norton criterion to any irreducible composition factor of the cyclic algebra ⟨a⟩ on V. The irreducible composition factors of the representation δ_{V} of ⟨a⟩ are in one-to-one correspondence with the irreducible factors of the characteristic polynomial c(δ_{V} (a)) of δ_{V} (a), see [Lan93]. More precisely, if f is an irreducible factor of c(δ_{V} (a)), then we can define an irreducible matrix representation δ_{f} by setting δ_{f}(a) to be the companion matrix of the polynomial f and extending δ_{f} to ⟨a⟩. We denote the corresponding ⟨a⟩-module by S_{f}.

Lemma 2.2.8. Let V be an A-module and let a ∈ A. Let f be an irreducible factor of the characteristic polynomial of δ_{V} (a). If for each simple ⟨a⟩- submodule S_{f} corresponding to f it holds that S_{f}A = V and moreover there exists a simple (left) ⟨a⟩-submodule S_{f}^{∗} corresponding to f with AS_{f}^{∗} = V^{∗}, then V is a simple A-module.

The conclusion we can draw from this lemma is that it is easy to find elements b ∈ A, such that δ_{V} (b) is singular, namely we just set b := f(δ_{V} (a)) for an element a ∈ A and an irreducible factor f of the characteristic polynomial c(δ_{V} (a)). The important question we have to answer is the following: what is the probability of finding “good” elements a ∈ A, i.e., of finding elements with a small dimensional kernel.

Definition 2.2.9. Let V be an A-module and a ∈ A. An irreducible factor of the characteristic polynomial c(δ_{V} (a)) is called good if its multiplicity in the minimal polynomial of δ_{V} (a) is equal to its multiplicity in c(δ_{V} (a)). In other words the restriction of V to ⟨a⟩ has a unique uniserial summand, whose composition factors are isomorphic to S_{f}. Moreover, f is called very good if f is an unrepeated irreducible factor of c(δ_{V} (a)). This is equivalent to saying that the corresponding simple ⟨a⟩-module S_{f} appears exactly once in the restriction of V to ⟨a⟩.

Concerning the probability of very good factors in case that the representation is irreducible, Holt and Rees proved, see [HR94].

Theorem 2.2.10. Let S be a simple A-module such that End_{A}(S) has degree e over F. Then for at least 1/5 of the elements a ∈ A there is an unrepeated irreducible factor of degree e in the characteristic polynomial of δ_{S}(a).

Remark 2.2.11. Let S be a simple A-module and let a in A be an element such that the characteristic polynomial of δ_{S}(a) has a very good factor f of degree dim_{F} (End_{A}(S)). Furthermore, let S_{f} = Kernel(f(δ_{S}(a))) be the corresponding simple ⟨a⟩-submodule of S. Since S is a simple A-module, it follows that End_{A}(S) is a division ring and therefore End_{A}(S) has only one simple module, namely End_{A}(S)_{End_{A}(S)}, the regular End_{A}(S)-module. It follows that End_{A}(S) acts irreducibly on S_{f} and hence for any two given nonzero vectors v_{1} and v_{2} in S_{f} there is an automorphism ϕ ∈ End_{A}(S) mapping v_{1} to v_{2}. This shows that the matrix representation of A with respect to a standard basis determined by Algorithm 2.1.19 using any nonzero vector v in S_{f} does not depend on v. Note that we only need that dim_{F} (Kernel(f(δ_{S}(a)))) = dim_{F} (End_{A}(S)) and hence the statement holds more generally for a in A with dim_{F} (Kernel((δ_{S}(a)))) = dim_{F} (End_{A}(S)).

This leads to the following algorithm for determining the splitting field of a given simple A-module.

Algorithm 2.2.12 (splitting field).

INPUT: A matrix representation δ_{S} for a generating system of A on the simple A-module S. CALCULATION:

•Set posfac to dim_{F} (S).

•Run through the words a in A.

•Calculate the characteristic polynomial c of δ_{S}(a), using Algorithm 2.2.15.

•for each irreducible factor f of c

–Determine K = Kernel(f(δ_{S}(a))) –Set posfac to the gcd of posfac and dim_{F} (K) –if dim_{F} (K) divides posfac

∗choose a nonzero vector v in K and set B to [v] ∗Put S into standard form using Algorithm 2.1.19 with v as input. ∗Set E = [id_{K}] ∗while B does not span K do

·Choose a vector w in K \ ⟨B⟩_{F} ·put S into standard form using w. ·if the (standard) matrix representations with respect to v and w are the same ·Restrict the bases change to K and append it to E. ·Determine the closure of B and w with respect to E. ·Replace B by the basis of the resulting subspace. ·else ·Take the next irreducible factor, ·i.e., go back to the for loop. ·fi;

∗od; ∗stop; we have proved that dim_{F} (K) is the degree of the splitting field.

–fi;

•od;

OUTPUT: The degree e of End_{A}(S) over F, a word a ∈ A and an irreducible factor f of the characteristic polynomial of δ_{S}(a) such that δ_{S}(f(a)) has a kernel of dimension e.

Algorithm 2.2.13 (equivalence test). INPUT: A matrix representation δ_{S} given for a generating system of A on the simple A-modules S, a word a ∈ A such that Kernel(δ_{S}(a)) is of dimension dim_{F} (End_{A}(S)) and a matrix representation δ_{T} for the generating system on a simple A-module T. CALCULATION:

•Calculate Kernel(δ_{S}(a));

•Determine a base change B to the standard basis of S using algorithm 2.1.19 with any nontrivial vector of the kernel as input;

•Calculate Kernel(δ_{T} (a));

•Determine a base change C to the standard basis of T using algorithm 2.1.19 with any nontrivial vector of the kernel as input;

•for all generators b of A do

–if (Bδ_{S}(b)B^{−1} ≠ Cδ_{T} (b)C^{−1})

–print S and T are not equivalent; stop;

–fi;

•return BC^{−1};

OUTPUT: The matrix BC^{−1} transforming the representation δ_{S} into δ_{T} or the information that δ_{S} and δ_{T} are not equivalent.

Before we give a summary of the C-MeatAxe implementation of the modified Norton criterion, we will first describe the algorithm for calculating the characteristic polynomial of a given square matrix M. In order to determine the characteristic polynomial we first consider the problem of finding for a given vector v the monic polynomial p_{v} of lowest degree such that v(p_{v}(M)) = 0. The polynomial p_{v} will be called the order polynomial of v with respect to M. The following algorithm solves the problem of finding the order polynomial in the more general situation where we work modulo a given subspace S of V.

Algorithm 2.2.14 (orderpol).

INPUT: An n by n matrix M, a semi-echelonized basis B of a subspace S of V, a nonzero vector vec ∈ V \ S. CALCULATION:

D := [vec]; C := []; card := 0; pivotindex := []; pivotfactor := []; \underline{repeat} card := card + 1; C[card] := [0, …, 0]; C[card][card] := 1; # first clean the new vector D[card] := clean(B, vec); \underline{for} i := 1 \underline{to} card − 1 \underline{do}

C[card] := C[card] − D[card][pivotindex[i]]/pivotfactor[i] ∗ C[i];

D[card] := D[card] − D[card][pivotindex[i]]/pivotfactor[i] ∗ D[i];

\underline{od}; \underline{if} D[card] ≠ 0

\underline{then}

index := firstnonzero(D[card]);
pivotindex[card] := index;
pivotfactor[card] := D[card][index];
vec := vec ∗ M :

\underline{fi}; \underline{until} D[card] = 0;

OUTPUT: C[card] holds the coefficients of the order polynomial of vec modulo the subspace S.

The subroutine version of the algorithm is given the matrix M, the vector vec and the matrix B. It returns the order polynomial f of vec and the semi-echelonized matrix D, whose rows span the subspace ⟨vec ∗ M^{i}|i = 1, …, deg(f) − 1⟩ modulo S.

Algorithm 2.2.15 (charpol).

INPUT: An n by n matrix M. CALCULATION:

polys := []; B := []; num := 0; \underline{for} i := 1 \underline{to} n \underline{do}

vec := [0, …, 0];

vec[i] := 1;

\underline{if} vec ∉ ⟨B⟩_{F} \underline{then}

num := num + 1;

polys[num] := orderpol(M, vec, B);

# orderpol also returns a matrix D, see above B := merge(B, D); # merge the two bases B and D into B

\underline{fi}; \underline{od};

OUTPUT: A list of polynomials polys whose product is the characteristic polynomial of M. Observe that the first polynomial is the order polynomial of the first canonical basis vector.

Remark 2.2.16. The reason for keeping the list of polynomials is twofold. We can take advantage of the factorization, when we want to determine the irreducible factors of the characteristic polynomial. Secondly, it is quite likely that the first factor, i.e., the order polynomial p_{1} of the first basis vector v_{1} already contains all the irreducible factors of the characteristic polynomial of M. Moreover, for any irreducible factor f of p_{1}, the vector v_{1}((p_{1}/f)(M)) is in the kernel of f(M). So in the case that f is a very good factor of the characteristic polynomial, we have already determined a vector in the kernel of f(M). This vector can therefore be used to test irreducibility using the Norton criterion.

Algorithm 2.2.17 (chop). Instead of giving the complete outline of the (complex) algorithm for determining the composition factors of a given matrix representation of A we explain the fundamental steps. The algorithm for determining the composition factors is a recursive procedure: given a matrix representation δ, we run through the words δ(a) in the given matrix generators and determine the characteristic polynomial of δ(a) using the algorithm above. We obtain a sequence of polynomials [p_{1}, …, p_{r}] such that p_{1} is the order polynomial of the first canonical basis vector v_{1}. This implies that for all irreducible factors f of p_{1} we have a vector v in Kernel(δ(f(a)). We use Algorithm 2.1.23 to determine vA. If vA is smaller than V then we enter the recursion with the matrix representations on vA and V/vA returned by Algorithm 2.1.23. Otherwise, if f is a very good factor of the characteristic polynomial of c(δ(a)) we use the subroutine orderpol to determine the order polynomial p_{1}^{∗} of the column vector [1, 0, …, 0]^{t} and the matrix δ(a). If f is not dividing p_{1}^{∗} we give up and take the next irreducible factor of c(δ(a)). Otherwise we calculate the vector v^{∗} such that f(δ(a))v^{∗} = 0 and use Algorithm 2.1.23 with the transposed matrix generators and the transposed vector (v^{∗})^{t}. If a nontrivial subspace is returned, then we enter the recursion with the transposed matrix generators on the subspace and the quotient space. Otherwise we mark the representation δ as being irreducible and we now call Algorithm 2.2.12 for determining the splitting field, which also returns a word in the generators of A whose kernel in δ is minimally nonzero. Finally we run through all irreducible representations δ′ found so far and check whether the representation δ and δ′ are equivalent using Algorithm 2.2.13 and the word for δ returned by the splitting field algorithm. If this turns out to be true then we increase the multiplicity of the representation δ′ by one. Otherwise we increase the list of known irreducible representations by one and set the multiplicity of the new irreducible representation δ to one.

Remark 2.2.18. In [HR94] it is shown that the above method has a great chance of finding an A-submodule given a reducible A-module in all but one case. This exceptional situation is given by an A-module whose composition factors are all isomorphic and which is simple modulo its radical. Holt and Rees also give an explicit example, which illustrates that such modules really arise. In Chapter 3 we will show that by taking a closer look at the nullspace of words in A one can actually solve this problem.

38 CHAPTER 2. DIRECT METHODS

Condensation and Module Categories

In the last chapter we were dealing with the direct methods of analyzing a given matrix representation of a finite-dimensional algebra A. The purpose of this chapter is the description of the methods based on the relationship between the module category of A and the module category of a subalgebra of A.

Such techniques have been applied algorithmically for the first time by R.A. Parker and they have been documented by Thackray in his Ph.D. thesis, see [Tha81]. Thackray calls them the fixed point reduction method and he uses the fixed point reduction as an alternative to the Norton criterion in order to determine A-invariant subspaces.

In this chapter we will first summarize the mathematical theory underlying the algorithms. Connections between module categories have been exploited for quite a while in the theory of representations of finite-dimensional algebras. The best relationship between module categories, one hopes for, is an equivalence of the categories, see Definition 3.1.1, also called a Morita equivalence. Morita had studied conditions that guarantee that the module categories for two F -algebras A and B are equivalent. He had found a nice way of describing an existing equivalence in terms of A-B bimodules. Morita equivalence will play the most important role in this chapter. Our aim is to use Morita’s theorems to gain knowledge on A-modules and it will turn out that this approach can be much more effective than the application of the direct methods. Even better, some questions about large dimensional A-modules, which we can answer using Morita equivalence, are out of reach for the direct methods.

We want to point out that there is a more general situation which is naturally of interest. For a given A-module V we can study the functor Hom_{A}(V, −) from the category of finitely generated (right) A-modules to the category of finitely generated (right) End_{A}(V) modules. If we put the following two conditions on the A-module V and its endomorphism ring End_{A}(V)

• a simple A-module is contained in V/ Rad(V) if and only if it is contained in the socle of V,

• End_{A}(V)_{End_{A}(V)} is an injective End_{A}(V)-module, i.e., End_{A}(V) is a quasi-Frobenius algebra,

then the functor Hom_{A}(V, −) defines at least a bijective correspondence between the simple A-modules in the socle of V and the simple End_{A}(V)- modules.

Moreover, M. Cabanes, see [Cab90], has shown that Hom_{A}(V, −) defines an equivalence between a certain (additive) subcategory of RMod(A) and the category RMod(End_{A}(V)) under the assumptions on V and End_{A}(V). However, the two conditions are not easy to verify in general. From the algorithmic point of view, we have to be able to determine at least a generating set for End_{A}(V) and an F -basis for Hom_{A}(V, M) if we want to investigate the structure of the End_{A}(V)-module Hom_{A}(V, M) for a given A-module M. Although there are now efficient algorithms available for solving both problems, these algorithms require a matrix representation of A on both M and V and hence are of limited use.

Since we are mainly interested in studying the category RMod(F G) for a finite group G, we can try to choose an F G-module V that enables us to solve these problems more efficiently. For example, if H is a subgroup of G and V is the transitive F G-permutation module 1_{H}^{G} then an explicit description of a natural basis for End_{F G}(1_{H}^{G}), discovered by I. Schur, in terms of the permutation action of G on the cosets of H is known. More precisely, the Schur basis of End_{F G}(1_{H}^{G}) is in one-to-one correspondence with the double cosets of G with respect to H. We abbreviate the basis element of the Schur basis corresponding to HgH by a_{HgH}. Given an F G-module M we can define an action of End_{F G}(1_{H}^{G}) on M^{H}, the fixed points of H on M, since M^{H} ≅ Hom_{F G}(1_{H}^{G}, M) as F -vector spaces by Frobenius reciprocity. The advantage of this action is the explicit description of M^{H} as a subspace of M in contrast to the original action of End_{F G}(1_{H}^{G}) on Hom_{F G}(1_{H}^{G}, M). In [CR87] an explicit formula for the action of a Schur basis element a_{HgH} on the fixed space M^{H} is given. This formula requires however the explicit knowledge of a set of coset representatives for the cosets of H ∩ H^{g} in H.

If M is a transitive F G-permutation module 1_{U}^{G}, too, then even the determination of M^{H} can be done very efficiently. In Section 3, we will give an explicit formula for a certain multiple of a Schur basis element a_{HgH} in its action on (1_{U}^{G})^{H}, which will only involve the H-orbits on the cosets of U and avoids the explicit calculation of the coset representatives mentioned above.

Finally, we will restrict ourselves to the case that the characteristic of F is not dividing |H| or (what amounts to the same) that the F G-permutation module V = 1_{H}^{G} is a projective F G-module. Then V is isomorphic to e_{H}F G, where e_{H} is the idempotent \frac{1}{|H|} ∑_{h∈H} h ∈ F G, and End_{F G}(V) is isomorphic to the subalgebra e_{H}F Ge_{H}. Moreover, the functor Hom_{F G}(e_{H}F G, −) is naturally equivalent to the restriction functor res_{e_{H}F Ge_{H}}^{F G} from RMod(F G) to RMod(e_{H}F Ge_{H}). For an F -algebra A the properties of the restriction functor res_{eAe}^{A} from RMod(A) to RMod(eAe) with e an idempotent in A have been studied by J.A. Green in [Gre80]. An important fact that we are going to use is that the two module categories RMod(A) and RMod(eAe) are Morita equivalent if and only if AeA is equal to A. In the case of A being the group algebra F G and e = e_{H} as defined above we will give a character theoretic reformulation of this condition, which we can check algorithmically using the permutation characters of G and the Brauer character table of G. In Chapter 6 we will list suitable subgroups H for the sporadic simple groups G up to order 10^{9}, such that the principal block B of F G and e_{H}Be_{H} are Morita equivalent. In Chapter 4 we will determine the layers of the socle series of projective indecomposable modules in some interesting cases using the Morita equivalence.

In the following we will give descriptions of the various algorithms that implement the basic ideas. Up to now, there are two situations we can handle using the C-MeatAxe namely the case that M is a permutation module and the case that M is the tensor product of two given, smaller dimensional F Gmodules. In both situations, there is an efficient way of determining the fixed space M^{H} that avoids working explicitly in a matrix representation of F G on M. However, the algorithmic approach used to determine the fixed space can still depend on the method used for creating the permutation representation. We also have to address the question how we can prove that a set of elements in e_{H}F Ge_{H} forms a generating set of e_{H}F Ge_{H}. We show that this can be answered if we can verify a certain condition for the regular e_{H}F Ge_{H}-module e_{H}F Ge_{H}. This criterion was extended by M. Wiegelmann in [Wie94] and increased the number of cases we can investigate algorithmically, especially in characteristic 2.

3.1 Morita Equivalence

Definition 3.1.1. Let C and D be two categories. C and D are called equivalent, if there exist covariant functors

Φ: C → D and Ψ: D → C,

such that Ψ ∘ Φ and Φ ∘ Ψ are naturally equivalent to the identity functors id_{C} and id_{D} of C and D, respectively. In the case that A and B are F - algebras, C = RMod(A), and D = RMod(B), then A and B are called Morita equivalent algebras and we say that Φ and Ψ form a Morita equivalence.

The following lemma gives a list of properties preserved by a Morita equivalence.

Lemma 3.1.2. Let A and B Morita equivalent algebras. Then with the notation as above the following holds for an A-module V.

a) V is simple if and only if Φ(V) is simple.

b) V is indecomposable if and only if Φ(V) is indecomposable.

c) V is projective if and only if Φ(V) is projective.

d) V is a projective generator if and only if Φ(V) is a projective generator.

e) End_{A}(V) ≅ End_{B}(Φ(V)) as rings.

Furthermore Φ preserves (split) exact sequences of modules and thus the lattices of submodules M(V) and M(Φ(V)) are isomorphic. This implies that Φ(Soc(V)) = Soc(Φ(V)), Φ(Rad(V)) = Rad(Φ(V)) and hence Φ maps the socle and radical series of V to the socle and the radical series of Φ(V).

Proof: For a detailed proof, see [AF74], page 254. ■

Lemma 3.1.3. Let A and B be Morita equivalent algebras. Then the number of isomorphism classes of simple modules is the same for A and B.

Proof: The statement follows immediately from part a) of the preceding lemma. ■

The types of functors we can naturally use for defining a Morita equivalence are the following.

Definition 3.1.4. Let A and B be F -algebras, let M be an A-B bimodule and let N be a B-A bimodule. a) The tensor functor − ⊗_{A} M: RMod(A) → RMod(B) is defined by

V ↦ V ⊗_{A} M

for V ∈ RMod(A) and

α ↦ (V ⊗_{A} M → W ⊗_{A} M: v ⊗ m ↦ α(v) ⊗ m)

for V, W ∈ RMod(A) and α ∈ Hom_{A}(V, W), m ∈ M.

b) The Hom functor Hom_{A}(N, −): RMod(A) → RMod(B) is defined by

V ↦ Hom_{A}(N, V)

and

α ↦ (Hom_{A}(N, V) → Hom_{A}(N, W): λ ↦ αλ)

for V, W ∈ RMod(A) and α ∈ Hom_{A}(V, W), λ ∈ Hom_{A}(N, V).

Remark 3.1.5. In general, it seems to be a difficult task to determine Hom_{A}(N, V) and the B-module structure on Hom_{A}(N, V). However, there are special situations, which we will describe later, where there is an elegant and efficient way of solving these problems algorithmically.

We now can state a criterion that ensures that a certain pair of tensor functors realizes a Morita equivalence.

Theorem 3.1.6. Let A and B be F -algebras, let P be an A-B bimodule and let Q be a B-A bimodule. Then the following statement holds. If P ⊗_{B} Q ≅ A as A-A bimodules and Q ⊗_{A} P ≅ B as B-B bimodules, then the functors − ⊗_{A} P and − ⊗_{B} Q define a Morita equivalence between A and B.

Proof: We have to show that (− ⊗_{A} P)(− ⊗_{B} Q) is naturally equivalent to id_{RMod(B)} and that (− ⊗_{B} Q)(− ⊗_{A} P) is naturally equivalent to id_{RMod(A)}. Now, let V be a B-module. Then

(− ⊗_{A} P)(− ⊗_{B} Q)(V) = (− ⊗_{A} P)(V ⊗_{B} Q) = (V ⊗_{B} Q) ⊗_{A} P ≅
V ⊗_{B} (Q ⊗_{A} P) ≅ V ⊗_{B} B ≅ V

and since all isomorphisms are natural, it follows that (− ⊗_{A} P)(− ⊗_{B} Q) is naturally equivalent to id_{RMod(B)}. Analogously we get that (−⊗_{B} Q)(−⊗_{A}P) is naturally equivalent to id_{RMod(A)}. ■

Corollary 3.1.7. Let Q be a projective generator for RMod(A) and let P = Hom_{A}(Q, A). Then A is Morita equivalent to E = End_{A}(Q) and an equivalence is given by Φ = − ⊗_{A} P and Ψ = − ⊗_{E} Q.

Proof: Using the elementary properties of projective generators in a module category, see [AF74] page 195, it follows that E = Hom_{A}(Q, Q) = Q ⊗_{A} P as E-E bimodules and A ≅ Hom_{E}(Q, Q) = P ⊗_{E} Q as A-A bimodules. ■

In his studies on Morita equivalent algebras, Morita showed a converse to the theorem above.

Theorem 3.1.8. Let A and B be Morita equivalent algebras and let

Φ: RMod(A) → RMod(B) and Ψ: RMod(B) → RMod(A)

form a Morita equivalence between A and B. Then P = Φ(A) is an A-B bimodule, Q = Ψ(B) is a B-A bimodule, such that P ⊗_{B} Q ≅ A as A-A bimodules and Q ⊗_{A} P ≅ B as B-B bimodules. Furthermore, the following statements hold.

i) P is a projective generator for RMod(B) and LMod(A) and Q is a projective generator for RMod(A) and LMod(B).

ii) There are bimodule isomorphisms

Q ≅ Hom_{A}(P, A) ≅ Hom_{B}(P, B),
P ≅ Hom_{A}(Q, A) ≅ Hom_{B}(Q, B).

iii) There are ring isomorphisms

A ≅ Hom_{B}(P, P) ≅ Hom_{B}(Q, Q),
B ≅ Hom_{A}(P, P) ≅ Hom_{A}(Q, Q).

iv) The functors Φ′ = −⊗_{A}P and Ψ′ = −⊗_{B}Q define a Morita equivalence between RMod(A) and RMod(B).

v) The functors Φ̂ = Hom_{A}(Q, −) and Ψ̂ = Hom_{B}(P, −) define a Morita equivalence between RMod(A) and RMod(B).

Proof: The proof uses basic facts about projective generators in a module category and it also uses elementary properties of tensor functors and categorical equivalences. A detailed proof can be found in [AF74], page 262.

■

We add the following related statements about module categories, which deal with a more general situation than Morita equivalence. They will however not be needed in the sequel. The following theorem, proved by M. Cabanes in [Cab90] determines a subcategory of RMod(A) that is equivalent to RMod(End_{A}(V)) for a given A-module V, if End_{A}(V) is a quasi-Frobenius algebra.

Theorem 3.1.9. Let A be an F -algebra, let V be an A-module such that End_{A}(V) is a quasi-Frobenius algebra. Consider the full subcategory A(V) of RMod(A) whose objects are the A-modules that are isomorphic to a submodule of a direct sum of copies of V and that are also isomorphic to a quotient of a direct sum of copies of V. Then Hom_{A}(V, −) induces a categorical equivalence between A(V) and RMod(End_{A}(V)).

Remark 3.1.10. As M. Cabanes points out in [Cab90], A(V) is generally not a subcategory of RMod(A) in the sense of abelian categories. More precisely, if M and N are in A(V) and N is an A-submodule of M, then the quotient M/N does not necessarily belong to A(V).

The following result goes back to Fitting. It studies the subcategory of RMod(A) that is equivalent to the category of projective End_{A}(V)-modules for a given A-module V.

Theorem 3.1.11. Let A be an F -algebra, and let Add_{A}(V) be the full subcategory of RMod(A), whose objects are the A-modules isomorphic to direct summands of direct sums of copies of V. Then Hom_{A}(V, −) restricted to Add_{A}(V) induces an equivalence of Add_{A}(V) and the category of projective End_{A}(V)-modules.

Proof: See [AF74], page 324. ■

3.2 Condensation

By Corollary 3.1.7 we know that A and End_{A}(V) are Morita equivalent in the case that V is a projective generator for RMod(A). We will restrict our attention to the situation where V is isomorphic to eA for a suitable idempotent e ∈ A and hence End_{A}(V) is isomorphic to the subalgebra eAe of A. The reason for studying this restricted situation lies in the fact, that finding nontrivial idempotents in A is easier to solve algorithmically than determining explicitly a nontrivial projective generator for RMod(A).

Definition 3.2.1. Let A be an F -algebra and let e ∈ A be an idempotent. We call the subalgebra eAe of A the condensation algebra corresponding to e and define the restriction functor res_{eAe}^{A}: RMod(A) → RMod(eAe) by

V ↦ V e

for V ∈ RMod(A) and for α ∈ Hom_{A}(V, W)

α ↦ α|_{V e} ∈ Hom_{eAe}(V e, W e).

Remark 3.2.2. If A is an F -algebra and e is an idempotent, then eA ⊗_{A} Ae ≅ eAe as eAe-eAe bimodules and hence the composition of the two tensor functors (− ⊗_{A} Ae) and (− ⊗_{eAe} eA) is naturally equivalent to id_{RMod(eAe)}. Moreover, the functor −⊗_{A}Ae and the restriction functor res_{eAe}^{A} are naturally equivalent and we can therefore replace the functor − ⊗_{A} Ae by res_{eAe}^{A}.

The following theorem characterizes under which conditions the two tensor functors (− ⊗_{eAe} eA) and (− ⊗_{A} Ae) form a Morita equivalence for A and eAe. Corollary 3.2.5 below shows that in this situation the functor −⊗_{eAe} eA when applied to the eAe-module V e (V an A-module) can be described intrinsically in V as a generalized induction process.

Theorem 3.2.3. Let e ∈ A be an idempotent, let e_{1}A, …, e_{r}A be representatives for the isomorphism classes of projective indecomposable A-modules, and let S_{1}, …, S_{r} be representatives of the isomorphism classes of simple Amodules, such that e_{i}A/ Rad(e_{i}A) ≅ S_{i}. Then the following statements are equivalent.

a) The functors (− ⊗_{eAe} eA) and (− ⊗_{A} Ae) form a Morita equivalence for A and eAe.

b) AeA is equal to A.

c) For all 1 ≤ i ≤ r, the A-module e_{i}A is a summand of eA.

d) S_{i}e ≠ {0} for all 1 ≤ i ≤ r.

Proof: a)⇒ d): If S is a simple A-module then it follows from Lemma 3.1.2 that Se ≅ S ⊗_{A} Ae is nonzero.

d)⇒ c): Since we have {0} ≠ S_{i}e ≅_{F} Hom_{A}(eA, S_{i}) it follows that e_{i}A must be a summand of eA.

c)⇒ b): Let e′ be a primitive idempotent in A, then eA ≅ e_{i}A for some i = 1, …, r. Hence there exist a, a′ ∈ A with e′ = ae_{i}a′ and so e′ is contained in AeA, and thus AeA = A.

b)⇒ c): e_{i}A is a summand of eA if and only if Hom_{A}(eA, S_{i}) ≅ S_{i}e is nonzero. Since

(S_{i}e)A = (S_{i}A)eA = S_{i}A = S_{i},

we get that S_{i}e is nonzero.

c)⇒ a): Since e_{i}A is a summand of eA for all i = 1, …, r it follows that eA is a projective generator for RMod(A) and Corollary 3.1.7 implies that ⊗_{A}Ae and − ⊗_{eAe} eA form a Morita equivalence between A and eAe. ■

Definition 3.2.4. Let A be an F -algebra and let e be an idempotent in A. If e fulfills one of the conditions of the previous theorem, then e is called a faithful idempotent of A.

Corollary 3.2.5. Let e be a faithful idempotent and let V be an A-module. Then V e ⊗_{eAe} eA ≅_{A} V = V e · A.

Proof: The natural equivalence of the functors id_{RMod(A)} and (− ⊗_{eAe} eA)(− ⊗_{A} Ae) implies

V = id_{RMod(A)}(V) ≅ (−⊗_{eAe}eA)(−⊗_{A}Ae)(V) ≅ (−⊗_{eAe}eA)(V e) = V e⊗_{eAe}eA.

Furthermore, if e is faithful, then V = V · A = V · AeA = V e · A. ■

Definition 3.2.6. An F -algebra A is called a basic algebra if A/ Rad(A) is a direct sum of skew fields.

Remark 3.2.7. Let e_{1}, …, e_{r} be a system of representatives for the primitive idempotents of an algebra A and let e be the idempotent e_{1} + … + e_{r}. From the previous theorem it follows that A is Morita equivalent to eAe. Moreover, since e = e_{1} + … + e_{r} is a decomposition of e into primitive orthogonal idempotents in eAe, it follows that eAe/ Rad(eAe) is direct sum of sqew fields. If an algebra A is a basic algebra and all simple A-modules are absolutely irreducible, then all simple A-modules are one-dimensional.

Proposition 3.2.8. If A and B are basic algebras, then A and B are Morita equivalent if and only if they are isomorphic.

Proof: See [Erd90], Lemma I.2.6.5. ■

Remark 3.2.9. In the representation theory of finite-dimensional algebras it is usually assumed that the algebra one studies is already a basic algebra, but in general we do not have a good strategy for finding nontrivial idempotents e in a given algebra A such that the algebra eAe is Morita equivalent to A. However, we will see in the applications that there are methods which we can use in the case of group algebras F G. These enable us to find idempotents e such that eF Ge is Morita equivalent to F G. Moreover, in the examples we will consider the algebra eF Ge will have a much smaller dimension than F G. As mentioned above, a general consequence of the Morita equivalence between A and eAe is that the lattice of submodules of an A-module M is the same as the one of Me. This turns out to be quite useful for practical purposes. Observe that the dimension of the A-module M is generally much larger than the dimension of Me. In the case that eAe is a basic algebra and the underlying field F is a splitting field for all simple A-modules, then the F -dimension of Me is just given by the number of composition factors of the A-module M.

Definition 3.2.10. Let e be an idempotent in A. Then the restriction functor res_{eAe}^{A} from RMod(A) to RMod(eAe) is called a condensation functor. If M is an A-module, then the eAe-module Me is called the condensed module and for an eAe-submodule N of Me we call NA, the A-module generated N, the uncondensed module.

Remark 3.2.11. If e is not a faithful idempotent, then eA is not a projective generator, and RMod(A) and RMod(eAe) are not Morita equivalent. But since the restriction functor res_{eAe}^{A} is still an exact functor, it preserves composition series of A-modules. This property has been successfully applied to determine the decomposition numbers for sporadic simple groups, see for example [Jan95] or [LP91]. A modified version of Corollary 3.2.5 still holds, namely for a given A-module V it follows that V e · A is an epimorphic image of V e ⊗_{eAe} eA.

3.3 Group Algebras and Permutation Modules

We focus on the situation where A is a group algebra or more generally a block of a group algebra. We will review an observation of I. Schur that gives a description of a ℤ-basis for the integral endomorphisms of a transitive permutation module 1_{H}^{G}. We also describe the action of the integral endomorphisms of the permutation module 1_{H}^{G} on the fixed space of H in a a given ℤG-module. The definition of this action goes back to J.A. Green, see [Gre78].

Lemma 3.3.1. If H is a subgroup of G, then (∑_{h∈H} h)ℤG = 1_{H}^{G}, is the (integral) permutation module of G on the cosets of H. A ℤ-basis for the integral endomorphism ring End_{ℤG}((∑_{h∈H} h)ℤG) can be described as follows. For each double coset HgH in G , let a_{HgH} be the unique ℤG-endomorphism that maps

(∑_{h∈H} h)y to ( ∑_{x∈HgH} x)y = ∑_{h∈H,k∈H/H∩H^{g}} hgky for all y ∈ G.

The so defined basis of End_{ℤG}((∑_{h∈H} h)ℤG) is called the Schur basis. Let us denote by e_{H} the idempotent \frac{1}{|H|}(∑_{h∈H} h). We define the ℚ-algebra homomorphism μ: e_{H}ℚGe_{H} → End_{ℚG}((∑_{h∈H} h)ℤG) by

μ(e_{H}ge_{H})((∑_{h∈H} h)y) = (e_{H}ge_{H})((∑_{h∈H} h)y) = \frac{1}{|H|} ∑_{h∈H,k∈H}(hgk)y

for all g ∈ ℚG and all y ∈ ℤG. This homomorphism is obviously injective and furthermore

μ(e_{H}ge_{H}) = \frac{|H ∩ H^{g}|}{|H|}a_{HgH}.

Hence μ is an isomorphism from e_{H}ℚGe_{H} onto End_{ℚG}((∑_{h∈H} h)ℤG)).

Lemma 3.3.2. Let H be a subgroup of G and let M be a ℤG-module. Then the following statements hold.

a) Hom_{ℤG}((∑_{h∈H} h)ℤG, M) is a (right) End_{ℤG}((∑_{h∈H} h)ℤG))-module. Furthermore, there exists an isomorphism κ: Hom_{ℤG}((∑_{h∈H} h)ℤG, M) → M^{H}. Hence, we can define a right action of End_{ℤG}((∑_{h∈H} h)ℤG) on M^{H} derived from the natural action on Hom_{ℤG}((∑_{h∈H} h)ℤG, M).

b) For a Schur basis element a_{HgH} of End_{ℤG}((∑_{h∈H} h)ℤG) the action on M^{H} is given as follows.

ma_{HgH} = ∑_{h∈H/H∩H^{g}} mgh for all m ∈ M^{H}.

c) ℚ ⊗ M^{H} is a e_{H}ℚGe_{H}-module and we get that me_{H}ge_{H} = mμ(e_{H}ge_{H}) for all m ∈ ℚ ⊗ M^{H} and all g ∈ G, i.e., the isomorphism μ defined in the previous lemma is compatible with the actions of e_{H}ℚGe_{H} and End_{ℚG}((∑_{h∈H} h)ℤG) on ℚ ⊗ M^{H}.

Proof: a): Hom_{ℤG}((∑_{h∈H} h)ℤG, M) is obviously a (right) End_{ℤG}((∑_{h∈H} h)ℤG) module given by the composition of homomorphisms. The second statement is an explicit form of Frobenius reciprocity. We claim that the map κ from Hom_{ℤG}((∑_{h∈H} h)ℤG, M) to M^{H} defined by κ(ϕ) = ϕ(∑_{h∈H} h) for all ϕ ∈ Hom_{ℤG}((∑_{h∈H} hℤG, M) is an isomorphism of abelian groups. κ is obviously a homomorphism of abelian groups and κ(ϕ) ∈ M^{H}, if ϕ is a ℤG-homomorphism. Moreover, κ is injective, since if κ(ϕ) = 0 it follows that ϕ(∑_{h∈h} h) = 0 and hence ϕ((∑_{h∈H} h)ℤG) = {0}, i.e., ϕ is the zero homomorphism. Conversely, given an element m ∈ M^{H} we can define a ℤGhomomorphism ϕ_{m} with κ(ϕ_{m}) = ϕ_{m}(∑_{h∈H} h) = m and ϕ_{m}((∑_{h∈H} h)x) = mx for all x ∈ ℚG. The action of a given α ∈ End_{ℚG}((∑_{h∈H} h)ℤG) on M^{H} is mα = (ϕ_{m}α)(∑_{h∈H} h) for all m ∈ M^{H}.

b): By part a) we get

ma_{HgH} = (ϕ_{m}a_{HgH})(∑_{h∈H} h) = ϕ_{m}( ∑_{h∈H,k∈H/(H∩H^{g})} hgk) =

∑_{k∈H/(H∩H^{g})} ϕ_{m}(∑_{h∈H} h)gk = ∑_{k∈H/(H∩H^{g})} mgk.

c): Since ℚ ⊗ M^{H} = ℚ ⊗ Me_{H} it follows that ℚ ⊗ M^{H} is an e_{H}ℚGe_{H}- module. Moreover,

me_{H}ge_{H} = mge_{H} = ∑_{k∈H/(H∩H^{g})} \frac{|H ∩ H^{g}|}{|H|}mgk =

m\frac{|H ∩ H^{g}|}{|H|}a_{HgH} = mμ(e_{H}ge_{H}).

■

Knowing the intersection H ∩ H^{g} and a set of coset representatives for the cosets of H ∩H^{g} in H we could explicitly determine the action of a Schur basis element a_{HgH}. Since the endomorphism ring of the F G-permutation module 1_{H}^{G} over any finite field F of characteristic p is the reduction modulo p of the integral endomorphism ring End_{ℤG}(1_{H}^{G}), see [Gra85], the reduction mod p of the Schur basis gives us a basis for End_{F G}(1_{H}^{G}).

It is however not an easy matter to determine the intersection H ∩ H^{g} and the coset representatives. We will see that if the characteristic of F is not dividing the order of H, then we can avoid to calculate the intersections. This is based on the following observation.

Lemma 3.3.3. Let H be a subgroup of G, let e_{H} = \frac{1}{|H|}(∑_{h∈H} h) and let Ω be a G-set with H-orbits O_{1}, …, O_{r}. Let M be the corresponding ℤGpermutation module with permutation basis b_{k}, k ∈ Ω. Then a ℤ-basis for the fixed space of H on M is given by \overline{O_{1}} = ∑_{k∈O_{1}} b_{k}, …, \overline{O_{r}} = ∑_{k∈O_{r}} b_{k}. Moreover, the action of e_{H}ge_{H} for g ∈ G on ℚ ⊗ M^{H} with respect to this basis is given by

\overline{O_{i}}e_{H}ge_{H} = ∑_{j=1}^{r} c_{ij} \frac{1}{|O_{j}|}\overline{O_{j}}

where c_{ij} = |{k ∈ O_{i} | k^{g} ∈ O_{j}}|.

Proof: \overline{O_{1}}, …, \overline{O_{r}} are obviously fixed by all elements h ∈ H. Conversely, if m = ∑_{k∈Ω} z_{k}b_{k}, z_{k} ∈ ℤ, is a fixed vector, then z_{k_{1}} = z_{k_{2}} for all k_{1}, k_{2} ∈ O_{i}. Note further that if b_{k} is an element of the permutation basis such that k ∈ O_{j}, then the product

b_{k}e_{H} = \frac{1}{|H|} ∑_{h∈H} b_{k}h = \frac{1}{|O_{j}|}\overline{O_{j}}.

Hence

\overline{O_{i}}e_{H}ge_{H} = \overline{O_{i}}ge_{H} = (∑_{k∈O_{i}} b_{k})ge_{H} =

∑_{k∈O_{i}}(b_{k}g)e_{H} = ∑_{k∈O_{i}} b_{k^{g}}e_{H} =

= ∑_{j=1}^{r} c_{ij} \frac{1}{|O_{j}|}\overline{O_{j}}.

■

Remark 3.3.4. The lemma above shows that idempotents of the form e_{H} are particularly easy to work with. Moreover, the action of an element e_{H}ge_{H} on ℚ ⊗ Me_{H} can be determined without knowing \frac{|H|}{|H∩H^{g}|}. Let ℤ_{p} denote the p-local integers \{\frac{a}{b}|a, b ∈ ℤ, p ∤ b\} and let |H|_{p}, |H|_{p′} be the p-part and the p′-part of the order of H, then |H|_{p}ege is an element in End_{ℤ_{p}G}(1_{H}^{G}) and \overline{|H|_{p}e_{H}ge_{H}}, the reduction mod p, gives an element in End_{F G}(1_{H}^{G}). So the above formula gives us a way of producing elements in End_{F G}(1_{H}^{G}). However, if we want to know the action of the full endomorphism ring, i.e., the action of the Schur basis elements \overline{a_{HgH}}, reduced mod p, on M^{H}, we have to be able to determine \frac{|H|}{|H∩H^{g}|} for a given g ∈ G.

The next lemma characterizes the part of End_{F G}(1_{H}^{G}) the action of which we can compute using the formula above.

Lemma 3.3.5. Let H be a subgroup of G and let F be a field of prime characteristic p. Then the following statements hold.

i) The Schur basis elements a_{HgH} reduced mod p such that |H ∩ H^{g}| is coprime to p are an F -basis for the image of the trace map Tr from End_{F} (1_{H}^{G}) to End_{F G}(1_{H}^{G}), i.e., for the two-sided ideal of projective endomorphisms of 1_{H}^{G}.

ii) The subalgebra ⟨\overline{|H|_{p}e_{H}ge_{H}}|g ∈ G⟩ of End_{F G}(1_{H}^{G}) is equal to the twosided ideal of projective endomorphisms of 1_{H}^{G}. If |H| is coprime to p, then eF Ge is the full F G-endomorphism ring End_{F G}(1_{H}^{G}).

Proof: Let r be the index of H in G and let us choose coset representatives g_{1}, …, g_{r} for the cosets of H in G. Furthermore, let us denote by f = ∑_{h∈H} h. Then fg_{1}, …, fg_{r} is a basis of the F G-permutation module 1_{H}^{G} and the F -linear maps δ_{i,j}, i, j = 1, …, r, defined by δ_{i,j}(fg_{k}) = 0 if k ≠ i and δ_{i,j}(fg_{i}) = fg_{j} form an F -basis of the F -endomorphism ring of 1_{H}^{G}. To calculate the image of the trace map Tr, we now apply Tr to the chosen basis δ_{i,j}, i, j = 1, …, r, and evaluate Tr(δ_{i,j}) on f, which gives

Tr(δ_{i,j})(f) = ∑_{g∈G} δ_{i,j}(fg)g^{−1}

= ∑_{g∈Hg_{i}} fg_{j}g^{−1} = ∑_{k∈H} fg_{j}g_{i}^{−1}k

= |H ∩ H^{g_{j}g_{i}^{−1}}| \overline{a_{Hg_{j}g_{i}^{−1}H}}(f).

Since an F G-endomorphism ϕ is uniquely determined by ϕ(f) it follows that

Tr(δ_{i,j}) = |H ∩ H^{g_{j}g_{i}^{−1}}| \overline{a_{Hg_{j}g_{i}^{−1}H}}.

Statement ii) follows from

|H|e_{H}ge_{H} = |H|_{p}|H|_{p′}e_{H}ge_{H} = |H ∩ H^{g}|a_{HgH}.

Hence, \overline{a_{HgH}} is a nontrivial projective endomorphism, if and only if \overline{|H|_{p}e_{H}ge_{H}} is a nontrivial projective endomorphism. ■

Remark 3.3.6. The implementation of the formula above in the C-MeatAxe is split into two parts. The program zmo determines the orbits of the subgroup H on the G-set underlying the permutation representation. The program zkd uses the orbit sizes |O_{j}| and the permutation for a given g ∈ G to determine the coefficients c_{ij} weighted by \frac{|H|_{p}}{|O_{j}|} reduced mod p.

The above lemma indicates why condensation of permutation modules was the starting point of condensation. In [Tha81], Thackray called this special form of condensing a module the fixed point reduction of the module and used it for analyzing permutation modules of some sporadic simple groups. It is obvious that the determination of the fixed space M^{H} for a given F Gmodule M can be done by just looking at the vectors fixed by a generating set of H.

For the idempotent of type e_{H} for a subgroup H of order coprime to the characteristic of F we can give a character theoretic condition, which ensures that e_{H}F Ge_{H} is Morita equivalent to F G.

Lemma 3.3.7. Let IBr(G) be the irreducible Brauer characters of G. Then the tensor functors − ⊗_{e_{H}F Ge_{H}} e_{H}F G and − ⊗_{F G} F Ge_{H} form a Morita equivalence between RMod(F G) and RMod(e_{H}F Ge_{H}) if and only if for all ϕ ∈ IBr(G) the character theoretic scalar product of the restriction of ϕ ∈ IBr(G) to H with the trivial character 1_{H} is nonzero .

Proof: From Theorem 3.2.3 we get that the two functors form a Morita equivalence between e_{H}F Ge_{H} and F G if and only if Se_{H} is nonzero for all simple F G-modules S. Furthermore, since Se_{H} = S^{H}, this just means that the restriction of S to H contains the trivial F H-module. Finally, note that the multiplicity of the trivial F H-module in the restriction of S to H is given by the character theoretic scalar product of the trivial character 1_{H} with the restriction of the Brauer character ϕ of S. ■

Remark 3.3.8. If B is a block of F G, then the tensor functors − ⊗_{e_{H}F Ge_{H}} e_{H}F G and −⊗_{F G} F Ge_{H} form a Morita equivalence between B and the block e_{H}Be_{H} of e_{H}F Ge_{H} if the condition of the previous lemma is satisfied for all irreducible Brauer characters belonging to the block B.

We now study the condensation algebra e_{H}F Ge_{H} for e_{H} = \frac{1}{|H|} ∑_{h∈H} h in further detail. The following theorem, whose proof was given by M. Wiegelmann in his diploma thesis [Wie94], gives an important criterion which ensures that a set of elements in e_{H}F Ge_{H} generates the algebra e_{H}F Ge_{H}.

Theorem 3.3.9. Let N ≤ G, H ⊴ N and suppose that N/H is a p-group generated by {n_{1}H … n_{l}H}, l ∈ ℕ. Let f = ∑_{n∈N} n ∈ F G and denote by V = fF G the permutation module of G on the cosets of N. Given elements g_{1}, …, g_{k}, k ∈ ℕ, consider the subalgebra B of e_{H}F Ge_{H} generated by e_{H}g_{1}e_{H}, …, e_{H}g_{k}e_{H}. If fB = V e_{H} then

{e_{H}n_{1}e_{H}, …, e_{H}n_{l}e_{H}, e_{H}g_{1}e_{H}, …, e_{H}g_{k}e_{H}}

is a generating set for e_{H}F Ge_{H}.

Proof: The collection of double cosets {NgH | g ∈ G} of N and H in G corresponds to the natural F -basis {fge_{H} | g ∈ G} of V e_{H} = fF Ge_{H}. Let us first fix one such double coset D = NgH and the corresponding element fge_{H} ∈ V e_{H}. The condition fB = V e_{H} implies that there exists an element b_{D} ∈ B such that fb_{D} = fge_{H}.

Let R be a set of double coset representatives for H in G. Then {e_{H}re_{H} | r ∈ R} forms an F -basis for e_{H}F Ge_{H} and we can write b_{D} uniquely as b_{D} = ∑_{r∈R} k_{r}e_{H}re_{H} with k_{r} ∈ F. By the choice of b_{D} the coefficients of the above sum satisfy the following conditions: ∑ k_{r} = 1 if the sum is taken over all r ∈ R with HrH ⊆ D, and ∑ k_{r} = 0 if the sum is taken over all r ∈ R with HrH ⊆ D′, where D′ is a double coset for N and H in G which is different from D.

Now consider e_{H}F Ge_{H} as a (left) permutation module for the algebra e_{H}F Ne_{H} ≅ F (N/H). Since N/H is a p-group, the radical of e_{H}F Ge_{H} as an F (N/H)-module is given by

{∑_{r∈R} l_{r}e_{H}re_{H} | ∑_{r:HrH∈NgH} l_{r} = 0 , ∀g ∈ G}.

It follows, that the radical together with all the elements b_{D} ∈ B (D runs over all double cosets of N and H in G) spans e_{H}F Ge_{H} as an e_{H}F Ne_{H}-module. By Nakayama’s lemma, see [CR81], the radical can be omitted from the generating set and we conclude that the b_{D} generate e_{H}F Ge_{H} considered as an e_{H}F Ne_{H}-module. Since e_{H}n_{1}e_{H},e_{H}n_{2}e_{H}, …, e_{H}n_{l}e_{H} generate the algebra e_{H}F Ne_{H} we are done. ■

Remark 3.3.10. In most of the cases we will consider in Chapter 4, we will apply the theorem with N = H. We will have to construct the F Gpermutation module 1_{H}^{G} on the cosets of H in G and then we will have to verify that the vector ∑_{h∈H} h spans the whole space (1_{H}^{G})e_{H} with respect to the elements e_{H}g_{1}e_{H}, …, e_{H}g_{k}e_{H} we have chosen. This can be done explicitly by using Algorithm 2.1.18.

For computational purposes there are obvious advantages of using the idempotents of the form e_{H}. First of all, a good knowledge of the subgroup structure of G enables us to search systematically for faithful idempotents of the form e_{H}. Secondly, in order to apply the condensation functor to a given F G-permutation module M we do not have to consider the whole module M and then multiply by e_{H}. Instead, we just have to determine the H-orbits on the underlying G-set. In principle we could use idempotents of other types but they more or less all lack the above mentioned properties of e_{H}, i.e., they are either difficult to find or we have to consider the whole module in order to be able to evaluate the condensation functor.

In the next chapter we will illustrate the power of using idempotents of the form e_{H} by constructing a Morita equivalence for blocks of several simple groups.

3.4 Peakword Condensation

In this section we will exploit the functorial relationship between RMod(A) and RMod(eAe) in a new direction. Instead of aiming at a Morita equivalence we will choose e to be a primitive idempotent. Then there will be only one simple eAe-module, eAe is a local algebra and we will see that the module category RMod(eAe) controls the so-called S-radical A-modules. This approach is taken by Müller, Ringe, and Lux, see [LMR94], for determining submodule lattices and also by Lux and Wiegelmann, see [LW96a], for determining the socle series and the radical series of a given A-module. In a forthcoming paper by M. Szőke and the author it will also exploited to devise a new algorithm for determining the homomorphisms between A-modules.

All of the approaches are based on special words in A called peakwords, which were first introduced in [LMR94]. These words enable us to find primitive idempotents in A.

So, let S be a simple A-module and let V be an A-module. For a ∈ A we denote by Kernel_{V} (a) the kernel of the F -endomorphism induced on V by a. We start with giving a short overview of the notation and main results from [LMR94].

Definition 3.4.1. An A-module V is called S-local if V/ Rad(V) ≅ S, and an idempotent e ∈ A is called S-primitive if the module eA is S-local. Furthermore, an element a_{S} ∈ A is called an S-peakword with respect to V, if the following conditions are fulfilled.

a) Kernel_{T} (a_{S}) = {0} for all composition factors T of V not isomorphic to S.

b) dim_{F} (Kernel_{S}(a_{S}^{2})) = [End_{A}(S) : F].

The following lemma used in [LMR94] follows easily from the adjointness of the restriction functor res_{eAe}^{A} and the tensor functor − ⊗_{eAe} eA.

Lemma 3.4.2. Let e ∈ A and let W be an eAe-module. Then the following holds. If S is a constituent of W̄ = W ⊗_{eAe} eA/ Rad(W ⊗_{eAe} eA) then Se is nonzero. Hence, if e is an S-primitive idempotent, then W̄ has S as its only constituent.

Proof: Let T be a simple A-module. Then T e ≅_{eAe} Hom_{A}(eA, T) and the assertions follow immediately from

Hom_{eAe}(W, T e) ≅ Hom_{A}(W ⊗_{eAe} eA, T).
■

Definition 3.4.3. Let S be a simple A-module and let M(V) be the lattice of submodules of the A-module V. We denote by L_{S}(V) the set of S-local A-submodules W and we denote by M_{S}(V) the subset of A-submodules W, such that W/ Rad(W) is isomorphic to a direct sum of copies of S. An element of M_{S}(V) is called an S-radical module.

The following theorem shows that M_{S}(V) is controlled locally.

Theorem 3.4.4. Let e ∈ A be an S-primitive idempotent. Then the mapping

κ: M_{S}(V) → M(V e): W ↦ W e

is an isomorphism of lattices. Its inverse is given by

κ^{−1}: M(V e) → M_{S}(V): W̄ ↦ W̄ A.

Proof: See [LMR94]. ■ Moreover, we also get the following description of L_{S}(V).

Theorem 3.4.5. Let e ∈ A be an S-primitive idempotent. Then the following statements hold.

a) The set of local eAe-submodules of V e is the set of cyclic eAe-submodules of V e, i.e., the set of submodules generated by some nonzero vector v ∈ V e.

b) L_{S}(V) is the set of A-submodules which are of the form vA for some nonzero v ∈ V e.

Proof: a): Since e is a primitive idempotent, it follows that eAe is a local algebra. So the local eAe-modules are the quotients of the regular eAemodule eAe_{eAe}, which, on the other hand, are exactly the cyclic eAe-modules.

b): Let W be an S-local submodule of V and take a vector v ∈ W e \ Rad(W)e. Suppose that vA ≤ Rad(W). Then v ∈ vAe ≤ Rad(W)e, a contradiction. Conversely, for a given vector v ∈ V e it holds that vA is an epimorphic image of the S-local A-module veAe ⊗_{eAe} eA and hence V e is S-local. ■

Remark 3.4.6. In [LMR94] it is shown that the set L_{S}(V) plays an important role in determining the complete lattice of submodules of a given A-module V. A theorem of D. Benson and J.H. Conway, see [BC85], states that the knowledge of L_{S}(V) for all constituents S of V, the inclusion relation between local submodules and a further relation in L_{S}(V) are sufficient to construct M(V).

We now address to the problem of finding idempotents and especially S-primitive idempotents.

Theorem 3.4.7. Let V be a faithful A-module and let a ∈ A. Then there exists a uniquely determined idempotent e ∈ A that induces the Fitting decomposition of a on V, that means, if V = Kernel_{V} (a^{r}) ⊕ Image_{V} (a^{r}) is the Fitting decomposition of V with respect to the endomorphism induced by a, then Kernel_{V} (a^{r}) = V e and Image_{V} (a^{r}) = V (1 − e). If a is an S-peakword for the composition factor S of V then e is an S-primitive idempotent with respect to V.

Proof: Let m_{K} ∈ F [X] be the minimal polynomial of a on Kernel_{V} (a^{r}) and let m_{I} ∈ F [X] be the minimal polynomial of a on Image_{V} (a^{r}). Then m_{K} = X^{r} and there are polynomials p, q ∈ F [X] with 1 = p ∗ m_{I} + qm_{K} ∈ F [X]. Then e := p(a)m_{I}(a) ∈ A fulfills all the requirements. Moreover, if S is composition factor of V, we can write S = Kernel_{S}(a^{s}) ⊕ Image_{S}(a^{s}). We claim the Se = Kernel_{S}(a^{s}). Let m_{k} ∈ F [X] be the minimal polynomial of a on Kernel_{S}(a^{s}) and let m_{i} ∈ F [X] be the minimal polynomial of a on Image_{V} (a^{s}). Then m_{k} = X^{s} and since S is a composition factor of V, it follows that m_{k} divides m_{K} and m_{i} divides m_{I}. Hence

1 = p ∗ \frac{m_{I}}{m_{i}} + q \frac{m_{K}}{m_{k}} ∈ F [X].

Moreover, Kernel_{S}(a^{s}) = S(p(a)\frac{m_{I}}{m_{i}} (a)m_{i}(a)) = S(p(a)m_{I}(a)) = Se. ■

This leads to the following straightforward algorithm that given b ∈ A determines the action of ebe on Kernel_{V} (a^{r}) = V e.

Algorithm 3.4.8 (Direct condensation).

INPUT: A matrix representation of A on an A-module V, an element a ∈ A acting singularly on V, an element b ∈ A. CALCULATION:

•Determine the Fitting decomposition, i.e., bases B and C for Kernel_{V} (a^{r}) and Image_{V} (a^{r}).

•Determine the matrix of ebe on the stable nullspace Kernel_{V} (a^{r}) with respect to the basis B.

OUTPUT: The matrix of ebe in its action on V e with respect to B, where e ∈ A is the idempotent we get from the lemma above applied to a.

Remark 3.4.9. This algorithm has been implemented as part of the C-MeatAxe using Algorithm 2.1.21. Furthermore, the C-MeatAxe program pwkond runs systematically through the words of A and checks whether a word a is a peakword for a simple A-module S with respect to a given Amodule V. Following Holt and Rees, see [HR94], it first determines the irreducible factors of the characteristic polynomial of a in S. For each factor f it checks whether the nullspace of f(a)^{2} in S is minimal and finally it checks that f(a) is invertible for all the simple constituents T not isomorphic to S.

We now describe a new algorithm for determining the socle series of a given A-module, which is also based on the use of peakwords. Let X(a) be the matrix of a ∈ A with respect to the standard basis for an arbitrarily chosen vector v ∈ Kernel_{S}(a_{S}). Furthermore, from the output of Algorithm 2.1.19 we can determine expressions w_{1}, w_{2}, …, w_{dim_{F} (S)} in A that define the standard basis for S derived from v recursively. We define the sequence SB(w, a_{S})_{1}, …, SB(w, a_{S})_{dim_{F} (S)} for a given vector w ∈ V by applying the recursive definitions to w instead of v. We can use Theorem 3.4.5 to characterize the vectors in Kernel_{V} (a_{S}) that generate simple submodules isomorphic to S.

Theorem 3.4.10. For w ∈ Kernel_{V} (a_{S}) the following statements are equivalent.

i) w generates a simple submodule isomorphic to S.

ii) The following equations hold for 1 ≤ i ≤ dim_{F} (S) and all elements a ∈ A:

SB(w, a_{S})_{i}a = ∑_{j=1}^{dim_{F} (S)} X_{ij}(a)SB(w, a_{S})_{j}.

Proof: i) implies ii) trivially, see Remark 2.2.11.

On the other hand, since w is in Kernel_{V} (a_{S}) it generates an S-local submodule by Theorem 3.4.5. Moreover, this submodule is contained in the A-submodule ⟨SB(w, a_{S})_{1} = w, …, SB(w, a_{S})_{dim_{F} (S)}⟩_{F}. This submodule is of dimension at most dim_{F} (S). Since w ≠ 0 this implies i). ■

Corollary 3.4.11. Let L_{S} be the subspace of vectors in Kernel_{V} (a_{S}) satisfying part ii) of Theorem 3.4.10. Then the S-homogeneous part of the socle of V is the A-submodule generated by L_{S}.

Proof: From Theorem 3.4.10 it follows that the A-submodule generated by L_{S} is a semisimple submodule which is a direct sum of copies of the simple module S. On the other hand for a given simple submodule T of M which is isomorphic to S there is a vector v ∈ Kernel_{T} (a_{S}) generating T and satisfying the condition ii) of Theorem 3.4.10. Hence this vector is contained in L_{S}. This implies that T = vA is contained in the A-submodule generated by L_{S}. ■

We now can state the resulting algorithm for calculating the socle of an A-module V.

Algorithm 3.4.12 (socle).

INPUT:

•A matrix representation δ of A given on a generating system E of A.

•For each composition factor S of V a peakword a_{S} for S and the words produced by the spinning algorithm for S with respect to a nonzero vector in Kernel_{S}(a_{S}). The matrices for the generating system E on S with respect to the standard basis for S.

CALCULATION:

•Set socbasis to the empty sequence;

•For all (up to isomorphism) composition factors S of V do

–Set locbas to the empty sequence;

–Determine a basis B_{S} of L_{S};

–For all vectors v in B_{S} do

∗If v is not in the F -span of locbas ·Append SB(v, a_{S}) to locbas;

∗fi;

od;

–Append locbas to socbas;

od;

OUTPUT:

•The basis socbas for the socle of V.

Determining the basis B_{S} of L_{S} is done by solving the system of linear equations we get from Theorem 3.4.10. Note that the number of unknowns of this system is the dimension of Kernel_{V} (a_{S}), which can be at most dim_{F} (End_{A}(S)) times the multiplicity of S as a composition factor of V. Note further that we only have to process the equations we get from a generating set for A. The algorithm as stated above can easily be extended to calculate the whole socle series of V by applying the algorithm recursively and keeping track of the base changes involved. It has been implemented in form of the C-MeatAxe program socleser by M. Wiegelmann and by M. Szőke, see [Wie94].

3.5 An Extension of the MeatAxe

In Chapter 2 we mentioned that for a uniserial A-module V of length 2 whose two composition factors are isomorphic the Norton criterion might fail to produce the nontrivial A-invariant subspace in a reasonable amount of time. We want to apply condensation in order to solve this problem. More precisely, we want to study the condensed A-module V e using an idempotent e belonging to a word a ∈ A. We may assume without loss of generality that the A-module V is a faithful A-module. Hence A is a local algebra and it follows from Theorem 3.2.3 that A and eAe are Morita equivalent. This implies that having found an eAe-invariant subspace W of V e the corresponding A-submodule W A will be the proper A-submodule of V. But, on a first glance, it looks as if we are facing the original problem, maybe for a module of smaller dimension. However, when e is a primitive idempotent, we get the following.

Lemma 3.5.1. Let F be a finite field and let A be an F -algebra. Let V be a faithful, uniserial A-module whose composition factors are isomorphic to the simple A-module S and let e ∈ A be an S-primitive idempotent. Then eAe/ Rad(eAe) ≅ End_{A}(S) and hence this quotient algebra is commutative.

This leads to a modified approach in this situation, which we now are going to describe.

By Theorem 2.2.10 at least 1/5 of the elements a in A have a characteristic polynomial c(δ_{S}(a)) with a very good factor f of degree [End_{A}(S) : F]. If we take an element a with this property, then it follows that f(a) obviously is a peakword for the simple A-module S. This shows that there is a good chance of finding an S-primitive idempotent for S. Fix an element a in A whose characteristic polynomial c(δ_{S}(a)) has a very good factor f of degree [End_{A}(S) : F] and let us denote the corresponding S-primitive idempotent by e. We are now going to study the subalgebra eAe, which is Morita equivalent to A, and its action on the eAe-module V e. From Section 2 it follows that the eAe-module V e is again a faithful, uniserial eAe-module of length 2 with composition factors isomorphic to Se. Moreover, by the lemma above, the quotient algebra eAe/ Rad(eAe) is isomorphic to End_{A}(S) and hence is commutative, since F is finite. It also follows from Section 2 that the unique eAe-invariant subspace of V e spans the unique proper subspace of V invariant under A.

We now distinguish the following two cases.

1. eAe is a cyclic algebra, i.e., generated by one element.

2. eAe is generated by more than one element.

In the first case, it follows that a generator x ∈ eAe does not act semisimple on V e since V e is uniserial. Since eAe acts irreducibly on Se this implies that the characteristic polynomial of x on Se is an irreducible polynomial p over F. Furthermore, the minimal polynomial of x on V e is equal to the characteristic polynomial p^{2} on V e, and the kernel of p(x) gives the unique eAe-invariant subspace. Since eAe is isomorphic to F [X]/(p^{2}), it is easy to see that the probability that a randomly chosen element in eAe is a generator is more than 1/2.

In the second case we first study the element a defined above. It is obviously acting irreducibly on Se and if it is not acting semisimple on V e, then it follows from the discussion in the first case that the kernel of f(a) is the unique nontrivial eAe-submodule of V e. Assume now that the element a is acting semisimple on V e. Let us choose an F -basis of V e such that the matrix of the a has the following form

( \begin{matrix}m_{f} & 0 \\ 0 & m_{f} \end{matrix} )

where m_{f} is the companion matrix of the polynomial f and where the first dim_{F} (Se) basis vectors are a basis for the unique simple aAe-submodule of V e.

The centralizer algebra in the full matrix algebra has the form

( \begin{matrix}f_{1} & f_{2} \\ f_{3} & f_{4} \end{matrix} )

where all the f_{i} are matrices of elements of End_{A}(S) in its regular representation. Intersecting the centralizer algebra with the algebra eAe, we get that the elements of eAe commuting with a are of the form

( \begin{matrix}f_{1} & 0 \\ f_{3} & f_{1} \end{matrix} ) .

This implies that an element in eAe commuting with the element a is acting semisimple if and only if f_{3} is zero, i.e., if the element lies in the subalgebra generated by a. Since by assumption the algebra eAe is not cyclic, we see that the probability of finding an element that has a unique invariant subspace or which does not commute with a is more than 1/2. Note that the image of the matrix of the commutator of a with an element not commuting with it, lies in the unique eAe-invariant subspace.

This leads to the following approach when V is a uniserial A-module of length 2.

Algorithm 3.5.2.

i) For all words a in A.

ii) Determine the characteristic polynomial c of a

iii) For the irreducible factors p of the order polynomial f_{1} of the first canonical basis vector, check whether p divides the quotient c/f_{1}. If not, then the vector v with v(p(a)) = 0 lies in the unique A-invariant subspace of V.

iv) Otherwise, use Algorithm 3.4.8 to condense elements x from A to eAe. If there is a factor f of the minimal polynomial of exe whose multiplicity in the minimal and the characteristic polynomial is the same then the image of f(exe) lies in the unique A-invariant subspace. If not, calculate the commutator of exe with a. If this is nontrivial, then the image of the commutator lies in the unique A-invariant subspace.

3.6 Implementations of the Condensation Method

As mentioned in the beginning of this chapter the first implementation of condensation was written by R.A. Parker and applied by J. Thackray. It dealt with F G-permutation representations and used idempotents of the form e_{H}. The C-MeatAxe program zkd, mentioned in Section 3, is a re-implementation of the original program written by R.A. Parker.

In [Ryb90] A. Ryba describes his implementation of the condensation method which given an F G-module V determines the condensation of the n-th exterior power Λ^{n}(V). His program is however restricted to the case that the condensation subgroup H is acting monomially on the F G-module V. He also proceeds in two steps: a first one in which the fixed space Λ^{n}(V)^{H} is determined and a second one where the action of e_{H}ge_{H} for a given g ∈ G is computed.

The more general case of condensing the tensor product of two F Gmodules V and W was studied in [LW96a]. In the sequel we give an outline of the basic steps of the resulting algorithm. There are two steps: determining the fixed space in the tensor product of V and W and the actual computation of the action of an element e_{H}ge_{H} on (V ⊗ W)^{H}. Calculating the fixed space is based on the following.

3.6. IMPLEMENTATIONS OF THE CONDENSATION METHOD 63

Proposition 3.6.1. Let V and W be F G-modules and let H be a p′-subgroup of G. The restrictions of V and W to the semisimple subalgebra F H can be decomposed into a direct sum of simple F H-submodules, say

V_{F H} = ⨁_{S} ⨁_{i=1}^{m_{S}} S_{i} and W_{F H} = ⨁_{S} ⨁_{j=1}^{n_{S^{♯}}} T_{j},

where S runs through a set of representatives for the isomorphism types of simple F H-modules, S^{♯} denotes the contragredient F G-module to S, m_{S} and n_{S^{♯}} are the multiplicities of S respectively S^{♯} as a constituent of V_{F H} respectively W_{F H}, and S ≅ S_{i}, S^{♯} ≅ T_{j} for 1 ≤ i ≤ m_{S} and 1 ≤ j ≤ n_{S^{♯}}. Note that we allow m_{S} and n_{S^{♯}} to be zero. Then

(V ⊗ W)e_{H} = ⨁_{S} ⨁_{i=1,j=1}^{m_{S},n_{S^{♯}}}(S_{i} ⊗ T_{j})e_{H}.

Algorithm 3.6.2 (tensor fixed space). The C-MeatAxe program socleser can can be used to determine F -bases for the restrictions of the A-modules V and W that give the decomposition of the restrictions into their simple constituents. The determination of the fixed space is therefore reduced to the case of the tensor product of a simple F Hmodule S and the contragredient F H-module S^{♯}. By Section 4 it follows that the Fitting decomposition of a peakword for the trivial F H-module in S ⊗ S^{♯} is the decomposition (S ⊗ S^{♯})e_{H} ⊕ (S ⊗ S^{♯})(1 − e_{H}). So it is straightforward to determine the action of e_{H} on S ⊗S^{♯} if a peakword for the trivial F H-module is known. Moreover, in this way we can avoid building the idempotent explicitly which would mean summing over all elements h ∈ H.

Algorithm 3.6.3 (tensor condense element). Let g be an element in G. In order to determine the action of ege on (V ⊗W)^{H} we can use the decomposition from Proposition 3.6.1. We only have to deal with simple F H-constituents S of V, such that the contragredient F H-module appears in W. For the technical details we refer to [LW96a].

As mentioned above, one of the major reasons why we use condensation is the reduction in the dimension of the modules we have to look at. A possible disadvantage of condensation is the fact that we no longer work with a group algebra but a subalgebra of the form eF Ge. In order to make efficient use of the smaller dimension of the condensed modules one should avoid to work explicitly with a matrix representation of F G on the module given. A typical example is given by an F G-permutation module. Such a module can be described by the underlying permutations of the group generators. Moreover, the condensed module with respect to a given condensation subgroup H is easily described as the subspace spanned by the H-orbit sums. More generally, we can handle the following cases along theses lines:

1) Permutation representations.

2) Monomial representations.

3) Tensor products.

4) Permutation representations given by vector orbits.

5) Exterior powers.

We would like to extend the approach to following situations:

1) Induced modules.

2) General symmetrizations of a given module.

3) Condensation with idempotents corresponding to nontrivial one dimensional representations λ of the condensation subgroup H.

In the moment the various condensation programs have been implemented in different systems and different programming languages. It would be of great use to put them in a unified fashion into a system like the C-MeatAxe. The monomial condensation programs have been written by A. Ryba in PAS-CAL, see [Ryb90], the tensor product condensation has been implemented by M. Wiegelmann as several C-MeatAxe programs, see [LW96a]. The direct condensation program for vector orbits was developed by R.A. Parker. Finally, the condensation of symmetric powers was developed and used by L. Soicher for working out polynomial invariants of a given matrix group. The condensation with nontrivial idempotents requires a modified approach to calculating the subspace V e.

The induced module condensation program would surely be of interest. As special cases it would contain permutation condensation and monomial condensation. It is therefore quite likely that in order to make it an efficient tool one would have to restrict the class of modules one wants to induce condense. Tensor and symmetrization condensation complement induce condense quite nicely. If one is looking for modules that do not have too many composition factors, one usually should study tensor products and symmetrizations of simple modules, see for example [Jan95].

Examples for Explicit Morita Equivalences

4.1 The Methods

In the following we will give explicit examples for Morita equivalences between a block B of a group algebra F G and a block e_{H}Be_{H} of a subalgebra of the form e_{H}F Ge_{H}, where H is a subgroup of G of order coprime to the characteristic of F. We will use the explicit equivalence to determine the layers of the socle series of the projective indecomposable B-modules. The examples we will study will be the principal blocks, since these are the most difficult cases to analyze both theoretically and computationally.

Lemma 3.3.7 provided us with a a character theoretic criterion that ensures that B and e_{H}Be_{H} are Morita equivalent. We now give a slightly different formulation of this criterion.

Lemma 4.1.1. Let S_{1}, …, S_{n} be the simple B-modules up to isomorphism. Furthermore, let Φ_{1}, …, Φ_{n} be the projective indecomposable characters corresponding to the projective covers of S_{1}, …, S_{n} and let H be a p′-subgroup. Then e_{H} is a faithful idempotent if and only if the permutation character 1_{H}^{G} contains each of the Φ_{1}, …, Φ_{n} at least once. Moreover, the dimension of the simple e_{H}Be_{H}-module S_{i}e_{H} is given by the multiplicity of Φ_{i} in 1_{H}^{G}.

Remark 4.1.2. The above criterion can easily be checked using the permutation character 1_{H}^{G} and the Brauer character table of a given finite group G. The GAP-procedure faithfulkondburn, written by the author, searches systematically for the subgroups H that fulfill the conditions above. A listing of the procedure is given in the Appendix, page 135. Faithfulkondburn takes as input the ordinary character table, the Brauer character table and the table of marks of G. For all sporadic simple groups G whose order is less than 10^{9} all the required data can be accessed in GAP, [S^{+}94]. The character tables given in [CCN^{+}85] are part of the ordinary character table library of GAP, and the Brauer character tables given in [JLPW95] form part of the Brauer character table library of GAP. G. Pfeiffer, see [Pfe91], has developed procedures in GAP to construct and to work with a table of marks. He has successfully applied the procedures to sporadic simple groups up to order 10^{9} whose tables of marks he has implemented in form of a GAP-library. For our purposes we use the GAP-command PermutationCharsTomTable, which derives the list of transitive permutation characters of G from the table of marks of G. For the sporadic simple groups up to order 10^{9}, the output of the GAP-procedure faithfulkondburn was used to determine the optimal choice for |H|. In Chapter 6 we list the subgroups H determined by the procedure for the sporadic simple groups up to order 10^{9} which we do not treat in the following example sections. We use the newest version of the GAPlibrary of table of marks installed by T. Merkwitz. This version also includes generators for all representatives of the conjugacy classes of subgroups as words in the standard generators of G, see [Wil96a], in the case that G is a sporadic simple group up to order 10^{9}. Furthermore, information on the modular representations in the principal block of G like dimensions of the simple modules, dimensions of the corresponding condensed modules, dimensions of the projective indecomposable modules and finally the dimensions of the condensed projective indecomposable modules is given as well.

Having found a suitable subgroup H we have to choose elements in the algebra e_{H}F Ge_{H} that generate the whole algebra. This can be done as follows.

Remark 4.1.3. Since (1_{H}^{G})e_{H} = (e_{H}F G)e_{H} it follows that (1_{H}^{G})e_{H} is the regular e_{H}F Ge_{H}-module. In order to prove that {a_{1}, …, a_{n}} is a generating set of e_{H}F Ge_{H} it is therefore sufficient to show that a chosen vector v ∈ (1_{H}^{G})e_{H} spans (1_{H}^{G})e_{H} under the action of {a_{1}, …, a_{n}}. The dimension of (1_{H}^{G})e_{H} can be easily computed since it is given by the character theoretic scalar product (1_{H}^{G}, 1_{H}^{G}) of the permutation character 1_{H}^{G} with itself.

A typical workstation has about 32 MBytes of main memory and this restricts the dimension of e_{H}F Ge_{H} we are able to study. In order to be able to verify that we have a generating set for e_{H}F Ge_{H} the dimension of (1_{H}^{G})e_{H} should not be larger than a few thousands depending on the underlying field size. This implies that |H| should not be too small. Furthermore, we also have to guarantee that e_{H} is a faithful idempotent in F G, which on the other hand indicates that H should not be too large.

Remark 4.1.4. In the case that the subgroup H is a normal subgroup of N ≤ G with N/H a p-group, then we can use Theorem 3.3.9 way to reduce the dimension of the condensed module we have to study. If n_{1}, …, n_{r} are generators for a complement of H in N and e_{H}g_{1}e_{H}, …, e_{H}g_{s}e_{H} are elements in e_{H}F Ge_{H} such that ∑_{n∈N} n spans the whole space (1_{N}^{G})e_{H} under the action of e_{H}g_{1}e_{H}, …, e_{H}g_{s}e_{H}, then e_{H}g_{1}e_{H}, …, e_{H}g_{s}e_{H}, e_{H}n_{1}e_{H}, …, e_{H}n_{r}e_{H} generate e_{H}F Ge_{H}.

Remark 4.1.5. The above criterion is very useful for sporadic simple groups in characteristic 2 where one only finds very small subgroups H of odd order. However, it usually possible to find a larger subgroup N normalizing H such that N/H is a 2-group. This simplifies the task of verifying that given elements generate the whole algebra e_{H}F Ge_{H}. A typical example is given by the sporadic simple group J_{2} in characteristic 2, where the condensation subgroup H has only order 5. However, in this case there exists a subgroup N normalizing H such that |N/H| = 8, see page 82.

Remark 4.1.6. In order to determine the layers of the socle series for projective indecomposable F G-modules, we now proceed as follows. Having found a generating set for the algebra e_{H}F Ge_{H}, we use the fact that the block B of F G is Morita equivalent to the block e_{H}Be_{H} of e_{H}F Ge_{H}. We analyze the regular e_{H}F Ge_{H}-module using the C-MeatAxe program chop, which gives us the constituents and their multiplicities. Since we know the dimensions of the condensed simple modules, see Lemma 4.1.1, it is straightforward to determine the one-to-one correspondence between the simple modules in B and the simple modules in e_{H}Be_{H}. Moreover, the regular e_{H}F Ge_{H}-module contains all the projective indecomposable e_{H}F Ge_{H}-modules, and hence we only have to be able to find the projective indecomposable summands of e_{H}Be_{H} in the regular e_{H}F Ge_{H}-module. Since B and e_{H}Be_{H} are Morita equivalent, we know that the layers of the socle series for a projective indecomposable module Q of B can be derived easily from the socle layers of the projective indecomposable module Qe_{H} of e_{H}F Ge_{H}: we just have to replace the condensed simple modules by the corresponding simple B-modules.

So the last problem which we have to solve is to find Qe_{H} as a direct summand of e_{H}F Ge_{H}. The following lemma describes a method that given an A-module V finds the projective indecomposable summands of V.

Lemma 4.1.7. Let A be an F -algebra, V be an A-module and S be a composition factor of V. Let π be an A-epimorphism from V onto the S-local A-module M and let e_{S} be an S-primitive idempotent. Then the image of π(v) of v ∈ V e_{S} does not generate M if and only if π(v) ∈ Rad_{A}(M) ∩ Me_{S} = Rad_{e_{S}Ae_{S}}(Me_{S}). Hence the subspace of vectors v in V e_{S}, for which π(v) does not generate M, has codimension dim_{F} (End_{A}(S)) = dim_{F} (Se_{S}). Furthermore, if M ≅ P (S), the projective cover of S, then the ratio of vectors in V e_{S} that do not generate an A-submodule isomorphic to P (S) to the vectors in V e_{S} is 1/| End_{A}(S)|.

Proof: Since multiplication by e_{S} preserves exact sequences, it follows that π maps V e_{S} onto Me_{S}. Furthermore, M is S-local and hence it follows that Me_{S} is an indecomposable e_{S}Ae_{S}-module with Me_{S}/ Rad(Me_{S}) ≅_{e_{S}Ae_{S}} Se_{S}. Since dim_{F} (Se_{S}) = dim_{F} (End_{A}(S)) and since a vector m ∈ Me_{S} does not generate Me_{S} if and only if m ∈ Rad(Me_{S}), the result follows.

Applying this to the special case that M ≅ P (S) we conclude that all v, such that π(v) generates M, generate an S-local submodule with quotient P (S). Since P (S) is projective this implies that v · A ≅ P (S). ■

The lemma above tells us that a randomly chosen vector from V e_{S} will generate P (S) with probability at least 1/2. Moreover, a module generated by a vector in V e_{S} is the projective cover P (S), if and only if it has the same dimension as P (S), something that is easily checked.

Summarizing we get the following algorithm for determining the layers of socle series of the projective indecomposable F G-modules in a block B.

Algorithm 4.1.8 (condensed socle series).

a) Find a p′-subgroup H fulfilling the conditions of Lemma 4.1.1.

b) Construct the permutation representation of G on the cosets of the subgroup H.

c) Condense elements g_{1}, …, , g_{s} of G and check that e_{H}g_{1}e_{H}, …, e_{H}g_{s}e_{H} generate the algebra e_{H}F Ge_{H}.

d) Analyze the regular e_{H}F Ge_{H}-module using Lemma 4.1.7.

Remark 4.1.9. The construction of a permutation representation for G on the cosets of a subgroup H can be achieved with different techniques. Given G and H in some permutation representation, there is a standard algorithm for determining coset representatives for the cosets of H in G, from which we can easily determine the required permutation representation of G on the cosets of H. If H is properly contained in a subgroup M < G, then it is a good idea to factorize the calculation, i.e., work with cosets of M in G and the cosets of H in M. In the Appendix we give the listing of the GAP-procedure imprimperm that implements this idea. It was the standard tool for constructing the permutation representations used in the examples. Alternatively, we could try to find a matrix representation of G over some field such that H is the stabilizer of a one dimensional subspace. In this case, we can use C-MeatAxe program zvp that implements the standard orbit algorithm for determining the permutation representation of G on the cosets of H.

The following table gives the known results concerning the layers of the socle series of projective indecomposable modules of sporadic simple groups in the principal block. We only list the results concerning blocks of noncyclic defect, since blocks of cyclic defect are easy to handle, see [HL89].

Primes 2  3  5
M_{11}  [Sch79] [Wak93a]
M_{12}  [Sch93a] [Sch79]
J_{1}  [LM78]
M_{22}  [LS94] [Wak93a]
J_{2}  page 82 page 92 page 95
M_{23}  [LW96a] [Wak93a]
HS  ?  [Wak93b] page 71
J_{3}  ?  page 97
M_{24}  ?  page 77

A question mark in the above table indicates that the layers of the socle series have not yet been determined for the case marked.

In the following sections and in Chapter 6 we use a special generating system of the group G in question, namely a so-called system of standard generators as defined by R.A. Wilson, see [Wil96a]. For the groups we are interested in, R.A. Wilson has constructed permutation representations and matrix representations in the standard generators. These representations are available as a MeatAxe-library and in the near future they will also be accessible as a GAP-library. This library also includes words in the standard generators that give generators for maximal subgroups of the group G. It is very convenient for us that T. Merkwitz, a student at Lehrstuhl D für Mathematik, has extended this collection of words to contain words for the generators of all subgroups up to conjugacy of sporadic simple groups of order up to 10^{9}. These words will be used in all the examples we deal with. Finally, given two elements a, b in a group G we use standard words called z1(a, b), …, z10(a, b), whose definition is given in the Appendix, see page 141, to form new elements in G derived from a and b. For example, when looking for a set of condensed elements that generate the condensation algebra we start by looking at z1, …, z10 in the standard generators.

Moreover, in each of the following sections we proceed as follows. We first identify the subgroup H we are going to use for condensation. We do this in the following way. We give the number of the G-conjugacy class of subgroups in the GAP-library of tables of marks in which the subgroup H is contained, and we also list the decomposition of the ordinary permutation character 1_{H}^{G} into the ordinary irreducible characters of G following the Atlas notation, see [CCN^{+}85]. More precisely, let an ordinary irreducible character be written as Xx in Atlas notation, i.e., X is the degree and x is a letter of the alphabet, then if the multiplicity n of Xx in 1_{H}^{G} is one, two or three, we write Xx, Xxx and Xxxx, otherwise we write Xx^{n}. Furthermore, we also give the scalar product (1_{H}^{G}, 1_{H}^{G}). As mentioned above, this scalar product is equal to the dimension of the condensation algebra e_{H}F Ge_{H}.

We then continue by listing the dimensions of the irreducible modular representations of G in the principal block B taken from the GAP-library of Brauer characters and we list the corresponding condensed dimensions, i.e., the H-fixed spaces of the modular irreducible representations. Note that these are equal to the dimensions of the irreducible representations in e_{H}Be_{H}. This table is followed by the Cartan matrix of B which gives the multiplicities of the simple B-modules in the projective indecomposable B-modules. Furthermore, for the projective indecomposable modules in B we also list the dimensions and the dimensions of the condensed projective indecomposable modules in e_{H}Be_{H}.

The final part of each section consists in applying Algorithm 4.1.8. More precisely, we first determine a generating set for e_{H}F Ge_{H}, where F is a splitting field for all simple B-modules, and then we explicitly construct the condensed projective indecomposable modules for e_{H}Be_{H} and their socle series. Finally, we use the explicit Morita equivalence to determine the layers of the projective indecomposable modules in B. Note that our convention for listing the layers [Soc_{1}(V), Soc_{2}(V)/ Soc_{1}(V), …, Soc_{r}(V)/ Soc_{r−1}(V)] of the socle series of a given F G-module V is as follows.

Soc_{1}(V)
Soc_{2}(V)/ Soc_{1}(V)
⋮
Soc_{r}(V)/ Soc_{r−1}(V)

If a given layer is the direct sum ⨁_{i=1}^{r} n_{i}S_{i} with simple F G-modules S_{i}, it is written as S_{1}^{n_{1}} ⊕ S_{2}^{n_{2}} ⊕ ⋯ ⊕ S_{r}^{n_{r}}.

4.2. THE P.I.M.S OF HS IN CHAR. 5 71

4.2 The P.I.M.s of HS in Char. 5

We first find a suitable subgroup H such that e_{H} is a faithful idempotent for the principal block. Using the GAP-procedure faithfulkondburn, listed in the Appendix, we see that the largest subgroup we can use is the subgroup whose number in the table of marks of G is 456. The order of the subgroup H is 192 and the scalar product (1_{H}^{G}, 1_{H}^{G}) is 1430. The permutation character of 1_{H}^{G} decomposes as

1a + 22aa + 77a^{4} + 154a^{4}bbcc + 175a^{5} + 231aa + 693a^{8} + 770a^{10}bbbccc +825a^{10} + 896a^{5}b^{5} + 1056a^{10} + 1386a^{7} + 1408a^{9} + 1750a^{8} + 1925a^{10}b^{10} +2520a^{16} + 2750a^{7} + 3200a^{15}.

The dimensions of the modular irreducible representations in the principal block and their condensed dimensions are:

1 21 55 98 133 133 210 280 280 518 1 1 2 1 1 1 1 1 1 3

The Cartan matrix of the principal block is:

7 6 4 5 1 1 2 1 1 2
6 16 5 12 6 6 8 3 3 7
4 5 5 6 2 2 1 0 0 2
5 12 6 18 6 6 8 5 5 11
1 6 2 6 4 3 3 1 1 3
1 6 2 6 3 4 3 1 1 3
2 8 1 8 3 3 10 6 6 6
1 3 0 5 1 1 6 6 5 5
1 3 0 5 1 1 6 5 6 5
2 7 2 11 3 3 6 5 5 10

The dimensions of the projective indecomposable modules and their condensed dimensions are:

3125 10375 2750 14125 4500 4500 10375 7750 7750 11375 38  91  36 110 38 38  66  43 43  76

We first construct the permutations for G on H, whose index in G is 231000, as follows: Since the permutation representation is imprimitive, we factorize the cosets with respect to the (maximal) subgroup 4^{3}: L_{3}(2) of index 4125 in G and use the GAP-procedure imprimperm given in the Appendix to construct the permutations.

We check that the condensations of the first six elements z1,z2, …, z6, see page 141 for their definition, in the standard generators of G generate the whole algebra e_{H}F Ge_{H} by spinning up the first standard basis vector and showing that it spans the whole space under their action.

We are now able to analyze the regular representation of the condensation algebra using the C-MeatAxe program chop. In the table below, the first column given the names of the composition factors, where the name is defined by the dimension and a letter of the alphabet distinguishing composition factors of the same dimension, the column SF gives the degree of the splitting field of the composition factors and the fingerprint lists the dimensions of the nullspaces of the 6 standard words defined in [Par84] evaluated in the composition factors. The result is:

Chopping completed: 16 different composition factors
Writing reg.cfinfo
Name   Mult  SF  Fingerprint
1a       91   1  1,0,0,0,0,0
1b       66   1  0,0,0,0,0,0
1c       38   1  0,0,0,0,0,1
1d       38   1  0,0,0,0,0,0
1e       38   1  0,0,1,0,0,0
1f       43   1  0,0,1,1,1,0
1g       43   1  0,0,0,0,0,0
1h      110   1  0,0,0,0,0,0
2a       36   1  0,1,0,0,0,1
3a       76   1  0,0,0,0,1,0
5a       25   1  0,0,1,0,0,0
5b       20   1  0,1,0,0,1,0
5c       15   1  0,1,0,0,1,0
7a        7   1  0,1,1,1,0,1
8a        8   1  1,1,0,1,1,0
10a      25   1  0,0,0,0,0,0

We next determine peakwords for the composition factors and evaluate powers of these words in the above representation until their nullspace is stable using the C-MeatAxe program pwkond. The printed output of pwkond lists for each composition factor the following information: the number of a word w in the generators as defined by the C-MeatAxe and an irreducible factor f of the characteristic polynomial of w such that f(w) is a peakword for the composition factor.

*** PEAK WORD KONDENSATION ***
Peak word for reg1a is 2 (x+4), nullity=1
Peak word for reg5c is 4 (x+2), nullity=1
Peak word for reg7a is 5 (x+2), nullity=1
Peak word for reg8a is 15 (x+3), nullity=1
Peak word for reg1h is 32 (x+1), nullity=1
Peak word for reg3a is 34 (x+1), nullity=1
Peak word for reg1e is 48 (x+3), nullity=1
Peak word for reg1c is 57 (x), nullity=1
Peak word for reg5b is 70 (x+3), nullity=1
Peak word for reg2a is 73 (x+3), nullity=1
Peak word for reg1f is 79 (x+2), nullity=1
Peak word for reg1b is 86 (x+4), nullity=1
Peak word for reg1g is 90 (x), nullity=1
Peak word for reg1d is 123 (x+1), nullity=1
Peak word for reg5a is 306 (x+3), nullity=1
Peak word for reg10a is 362 (x+1), nullity=1

We spin up a random vector in the stable nullspace of each peakword according to Lemma 4.1.7. This gives us the projective indecomposable modules for the condensed algebra in a straightforward way. The last step now requires again an analysis of the composition factors of the projective indecomposable modules and peakwords for their composition factors. Finally, we apply the C-MeatAxe program socleser to determine the socle series of the projective indecomposable modules and replace the simple e_{H}F Ge_{H}-modules by the corresponding simple F G-modules. The correspondence can be easily determined since the multiplicity of each of the simple F G-modules in 1_{H}^{G} can be derived from the Brauer character table. The multiplicities are given in the following table.

\begin{matrix}1a & 21a & 55a & 98a & 133a & 133b & 175a & 210a & 280a & 280b & 518a \\ 38 & 91 & 36 & 110 & 38 & 38 & 15 & 66 & 43 & 43 & 76 \end{matrix}

\begin{matrix}650a & 1275a & 1750a & 1925a & 2750a \\ 20 & 25 & 8 & 25 & 7 \end{matrix}

These multiplicities have to match with the multiplicities of the corresponding condensed modules in the regular representation of e_{H}F Ge_{H}. This determines almost uniquely the correspondence we look for up to automorphisms.

The only problem left is to distinguish 1a from 133a and 133b. We can solve this problem by studying the condensed permutation module on 100 points, where obviously only 1a is a constituent. The resulting layers for the socle series of the projective indecomposable modules of G are as follows.

1a

98a ⊕ 21a 1a^{2} ⊕ 210a ⊕ 55a^{2} ⊕ 518a 98a^{2} ⊕ 21a^{3} ⊕ 280b ⊕ 280a

1a^{3} ⊕ 133b ⊕ 133a ⊕ 55a ⊕ 518a 98a ⊕ 21a 210a ⊕ 55a 98a ⊕ 21a

1a

21a

210a ⊕ 1a ⊕ 133b ⊕ 133a ⊕ 55a ⊕ 518a

21a^{5} ⊕ 133a^{4} ⊕ 280b ⊕ 280a 210a^{3} ⊕ 1a^{3} ⊕ 133b^{2} ⊕ 133a^{2} ⊕ 55a^{2} ⊕ 518a^{3}

21a^{5} ⊕ 133a^{4} ⊕ 280b ⊕ 280a 210a^{3} ⊕ 1a ⊕ 133b^{2} ⊕ 133a^{2} ⊕ 55a ⊕ 518a^{2}

21a ⊕ 21a^{3} ⊕ 133a^{4} ⊕ 280b ⊕ 280a 210a ⊕ 1a ⊕ 133b ⊕ 133a ⊕ 55a ⊕ 518a

21a

55a 98a ⊕ 21a

1a^{2} ⊕ 133b ⊕ 133a ⊕ 55a ⊕ 518a

98a^{3} ⊕ 21a^{2} 1a ⊕ 210a ⊕ 55a^{2} ⊕ 518a

98a ⊕ 21a 1a ⊕ 133b ⊕ 133a

98a ⊕ 21a 55a

98a

1a ⊕ 210a ⊕ 133a ⊕ 133b ⊕ 55a ⊕ 518a ⊕ 518a

98a^{6} ⊕ 21a^{4} ⊕ ⊕280b^{2} ⊕ 280a^{2} 1a^{2} ⊕ 210a^{3} ⊕ 133a^{2} ⊕ 133b^{2} ⊕ 55a^{3} ⊕ 518a^{5}

98a^{6} ⊕ 21a^{4} ⊕ 280b^{2} ⊕ 280a^{2} 1a ⊕ 210a^{3} ⊕ 133a^{2} ⊕ 133b^{2} ⊕ 55a ⊕ 518a^{2}

98a^{4} ⊕ 21a^{4} ⊕ 280b ⊕ 280a

1a ⊕ 210a ⊕ 133a ⊕ 133b ⊕ 55a ⊕ 518a ⊕ 518a 98a

133a 21a ⊕ 98a

210a ⊕ 133b ⊕ 55a ⊕ 518a

21a^{2} ⊕ 98a^{2} ⊕ 280a 133a^{2} ⊕ 210a ⊕ 1a ⊕ 133b ⊕ 518a

21a^{2} ⊕ 98a^{2} ⊕ 280b 210a ⊕ 133b ⊕ 55a ⊕ 518a

21a ⊕ 98a 133a

133b 98a ⊕ 21a

210a ⊕ 133a ⊕ 55a ⊕ 518a

98a^{2} ⊕ 21a^{2} ⊕ 280b 133b^{2} ⊕ 210a ⊕ 1a ⊕ 133a ⊕ 518a

98a^{2} ⊕ 21a^{2} ⊕ 280a 210a ⊕ 133a ⊕ 55a ⊕ 518a

98a ⊕ 21a 133b

210a

98a ⊕ 21a ⊕ 280b ⊕ 280a 210a^{2} ⊕ 1a ⊕ 133a ⊕ 133b ⊕ 518a^{2}

98a^{2} ⊕ 21a^{3} ⊕ 280b^{2} ⊕ 280a^{2} 210a^{4} ⊕ 133a ⊕ 133b ⊕ 55a ⊕ 518a^{2}

98a^{3} ⊕ 21a^{3} ⊕ 280b^{2} ⊕ 280a^{2} 210a^{2} ⊕ 1a ⊕ 133a ⊕ 133b ⊕ 518a^{2}

98a ⊕ 21a ⊕ 280b ⊕ 280a 210a

280a
210a ⊕ 518a

280a ⊕ 98a^{2} ⊕ 21a ⊕ 280b^{2} 210a^{2} ⊕ 1a ⊕ 133b ⊕ 518a^{2} 280a^{2} ⊕ 98a^{2} ⊕ 21a ⊕ 280b

210a^{2} ⊕ 133a ⊕ 518a 280a ⊕ 98a ⊕ 21a ⊕ 280b^{2}

210a ⊕ 518a

280a
280b
210a ⊕ 518a

280b ⊕ 98a^{2} ⊕ 21a ⊕ 280a^{2} 210a^{2} ⊕ 1a ⊕ 133a ⊕ 518a^{2} 280b^{2} ⊕ 98a^{2} ⊕ 21a ⊕ 280a

210a^{2} ⊕ 133b ⊕ 518a 280b ⊕ 98a ⊕ 21a ⊕ 280a^{2}

210a ⊕ 518a

280b
518a

280b ⊕ 21a ⊕ 280a ⊕ 98a^{2} 210a^{2} ⊕ 1a ⊕ 133b ⊕ 133a ⊕ 55a ⊕ 518a^{3}

280b^{2} ⊕ 21a^{3} ⊕ 280a^{2} ⊕ 98a^{5} 210a^{2} ⊕ 1a ⊕ 133b ⊕ 133a ⊕ 55a ⊕ 518a^{3}

280b ⊕ 21a^{2} ⊕ 280a ⊕ 98a^{2} 210a^{2} ⊕ 133b ⊕ 133a ⊕ 518a^{2} 280b ⊕ 21a ⊕ 280a ⊕ 98a^{2}

518a

4.3. THE P.I.M.S OF M_{24} IN CHAR. 3 77

4.3 The P.I.M.s of M_{24} in Char. 3

We get faithful condensation for the subgroup H with number 1234 in the table of marks. The order of the subgroup H is 320.

The dimension of the condensed module (1_{H}^{G})e_{H} is 2884. The permutation character 1_{H}^{G} decomposes as

1a + 23aa + 231ab + 252a^{8} + 253aa + 483a^{7} + 770ab + 990ab + 1035a^{10}bc +1265a^{9} + 1771aaa + 2024a^{10} + 2277a^{8} + 3312a^{18} + 3520a^{20} + 5313a^{24} +5544a^{10} + 5796a^{18} + 10395a^{26}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 22 231 483 770 770 1243 1 1 1 7 1 1 8

The Cartan matrix of the block is:

5 4 3 3 4 4 2
4 8 6 3 5 5 4
3 6 9 3 6 6 3
3 3 3 4 4 4 2
4 5 6 4 8 7 3
4 5 6 4 7 8 3
2 4 3 2 3 3 3

The dimensions of the projective indecomposable modules and their condensed dimensions are:

10881 15687 16632 11340 18711 18711 10098 57  81  75  61  82  82  53

We first construct the permutation representation of G on the cosets of H. We check that H is contained in the representative M of the conjugacy class of the subgroup isomorphic to 2^{6} : 3S_{6} stored in the table of marks. The GAP-procedure imprimpermcan is then applied to construct the permutations for G on the cosets of H in G. Note that imprimpermcan assumes that the subgroup M is the stabilizer of the first point. Furthermore, it also assumes that it is given a small M-orbit on the underlying set, which is faithful. The generators h1, h2, h3 and h4 for H given in the table of marks are not a minimal generating set. Instead we take h3 ∗ h2 ∗ h4 and h4 ∗ h3 ∗ h2 ∗ h1 which we checked to generate H using GAP. Having a small generating set at our hands is useful since it cuts down the amount of time that we have to spend on the following calculations.

The first task is to find a generating set for the condensation algebra e_{H}F Ge_{H}. We do this in the usual manner by condensing elements of G and by generating the subspace spanned by e_{H} under the subalgebra generated by the condensed elements. The condensations of the elements z1, …, z10, see page 141, in the standard generators generate a subalgebra of dimension 2877. We now have to find an element in e_{H}F Ge_{H} that is not contained in the subalgebra. To check whether an element a is contained in the subalgebra can be easily accomplished by testing whether e_{H}a lies in the subspace and so can be done by using Gaussian elimination. Note that e_{H} is the first basis vector in the condensed permutation module by the construction of the underlying permutations. This method is obviously superior to the method of adding the element to generating set and checking whether the space spanned by e_{H} is the whole space. Using this new test we get the last generator which is the condensation of the word z7(z7, z8).

The correspondence between the composition factors of the original permutation module and the composition factors of the condensed module can easily be found by decomposing the permutation character 1_{H}^{G} into the irreducible Brauer characters. The multiplicities of the composition factors and the knowledge of their condensed dimensions uniquely determine this correspondence up to a choice for the irreducibles of degree 770.

As before we determine the peakwords for simple e_{H}F Ge_{H}-modules using the C-MeatAxe program pwkond. Then we proceed by taking random vectors in the stable nullspace of each peakword until we find the corresponding condensed projective indecomposable module. The C-MeatAxe program socleser finally determines the socle series of the condensed projective indecomposable modules. In contrast to the first part of the calculation, i.e., the process of finding generators for the condensed algebra and the construction of the permutations on 765072 points, this takes effectively no time at all.

Finally, we list the layers of the socle series for the projective indecomposable modules in principal block of G in characteristic 3.

1a
770b ⊕ 22a

1a ⊕ 231a ⊕ 483a ⊕ 1234a 770b ⊕ 22a ⊕ 770a^{2} 1a^{2} ⊕ 770b ⊕ 483a

231a ⊕ 483a 22a ⊕ 770a 770b ⊕ 1234a

231a 22a ⊕ 770a

1a

22a

231a ⊕ 1a ⊕ 1234a 22a^{2} ⊕ 770a ⊕ 770b

231a ⊕ 1a ⊕ 770b ⊕ 483a ⊕ 1234a

22a ⊕ 231a ⊕ 483a

22a ⊕ 770a^{2} 1a ⊕ 770b^{2} ⊕ 1234a

231a^{2} ⊕ 483a 22a^{2} ⊕ 770a^{2}

1a ⊕ 770b ⊕ 1234a 231a

22a

231a

231a ⊕ 770a ⊕ 22a 231a ⊕ 770b ⊕ 1a ⊕ 1234a

231a ⊕ 22a ⊕ 483a

770a^{2} ⊕ 22a 770b^{2} ⊕ 1a ⊕ 1234a

231a^{2} ⊕ 483a 770a^{2} ⊕ 22a^{2}

770b^{2} ⊕ 1a ⊕ 1234a 231a^{2} ⊕ 483a 770a ⊕ 22a

770b
231a

\begin{matrix}483a \\ 770a \\ 770b ⊕ 1a ⊕ 1234a \\ 231a ⊕ 22a ⊕ 483a \\ 770a ⊕ 22a ⊕ 1a \\ 770b ⊕ 1a \\ 231a ⊕ 483a \\ 770a ⊕ 22a \\ 770b ⊕ 1234a \\ 231a \\ 770a \\ 770b \\ 483a \end{matrix} \begin{matrix}770a \\ 770b ⊕ 1a ⊕ 1234a \\ 231a ⊕ 22a ⊕ 483a \\ 770a^{2} ⊕ 22a ⊕ 1a \\ 770b^{2} ⊕ 1a ⊕ 1234a \\ 231a^{2} ⊕ 483a \\ 770a^{2} ⊕ 22a^{2} \\ 770b^{2} ⊕ 1a ⊕ 1234a \\ 231a^{2} ⊕ 483a \\ 770a^{2} ⊕ 22a \\ 770b^{2} \\ 231a ⊕ 483a \\ 770a \end{matrix}

\begin{matrix}770b \\ 231a ⊕ 483a \\ 770a^{2} ⊕ 22a \\ 770b^{2} ⊕ 1a^{2} ⊕ 1234a \\ 231a^{2} ⊕ 483a \\ 770a^{2} ⊕ 22a^{2} \\ 770b^{2} ⊕ 1a ⊕ 1234a \\ 231a^{2} ⊕ 483a \\ 770a^{2} ⊕ 22a^{2} \\ 770b^{2} ⊕ 1a ⊕ 1234a \\ 231a ⊕ 483a \\ 770a \\ 770b \end{matrix} \begin{matrix}1234a \\ 770b ⊕ 22a \\ 1a ⊕ 231a ⊕ 483a \\ 22a ⊕ 770a \\ 770b ⊕ 1234a \\ 231a \\ 22a ⊕ 770a \\ 770b ⊕ 1a \\ 231a ⊕ 483a \\ 22a ⊕ 770a \\ 1234a \end{matrix}

4.4 The P.I.M.s of J_{2} in Char. 2

We get faithful condensation for the subgroup H with number 11 in the table of marks of J_{2}. This subgroup is cyclic of order five and its non-identity elements are contained in the conjugacy classes 5a and 5b.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 24228 and the permutation character 1_{H}^{G} decomposes as

\begin{matrix}1a + 14a^{4}b^{4} + 21a^{7}b^{7} + 36a^{4} + 63a^{15} + 70a^{16}b^{16} + 90a^{22} + 126a^{26} + 160a^{28} \\ +175a^{35} + 189a^{39}b^{39} + 224a^{44}b^{44} + 225a^{45} + 288a^{60} + 300a^{60} + 336a^{64}. \end{matrix}

The dimensions of the modular irreducible representations and their condensed dimensions are:

1 6 6 14 14 36 84 1 2 2 4 4 4 12

The Cartan matrix of the principal block is:

112 76 76 40 40 40 32
76 54 52 28 28 28 22
76 52 54 28 28 28 22
40 28 28 20 16 14 14
40 28 28 16 20 14 14
40 28 28 14 14 17 11
32 22 22 14 14 11 13

The dimensions of the projective indecomposable modules in the principal block and their condensed dimensions are:

6272 4352 4352 2560 2560 2304 2176 1280 888 888 520 520 464 432

Since the condensation subgroup H is of order 5, we use Theorem 3.3.9 to verify the Morita equivalence. We choose a subgroup M of isomorphism type 2×2×5: 2, which normalizes H. This subgroup lies inside a maximal subgroup of type A_{5} × D_{10} and in a maximal subgroup of type A_{4} × A_{5}. According to Theorem 3.3.9 we have to find generators in the condensed permutation module of J_{2} on the cosets of M and then add the condensations of three more elements generating a complement to H in M.

4.4. THE P.I.M.S OF J_{2} IN CHAR. 2 83

We proceed as follows. First, we construct generators for J_{2} out of J_{2}.2 by using the first generator of the standard generators of J_{2}.2 and the square of the word z4 in the standard generators as the second generator for J_{2}. In the permutation representation on 100 points we can easily check using GAP that these two elements do indeed generate J_{2}.

We construct generators for the intermediate subgroup M as words in the chosen generators for J_{2}. This is done as follows:

The subgroup M can be interpreted as a subgroup of the centralizer of a 2b-element, which is 2 × 2 × A_{5}. We use the (standard) dihedral group trick, see [Lin95], to construct a few elements of the centralizer in the following way:

We first choose an involution t of type 2b by taking the fifth power of the product z5 ∗ z1 in the generators of J_{2}. We now multiply t with conjugates of t in order to find centralizing involutions. As conjugating elements we use the elements z9 and z10 in the generators of J_{2}. It turns out that the product t ∗ t^{z9} has order 12 and hence its sixth power is a centralizing involution t_{1} of type 2a. The product t ∗ (t^{z9})^{z10} is of order 10 and its fifth power t_{2} is again an involution of type 2a. These two involutions t_{1} and t_{2} already generate a dihedral group of order 10 inside the A_{5} contained in 2×2×A_{5}. The element t^{z10} is a 2b involution t_{4} and it can be checked that it commutes with t. The subgroup generated by t_{1}, t_{2}, t, t_{4} of C_{J_{2}}(t) is of type 2 × D_{10} = M. Note that the product of the first two commuting involutions is of order five and generates the normal subgroup of order 5. We therefore choose the product as the generator of the condensation subgroup. Moreover, t_{2}, t and the fifth power t_{5} of the product of t_{2} ∗ t_{4} generate a subgroup of order 8, which is a complement to the normal subgroup of order 5.

We continue by constructing a transitive permutation representation of degree 15120 for J_{2} with point stabilizer M. We construct the composition factor of degree 63 of the permutation module of degree 100 of J_{2} over the field with 7 elements using the C-MeatAxe program chop. We restrict the representation to the generators of the subgroup M and intersect the eigenspaces of the generators for the eigenvalue 1. This results in a 3-dimensional subspace of vectors fixed under the subgroup M. The C-MeatAxe program zvp shows that the J_{2}-orbit of the first vector of the basis gives us the requested permutation representation of degree 15120.

The condensation subgroup of order 5 has 3021 regular orbits and 15 fixed points, so that the condensed module will have dimension 3036.

We produce elements in eF J_{2}e such that the smallest invariant subspace invariant under the elements (corresponding to the trivial coset) will be the whole condensed space.

It turns out that the second and the third condensed generator already suffice to span the whole space. So according to Theorem 3.3.9, they generate together with the elements t_{2}, t and t_{5} the full condensation algebra.

We now look at the permutations of J_{2} on the cosets of a subgroup of order 75 contained in the normalizer of a Sylow 5-subgroup. We can easily construct the permutations using the GAP program imprimperm, where we use the full normalizer of the Sylow 5-subgroup as the intermediate subgroup.

In the condensed module the word 164 is a peakword for the trivial module and its 32-th power yields a 112 dimensional nullspace. The 112-th basis vector of this nullspace generates a 1280 dimensional space, which therefore is the condensed projective cover of the trivial module.

In order to construct the other projective indecomposable modules in the principal block, we proceed as follows. Since there are no defect zero blocks, we first construct two projective indecomposable modules of degrees 896 and 512. They belong to a block with an elementary abelian defect group of order four whose involutions lie in the J_{2}-conjugacy class 2b. This implies that a permutation representation with respect to a subgroup which does not contain 2b elements is projective when restricted to the second block. In order to construct the projective indecomposable module of degree 896, we take the permutation representation of degree 8064 with point stabilizer 5^{2}: 4. From the Brauer character table we can derive that the 896 is appearing exactly once in the permutation representation. We therefore construct the projective indecomposable summand of degree 896 by condensing this permutation representation. It is straightforward to identify the direct summand corresponding to the 896. The closure of this subspace in the original 8064-dimensional permutation representation gives us the 896. The tensor product of the irreducible module 6a with the 896 decomposes into a direct sum of two copies of the projective indecomposable P (84a) and two copies of the projective indecomposable module in the second block of degree 512. This time we apply the C-MeatAxe programs for tensor condense to construct the projective indecomposable module of degree 512.

The tensor products of the 512, which also is the projective cover of 64a, with the irreducible modules 6a, 6b, 14a and 14b can be decomposed into projective indecomposable modules as follows.

6a ⊗ P (64a) = P (84a) ⊕ P (160a)
6b ⊗ P (64a) = P (14b) ⊕ P (64b)
14a ⊗ P (64a) = P (36a) ⊕ P (84a) ⊕ P (160a)^{3}
14b ⊗ P (64a) = P (6a) ⊕ P (64a)^{2} ⊕ P (160a)^{2}

The condensed tensor products have dimension 608, 624, 1424, 1448. We use the C-MeatAxe programs implementing tensor condensation as described in Section 3.6 to construct the action of e_{H}F J_{2}e_{H} on the condensed tensor products.

4.4. THE P.I.M.S OF J_{2} IN CHAR. 2 85

The layers of the socle series of the projective indecomposable modules in the principal block now read as follows.

1a

6a ⊕ 6b ⊕ 84a 1a^{3} ⊕ 14a^{2} ⊕ 14b^{2} ⊕ 36a^{2}

6a^{4} ⊕ 6b^{4} ⊕ 84a^{2} 1a^{9} ⊕ 14a^{3} ⊕ 14b^{3} ⊕ 36a^{2}

6a^{6} ⊕ 6b^{6} ⊕ 84a 1a^{11} ⊕ 14a^{2} ⊕ 14b^{2} ⊕ 36a^{2}

6a^{7} ⊕ 6b^{7} ⊕ 84a^{4} 1a^{10} ⊕ 14a^{6} ⊕ 14b^{6} ⊕ 36a^{6}

6a^{9} ⊕ 6b^{9} ⊕ 84a^{4} 1a^{15} ⊕ 14a^{4} ⊕ 14b^{4} ⊕ 36a^{4}

6a^{9} ⊕ 6b^{9} ⊕ 84a 1a^{15} ⊕ 14a^{4} ⊕ 14b^{4} 6a^{7} ⊕ 6b^{7} ⊕ 84a^{4}

1a^{6} ⊕ 14a^{5} ⊕ 14b^{5} ⊕ 36a^{6}

6a^{6} ⊕ 6b^{6} ⊕ 84a^{4} 1a^{6} ⊕ 14a^{2} ⊕ 14b^{2} ⊕ 36a^{6}

6a^{6} ⊕ 6b^{6} 1a^{12} ⊕ 14a^{2} ⊕ 14b^{2} 6a^{5} ⊕ 6b^{5} ⊕ 84a^{2}

1a^{5} ⊕ 14a^{2} ⊕ 14b^{2} ⊕ 36a^{2}

6a^{3} ⊕ 6b^{3} ⊕ 84a^{2} 1a^{2} ⊕ 14a ⊕ 14b ⊕ 36a^{2}

6a^{2} ⊕ 6b^{2} 1a^{4} ⊕ 14a ⊕ 14b 6a^{2} ⊕ 6b^{2} ⊕ 84a^{2}

1a^{2} ⊕ 14a^{2} ⊕ 14b^{2} ⊕ 36a^{2}

6a^{2} ⊕ 6b^{2} ⊕ 84a^{2} 1a^{2} ⊕ 14a ⊕ 14b ⊕ 36a^{2}

6a^{2} ⊕ 6b^{2} 1a^{4} ⊕ 14a ⊕ 14b 6a^{2} ⊕ 6b^{2} ⊕ 84a^{2}

1a^{2} ⊕ 14a^{2} ⊕ 14b^{2} ⊕ 36a^{2}

6a^{2} ⊕ 6b^{2} ⊕ 84a 1a^{2} ⊕ 36a^{2} 6a ⊕ 6b

1a

6a

1a ⊕ 14a ⊕ 36a 6a ⊕ 6b^{2} ⊕ 84a

1a^{4} ⊕ 14a ⊕ 14b^{2} ⊕ 36a

6a^{4} ⊕ 6b^{3} ⊕ 84a 1a^{6} ⊕ 14a^{2} ⊕ 14b^{2} ⊕ 36a

6a^{3} ⊕ 6b^{5} ⊕ 84a^{2} 1a^{7} ⊕ 14a^{3} ⊕ 14b^{3} ⊕ 36a^{3}

6a^{6} ⊕ 6b^{5} ⊕ 84a^{2} 1a^{9} ⊕ 14a^{3} ⊕ 14b^{2} ⊕ 36a^{3}

6a^{7} ⊕ 6b^{5} ⊕ 84a^{2} 1a^{9} ⊕ 14a^{3} ⊕ 14b^{4} ⊕ 36a^{2}

6a^{6} ⊕ 6b^{5} ⊕ 84a^{2} 1a^{7} ⊕ 14a^{3} ⊕ 14b^{2} ⊕ 36a^{3}

6a^{5} ⊕ 6b^{6} ⊕ 84a^{2} 1a^{6} ⊕ 14a^{2} ⊕ 14b^{3} ⊕ 36a^{3}

6a^{3} ⊕ 6b^{5} ⊕ 84a^{2} 1a^{6} ⊕ 14a^{2} ⊕ 14b^{2} ⊕ 36a^{2}

6a^{4} ⊕ 6b^{3} ⊕ 84a 1a^{5} ⊕ 14a ⊕ 14b^{2} ⊕ 36a

6a^{3} ⊕ 6b^{2} ⊕ 84a 1a^{3} ⊕ 14a^{2} ⊕ 36a 6a ⊕ 6b^{3} ⊕ 84a 1a^{2} ⊕ 14b^{2} ⊕ 36a

6a^{3} ⊕ 84a 1a^{2} ⊕ 14a^{2} ⊕ 36a

6b^{3} ⊕ 84a 1a^{2} ⊕ 14b^{2} ⊕ 36a

6a^{3} ⊕ 84a 1a^{2} ⊕ 14a^{2} ⊕ 36a

6b^{3} ⊕ 84a 1a^{2} ⊕ 14b^{2} ⊕ 36a

6a^{3} ⊕ 84a 1a^{2} ⊕ 14a ⊕ 36a

6b^{2} 1a ⊕ 36a

6a

6b

1a ⊕ 14b ⊕ 36a 6b ⊕ 6a^{2} ⊕ 84a

1a^{4} ⊕ 14b ⊕ 14a^{2} ⊕ 36a

6b^{4} ⊕ 6a^{3} ⊕ 84a 1a^{6} ⊕ 14b^{2} ⊕ 14a^{2} ⊕ 36a

6b^{3} ⊕ 6a^{5} ⊕ 84a^{2} 1a^{7} ⊕ 14b^{3} ⊕ 14a^{3} ⊕ 36a^{3}

6b^{6} ⊕ 6a^{5} ⊕ 84a^{2} 1a^{9} ⊕ 14b^{3} ⊕ 14a^{2} ⊕ 36a^{3}

6b^{7} ⊕ 6a^{5} ⊕ 84a^{2} 1a^{9} ⊕ 14b^{3} ⊕ 14a^{4} ⊕ 36a^{2}

6b^{6} ⊕ 6a^{5} ⊕ 84a^{2} 1a^{7} ⊕ 14b^{3} ⊕ 14a^{2} ⊕ 36a^{3}

6b^{5} ⊕ 6a^{6} ⊕ 84a^{2} 1a^{6} ⊕ 14b^{2} ⊕ 14a^{3} ⊕ 36a^{3}

6b^{3} ⊕ 6a^{5} ⊕ 84a^{2} 1a^{6} ⊕ 14b^{2} ⊕ 14a^{2} ⊕ 36a^{2}

6b^{4} ⊕ 6a^{3} ⊕ 84a 1a^{5} ⊕ 14b ⊕ 14a^{2} ⊕ 36a

6b^{3} ⊕ 6a^{2} ⊕ 84a 1a^{3} ⊕ 14b^{2} ⊕ 36a 6b ⊕ 6a^{3} ⊕ 84a 1a^{2} ⊕ 14a^{2} ⊕ 36a

6b^{3} ⊕ 84a 1a^{2} ⊕ 14b^{2} ⊕ 36a

6a^{3} ⊕ 84a 1a^{2} ⊕ 14a^{2} ⊕ 36a

6b^{3} ⊕ 84a 1a^{2} ⊕ 14b^{2} ⊕ 36a

6a^{3} ⊕ 84a 1a^{2} ⊕ 14a^{2} ⊕ 36a

6b^{3} ⊕ 84a 1a^{2} ⊕ 14b ⊕ 36a

6a^{2} 1a ⊕ 36a

6b

14a 6a ⊕ 84a

1a^{2} ⊕ 14a^{2} ⊕ 14b 6a ⊕ 6b^{2} ⊕ 84a

1a^{3} ⊕ 14a ⊕ 14b^{3} ⊕ 36a 6a^{2} ⊕ 6b^{2} ⊕ 84a 1a^{2} ⊕ 14a ⊕ 36a^{2}

6a^{3} ⊕ 6b^{3} 1a^{6} ⊕ 14a ⊕ 14b 6a^{3} ⊕ 6b^{2} ⊕ 84a^{2}

1a^{4} ⊕ 14a^{4} ⊕ 14b^{2} ⊕ 36a^{2}

6a^{3} ⊕ 6b^{4} ⊕ 84a^{2} 1a^{4} ⊕ 14a ⊕ 14b ⊕ 36a^{2}

6a^{3} ⊕ 6b^{2} 1a^{5} ⊕ 14a^{2} ⊕ 14b 6a^{2} ⊕ 6b^{3} ⊕ 84a 1a^{2} ⊕ 14b^{2} ⊕ 36a^{2} 6a^{2} ⊕ 6b^{2} ⊕ 84a 1a^{2} ⊕ 14a ⊕ 36a

6a ⊕ 6b^{2} 1a^{2} ⊕ 14a ⊕ 14b

6a^{2} ⊕ 84a 1a ⊕ 14a^{2} ⊕ 36a

6b^{2} ⊕ 84a 1a ⊕ 14b ⊕ 36a

6a^{2} 1a^{2} ⊕ 14a 6b^{2} ⊕ 84a

1a ⊕ 14b^{2} ⊕ 36a

6a^{2} ⊕ 84a 1a ⊕ 14a ⊕ 36a

6b^{2} 1a^{2} ⊕ 14b 6a ⊕ 84a

14a

14b 6b ⊕ 84a

1a^{2} ⊕ 14b^{2} ⊕ 14a 6b ⊕ 6a^{2} ⊕ 84a

1a^{3} ⊕ 14b ⊕ 14a^{3} ⊕ 36a 6b^{2} ⊕ 6a^{2} ⊕ 84a 1a^{2} ⊕ 14b ⊕ 36a^{2}

6b^{3} ⊕ 6a^{3} 1a^{6} ⊕ 14b ⊕ 14a 6b^{3} ⊕ 6a^{2} ⊕ 84a^{2}

1a^{4} ⊕ 14b^{4} ⊕ 14a^{2} ⊕ 36a^{2}

6b^{3} ⊕ 6a^{4} ⊕ 84a^{2} 1a^{4} ⊕ 14b ⊕ 14a ⊕ 36a^{2}

6b^{3} ⊕ 6a^{2} 1a^{5} ⊕ 14b^{2} ⊕ 14a 6b^{2} ⊕ 6a^{3} ⊕ 84a 1a^{2} ⊕ 14a^{2} ⊕ 36a^{2} 6b^{2} ⊕ 6a^{2} ⊕ 84a 1a^{2} ⊕ 14b ⊕ 36a

6b ⊕ 6a^{2} 1a^{2} ⊕ 14b ⊕ 14a

6b^{2} ⊕ 84a 1a ⊕ 14b^{2} ⊕ 36a

6a^{2} ⊕ 84a 1a ⊕ 14a ⊕ 36a

6b^{2} 1a^{2} ⊕ 14b 6a^{2} ⊕ 84a

1a ⊕ 14a^{2} ⊕ 36a

6b^{2} ⊕ 84a 1a ⊕ 14b ⊕ 36a

6a^{2} 1a^{2} ⊕ 14a 6b ⊕ 84a

14b

36a 6b ⊕ 6a

1a^{2} 6b ⊕ 6a

1a^{2} ⊕ 36a ⊕ 14a ⊕ 14b

6b ⊕ 6a ⊕ 84a^{2} 1a^{2} ⊕ 36a^{2} ⊕ 14a^{2} ⊕ 14b^{2}

6b^{3} ⊕ 6a^{3}

1a^{6} 6b^{3} ⊕ 6a^{3}

1a^{4} ⊕ 36a ⊕ 14a^{2} ⊕ 14b^{2} 6b^{2} ⊕ 6a^{2} ⊕ 84a^{3} 36a^{4} ⊕ 14a^{2} ⊕ 14b^{2}

6b^{3} ⊕ 6a^{3}

1a^{6} 6b^{3} ⊕ 6a^{3}

1a^{6} ⊕ 14a^{2} ⊕ 14b^{2} 6b^{2} ⊕ 6a^{2} ⊕ 84a^{2} 36a^{3} ⊕ 14a ⊕ 14b

6b ⊕ 6a

1a^{2} 6b ⊕ 6a

1a^{2} ⊕ 14a ⊕ 14b 6b ⊕ 6a ⊕ 84a^{2} 36a^{2} ⊕ 14a ⊕ 14b

6b ⊕ 6a

1a^{2} 6b ⊕ 6a

1a^{2} ⊕ 14a ⊕ 14b 6b ⊕ 6a ⊕ 84a^{2} 36a^{2} ⊕ 14a ⊕ 14b

6b ⊕ 6a

1a^{2} 6b ⊕ 6a

1a^{2} 6b ⊕ 6a

36a

84a 1a ⊕ 14a ⊕ 14b

6a ⊕ 6b 1a^{2} ⊕ 14a ⊕ 14b 6a ⊕ 6b ⊕ 84a^{2}

1a ⊕ 14a ⊕ 14b ⊕ 36a^{2}

6a^{2} ⊕ 6b^{2}

1a^{4} 6a^{2} ⊕ 6b^{2}

1a^{4} ⊕ 14a^{2} ⊕ 14b^{2} 6a^{2} ⊕ 6b^{2} ⊕ 84a^{3}

1a ⊕ 14a^{2} ⊕ 14b^{2} ⊕ 36a^{3}

6a^{2} ⊕ 6b^{2}

1a^{4} 6a^{2} ⊕ 6b^{2}

1a^{4} ⊕ 14a ⊕ 14b 6a^{2} ⊕ 6b^{2} ⊕ 84a^{2} 14a ⊕ 14b ⊕ 36a^{2}

6a ⊕ 6b

1a^{2} 6a ⊕ 6b

1a^{2} ⊕ 14a ⊕ 14b 6a ⊕ 6b ⊕ 84a^{2} 14a ⊕ 14b ⊕ 36a^{2}

6a ⊕ 6b

1a^{2} 6a ⊕ 6b

1a^{2} ⊕ 14a ⊕ 14b 6a ⊕ 6b ⊕ 84a^{2} 14a ⊕ 14b ⊕ 36a^{2}

6a ⊕ 6b

1a^{2} 6a ⊕ 6b

1a ⊕ 14a ⊕ 14b 84a

4.5 The P.I.M.s of J_{2} in Char. 3

We get faithful condensation for the subgroup H with number 76 in the table of marks. The order of H is 32.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 660. The permutation character 1_{H}^{G} decomposes as

\begin{matrix}1a + 14aabb + 21ab + 36a^{4} + 63a^{5} + 70aabb + 90a^{4} + 126a^{8} + 160a^{6} \\ +175a^{5} + 189aaabbb + 224a^{6}b^{6} + 225a^{5} + 288a^{10} + 300a^{10} + 336a^{12}. \end{matrix}

The dimensions of the modular irreducibles and their condensed dimensions are:

1 13 13 21 21 57 57 133 1 1 1 1 1 1 1 3

The Cartan matrix of the block is:

9 6 6 2 2 1 1 6
6 9 6 1 2 4 2 6
6 6 9 2 1 2 4 6
2 1 2 4 2 0 1 4
2 2 1 2 4 1 0 4
1 4 2 0 1 3 1 2
1 2 4 1 0 1 3 2
6 6 6 4 4 2 2 9

The dimensions of the projective indecomposable modules and their condensed dimensions are:

1161 1404 1404 756 756 594 594 1755 45 48 48 24 24 18 18 57

We take the condensations of the words z9 and z10 in the standard generators. Then these elements together the condensation of z7(z9, z10) and the condensations of standard generators generate the condensed algebra. The resulting socle series for the projective indecomposable modules are as follows.

1a
13b ⊕ 13a
1a^{2} ⊕ 133a^{2}

13b^{2} ⊕ 21a ⊕ 13a^{2} ⊕ 21b 1a^{3} ⊕ 57a ⊕ 57b ⊕ 133a^{2} 13b^{2} ⊕ 21a ⊕ 13a^{2} ⊕ 21b

1a^{2} ⊕ 133a^{2}

13b ⊕ 13a
1a
13a

1a ⊕ 57a ⊕ 133a 13a^{2} ⊕ 13b^{2} ⊕ 21b

1a^{2} ⊕ 57a ⊕ 57b ⊕ 133a^{2}

13a^{3} ⊕ 13b^{2} ⊕ 21a 1a^{2} ⊕ 57a ⊕ 57b ⊕ 133a^{2}

13a^{2} ⊕ 13b^{2} ⊕ 21b 1a ⊕ 57a ⊕ 133a

13a
13b

1a ⊕ 57b ⊕ 133a 13b^{2} ⊕ 21a ⊕ 13a^{2}

1a^{2} ⊕ 57b ⊕ 57a ⊕ 133a^{2}

13b^{3} ⊕ 13a^{2} ⊕ 21b 1a^{2} ⊕ 57b ⊕ 57a ⊕ 133a^{2}

13b^{2} ⊕ 21a ⊕ 13a^{2} 1a ⊕ 57b ⊕ 133a

13b
21a
133a

21a ⊕ 21b ⊕ 13b 1a ⊕ 57b ⊕ 133a

21a ⊕ 13a
1a ⊕ 133a
21b ⊕ 13b
133a
21a

21b 133a

21b ⊕ 21a ⊕ 13a 1a ⊕ 57a ⊕ 133a

21b ⊕ 13b 1a ⊕ 133a 21a ⊕ 13a

133a 21b

57a 13a 133a

13a ⊕ 21b ⊕ 13b 57a ⊕ 1a ⊕ 57b

13a ⊕ 13b 133a 13a 57a

57b 13b 133a

13b ⊕ 21a ⊕ 13a 57b ⊕ 1a ⊕ 57a

13b ⊕ 13a 133a 13b 57b

133a 21a ⊕ 13a ⊕ 21b ⊕ 13b 1a^{2} ⊕ 57a ⊕ 57b ⊕ 133a^{2} 21a ⊕ 13a^{2} ⊕ 21b ⊕ 13b^{2}

1a^{2} ⊕ 133a^{3} 21a ⊕ 13a^{2} ⊕ 21b ⊕ 13b^{2} 1a^{2} ⊕ 57a ⊕ 57b ⊕ 133a^{2} 21a ⊕ 13a ⊕ 21b ⊕ 13b

133a

4.6 The P.I.M.s of J_{2} in Char. 5

We get faithful condensation for the subgroup H with number 111 in the table of marks. The order of the subgroup H is 96.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 95. The permutation character 1_{H}^{G} decomposes as

\begin{matrix}1a + 14ab + 21ab + 36aaa + 63aaa + 70ab + 90aa + 126aa + 160aaa \\ +175aa + 189ab + 224aaabbb + 288aaa + 300aa + 336a^{4}. \end{matrix}

The dimensions of the modular irreducibles and their condensed dimensions are:

1 14 21 41 85 189 1 1 1 1 1 1

The Cartan matrix of the block is:

3 1 2 1 0 0
1 6 3 0 1 3
2 3 7 2 1 3
1 0 2 3 2 1
0 1 1 2 3 2
0 3 3 1 2 6

The dimensions of the projective indecomposable modules and their condensed dimensions are:

100 800 925 525 750 1450 7 14 18 9 9 15

The condensations of the words z1,z5 and z6 in the standard generators are a generating system for e_{H}F Ge_{H}. The resulting socle series for the projective indecomposable modules are as follows.

1a
21a
1a ⊕ 41a ⊕ 14a
21a
1a
14a
14a ⊕ 21a ⊕ 189a
14a^{2} ⊕ 21a ⊕ 189a ⊕ 1a ⊕ 85a
14a ⊕ 21a ⊕ 189a
14a

21a

21a ⊕ 41a ⊕ 189a ⊕ 1a ⊕ 14a

21a^{3} ⊕ 189a ⊕ 85a ⊕ 14a 21a ⊕ 41a ⊕ 189a ⊕ 1a ⊕ 14a

21a

41a 21a ⊕ 85a

41a ⊕ 189a ⊕ 1a

21a ⊕ 85a 41a

85a 189a ⊕ 41a

85a ⊕ 21a ⊕ 14a

189a ⊕ 41a 85a

189a

189a ⊕ 21a ⊕ 85a ⊕ 14a 189a^{2} ⊕ 21a ⊕ 41a ⊕ 14a 189a ⊕ 21a ⊕ 85a ⊕ 14a

189a

4.7 The P.I.M.s of J_{3} in Char. 3

We get faithful condensation for the subgroup H, whose number in the table of marks is 90. It has structure 2^{4}: 5.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 7926. The permutation character 1_{H}^{G} decomposes as

\begin{matrix}1a + 85aabb + 323a^{5}b^{5} + 324a^{4} + 646a^{7}b^{7} + 816a^{8} + 1140a^{18} + 1215a^{18}b^{18} \\ +1615a^{23} + 1920a^{24}b^{24}c^{24} + 1938a^{25}b^{25} + 2432a^{32} + 2754a^{31} + 3078a^{35}. \end{matrix}

The dimensions of the modular irreducibles over F_{9} and their condensed dimensions are:

1 18 18 84 84 153 153 934 1 1 1 1 1 1 1 12

The Cartan matrix of the block is:

62 26 26 47 47 40 40 19
26 16 14 23 23 23 22 10
26 14 16 23 23 22 23 10
47 23 23 40 39 37 37 16
47 23 23 39 40 37 37 16
40 23 22 37 37 40 38 14
40 22 23 37 37 38 40 14
19 10 10 16 16 14 14 8

The dimensions of the corresponding projective indecomposable modules over F_{9} and their condensed dimensions are:

38880 20655 20655 33777 33777 32076 32076 14823 516 267 267 438 438 405 405 195

We apply Theorem 3.3.9 to find generators for the condensation algebra with respect to the subgroup H = 2^{4}: 5. We first take the words given explicitly for the normalizer N = 2^{4}: 15 whose subgroup number in the table of marks of J_{3} is 113, see [Mer96]. The two generators n_{1} and n_{2} given in the table of marks are of order 2 and 15, and one verifies using GAP that the group generated by n_{1} and n_{2}^{3} is the condensation subgroup H. In order to calculate the imprimitive permutation representation for the standard generators of J_{3} on N we us the GAP procedure imprimpermcan given in the Appendix, page 139. More precisely, the maximal subgroup M of type L_{2}(16): 2 with subgroup number 136 contains N, so we can use it to factorize the calculation of the coset representatives. The procedure imprimpermcan is a special version of the GAP procedure imprimperm. It requires that M is the point stabilizer of the permutation representation of J_{3} given for the standard generators. (Moreover, it requires that the subgroup N is a subgroup of the point stabilizer.) It is easily verified using GAP that if we conjugate the standard generators of J_{3} by an appropriate permutation, then all the requirements are fulfilled and furthermore the point stabilizer acts faithfully on the first 86 points. (The permutation can explicitly constructed by sorting the orbits of the maximal subgroup M according to increasing length.)

In the resulting condensed module of dimension 2642 we prove using the MeatAxe program zsp, that the condensations of the following words in the standard generators for J_{3} generate the full condensed algebra: z3 and z5 and their product z3 ∗ z5, to which we have to add the second generator of N by Theorem 3.3.9, whose definition is best looked up in the GAP-library of table of marks.

The C-MeatAxe programs chop and pwkond are used to find the peakwords for the simple constituents. We calculate the stable nullspace for the peakword of the simple module 12a. The closure of a randomly chosen vector in the kernel gives a module of dimension 195, which must be the projective indecomposable module belonging to 12a, since the dimension of the projective indecomposable is exactly 195.

We next construct the permutation representation of J_{3} on a subgroup U of order 320, which is unique up to conjugation in J_{3}. This is done by constructing matrices for the generators of U in one of the 153-dimensional representation over F_{9}. We use the explicit description of the generators of U in form of words in the standard generators to show that there is a unique one-dimensional subspace fixed by U up to scalar multiplication. The stabilizer of this one-dimensional subspace is U in both of the 153-dimensional irreducible representations. We verify this by constructing the corresponding permutation representation by applying the C-MeatAxe program zvp to the matrices of J_{3} and a nonzero vector in the one-dimensional fixed space of U. We also construct permutations for the generators of H using the words in the standard generators given for the generators of H in the table of marks. The resulting condensed permutation module has dimension 2022 and by using chop and pwkond, we can determine the peakword for the 1dimensional simple condensed module that corresponds to the trivial F_{9}J_{3} module. A random vector taken from the stable nullspace gives us a (cyclic) module of dimension 516, which therefore has to be the condensed projective cover of the trivial module.

In the final step, we construct the condensed permutation module on H itself. This is done in an analogous fashion to the case of constructing the permutations on N. It takes about one hour of CPU time to determine the permutations for the standard generators of J_{3} and for the generators of N from which we can immediately derive generators for the condensation subgroup.

The condensed module has dimension 7926. The condensed permutation module on the cosets of N already contains all simple modules of the condensed algebra and so we already can determine peakwords for all the simple condensed modules. Now we proceed by determining explicitly the stable nullspaces for the peakwords of the three simple condensed modules of dimension 2. Random vectors in these kernels will then generate the corresponding projective indecomposable modules, since by Lemma 4.1.7 the probability that a random vector generates a projective indecomposable module, is quite high. Moreover, we can easily check whether the generated module is the projective cover in question. Its dimension is at most that of the projective cover and it is the projective cover if and only if its dimension is that of the projective cover. Note that the dimension of the projective cover can be read off from the tables given above.

Observe that these calculations take about 12 hours of CPU time and this is one of the reasons why we constructed the projective covers for the trivial and the 12-dimensional simple module as direct summands of smaller condensed modules.

The following information can be read off from the Brauer character table of J_{3}. All simple modules but 84a and 84b are self-dual and the simple modules 1a and 934a are the simple modules in the principal block fixed by the Frobenius automorphism. For each pair of simple F_{9}J_{3}-modules interchanged by the Frobenius automorphism, we only list the socle layers of one of them , since the layers of the socle series of the other can be derived from it by applying the Frobenius automorphism to the simple constituents of the socle layers.

1a
84a^{2} ⊕ 84b^{2}

1a^{7} ⊕ 153a ⊕ 153b ⊕ 934a^{3} 84a^{4} ⊕ 84b^{4} ⊕ 18a^{4} ⊕ 18b^{4} 1a^{7} ⊕ 153a^{3} ⊕ 153b^{3} ⊕ 934a^{2} 84a^{4} ⊕ 84b^{4} ⊕ 18a^{3} ⊕ 18b^{3} 1a^{7} ⊕ 153a^{4} ⊕ 153b^{4} ⊕ 934a 84a^{6} ⊕ 84b^{6} ⊕ 18a ⊕ 18b

1a^{10} ⊕ 153a^{4} ⊕ 153b^{4} ⊕ 934a^{2} 84a^{4} ⊕ 84b^{4} ⊕ 18a^{2} ⊕ 18b^{2} 1a^{5} ⊕ 153a^{4} ⊕ 153b^{4} ⊕ 934a^{2} 84a^{5} ⊕ 84b^{5} ⊕ 18a^{3} ⊕ 18b^{3} 1a^{6} ⊕ 153a^{4} ⊕ 153b^{4} ⊕ 934a^{2} 84a^{3} ⊕ 84b^{3} ⊕ 18a^{4} ⊕ 18b^{4}

1a^{4} ⊕ 153a^{4} ⊕ 153b^{4} 84a^{5} ⊕ 84b^{5} ⊕ 18a^{2} ⊕ 18b^{2} 1a^{2} ⊕ 153a^{3} ⊕ 153b^{3} ⊕ 934a^{2}

84a^{3} ⊕ 84b^{3} ⊕ 18a ⊕ 18b 1a^{2} ⊕ 153a^{2} ⊕ 153b^{2} ⊕ 934a 84a^{2} ⊕ 84b^{2} ⊕ 18a ⊕ 18b

1a^{4} ⊕ 153a^{2} ⊕ 153b^{2} ⊕ 934a^{2}

84a^{2} ⊕ 84b^{2} ⊕ 18a ⊕ 18b

1a^{2} ⊕ 153a^{2} ⊕ 153b^{2} 84a^{2} ⊕ 84b^{2} ⊕ 18a ⊕ 18b

1a^{2} ⊕ 153a^{2} ⊕ 153b^{2} ⊕ 934a^{2} 84a ⊕ 84b ⊕ 18a ⊕ 18b 1a^{2} ⊕ 153a^{2} ⊕ 153b^{2} 84a ⊕ 84b ⊕ 18a ⊕ 18b

153a ⊕ 153b

84a ⊕ 84b
153a ⊕ 153b

84a ⊕ 84b ⊕ 18a ⊕ 18b 153a ⊕ 153b

84a ⊕ 84b
1a

18a 153b ⊕ 934a

18a ⊕ 84a^{2} ⊕ 18b ⊕ 84b

1a^{4} ⊕ 153a^{2} ⊕ 934a

18b^{2} ⊕ 84b^{3} 153b^{2} ⊕ 1a^{3} ⊕ 153a ⊕ 934a

18a^{3} ⊕ 84a^{3} ⊕ 84b 153b^{3} ⊕ 1a 18a ⊕ 84a^{4}

153b ⊕ 1a^{2} ⊕ 153a^{3} ⊕ 934a 84a ⊕ 18b^{4} ⊕ 84b^{3} 1a^{3} ⊕ 153a^{4} ⊕ 934a 18a ⊕ 18b ⊕ 84b^{5}

153b^{4} ⊕ 1a^{4} ⊕ 153a ⊕ 934a

18a^{4} ⊕ 84a^{2} ⊕ 84b 153b^{4} ⊕ 1a^{2} 18a ⊕ 84a^{5}

153b ⊕ 1a ⊕ 153a^{3} ⊕ 934a

84a ⊕ 18b^{3} ⊕ 84b 1a ⊕ 153a^{3} 18b ⊕ 84b^{4}

153b ⊕ 1a ⊕ 153a ⊕ 934a 18a ⊕ 84b 153b^{2} ⊕ 1a 18a ⊕ 84a^{2}

153b ⊕ 1a ⊕ 934a

84a 1a ⊕ 153a 18b ⊕ 84b 153a ⊕ 934a

84b 153b ⊕ 1a 18a ⊕ 84a

153b 84a 153a 18b 153a 84b 153b 18a

84a

1a^{2} ⊕ 153a ⊕ 934a 84a ⊕ 18b^{2} ⊕ 84b^{3} ⊕ 18a 1a^{4} ⊕ 153a ⊕ 153b^{2} ⊕ 934a

84a^{3} ⊕ 84b ⊕ 18a^{3} 1a^{4} ⊕ 153a^{2} ⊕ 153b^{3} ⊕ 934a^{2} 84a^{4} ⊕ 18b^{3} ⊕ 84b^{3} ⊕ 18a 1a^{6} ⊕ 153a^{4} ⊕ 153b^{2} ⊕ 934a

84a^{3} ⊕ 18b^{4} ⊕ 84b^{5} 1a^{4} ⊕ 153a^{4} ⊕ 153b^{2} ⊕ 934a 84a^{2} ⊕ 18b ⊕ 84b^{5} ⊕ 18a^{3} 1a^{5} ⊕ 153a ⊕ 153b^{5} ⊕ 934a

84a^{5} ⊕ 84b ⊕ 18a^{5} 1a^{3} ⊕ 153a^{2} ⊕ 153b^{5} ⊕ 934a^{2} 84a^{6} ⊕ 18b^{2} ⊕ 84b^{2} ⊕ 18a 1a^{5} ⊕ 153a^{6} ⊕ 153b ⊕ 934a

84a ⊕ 18b^{5} ⊕ 84b^{4} 1a^{3} ⊕ 153a^{5} ⊕ 153b ⊕ 934a

18b ⊕ 84b^{6} ⊕ 18a 1a^{2} ⊕ 153a ⊕ 153b^{4} ⊕ 934a

84a^{2} ⊕ 84b ⊕ 18a^{4} 1a^{2} ⊕ 153b^{5} ⊕ 934a

84a^{6} ⊕ 18a 1a^{2} ⊕ 153a^{2} ⊕ 153b ⊕ 934a

84a ⊕ 18b^{2} ⊕ 84b^{2}

1a ⊕ 153a^{3}
18b ⊕ 84b^{3}

1a ⊕ 153a ⊕ 153b ⊕ 934a 84a ⊕ 84b ⊕ 18a 1a ⊕ 153b^{2} ⊕ 934a

84a^{2} ⊕ 18a
1a ⊕ 153a ⊕ 153b
84a ⊕ 18b
1a ⊕ 153a^{2}
18b ⊕ 84b
153a
84b
153b
18a
153b
84a

153a
18b ⊕ 84b

153a ⊕ 153b ⊕ 1a ⊕ 934a

84b ⊕ 18a^{2} ⊕ 84a^{2} 153a ⊕ 153b^{2} ⊕ 1a^{3} ⊕ 934a 18b^{2} ⊕ 84b^{2} ⊕ 18a ⊕ 84a^{3} 153a^{3} ⊕ 153b^{2} ⊕ 1a^{4} ⊕ 934a

18b^{3} ⊕ 84b^{4} ⊕ 84a^{2} 153a^{4} ⊕ 153b^{2} ⊕ 1a^{4} ⊕ 934a 18b ⊕ 84b^{4} ⊕ 18a^{3} ⊕ 84a^{2} 153a ⊕ 153b^{5} ⊕ 1a^{4} ⊕ 934a

84b ⊕ 18a^{4} ⊕ 84a^{5} 153a^{3} ⊕ 153b^{4} ⊕ 1a^{4} ⊕ 934a 18b^{4} ⊕ 84b^{2} ⊕ 18a ⊕ 84a^{5} 153a^{7} ⊕ 153b ⊕ 1a^{4} ⊕ 934a

18b^{4} ⊕ 84b^{6} ⊕ 84a 153a^{4} ⊕ 153b^{3} ⊕ 1a^{3} ⊕ 934a 18b ⊕ 84b^{5} ⊕ 18a^{3} ⊕ 84a 153a ⊕ 153b^{5} ⊕ 1a^{2} ⊕ 934a

84b ⊕ 18a^{3} ⊕ 84a^{4} 153a ⊕ 153b^{4} ⊕ 1a^{2} ⊕ 934a

18b ⊕ 18a ⊕ 84a^{5} 153a^{3} ⊕ 153b ⊕ 1a^{2} ⊕ 934a

18b^{2} ⊕ 84b^{2} ⊕ 84a 153a^{3} ⊕ 1a^{2} ⊕ 934a

18b ⊕ 84b^{3} 153a ⊕ 153b ⊕ 1a^{2} ⊕ 934a

84b ⊕ 18a ⊕ 84a

153b^{2} ⊕ 1a ⊕ 934a
18a ⊕ 84a^{2}
153a ⊕ 153b ⊕ 1a
18b ⊕ 84b ⊕ 84a
153a^{2} ⊕ 1a
18b ⊕ 84b^{2}
153a ⊕ 153b
84b ⊕ 18a
153b^{2}
18a ⊕ 84a
153a ⊕ 153b
18b ⊕ 84a
153a

934a

84a ⊕ 84b ⊕ 18b ⊕ 18a 1a^{3} ⊕ 153a ⊕ 153b ⊕ 934a 84a ⊕ 84b ⊕ 18b ⊕ 18a

1a^{2} ⊕ 153a ⊕ 153b 84a^{2} ⊕ 84b^{2} ⊕ 18b ⊕ 18a 1a ⊕ 153a ⊕ 153b ⊕ 934a

84a ⊕ 84b 1a^{2} ⊕ 153a ⊕ 153b

84a ⊕ 84b ⊕ 18b ⊕ 18a 1a^{2} ⊕ 153a ⊕ 153b ⊕ 934a^{2}

84a ⊕ 84b ⊕ 18b ⊕ 18a

1a^{2} ⊕ 153a ⊕ 153b 84a^{2} ⊕ 84b^{2} ⊕ 18b ⊕ 18a

153a ⊕ 153b ⊕ 934a

84a ⊕ 84b 1a^{2} ⊕ 153a ⊕ 153b

84a ⊕ 84b ⊕ 18b ⊕ 18a

1a ⊕ 153a ⊕ 153b

84a ⊕ 84b 1a^{2} ⊕ 153a ⊕ 153b

84a ⊕ 84b ⊕ 18b ⊕ 18a 153a ⊕ 153b ⊕ 934a

84a ⊕ 84b 1a^{2} ⊕ 153a ⊕ 153b

18b ⊕ 18a 153a ⊕ 153b

84a ⊕ 84b

153a ⊕ 153b 84a ⊕ 84b ⊕ 18b ⊕ 18a

934a

4.8. THE P.I.M.S OF W (E_{6}) IN CHAR. 3 105

4.8 The P.I.M.s of W (E_{6}) in Char. 3

We consider the Weyl group of type E_{6}, which is isomorphic to U_{4}(2) : 2, and its projective indecomposable modules in characteristic 3.

We define the condensation subgroup H to be a Sylow 5-subgroup of U_{4}(2): 2 which is cyclic of order 5. The scalar product (1_{H}^{G}, 1_{H}^{G}) is 2080.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 1 5 5 10 10 14 14 25 25 81 81 1 1 1 1 1 2 2 2 5 5 17 17

The Cartan matrix for G is:

6 1 4 2 2 1 2 3 0 1 0 0
1 6 2 4 1 2 3 2 1 0 0 0
4 2 10 5 5 4 4 2 4 2 0 0
2 4 5 10 4 5 2 4 2 4 0 0
2 1 5 4 12 9 2 4 4 5 0 0
1 2 4 5 9 12 4 2 5 4 0 0
2 3 4 2 2 4 6 1 2 0 0 0
3 2 2 4 4 2 1 6 0 2 0 0
0 1 4 2 4 5 2 0 5 2 0 0
1 0 2 4 5 4 0 2 2 5 0 0
0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 1

We use the generators for G given in the Atlas of group representations, see [Wil96a]. Note that there is currently no definition of standard generators for G and so we give the 5 by 5 matrices over F_{3} for the two generators listed there.

( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 2 & 2 \end{matrix} ) .

It is easily checked using the C-MeatAxe program zvp that the vector ( 1 2 2 0 0 ) gives rise to a G-orbit of length 36. We will use the permutations of degree 36 for the two generators in our GAP calculations.

As usual, we construct the permutation representation of G on the cosets of H by using the GAP-procedure imprimperm and we choose the normalizer of H in G as the required subgroup containing H. We verify that the condensations of the words z1, …, z6 in the words z7 and z8 of the above chosen generators generate the algebra e_{H}F Ge_{H}. We then use the C-MeatAxe program chop and pwkond to get the constituents and their peakwords.

There are two simple F G-modules 25^{+} and 25^{−} of degree 25 in the principal block with 25^{−} = 25^{+} ⊗ 1^{−}, where 1^{−} is the nontrivial extension of the trivial representation of U_{4}(2). The radical series of the projective cover of the 25^{+} is stated wrongly in [Ben91]. We can prove this easily using the Morita equivalence. We first pick a random vector in the stable nullspace of the peakword for 5a, the first five dimensional simple condensed module, and construct the corresponding projective indecomposable module using the C-MeatAxe program zsp. Note that 5a w.l.o.g. corresponds to the first simple F G-module 25^{+} of degree 25 by the table given above. All simple F Gmodules are self-dual as can be checked by investigating the Brauer character table. It follows that the layers of the socle series of 25^{+} are the layers of the radical series of 25^{+} and so we are allowed use the C-MeatAxe program socleser to determine the layers of the radical series. The resulting layers of the radical series are as follows.

5a
1a ⊕ 2a
1b ⊕ 2b ⊕ 2c ⊕ 5a
1a^{2} ⊕ 1c ⊕ 2a ⊕ 5b
1b ⊕ 2b ⊕ 2c ⊕ 5a
1a ⊕ 2a
5a
2a
2c
5b
2c
2a
5a

Applying the Morita equivalence we get the layers of the radical series for the projective cover of the 25^{+} in G.

25^{+}
10^{−} ⊕ 5^{−}
10^{+} ⊕ 14^{+} ⊕ 5^{+} ⊕ 25^{+}
10^{−} ⊕ 10^{−} ⊕ 1^{−} ⊕ 5^{−} ⊕ 25^{−}
10^{+} ⊕ 14^{+} ⊕ 5^{+} ⊕ 25^{+}
10^{−} ⊕ 5^{−}
25^{+}
5^{−}
5^{+}
25^{−}
5^{+}
5^{−}
25^{+}

So the radical length for this projective indecomposable module is 13 and not 9 as stated in [Ben91]. We also checked the layers of the radical series for the other projective indecomposable modules given in [Ben91].

108 CHAPTER 4. EXAMPLES FOR MORITA EQUIVALENCES

An Algorithm for H^{1}(G, V)

In this chapter we describe an algorithm for calculating an upper bound for the dimension of the first cohomology group H^{1}(G, V) for an F_{p}G-module V. In the examples we will give, we will verify that the upper bound is exact using the information about the p-modular decomposition matrix of G and the structure of certain reducible F_{p}G-modules.

5.1 The Algorithm

In order to describe the algorithm we use the following group theoretical interpretation of the first cohomology group. Let E = {(v, g)|v ∈ V, g ∈ G} be the semidirect product of the F_{p}G-module V with G. Note that the product in E of two elements (v_{1}, g_{1}) and (v_{2}, g_{2}) is given by (v_{1}g_{2} + v_{2}, g_{1}g_{2}). A complement to V in E is a subgroup K of the form {(d_{K}(g), g)|g ∈ G, d_{K}(g) ∈ V } and we consider d_{K} as a map from G to V.

Definition 5.1.1. Let V be an F_{p}G-module. A map d: G → V is called a derivation if d(gh) = d(g)h + d(h) for all g, h ∈ G. The set of all derivations Der(G, V) is a group under pointwise addition. Moreover, for a given v ∈ V the map d_{v}: G → V with d_{v}(g) = v(1 − g) is a derivation, called an inner derivation, and the subset of all inner derivations Ider(G, V) is a subgroup of Der(G, V). The quotient group H^{1}(G, V) = Der(G, V)/ Ider(G, V) is called the first cohomology group of G with coefficients in V.

Remark 5.1.2. Note that the map d_{K} defined above corresponding to a complement K is a derivation. In the algorithm for determining an upper bound for H^{1}(G, V) we will exploit the fact that a derivation is uniquely determined by its restriction to a generating set of G.

110 CHAPTER 5. AN ALGORITHM FOR H^{1}(G, V)

A description of all complements in terms of derivations can be given as follows.

Lemma 5.1.3. Let V be an F_{p}G-module and let V ⋊ G be the semidirect product of V and G. Then to each complement K in V ⋊ G there exists a unique d ∈ Der(G, V) such that K is of the form

K = {(d(g), g)|g ∈ G}.

Conversely, a derivation d defines a complement K = {d(g), g)|g ∈ G}. Moreover, two complements corresponding to the derivations d_{1} and d_{2} are conjugate in V ⋊ G if and only if d_{1} − d_{2} is an inner derivation.

Proof: See [Cel92]. ■

In order to determine the conjugacy classes of complements we first have to determine the derivations from G to V and then take the quotient modulo Ider(G, V). In order to reduce the amount of work we have to do, we use the fact mentioned above, that derivations are uniquely determined by their values on a generating set {g_{1}, …, g_{r}} of G.

Let us suppose for the moment that we are given a presentation of G in terms of abstract generators {x_{1}, …, x_{r}} and defining relations r_{1}(g_{1}, …, g_{r}) = 1, …, r_{m}(g_{1}, …, g_{r}) = 1. This means that G ≅ F/R, where F is a free group on x_{1}, …, x_{r} and R is generated by r_{1}, …, r_{m} as a normal subgroup of F. It is easy to see that a map d: G → V is a derivation if and only if the elements ĝ_{i} = (d(g_{i}), g_{i}) satisfy the relations r_{1}, …, r_{m}, i.e., if r_{j}(ĝ_{1}, …, ĝ_{r}) = 1_{V ⋊G} for j = 1, …, m. Let us consider a fixed relation r(ĝ_{1}, …, ĝ_{r}) = 1. Rewriting this equation using the definitions of ĝ_{i}, i = 1, …, r, and the multiplication in the semidirect product V ⋊ G, we get

((d(g_{1})w_{1}) + (d(g_{2})w_{2}) + … + (d(g_{r})w_{r}), r(g_{1}, …, g_{r})) = 1_{V ⋊G}

where w_{1}, …, w_{r} are words in g_{1}, …, g_{r}. Since G acts linearly on V, the above equation gives a linear equation in the d(g_{i}). In this way we have reduced the problem of determining the derivations Der(G, V) to a problem of linear algebra, namely solving the system of linear equations derived form r_{1}, …, r_{m}.

However, for the groups we want to study, we usually do not want to consider a presentation at all. The number of generators in presentations for simple groups G will be quite large and furthermore the F_{p}G-modules V we want to study are usually only given in terms of the matrices in the standard generators of G, see [Wil96a]. Instead of working with a presentation of G we use relations in G given by the orders of random products in the generators. These give necessary conditions on a function d: G → V to be a derivation.

Hence, for the price of feasibility of the calculations we only get upper bounds for H^{1}(G, V). In practice however these upper bounds turned out to be quite accurate and as already mentioned, in the examples, we can prove that they are actually exact.

Before we describe the complete algorithm, we use the following observation, already described in [Cel92], to reduce the number of unknowns in the system of linear equations we have to solve.

Theorem 5.1.4. Let G be generated by g_{1}, …, g_{r} and assume that g_{1} is of order coprime to p. Let Der_{0}(G, V) be the subgroup of Der_{0}(G, V) with d(g_{1}) = 0 and let Ider_{0}(G, V) be the corresponding subgroup of Ider(G, V). Then Der(G, V) = Der_{0}(G, V) + Ider(G, V). Hence

H^{1}(G, V) ≅ Der_{0}(G, V)/ Ider_{0}(G, V).

Note that a map d_{v} is in Ider_{0}(G, V), if and only if d_{v}(g_{1}) = (v − 1)g_{1} = v − vg_{1} = 0, i.e., if v is a fixed vector of g_{1}.

Hence, in the case of a group G generated by two elements g_{1} and g_{2} with g_{1} of order coprime to the characteristic then we only have to solve linear equations in dim_{F_{p}}(V) unknowns. This property is easily verified in all the examples given below. The resulting algorithm now reads as follows.

Algorithm 5.1.5 (first cohomology group).

INPUT: Two invertible n by n matrices M and N over F_{p}, the first one being of order coprime to p and the number limit of random elements used. CALCULATION:

T := [0, M]; S := [ID(n), N]; el := [ID(n), N]; boundaries := (nullspace(M − ID)) ∗ (N − ID); rel := []; count := 0; \underline{while} (count < limit) \underline{do}

count := count + 1;

# limit bounds the number of random elements generated flag := RandomInt(2); # RandomInt(2) generates (pseudo) randomly the numbers 1 and 2 \underline{if} (flag = 1) \underline{then}

semmul(el, T);

# semmul implements the product # in the semidirect product

\underline{else}

semmul(el, S);

\underline{fi};

n := order(el[2]);

# order returns the order of an invertible matrix

newrel := sempower(el, n);

# sempower implements the power of an element

# in the semidirect product

rel := semiechelonize(rel, newrel[1]);

# semiechelonize determines a semi-echelonized basis of the # space spanned by rel and newrel

\underline{od}; cyclespace := nullspace(transposed(rel));

OUTPUT: The difference dim_{F_{p}}(cyclespace) − dim_{F_{p}}(boundaries), which is an upper bound for the dimension of the first cohomology group of the group G generated by M and N.

5.2 Applications

We use Algorithm 5.1.5 to give upper bounds for the first cohomology groups of all simple F_{2}G modules for the sporadic simple groups J_{3}, HS, M_{24} and McL. Using the results about the socle series for the other sporadic simple groups up to order 10^{9} over F_{2}, this will verify in these cases Guralnick’s conjecture, i.e., that the dimension of the first cohomology group H^{1}(G, V) is at most 2 for an absolutely irreducible F_{p}G-module V.

We proceed as follows. First, we will construct all simple F_{2}G-modules by using the algorithms given in Chapter 2. The starting point are the irreducible representations over F_{2} given in R.A. Wilson’s Atlas of group representations, see [Wil96a]. On the other hand, we use the Brauer character table as given in [JLPW95] and in GAP to determine the necessary tensor products and symmetrizations that contain all the irreducible representations listed below.

There is an alternative interpretation of the first cohomology group: the elements of H^{1}(G, V) are in one-to-one correspondence with the equivalence classes of extensions of the trivial F_{p}G-module 1a with V, see for example [Ben91]. Moreover, in the case that V is a simple F_{p}G-module, then the dimension of H^{1}(G, V) is the product of dim_{F} (End_{F_{p}G}(V)) and the multiplicity of V in the the second layer of the radical series of the projective cover of the trivial module. So to prove that the bound the algorithm gives is correct, we merely have to construct an F_{p}G-module M such that M/ Rad(M) ≅ 1a and the multiplicity of V in Rad(M)/ Rad^{2}(M) is dim_{F_{p}}(H^{1}(G, V))/ dim_{F_{p}}(End_{F_{p}G}(V)). Observe that this also implies that we only have to investigate simple F_{p}G-modules in the principal block.

A second tool in proving a lower bound for H^{1}(G, V) is the following.

Lemma 5.2.1. Let R a complete, discrete valuation ring with maximal ideal π such that R/π = F_{p} and with quotient field K. Let P (M) be the projective cover of the simple F_{p}G-module M with projective indecomposable character Φ(M). Let χ_{1} and χ_{2} be ordinary K-characters of G such that Φ(M) = χ_{1} + χ_{2}. Then there exists an indecomposable R-lattice L with K-character χ_{1} such that Soc(L/πL) ≅ M.

Proof: See [Lan83], Corollary 17.4. ■

Remark 5.2.2. We will apply the preceding lemma in the case that G is a simple group and M is the trivial F_{p}G-module. Let Φ(M) = χ_{1} + χ_{2}, where χ_{1} is an ordinary irreducible character whose reduction mod p is χ̄_{1} = 1_{G} + ϕ_{2} + … + ϕ_{r}. Then we conclude that at least for one of the simple F_{p}G-modules corresponding to ϕ_{i} the first cohomology group is nontrivial. A typical application will be the case that all summands but one of χ̄_{1} are already known to have trivial cohomology by Algorithm 5.1.5.

The following table gives the upper bounds for the irreducible representations of M_{24} determined by Algorithm 5.1.5.

1a 11a 11b 44a 44b 120a 220a 220b
dim_{F_{2}}(H^{1}(M_{24}, V)) 0 0 1 1 1  0  1  0

\begin{matrix}252a & 320a & 320b & 1242a & 1792a \\ 1 & 0 & 0 & 1 & 0 \end{matrix}

The decompositions of the ordinary irreducible characters χ_{2} and χ_{4} show that 11b and 44a, 44b have nontrivial cohomology. In order to show that 220a and 252a have nontrivial cohomology we analyze the tensor product of 11a and 44b, which contains two uniserial subquotients with composition factors 1a and 220a resp. 1a and 252a. Finally, the tensor product of 44a and 44b contains a uniserial subquotient with composition factors 1242a and 1a, which shows that the bound for 1242a is correct.

The following table gives the upper bounds for the irreducible representations of HS.

1a 20a 56a 132a 518a 1000a
dim_{F_{2}}(H^{1}(HS, V)) 0 2 0  1  1  1

The decomposition of the ordinary irreducible character χ_{2} shows that the bound for 20a is correct. The exterior square of the 20a contains a uniserial submodule with composition factors 1a and 132a, which shows that the bound for 132a is correct. Since the permutation module of HS on the cosets of L_{3}(4): 2_{1} contains a uniserial subquotient with composition factors 518a and 1a, it follows that the bound for 518a is correct. Finally, we observe that the tensor product of 20a and 132a contains a uniserial subquotient whose composition factors are 1a and 1000a, which implies that the upper bound for 1000a is correct.

The following table gives the upper bounds for the irreducible representations over F_{2} of J_{3}.

1a 156a 80a 168a 244a 644a 966a
dim_{F_{2}}(H^{1}(J_{3}, V)) 0 0  0  2  1  2  0

Note that the splitting field for 156a, 168a and 644a is the field with four elements. The ordinary irreducible characters χ_{2}, χ_{3},χ_{4} and χ_{5} show that the bounds for 168a and 644a are exact. We now study the tensor product of two irreducible representations of the triple cover 3J_{3}, namely the representation 9a and a representation of degree 153 such that their tensor product is a module for the factor group J_{3}. This module has a uniserial subquotient with composition factors 244a and 1a, and this proves that the upper bound for 244a is exact.

The construction of the irreducible representations of McL is more complicated. We follow Thackray, see [Tha81]. In principle, the most difficult task is the construction of the representations of degree 2124 and 3584. The 3584a is a constituent of the exterior fourth power of the 22-dimensional irreducible representation, and we leave it to the C-MeatAxe program chop to analyze the composition factors completely. The representation of degree 2124 is a constituent of the permutation representation of McL on the cosets of the maximal subgroup 3_{+}^{1+4}: S_{5}. This subgroup fixes a vector in the 230dimensional representation and this vector is contained in the fixed space of an element of order 30, which is 10-dimensional. Hence the C-MeatAxe program zvp can produce the required permutation representation quite easily. We then condense with respect to a subgroup H of order 11, which unveils a subspace containing the condensed representation corresponding to the representation of degree 2124. Using the C-MeatAxe program zuk, it is possible to calculate the closure of this subspace in the original permutation representation. Since 748a and 748b, and 2124a and 2124b are conjugate under the outer automorphism, we only have to apply Algorithm 5.1.5 to 748a and 2124a.

1a 22a 230a 748a 748b 2124a 2124b 3584a
dim_{F_{2}}(H^{1}(McL, V)) 0 0  1  1  1  0  0  0

The ordinary irreducible characters χ_{3} and χ_{14} show that the bounds are exact.

We finally apply Algorithm 5.1.5 to the 4370-dimensional representation of the Baby Monster BM given on the standard generators of order 2 and 3, constructed by R.A. Wilson, see [Wil93]. The algorithm shows that the first cohomology group is at most 1-dimensional. On the other hand, the ordinary irreducible character χ_{2} which reduced mod 2 is the sum of the trivial character and the Brauer character of the irreducible representation of degree 4370 shows that the bound is exact. Hence, by the lemma above, it follows that H^{1}(BM, V) is 1-dimensional.

116 CHAPTER 5. AN ALGORITHM FOR H^{1}(G, V)

Faithful Condensation for Sporadic Groups

In this chapter, we determine a subgroup H suitable for condensation in the cases not covered in the example sections of Chapter 4. For the cases where the socle series of the projective indecomposable modules in the principal block B can be found in the literature we give an explicit generating set for the condensation algebra e_{H}F Ge_{H}. This allows us to recover the results on the layers of the socle series but it also allows us to use Morita equivalence when we want to study a given module in B. The cases where the layers of the socle series are not yet known could in principle be attacked with the method presented in Chapter 4 but lack of time and lack of computer power and memory have prevented us from doing so. In each of the following sections we proceed as described in chapter 4. Finally, we give an explicit generating set for the condensation algebra e_{H}F Ge_{H}. These generators are condensations of elements given as words in the standard generators of G and the definition of these words can be found in the Appendix, page 141. As mentioned above, this enables us given an F G-module V in terms of the matrices in the standard generators of G to construct explicitly the condensed e_{H}F Ge_{H}-module V e_{H} in terms of the matrices in the generating set for e_{H}F Ge_{H}.

6.1 Faithful Condensation for M_{11} in Char. 2

Faithful condensation for the subgroup H with number 13 in the table of marks. The group H is a Sylow 3-subgroup.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 112. The permutation character decomposes as

1a + 10aabbcc + 11aaa + 44a^{4} + 45a^{5} + 55a^{7}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 10 44 1 2 4

The Cartan matrix of the block is:

4 2 2
2 5 1
2 1 3

The dimensions of the projective indecomposable modules and their condensed dimensions are:

112 96 144 16 16 16

6.2. FAITHFUL CONDENSATION FOR M_{11} IN CHAR. 3 119

6.2 Faithful Condensation for M_{11} in Char. 3

Faithful condensation for subgroup H with number 6 in the table of marks. H is a Sylow 5-subgroup.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 320. The permutation character decomposes as

1a + 10aabbcc + 11aaa + 16a^{4}b^{4} + 44a^{8} + 45a^{9} + 55a^{11}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 5 5 10 10 10 24 1 1 1 2 2 2 4

The Cartan matrix of the block is:

5 3 3 0 2 2 1
3 4 3 1 1 2 2
3 3 4 1 2 1 2
0 1 1 2 0 0 1
2 1 2 0 3 1 1
2 2 1 0 1 3 1
1 2 2 1 1 1 2

The dimensions of the projective indecomposable modules and their condensed dimensions are:

99 126 126 54 81 81 99 23 26 26 10 17 17 19

6.3 Faithful Condensation for M_{12} in Char. 2

Faithful condensation for the subgroup H with number 35 in the table of marks. The order of H is 9.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 1216. The permutation character decomposes as

\begin{matrix}1a + 11aaabbb + 45a^{5} + 54a^{6} + 55a^{7}b^{7}c^{7} + 66a^{10} + 99a^{11} + 120a^{16} \\ +144a^{16} + 176a^{16}. \end{matrix}

The dimensions of the modular irreducibles and their condensed dimensions are:

1 10 44 1 2 4

The Cartan matrix of the block is:

16 16 12
16 20 14
12 14 14

The dimensions of the projective indecomposable modules and their condensed dimensions are:

704 832 768 96 112 96

The condensations of the elements z1, …, z5 in the standard generators generate e_{H}F Ge_{H}.

6.4 Faithful Condensation for M_{12} in Char. 3

Faithful condensation for the subgroup H with number 50 in the table of marks. The order of the subgroup is 16.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 424. The permutation character decomposes as

\begin{matrix}1a + 11aabb + 16aabb + 45aaa + 54a^{8} + 55a^{5}bbcc + 66a^{6} + 99a^{7} + 120a^{4} \\ +144a^{10} + 176a^{10}. \end{matrix}

The dimensions of the modular irreducibles and their condensed dimensions are:

1 10 10 15 15 34 45 45 1 1 1 1 1 2 1 1

The Cartan matrix of the block is:

14 4 4 5 5 5 2 2
4 4 2 1 1 2 2 1
4 2 4 1 1 2 1 2
5 1 1 4 3 2 2 2
5 1 1 3 4 2 2 2
5 2 2 2 2 3 1 1
2 2 1 2 2 1 3 2
2 1 2 2 2 1 2 3

The dimensions of the projective indecomposable modules and their condensed dimensions are:

594 297 297 378 378 297 351 351 46 19 19 22 22 21 16 16

The condensations of the words z1, …, z5 in the standard generators generate e_{H}F Ge_{H}.

6.5 Faithful Condensation for J_{1} in Char. 2

Faithful condensation for the subgroup H with number 21 in the table of marks. The order of H is 21.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 412. The permutation character decomposes as

1a + 56a^{4}b^{4} + 76a^{4}b^{4} + 77aaab^{5}c^{5} + 120a^{6}b^{6}c^{6} + 133a^{7}b^{5}c^{5} + 209a^{9}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 20 56 56 76 1 2 2 2 2

The Cartan matrix of the block is:

8 4 4 4 4
4 4 3 3 1
4 3 4 2 2
4 3 2 4 2
4 1 2 2 4

The dimensions of the projective indecomposable modules and their condensed dimensions are:

840 496 552 552 552 40 26 26 26 22

The condensations of the words z1,z2 and z3 in the standard generators generate e_{H}F Ge_{H}.

6.6 Faithful Condensation for M_{22} in Char. 2

Faithful condensation for the subgroup H with number 26 in the table of marks. The subgroup H is a Sylow 3-subgroup.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 5504. The permutation character decomposes as

\begin{matrix}1a + 21a^{5} + 45a^{5}b^{5} + 55a^{7} + 99a^{11} + 154a^{18} + 210a^{26} + 231a^{23} \\ +280a^{32}b^{32} + 385a^{41}. \end{matrix}

The dimensions of the modular irreducibles and their condensed dimensions are:

1 10 10 34 70 70 98 1 2 2 2 6 6 10

The Cartan matrix of the block is:

92 42 42 50 12 12 20
42 21 20 23 5 5 9
42 20 21 23 5 5 9
50 23 23 29 7 7 10
12 5 5 7 3 2 2
12 5 5 7 2 3 2
20 9 9 10 2 2 6

The dimensions of the projective indecomposable modules and their condensed dimensions are:

6272 2816 2816 3456 896 896 1408 704 320 320 384 96 96 160

Since the condensed module (1_{H}^{G})e_{H} is of dimension 5504, we prefer to use Theorem 3.3.9 in its stronger version, i.e., we take the condensed module on the cosets of the normalizer N of H in G, which is of order 72. The condensed permutation module has dimension 688 as can be derived from the permutation character 1_{N}^{G}. The coset N spans the whole space under the condensations of the words z1, z2, z3, z4 and z7 in the standard generators. These elements together with the condensations of the generators of the normalizer given in the table of marks therefore generate e_{H}F Ge_{H}.

6.7 Faithful Condensation for M_{22} in Char. 3

Faithful condensation for the subgroup H with number 106 in the table of marks. The order of the subgroup H is 64.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 145. The permutation character decomposes as

1a + 21aa + 55aaa + 99aa + 154a^{6} + 210aaa + 231a^{5} + 280aabb + 385a^{7}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 49 49 55 231 1 1 1 3 1

The Cartan matrix of the block is:

3 2 2 2 1
2 3 2 2 2
2 2 3 2 2
2 2 2 3 1
1 2 2 1 3

The dimensions of the projective indecomposable modules and their condensed dimensions are:

540 819 819 594 945 14 15 15 16 11

6.8 Faithful Condensation for M_{23} in Char. 2

Faithful condensation for the subgroup H with number 49 in the table of marks. The order of the subgroup H is 21.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 23212. The permutation character decomposes as

\begin{matrix}1a + 22a^{4} + 45aabb + 230a^{14} + 231a^{15}b^{9}c^{9} + 253a^{13} + 770a^{40}b^{40} \\ +896a^{40}b^{40} + 990a^{47}b^{47} + 1035a^{49} + 2024a^{96}. \end{matrix}

The dimensions of the modular irreducibles and their condensed dimensions are:

1 11 11 44 44 120 220 220 252 1 2 2 1 1 8 7 7 12

The Cartan matrix of the block is:

46 39 39 28 28 16 17 17 24
39 40 39 26 26 16 17 16 22
39 39 40 26 26 16 16 17 22
28 26 26 21 18 12 10 11 14
28 26 26 18 21 12 11 10 14
16 16 16 12 12 8 6 6 8
17 17 16 10 11 6 9 7 10
17 16 17 11 10 6 7 9 10
24 22 22 14 14 8 10 10 15

The dimensions of the projective indecomposable modules and their condensed dimensions are:

18816 17920 17920 11904 11904 7040 8064 8064 10880 912 872 872 582 582 348 384 384 524

There is a better choice for H than the one offered above. This subgroup has been used in [LW96a] to determine the socle series of the projective indecomposable modules for G in the principal block.

6.9 Faithful Condensation for M_{23} in Char. 3

Faithful condensation for the subgroup H with number 136 in the table of marks. The order of the subgroup H is 160.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 534. The permutation character decomposes as

\begin{matrix}1a + 22aaa + 230a^{7} + 231aaabbbccc + 253a^{4} + 770ab + 896a^{6}b^{6} \\ +990aaabbb + 1035a^{12} + 2024a^{14}. \end{matrix}

The dimensions of the modular irreducibles and their condensed dimensions are:

1 22 104 104 253 770 770 1 3 2 2 4 1 1

The Cartan matrix of the block is:

2 1 1 1 1 1 1
1 5 3 3 1 2 2
1 3 3 2 1 2 1
1 3 2 3 1 1 2
1 1 1 1 2 1 1
1 2 2 1 1 3 1
1 2 1 2 1 1 3

The dimensions of the projective indecomposable modules and their condensed dimensions are:

2025 4068 3150 3150 2277 3690 3690 15 36 27 27 18 21 21 6.10. FAITHFUL CONDENSATION FOR HS IN CHAR. 2 127

6.10 Faithful Condensation for HS in Char. 2

Faithful condensation for the subgroup H with number 146 in the table of marks. The order of the subgroup H is 21.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 100732. The permutation character decomposes as

1a + 22a^{4} + 77a^{7} + 154a^{8}b^{8}c^{8} + 175a^{11} + 231a^{15} + 693a^{33} + 770a^{40}b^{40}c^{40} +825a^{43} + 896a^{40}b^{40} + 1056a^{46} + 1386a^{66} + 1408a^{70} + 1750a^{80} +1925a^{91}b^{91} + 2520a^{120} + 2750a^{134} + 3200a^{150}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 20 56 132 518 1000 1 2 4 4 22 42

The Cartan matrix of the block is:

416 264 92 120 72 42
264 172 62 76 48 25
92 62 27 26 18 10
120 76 26 36 20 12
72 48 18 20 16 7
42 25 10 12 7 7

The dimensions of the projective indecomposable modules and their condensed dimensions are:

105984 67072 25600 30208 19968 13312 5140 3266 1244 1464 966 628

6.11 Faithful Condensation for HS in Char. 3

Faithful condensation for the subgroup H with number 482 in the table of marks. The order of the subgroup is 256.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 746. The permutation character decomposes as

1a + 22a + 77aaa + 154aaa + 175aaa + 231a + 693a^{7} + 770a^{4}bc + 825a^{5} +896a^{4}b^{4} + 1056a^{6} + 1386a^{5} + 1408a^{6} + 1750a^{8} + 1925a^{9}b^{5} + 2520a^{10} +2750a^{9} + 3200a^{12}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 22 154 321 748 1176 1253 1 1 3 2 3  5  2

The Cartan matrix of the block is:

4 1 1 0 2 2 2
1 4 1 1 2 1 2
1 1 3 1 0 0 2
0 1 1 2 0 1 2
2 2 0 0 3 2 1
2 1 0 1 2 3 2
2 2 2 2 1 2 4

The dimensions of the projective indecomposable modules and their condensed dimensions are:

6534 5742 3312 4500 5895 7875 9108 28 25 17 17 25 30 35

The condensations of the words z1, …, z5 and z8 in the standard generators generate the algebra e_{H}F Ge_{H}.

6.12. FAITHFUL CONDENSATION FOR J_{3} IN CHAR. 2 129

6.12 Faithful Condensation for J_{3} in Char. 2

Faithful condensation for the subgroup H with number 54 in the table of marks. The order of the subgroup H is 27.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 68932. The permutation character decomposes as

1a + 85a^{5}b^{5} + 323a^{11}b^{11} + 324a^{12} + 646a^{24}b^{24} + 816a^{32} + 1140a^{44} +1215a^{45}b^{45} + 1615a^{59} + 1920a^{72}b^{72}c^{72} + 1938a^{70}b^{70} + 2432a^{90} + 2754a^{102} +3078a^{114}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 78 78 80 84 84 244 322 322 966 1 2 2 2 4 4 10 10 10 34

The Cartan matrix of the block is:

84 16 16 24 32 32 26 20 20 14
16 7 6 8 9 9 8 3 4 6
16 6 7 8 9 9 8 4 3 6
24 8 8 19 18 18 12 4 4 12
32 9 9 18 21 20 12 6 6 12
32 9 9 18 20 21 12 6 6 12
26 8 8 12 12 12 13 6 6 7
20 3 4 4 6 6 6 7 4 2
20 4 3 4 6 6 6 4 7 2
14 6 6 12 12 12 7 2 2 9

The dimensions of the projective indecomposable modules and their condensed dimensions are:

42624 13184 13184 22912 24704 24704 18048 8832 8832 15616 1588 484 484 846 916 916 666 328 328 574

6.13 Faithful Condensation for M_{24} in Char. 2

Faithful condensation for the subgroup H whose number in the table of marks is 303. The order of the subgroup H is 27.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 336224. The permutation character decomposes as

1a + 23aaa + 45aaabbb + 231a^{7}b^{7} + 252a^{14} + 253a^{15} + 483a^{21} + 770a^{28}b^{28} +990a^{38}b^{38} + 1035a^{41}b^{37}c^{37} + 1265a^{53} + 1771a^{77} + 2024a^{78} + 2277a^{87} +3312a^{120} + 3520a^{132} + 5313a^{189} + 5544a^{210} + 5796a^{210} + 10395a^{385}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 11 11 44 44 120 220 220 252 320 320 1242 1792 1 1 1 2 2 6 6 6 8 8 8 46 64

The Cartan matrix of the block is:

342 361 361 245 245 208 120 120 160 50 50 46 18
361 399 396 264 263 228 130 130 172 54 56 49 21
361 396 399 263 264 228 130 130 172 56 54 49 21
245 264 263 182 179 152 87 87 114 34 35 33 13
245 263 264 179 182 152 87 87 114 35 34 33 13
208 228 228 152 152 132 74 74 100 32 32 28 12
120 130 130 87 87 74 47 45 56 19 18 18 8
120 130 130 87 87 74 45 47 56 18 19 18 8
160 172 172 114 114 100 56 56 80 25 25 20 8
50 54 56 34 35 32 19 18 25 11 9 8 4
50 56 54 35 34 32 18 19 25 9 11 8 4
46 49 49 33 33 28 18 18 20 8 8 10 4
18 21 21 13 13 12 8 8 8 4 4 4 3

6.13. FAITHFUL CONDENSATION FOR M_{24} IN CHAR. 2 131

The dimensions of the projective indecomposable modules and their condensed dimensions are:

269312 293888 293888 193536 193536 168960 102400 102400
10080 10992 10992 7264 7264 6320 3808 3808
125952 46080 46080 45056 21504
4704 1696 1696 1664  784

There is no generating system known for the condensed algebra e_{H}F Ge_{H}.

6.14 Faithful Condensation for McL in Char. 2

Faithful condensation for the subgroup H with number 245 in the table of marks. The order of the subgroup H is 243.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 15318. The permutation character decomposes as

1a + 22aa + 231aaa + 252a^{4} + 770a^{4}b^{4} + 896aabb + 1750a^{10} + 3520a^{18}b^{18} +4500a^{16} + 4752a^{20} + 5103a^{21} + 5544a^{26} + 8019a^{33}b^{33} + 8250a^{36}b^{36} +9625a^{39} + 9856a^{37}b^{37} + 10395a^{43}b^{43}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 22 230 748 748 2124 2124 3584 1 2 2 2 2  6  6  14

The Cartan matrix of the block is:

36 32 18 20 20 6 6 14
32 88 24 36 36 20 20 24
18 24 17 10 10 7 7 7
20 36 10 20 16 8 6 12
20 36 10 16 20 6 8 12
6 20 7 8 6 7 5 5
6 20 7 6 8 5 7 5
14 24 7 12 12 5 5 9

The dimensions of the projective indecomposable modules and their condensed dimensions are:

110464 232320 74240 102784 102784 55936 55936 73600 484  976  322  436  436  230 230 310

6.15. FAITHFUL CONDENSATION FOR MCL IN CHAR. 3 133

6.15 Faithful Condensation for McL in Char. 3

Faithful condensation for the subgroup H with number 179 in the table of marks. The order of the subgroup H is 100.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 89941. The permutation character decomposes as

1a + 22aa + 231aaa + 252a^{6} + 770a^{6}b^{6} + 896a^{9}b^{9} + 1750a^{22} + 3520a^{38}b^{32} +4500a^{48} + 4752a^{46} + 5103a^{56} + 5544a^{53} + 8019a^{79}b^{79} + 8250a^{82}b^{82} +9625a^{100} + 9856a^{99}b^{99} + 10395a^{102}b^{102}.

The dimensions of the modular irreducibles and their condensed dimensions are:

1 21 104 104 210 560 605 605 1498 2794 1 1 1 1 2 4 6 6 16 27

The Cartan matrix of the block is:

262 174 134 134 174 226 40 40 51 39
174 133 91 91 117 144 21 21 39 26
134 91 70 69 90 116 20 20 27 20
134 91 69 70 90 116 20 20 27 20
174 117 90 90 129 162 27 27 33 27
226 144 116 116 162 211 37 37 42 36
40 21 20 20 27 37 9 8 6 6
40 21 20 20 27 37 8 9 6 6
51 39 27 27 33 42 6 6 13 8
39 26 20 20 27 36 6 6 8 8

The dimensions of the projective indecomposable modules and their condensed dimensions are:

428652 283581 220887 220887 296703 387828 67068 67068
4305 2877 2220 2220 2958 3858 663 663
86022 72171
874  719

6.16 Faithful Condensation for McL in Char. 5

Faithful condensation for the subgroup H with number 245 in the table of marks. The order of the subgroup H is 243.

The scalar product (1_{H}^{G}, 1_{H}^{G}) is 15318. The permutation character decomposes as

1a + 22aa + 231aaa + 252a^{4} + 770a^{4}b^{4} + 896aabb + 1750a^{10} + 3520a^{18}b^{18} +4500a^{16} + 4752a^{20} + 5103a^{21} + 5544a^{26} + 8019a^{33}b^{33} + 8250a^{36}b^{36} +9625a^{39} + 9856a^{37}b^{37} + 10395a^{43}b^{43}

The dimensions of the modular irreducibles and their condensed dimensions are:

1 21 210 230 560 896 896 1200 1200 3038 3245 3245 1 1 2 2 2 2 2  7  7  12 11 11

The Cartan matrix of the block is:

9 11 3 2 8 8 8 5 5 3 1 1
11 24 8 5 14 15 15 8 8 10 2 2
3 8 7 1 8 5 5 3 3 3 0 0
2 5 1 3 0 1 1 0 0 2 0 0
8 14 8 0 23 15 15 12 12 6 2 2
8 15 5 1 15 16 15 9 9 7 2 2
8 15 5 1 15 15 16 9 9 7 2 2
5 8 3 0 12 9 9 8 7 3 2 1
5 8 3 0 12 9 9 7 8 3 1 2
3 10 3 2 6 7 7 3 3 6 1 1
1 2 0 0 2 2 2 2 1 1 2 0
1 2 0 0 2 2 2 1 2 1 0 2

The dimensions of the projective indecomposable modules and their condensed dimensions are:

47750 100625 31625 8875 101750 93625 93625 60500 60500
206  425  141 43  428  381 381 253 253
49125 17875 17875
199  70  70

Appendix

The GAP-Procedure Faithfulkondburn

#
# we find all subgroups that lead to faithful
# condensation.
#
# input is the name of the group, a prime, and the block
# we want to study.
#
# we get the table of marks, construct the permtuation characters
# on  all p'-subgroups
# and only keep those which contain all the p.i.m.s
#
findfaithburn:=function(group,prime,blocknumber)
    local i,t,ct,modtafel,ordchars,modchars,decblock,
          decblockt,fus,permchars,exp,order,
          candidates,mat,projectives,fil,norm,chars,x,
          cartan,dimfixmod,dimfixproj,
          numcandidates,numfil;
    t:=TableOfMarks(group);
    ct:=CharTable(group);
    modtafel:=CharTable(ConcatenationString(group,
                        "mod",String(prime)));
    ordchars:=ct.irreducibles{modtafel.
                blocks[blocknumber].ordchars};
    modchars:=modtafel.irreducibles
                {modtafel.blocks[blocknumber].modchars};
    decblock:=Decomposition(modchars,
               Restricted(ct,modtafel,ordchars),"nonnegative");
    decblockt:=Transposed(decblock);
    fus:=FusionCharTableTom(ct,t);

if IsList ( fus[1] ) then
    fus :=fus[1];
    Print("warning there is more than one possible fusion\n");
fi;
permchars:=PermCharsTom(t,fus);
exp:=0;
order:=ct.size;
while ( order mod prime = 0 ) do
    exp:=exp+1;
    order:=order/prime;
od;
#
# find the permutation characters belonging to p'-subgroups
#
candidates:=[];
numcandidates:=[];
for i in [1..Length(permchars)] do
    if ( permchars[i][1]  mod prime^exp = 0 ) then
        Add(candidates,permchars[i]);
        Add(numcandidates,i);
    fi;
od;
#
mat:=MatScalarProducts(ct,ordchars,candidates);
projectives:=Decomposition(decblockt,mat,5);
fil:=[];
numfil:=[];
dimfixmod:=[];
dimfixproj:=[];
cartan:=decblockt*decblock;
for i in [1..Length(candidates)] do
    if Position(projectives[i] ,0) = false  then
        Add(fil,candidates[i]);
        Add(numfil,numcandidates[i]);
        Add(dimfixmod,projectives[i]);
        Add(dimfixproj,cartan*projectives[i]);
    fi;
od;
norm:=List(fil,x->MatScalarProducts(ct,[x],[x]));
return(rec(permchars:=fil,numbersinburn:=numfil,

cartan:=cartan,dimmodchars:=List(modchars,
x->x[1]),dimfixmod:=dimfixmod,dimfixproj:=dimfixproj));
end;

The GAP-Procedure Imprimperm

#
# we want to construct the permutation representation
# of the group G on the cosets of  the subgroup U2 . We
# are also given the group U1, with U2 < U1 < G.
# we factorize our problem.
# i.e. we get transversals for G in U1 and U1 in U2.
#
imprimperm:=function(g,u1,u2,list)
    local t1, t2, newlist, i, j, cosetrep, imcoset, imcosetrep,
          difference, pos, k, imcoset2,index1,index2,start;
    t1:=CanonicalRightTransversal(g,u1);
    index1:=Length(t1);
    t2:=CanonicalRightTransversal(u1,u2);
    index2:=Length(t2);
    newlist:=[];
    for i in [1..Length(list)] do
        newlist[i]:=[];
        for j in [1..Length(t1)] do
            cosetrep:=t1[j];
            imcoset:=cosetrep*list[i];
            imcosetrep:=CanonicalCosetElement(u1,imcoset);
            difference:=imcoset*imcosetrep^-1;
            pos:=Position(t1,imcosetrep);
            start:=(pos-1)*index2;
            for k in [1..Length(t2)] do
                imcoset2:=CanonicalCosetElement(u2,t2[k]*difference);
                Add(newlist[i],start+Position(t2,imcoset2));
            od;
        od;
    od;
    return newlist;
end;

The GAP-Procedure Imprimpermcan

#
# we want to construct the permutation representation
# of the group G on the cosets of the subgroup U2 . We
# are also given the group U1, with U2 < U1 < G.
# we factorize our problem.
# i.e. we get transversals for G in U1 and U1 in U2.
#
# we als  assume that the the subgroup $U1$ is the
# stabilizer of the first base point and that  the
# first n points are a faithful U1-orbit.
#
imprimpermcan:=function(g,u1,u2,n,list)
    local t1, t2, newlist, i, j, cosetrep, imcoset, imcosetrep,
          difference, pos, k, imcoset2,index1,index2,start,base,
          resdifference,p,resu1,resu2;
#    t1:=CanonicalRightTransversal(g,u1);
base:=Base(g)[1];
    index1:=Length(g.orbit);
#    index1:=Length(t1);
# we use largest moved point here.
#
   resu1:=Group(List(u1.generators,gen->
    PermList(List([1..n],i->i^gen))),());
   resu2:=Group(List(u2.generators,gen->
    PermList(List([1..n],i->i^gen))),());
    t2:=CanonicalRightTransversal(resu1,resu2);
    index2:=Length(t2);
    newlist:=[];
    for i in [1..Length(list)] do
        newlist[i]:=[];
        for j in [1..index1] do
# first find element corresponding to schreier vector
# i.e . which maps 1 to j.
#
        cosetrep:=();
while  (j^cosetrep <> base ) do
cosetrep:=cosetrep*g.transversal[j^cosetrep];
od;
#

# invert to get rep
#
cosetrep:=cosetrep^-1;
#
# the image coset is just the image under list[i]
#
            imcoset:=j^list[i];
difference:=();
#
# divide out the schreier vector which map imcoset to 1
#
while  (imcoset^difference <> base ) do
difference:=difference*g.transversal[imcoset^difference];
od;
difference:=cosetrep*list[i]*difference;
   resdifference:=PermList(List([1..n],p->p^difference));
            pos:=imcoset;
            start:=(pos-1)*index2;
            for k in [1..Length(t2)] do
                imcoset2:=CanonicalCosetElement(resu2,
t2[k]*resdifference);
                Add(newlist[i],start+Position(t2,imcoset2));
            od;
        od;
    od;
    return newlist;
end;

The Definition of the Words z1, …, z10.

Given two elements a and b in an F -algebra A we define the words z1(a, b), …, z10(a, b) recursively following the C-MeatAxe implementation.

\begin{matrix}z1(a, b) & a \\ z2(a, b) & b \\ z3(a, b) & z1(a, b) ∗ z2(a, b) \\ w(a, b) & z3(a, b) ∗ z2(a, b) \\ z4(a, b) & z3(a, b) ∗ w(a, b) \\ z5(a, b) & z3(a, b) ∗ z4(a, b) \\ z6(a, b) & z3(a, b) ∗ z5(a, b) \\ z7(a, b) & z4(a, b) ∗ z5(a, b) \\ z8(a, b) & z3(a, b) ∗ z6(a, b) \\ z9(a, b) & z7(a, b) ∗ w(a, b) \\ v(a, b) & z4(a, b) ∗ w(a, b) \\ z10(a, b) & v(a, b) ∗ z5(a, b) \end{matrix}

142

List of Important Algorithms

add matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 multiply matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 clean vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 semi-echelonize matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 nullspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 inverse of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 clean and quotient space . . . . . . . . . . . . . . . . . . . . . . . . . . 25 split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 splitting field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 equivalence test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 order polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 35 chop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 direct condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 socle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 tensor fixed space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 tensor condense element . . . . . . . . . . . . . . . . . . . . . . . . . . 63 first cohomology group . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 144

Frequently Used Symbols

G  a finite group
H, U  subgroups of G
F  a finite field
F G  the group algebra of G over F
A  an F -algebra
RMod(A)  the category of (right) A-modules
e  an idempotent in A
e_{H}  the idempotent \frac{1}{|H|} ∑_{h∈H} h in F H
V, W  A-modules
Hom_{A}(V, W) the A-homomorphisms from V to W
End_{A}(V)  the A-endomorphism ring of V
ϕ,ψ  A-homomorphisms
Kernel(ϕ)  the kernel of ϕ
Image(ϕ)  the image of ϕ
⊗_{A}  the tensor product over A

List of Notation

(1_{H}^{G}, 1_{H}^{G}), 66 − ⊗_{A} M, 43 1_{H}^{G}, 40, 48 A, 9 A(V), 45 A_{A}, 10 F G, 9 F_{p}[X], 15 Hom_{F} (V, F), 27 IBr(G), 52 P (S), 11 SB(w, a_{S})_{i}, 58 S^{♯}, 63 V ⋊ G, 110 V^{∗}, 27 W^{0}, 27 Der(G, V), 109 End_{A}(V), 10 End_{ℤG}((∑_{h∈H} h)ℤG), 48 H^{1}(G, V), 109 Hom_{A}(N, −), 43 Hom_{A}(V, W), 10 Ider(G, V), 109 Λ^{n}(V), 62 Φ, 42 Ψ, 42 RMod(A), 11 res_{eAe}^{A}, 48 res_{e_{H}F Ge_{H}}^{F G}, 41 Tr, 12 δ_{V}, 12 δ_{V,B}, 12 ⟨a⟩, 31

ℤ_{p}, 51
C, 42
D, 42
L_{S}(V), 55
M(V), 28, 42, 55
M_{S}(V), 55
|H|_{p}, 51
|H|_{p′}, 51
Rad(V), 10
Rad_{i}(V), 11
Soc(V), 10
Soc_{i}(V), 11
a_{HgH}, 48
c(δ_{V} (a)), 31
c_{n}, 16
d_{K}, 109
e_{H}, 41, 48
id_{C}, 42
id_{D}, 42

Index

absolutely irreducible A-module, 11 A-endomorphism ring, 10 A-homomorphism, 10 A-module, 9 annihilator, 27 A-representation, 12 A-submodule, 10

basic algebra, 47 bimodule, 10 block, 53

characteristic polynomial, 31, 32 complement, 109 condensation algebra, 45 condensation functor, 48 condensed module, 48 contragredient F G-module, 63 Conway polynomial, 16 cyclic algebra, 31

derivation, 109 dual vector space, 27

End_{A}(V) − A bimodule, 10 equivalence of categories, 42 exterior power, 62

faithful idempotent, 47 first cohomology group, 109 Fitting decomposition, 56

good factor, 32 group algebra, 9

Hom functor, 43 indecomposable A-module, 10 inner derivation, 109 irreducible A-module, 10 irreducible A-representation, 12

lattice of submodules, 28, 42, 55,

56 local A-module, 67

matrix representation, 12 minimal polynomial, 32 module category, 11 Morita equivalence, 42, 46, 65

criterion, 65 properties, 42 Morita theorem, 44

Norton criterion, 28, 31, 60

order polynomial, 34

peakword, 55

permutation module, 48

pivotfactor, 19 pivotindex, 19 projective cover, 11 projective endomorphism, 12 projective generator, 11, 43, 44 projective indecomposable A-module,

10

radical, 10

radical series, 11, 42

R-algebra, 9

reducible A-module, 10 restriction functor, 41, 48

Schur basis, 48 semi-echelonized matrix, 19 semidirect product, 109 semisimple A-module, 10 simple A-module, 10 socle, 10, 58 socle layers, 11 socle length, 11 socle series, 11, 42, 58 splitting field, 11 S-primitive idempotent, 55, 60 S-radical module, 55 standard basis, 24, 32, 34, 58

tensor functor, 43 trace map, 12, 51

uncondensed module, 48 uniserial module, 10

very good factor, 32, 60

Bibliography

[AF74] F.W. Anderson and K.R. Fuller. Rings and categories of modules, volume 13 of Graduate Texts in Mathematics. Springer-Verlag, New York Heidelberg Berlin, 1974.

[AG84] M. Aschbacher and R. Guralnick. Some applications of the first cohomology group. J. Algebra, 90:446–460, 1984.

[AS85] M. Aschbacher and L. Scott. Maximal subgroups of finite groups. J. Algebra, 92:44–80, 1985.

[BC85] D. Benson and J.H.C. Conway. Diagrams for modular lattices. J. Pure Appl. Algebra, 37:111–116, 1985.

[Ben83a] D. Benson. The Loewy structure of the projective indecomposable modules for A_{8} in characteristic 2. Comm. Algebra, 11(13):1395– 1432, 1983.

[Ben83b] D. Benson. The Loewy structure of the projective indecomposable modules for A_{9} in characteristic 2. Comm. Algebra, 11(13):1433– 1453, 1983.

[Ben89] D. Benson. Projective modules for the group of twenty-seven lines on a cubic surface. Comm. Algebra, 17(5):1017–1068, 1989.

[Ben91] D. Benson. Representations and cohomology I: Basic representation theory of finite groups and associative algebras, volume 30 of Cambridge studies in advanced mathematics. Cambridge University Press, 1991.

[Cab90] M. Cabanes. A criterion for complete reducibility and some applications. Asterisque, 181-182:93–112, 1990.

[CCN^{+}85] J. H. Conway, R. S. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson. An Atlas of Finite Groups. Clarendon Press, Oxford, 1985.

[Cel92] F. Celler. Kohomologie und Normalisatoren in GAP. Diplomarbeit, Lehrstuhl D für Mathematik, RWTH Aachen, 1992.

[Coh66] P.M. Cohn. Morita equivalence and duality. Mathematics Notes. Queen Mary College, London, 1966.

[Coh93] H. Cohen. A Course in Computational Algebraic Number Theory, volume 138 of Graduate Texts in Mathematics. Springer-Verlag, 1993.

[CR81] C. W. Curtis and I. Reiner. Methods of Representation Theory I. Pure and Applied Mathematics. Wiley-Interscience, 1981.

[CR87] C.W. Curtis and I. Reiner. Methods of Representation Theory II. Pure and Applied Mathematics. Wiley-Interscience, 1987.

[Erd90] K. Erdmann. Blocks of Tame Representation Type and Related Algebras, volume 1428 of Lecture Notes in Mathematics. Springer-Verlag, 1990.

[Fei82] W. Feit. The Representation Theory of Finite Groups. North-Holland Mathematical Library. North-Holland, 1982.

[FMR^{+}95] P. Fleischmann, G.O. Michler, P. Roelse, J. Rosenboom, R. Staszewski, C. Wagner, and M. Weller. Linear Algebra over Small Finite Fields on Parallel Machines. Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, Heft 23. Universität GH Essen, 1995.

[Gor82] D. Gorenstein. Finite Simple Groups. The University Series in Mathematics. Plenum Press, New York, 1982.

[Gra85] J. Grabmeier. Unzerlegbare Moduln mit trivialer Youngquelle und Darstellungstheorie der Schuralgebra. Bayreuth. Math. Schr., 20:9–152, 1985.

[Gre78] J.A. Green. On a theorem of Sawada. J. London Math. Soc., 18(2):247–252, 1978.

[Gre80] J.A. Green. Polynomial representations of GL_{n}, volume 830 of Lecture Notes in Mathematics. Springer-Verlag, Heidelberg, New York Berlin, 1980.

[Gur86] R. Guralnick. The dimension of the first cohomology group. In V.Dlab, P. Gabriel, and G. Michler, editors, Representation Theory II, Groups and Orders, volume 1178 of Lecture Notes in Mathematics, pages 94–97. Springer-Verlag, Berlin Heidelberg New York Tokyo, 1986.

[HJLP97] G. Hiss, C. Jansen, K. Lux, and R.A. Parker. Computational Modular Character Theory. Manuscript, 1997.

[HL88] G. Hiss and K. Lux. The Brauer characters of the Hall-Janko group. Comm. Algebra, 16(2):357–389, 1988.

[HL89] G. Hiss and K. Lux. Brauer Trees of Sporadic Groups. Clarendon Press, Oxford, 1989.

[HLP91] G. Hiss, K. Lux, and R.A. Parker. The 5-modular characters of the McLaughlin group and its covering group. Manuscripta Math., 73:91–114, 1991.

[HR94] D. F. Holt and S. Rees. Testing modules for irreducibility. J. Austral. Math. Soc. Ser. A, 57:1–16, 1994.

[Jam73] G. D. James. The modular characters of the Mathieu groups. J. Algebra, 27:57–111, 1973.

[Jan66] Z. Janko. A new finite simple group with abelian Sylow 2subgroups and its characterization. J. Algebra, 3(2):147–186, 1966.

[Jan95] C. Jansen. Ein Atlas 3-modularer Charaktertafeln. PhD thesis, Lehrstuhl D für Mathematik, RWTH Aachen, 1995.

[Jan96] C. Jansen. The minimal degrees of faithful, irreducible representations for sporadic simple groups and their Schur covers. Private communication, 1996.

[JLPW95] C. Jansen, K. Lux, R. A. Parker, and R. A. Wilson. An Atlas of Brauer Characters. London Mathematical Society Monographs New Series. Oxford University Press, 1995.

[Kas77] F. Kasch. Moduln und Ringe. Mathematische Leitfäden. B.G. Teubner Stuttgart, 1977.

[Kos87] S. Koshitani. The Loewy structure of the projective indecomposable modules for SL(3, 3) and its automorphism group in characteristic 3. Comm. Algebra, 15(6):1215–1253, 1987.

[Kos94] S. Koshitani. Projective modules of finite groups with elementary abelian Sylow 3-subgroups of order 9 in characteristic 3. Proc. Roy. Soc. Edinburgh Sect. A, 124(1):161–168, 1994.

[Lan83] P. Landrock. Finite Group Algebras and their Modules, volume 84 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1983.

[Lan93] S. Lang. Algebra. Addison Wesley, 3rd edition, 1993.

[Lin95] S. Linton. The art and science of computing in large groups. In W. Bosma et al., editor, Computational algebra and number theory, volume 325 of Math. Appl., pages 91–109. Kluwer Academic Publishers, 1995.

[LM78] P. Landrock and G.O. Michler. Block structure of the smallest Janko group. Math. Ann., 232:205–238, 1978.

[LMR94] K. Lux, J. Müller, and M. Ringe. Peakword condensation and submodule lattices: An application of the Meat-Axe. J. Symb. Comput., 17:529–544, 1994.

[LP91] K. Lux and H. Pahlings. Computational aspects of representation theory of finite groups. In G.O. Michler and C.M. Ringel, editors, Representation Theory of Finite Groups and Finite-Dimensional Algebras, pages 37–64. Birkhäuser, Basel, 1991.

[LS94] W. Lempken and R. Staszewski. The structure of the projective indecomposable module of { \atop 3M_{22}} in characteristic 2. Math. Comp., 62(206):841–850, 1994.

[LW96a] K. Lux and M. Wiegelmann. An application of condensation, the P.I.M. structure of the Mathieu group M_{23}. In preparation, 1996.

[LW96b] K. Lux and M. Wiegelmann. Condensing tensor product modules. In R. T. Curtis and R. A. Wilson, editors, The Atlas, Ten Years On. Cambridge University Press, 1996. To appear.

[Mer96] T. Merkwitz. Beiträge zur den Markentafeln sporadischer, einfacher Gruppen. Private communication, December 1996.

[Nic88] W. Nickel. Endliche Körper in dem gruppentheoretischen Programmsystem GAP. Diplomarbeit, Lehrstuhl D für Mathematik, RWTH Aachen, 1988.

[Nor79] S. Norton. The construction of J_{4}. In Bruce Cooperstein and Geoffrey Mason, editors, The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, June 25–July 20, 1979, volume 37 of Proceedings of Symposia in Pure Mathematics, pages 271–277. American Mathematical Society, Providence, RI, 1979.

[NPP84] J. Neubüser, H. Pahlings, and W. Plesken. CAS: Design and use for the handling of characters of finite groups. In M. Atkinson, editor, Computational Group theory, pages 195–247. Academic Press, 1984.

[Pah89] H. Pahlings. Course on representation theory. 1989.

[Par84] R. A. Parker. The computer calculation of modular characters (the Meat-Axe). In M. D. Atkinson, editor, Computational Group Theory, pages 267–274. Academic Press, 1984.

[Pfe91] G. Pfeiffer. Von Permutationscharakteren und Markentafeln. Diplomarbeit, Lehrstuhl D für Mathematik, RWTH Aachen, 1991.

[Pie82] R.S. Pierce. Associative Algebras, volume 88 of Graduate Texts in Mathematics. Springer-Verlag, New York Heidelberg Berlin, 1982.

[Rei75] I. Reiner. Maximal Orders, volume 5 of L.M.S Monographs. Academic Press, London New York San Francisco, 1975.

[Rin94] M. Ringe. The C-MeatAxe, Release 2.2.0. RWTH Aachen, 1994.

[Ryb90] A.J. Ryba. Condensation programs and their application to the decomposition of modular representations. J. Symb. Comput., 9:591–600, 1990.

[S^{+}94] M. Schönert et al. GAP – Groups, Algorithms and Programming. Lehrstuhl D für Mathematik, RWTH Aachen, 3.4 edition, 1994.

[Saw77] H. Sawada. A characterization of split (B, N)-pairs. Math. Z., 155:29–41, 1977.

[Sch11] I. Schur. Über die Darstellungen der symmetrischen und der alternierenden Gruppen durch gebrochene lineare Substitutionen. J. Reine Angew. Math., 139:155–250, 1911.

[Sch79] G.J.A. Schneider. The Mathieu group M_{11}, vertices, Greencorrespondents and socle series of the projective indecomposable modules. Master’s thesis, St. Peter’s College, Oxford University, 1979.

[Sch93a] G.J.A. Schneider. The structure of the projective indecomposable modules of the Mathieu group M_{12} in characteristic 2 and 3. Arch. Math., 60(4):321–326, 1993.

[Sch93b] G.J.A. Schneider. The structure of the projective indecomposable modules of the Suzuki group Sz(8) in characteristic 2. Math. Comp., 60(202):779–786, 1993.

[Sie91] S. Siegel. Projective modules for A_{9} in characteristic three. Comm. Algebra, 19(11):3099–3117, 1991.

[Tha81] J. G. Thackray. Modular representations of some finite groups. PhD thesis, University of Cambridge, 1981.

[Thé95] J. Thévenaz. G-Algebras and Modular Representation Theory. Oxford Mathematical Monographs. Oxford University Press, 1995.

[vdW85] B.L. van der Waerden. A History of Algebra. Springer-Verlag, Berlin Heidelberg New York Tokyo, 1985.

[Wak93a] K. Waki. The Loewy structure of the projective indecomposable modules for the Mathieu groups in characteristic 3. Comm. Algebra, 21(5):1457–1485, 1993.

[Wak93b] K. Waki. The projective indecomposable modules for the Higman-Sims group in characteristic 3. Comm. Algebra, 21(10):3475–3487, 1993.

[Wie94] M. Wiegelmann. Fixpunktkondensation von Tensorproduktmoduln. Diplomarbeit, Lehrstuhl D für Mathematik, RWTH Aachen, 1994.

[Wil93] R.A. Wilson. A new construction of the Baby Monster, and its applications. Bull. London Math. Soc., 25:431–437, 1993.

[Wil96a] R.A. Wilson. An Atlas of Group Representations. http://www.mat.bham.ac.uk/R.A.Wilson/mtx-atlas, 1996.

[Wil96b] R.A. Wilson. Standard generators for sporadic simple groups. J. Algebra, 184:505–515, 1996.