Computational Modular Character Theory

G. HISS  C. JANSEN  K. LUX  R. PARKER

November 11, 1993

To Joachim Neubüser and Herbert Pahlings Acknowledgements

During the development of the MOC-system and the preparation of this book we were supported by various mathematicians and institutions.

Above all, we wish to thank Joachim Neubüser and Herbert Pahlings who brought us together at Aachen. Pahlings was the supervisor of the PhD-thesis of Lux and of the Diploma thesis of Jansen, which were concerned with extensions and applications of the MOC-system. Neubüser and Pahlings provided a great working atmosphere at their institute, the Lehrstuhl D für Mathematik, and supported our project during the whole time. We therefore dedicate this book to them.

It is a great pleasure for us to thank H. W. Lenstra for his immediate interest in a particular number theoretical problem related to the MOC-system, for solving this problem and for allowing us to include his proof of this previously unpublished result in our book. Meinolf Geck read a preliminary version of the manuscript with great care. His various remarks and observations concerning the theory of modular characters have been incorporated. Together with his detailed comments and suggestions on the style of the exposition, this lead to many improvements and added very much to the legibility of our book. We wish to thank him for all his work. Finally we thank Thomas Breuer for suggestions concerning abelian number fieds.

We also gratefully acknowledge the financial support we received from the Deutsche Forschungsgemeinschaft and from the Bundesland Nordrhein-Westfalen in form of the Bennigsen-Foerder-Preis.

Preface

This book introduces some new ideas for the computer calculation of modular characters of finite groups. These ideas underly the computer algebra system MOC, which has been developed by the authors. Since there have been quite a few results obtained with the aid of MOC, it seems appropriate to publish the concepts and algorithms on which it is built. The sources of the programs of this system are freely available. Should anyone wish to use them for any purpose, apply to any of the authors.

This book consists of six chapters and an appendix. In the first chapter we recall and collect known results about modular character tables of finite groups and introduce the MOC-system. In the second chapter we offer a quick introduction, mostly without proofs, to the modular character theory of finite groups. It is our intention to lead the beginner and non-expert quickly to a state of being able to produce new results with MOC. In Chapter 3 we describe the computational concepts underlying our system. They allow us to transform the problems we are concerned with into problems of solving systems of integral linear equations and inequalities. These concepts are of independent interest as they can also be applied in a generic way to the modular representation theory of finite groups of Lie type. In the fourth chapter we introduce some principal data structures. In particular we suggest a new character table format in which all entries are integers. We also present a previously unpublished result due to H. W. Lenstra on certain integral bases for abelian number fields. The heart of our book is the fifth chapter, where the algorithms are introduced. Some of them should be of independent interest, for example our algorithms for dealing with rational linear equations. Finally, in Chapter 6 we have worked out a non-trivial example in detail, namely the calculation of the 7-modular decomposition numbers of the double covering group of Conway’s largest group.

In the appendix we have collected some results obtained in the course of examples throughout the book. These results include the 5-modular decomposition numbers of the Conway group Co_{2}, including the Brauer character table and the 7-modular decomposition numbers of 2Co_{1}. Both results are published for the first time.

Finally we give a bibliography, which is not intended to be exhaustive, of papers and monographs cited in our book and also of some related work.

We hope that even the expert in the field of modular representation theory will find our book interesting, because of our different point of view—we must compute with the objects.

Aachen, Heidelberg, Shepton Mallet November 1993 G. H., C. J., K. L., R. P.

List of notations

1. Numbers

|M| Number of elements of the set M p A prime number gcd(n, m) Greatest common divisor of the integers n und m n_{p}, n_{p′} p-part respectively p-free part of the integer n ℤ Integers ℕ Positive integers ℕ_{0} ℕ ∪ {0} ℚ Rational numbers ℚ_{n} ℚ(ζ_{n}), where ζ_{n} denotes a primitive n-th roots of

unity

ℝ Real numbers ℂ Complex numbers z̄ Complex conjugate of z ∈ ℂ 𝔽_{q} Finite field with q elements G(L/K) Galois group of Galois field extension K ⊆ L

2. Matrices

X^{t} Transpose of matrix X det(X) Determinant of X E_{n} n × n-Identity matrix

3. Groups

G Finite group H ≤ G H is a subgroup of G C_{G}(X) Centralizer of X ⊆ G N_{G}(H) Normalizer of subgroup H in G G_{p}, G_{p′} Set of p- respectively p′-elements of G |x| Order of x ∈ G

6. Rings and modules

R Commutative ring with 1 R^{∗} Group of units of R rank_{R}(X) R-rank of the free R-module X RG Group algebra of G over R Hom_{RG}(X, Y) Set of RG-homomorphisms between the RG-modules

X and Y

[X, Y] rank_{R}(Hom_{RG}(X, Y)) for RG-modules X and Y X^{G} Induced module X_{H} Restricted module X^{∗} RG-module dual to X (K, R, F) p-modular system:

R: Ring of algebraic integers K: Field of fractions of R F: Residue class field of R of characteristic p ≠ 0

8. Characters

ϑ_{X} Brauer character afforded by the representation X Φ_{φ} Projective indecomposable character corresponding

to the irreducible Brauer character φ

PIM Projective indecomposable character ϑ^{2+} Symmetric square of character ϑ ϑ^{2−} Skew square of character ϑ ω_{χ} Central character corresponding to χ ∈ Irr(G) Irr(G) Set of irreducible K-characters of G IBr(G) Set of irreducible Brauer characters of G with respect

to (K, R, F)

IPr(G) Set of projective indecomposable characters of G B A union of blocks of G Irr(B) Set of irreducible K-characters of G belonging to B IBr(B) Set of irreducible Brauer characters of G with respect

to (K, R, F) belonging to B IPr(B) Set of projective indecomposable characters of G be-

longing to B

G_{0}(KG) Set of all ℤ-linear combinations of Irr(G) G_{0}(F G) Set of all ℤ-linear combinations of IBr(G) K_{0}(F G) Set of all ℤ-linear combinations of IPr(G) G_{0}(B) Set of all ℤ-linear combinations of Irr(B) G_{0}(B)^{+} Set of all ℕ-linear combinations of Irr(B) G_{0}(B̂) Set of all ℤ-linear combinations of IBr(B) G_{0}(B̂)^{+} Set of all ℕ-linear combinations of IBr(B) K_{0}(B) Set of all ℤ-linear combinations of IPr(B) C(G, B) Set of class functions of G with values in K, spanned

by Irr(B)

C_{p′}(G) Set of class functions of G which vanish on p-singular

elements

C_{p′}(G, B) Set of class functions of G with values in K, spanned

by IBr(B)

ψ_{B} Projection of class function ψ to C(G, B) respectively

C_{p′}(G, B)

C(G) Set of class functions of G with values in K

[M] Matrix of values of M ⊂ C(G) χ_{H} Class function restricted to subgroup H χ^{G} Class function induced to G χ̄ Class function complex conjugate to χ M̄ Set of complex conjugates of M ⊂ C(G) ⟨ , ⟩ Inner product of C(G) ⟨M, N⟩ Matrix of inner products between the elements of the

sets M, N ⊂ C(G)

ψ̂ Restriction of class function ψ to the p′-elements Îrr(B) {χ̂ | χ ∈ Irr(B)} D_{B} Decomposition matrix of B d_{χφ} Decomposition number BS Basic set of Brauer characters PS Basic set of projective characters BS_{0} Special basic set of Brauer characters BA Basis of Brauer atoms dual (with respect to ⟨ , ⟩) to

PS

PA Basis of projective atoms dual to BS PA_{0} Basis of projective atoms dual to BS_{0}

Contents

Acknowledgements vii

Preface ix

List of notations xi

1 Introduction 1 1.1 Modular characters . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Known results . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 MOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Series of groups . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Special defect groups . . . . . . . . . . . . . . . . . . . . . 6

2 Background from modular character theory 9 2.1 Brauer characters . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The decomposition matrix . . . . . . . . . . . . . . . . . . 10 2.3 Projective characters and orthogonality relations . . . . . 12 2.4 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Generation of characters . . . . . . . . . . . . . . . . . . . 17

2.5.1 Restriction and induction . . . . . . . . . . . . . . 17 2.5.2 Restriction to p-regular classes . . . . . . . . . . . 18 2.5.3 Restriction to blocks . . . . . . . . . . . . . . . . . 18 2.5.4 Tensor products . . . . . . . . . . . . . . . . . . . 19 2.5.5 Symmetrizations . . . . . . . . . . . . . . . . . . . 19 3 Computational Concepts 23

3.1 Basic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 The first fundamental problem . . . . . . . . . . . . . . . 29 3.4 The second fundamental problem . . . . . . . . . . . . . . 34 3.5 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.6 Special basic sets . . . . . . . . . . . . . . . . . . . . . . . 36 3.7 Triangular decomposition matrices . . . . . . . . . . . . . 39

4 Data structures 41 4.1 Long integers . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Integers in abelian number fields . . . . . . . . . . . . . . 43 4.4 Character tables . . . . . . . . . . . . . . . . . . . . . . . 44 4.5 Lenstra’s theorem on integral bases . . . . . . . . . . . . . 47

5 Algorithms 51 5.1 Integral linear equations . . . . . . . . . . . . . . . . . . . 51 5.2 Improving basic sets . . . . . . . . . . . . . . . . . . . . . 55

5.2.1 Proving indecomposability . . . . . . . . . . . . . . 56 5.2.2 Subtracting indecomposables . . . . . . . . . . . . 64 5.2.3 Triangular matrix of scalar products . . . . . . . . 69 5.2.4 Proving decomposability . . . . . . . . . . . . . . . 72 5.2.5 Solving the second fundamental problem . . . . . . 73 5.2.6 An all-integer integer simplex algorithm . . . . . . 75

5.3 Automatic proofs . . . . . . . . . . . . . . . . . . . . . . . 81 5.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.5 Advanced methods . . . . . . . . . . . . . . . . . . . . . . 86

5.5.1 Associativity of the tensor product . . . . . . . . . 89 5.5.2 Adjointness . . . . . . . . . . . . . . . . . . . . . . 89 5.5.3 Nakayama relations . . . . . . . . . . . . . . . . . 89 5.5.4 Self duality . . . . . . . . . . . . . . . . . . . . . . 90 5.5.5 Trivial source modules . . . . . . . . . . . . . . . . 90 5.5.6 Lattices . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5.7 Fong’s lemma . . . . . . . . . . . . . . . . . . . . . 91 5.5.8 Theorem of Benson and Carlson . . . . . . . . . . 92 5.5.9 Blocks with cyclic defect groups . . . . . . . . . . 92 5.5.10 Webb’s theorem . . . . . . . . . . . . . . . . . . . 93 6 Calculating the 7-modular decomposition matrices of the

Conway group 95 6.1 The faithful block of maximal defect . . . . . . . . . . . . 95 6.2 The principal block . . . . . . . . . . . . . . . . . . . . . . 96 6.3 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

A  109

A.1 The 5-decomposition numbers of Co_{2} . . . . . . . . . . . . 109 A.2 The 7-decomposition numbers of 2Co_{1} . . . . . . . . . . . 110

A.2.1 Decomposition matrix of Block 1 . . . . . . . . . . 110 A.2.2 Decomposition matrix of Block 7 . . . . . . . . . . 111 A.2.3 Degrees of irreducible Brauer characters . . . . . . 112

B  113

B.1 The 5-modular character table of Co_{2} . . . . . . . . . . . 113

Bibliography 121

Index 133

Chapter 1

Introduction

1.1 Modular characters

Representation theory is an effective and indispensable tool in the investigation of finite groups. The table of ordinary irreducible characters of a finite group G encodes an enormous amount of information about G. For example, it can be used in many cases to show that G is a Galois group over some abelian extension of the rational numbers [101]. The beautiful book of Isaacs [70] gives an impression of the innumerable applications of character theory.

Ordinary characters for arbitrary finite groups where introduced by Frobenius in generalization of characters of abelian groups. Frobenius himself already calculated the character tables of the symmetric and alternating groups and of the simple Mathieu groups. To accomplish this work he was led to invent the method of inducing characters [42]. Schur not much later gave another proof for the character tables of the symmetric and alternating groups, and continued his investigations to complete the character tables of their covering groups [112]. The ordinary character tables of the finite simple groups of Lie type are now almost known by the work of Deligne and Lusztig [97]. The famous “Atlas of Finite Groups” [19] collects, among a wealth of other valuable information, the character tables of the most interesting finite simple groups up to a certain order, in particular those for the sporadic simple groups.

Brauer started the investigation of representations of a finite group over a field whose characteristic divides the group order. In [12], he introduced the idea of modular characters, now called Brauer characters, and the idea of decomposition numbers. The knowledge of decomposition numbers is equivalent to the knowledge of the irreducible Brauer characters. One of the main tasks of modular representation theory is to find methods for the calculation of the decomposition numbers for a given finite group.

Let us mention a few of the various applications of modular representation theory. To begin with, it is useful for the determination of the maximal subgroups of a group. For example, building upon Aschbacher’s subgroup theorem [2] and some results on modular representations of finite simple groups, Kleidman and Liebeck were able to determine the maximal subgroups of most of the finite classical groups in [87]. There are also applications outside the theory of finite groups, for example in geometry and combinatorics, where finite geometries are studied via their automorphism groups (see for example [109]) or in topology, where topological spaces are investigated via their fundamental groups (see for example [62]). In many potential applications the problem can be reduced to the case of the simple groups, their automorphism groups and covering groups. These groups are known by the classification of the finite simple groups. It would therefore be very desirable to have a description of all irreducible Brauer characters for this class of groups. Such a large collection of examples would doubtlessly lead to new conjectures and theorems in representation theory and then in turn perhaps to new and simpler ways of obtaining these Brauer characters. As Michler has indicated in his survey article [103], such a collection of modular character tables is also extremly useful in checking various famous conjectures of Brauer and Alperin.

1.2 Known results

Let us now describe some of the history of the Brauer character tables which are known up to the present. Brauer and Nesbitt in [12] already calculated the modular irreducible characters for the groups PSL(2, q) in the defining characteristic case, i.e., for fields of characteristic p dividing q. Janko in [81] found the Brauer characters for his first sporadic group J_{1} in every odd characteristic and used one of the 11-modular irreducible characters to construct his group as a matrix group of degree 7 over a field of characteristic 11. Fong in [36] determined the 2-decomposition numbers for this group with some ambiguities. In [71] James calculated all but one of the irreducible Brauer characters for the Mathieu groups, the only exception being one character in characteristic 2 for M_{24}. This last character was found only much later with the help of the Meat-Axe [106]. Humphreys in [65] and Benson in [4] investigated the covering groups of the Mathieu groups. The Higman–Sims group has been studied by Humphreys in [66], and Thackray in his thesis [123] determined the 2-modular characters of the McLaughlin group. In [32] Feit introduced some new methods by utilizing Green correspondence in the determination of decomposition numbers, thereby calculating many substantial examples, including some for the triple covering group of the McLaughlin group and the third Janko group J_{3}. Feit’s methods have been extended and led to the results of [56]. By the combined efforts of various authors the decomposition numbers are now completely known for all groups appearing in the Atlas up to page 100, the largest of which is the McLaughlin group. These results are collected in the first part of a modular Atlas of finite groups [82].

1.3 MOC

The computer algebra system MOC (for MOdular Characters) which we are going to describe in this book has been developed by the authors in order to automate some of the more elementary methods known for calculating decomposition numbers. Parts of the material presented here has already been sketched in [99]. An account of the methods we have in mind has been given by James and Kerber in [80, §6.3]. In a sense, modular character theory has been a part of “computational algebra” right from the beginning, and most of the elementary methods of [loc. cit.] have already been described by Brauer and Nesbitt in [12]. Many of the algorithms of MOC which are mostly concerned with systems of integral equations or inequalities are not new, but do not seem to be widely known or used. A subsidiary purpose of our book is therefore to show how powerful these methods can be, particularly the methods of solving systems of integral linear equations by p-adic expansion and the Gomory-algorithm for solving integral linear inequalities.

Some of the concepts developed for MOC have turned out to be of theoretical interest in the investigation of other problems, too. The work of Fong–Srinivasan and Dipper–James has led Geck and one of the authors to the formulation of a theorem on basic sets of groups of Lie type [46]. The concept of basic set as used in this paper was introduced in MOC in order to mechanize the calculation of decomposition numbers for individual finite groups. It is now clear from our experience that many concepts and algorithms can be used also for series of groups of Lie type, since the methods work for generic character tables not just for individual groups.

Some other aspects of our computational work are worth mentioning. First of all, our need for simple data structures for calculation in cyclotomic fields led to a conjecture that abelian number fields always have a nice integral basis consisting of sums (over an orbit) of roots of unity. This conjecture has now been proved by H. W. Lenstra. We are grateful to him for allowing us to present his proof in our book. These special bases for abelian number fields are now also used in the Aachen GAP system [111].

The principal design of MOC is at present different from other computer algebra systems available. MOC consists of a collection of small one-purpose FORTRAN-programs. We have, for example one program for matrix multiplication and one for solving integral linear equations. The various programs communicate via files. More complex tasks are achieved by using Bourne-shell scripts calling a sequence of MOC programs. The shell scripts are an essential part of the MOC system. One advantage of this design is the simple extendability of the system. It allows the easy addition of new commands to MOC. A further advantage is the portability to other UNIX machines. Another important feature of MOC is supported by this design. Every run of a program is recorded in a file which is attached to a particular group and prime. The MOC programs also write their output onto this log-file automatically. This makes it possible to repeat particular calculations and, what is most important, to write down proofs for the results obtained by the machine. Even the proof writing can be done automatically to a certain extent. This feature of MOC has been used to obtain most of the proofs of [56].

We emphasize the fact that MOC only works with Brauer characters and not with representations (c.f. the Meat-Axe [106]) and uses only elementary methods to the extent described in [80, § 6.3]. Nevertheless, in connection with the more advanced methods described in Section 5.5, one can get results which are out of reach of systems working with representations because of the large degrees of the representations involved. On the other hand, for some groups, the Meat-Axe and Condensation (see [110, 98]) are used successfully to solve problems left open by MOC. This applies in particular for calculations in characteristics 2 and 3, where elementary methods are much less efficient than for larger primes. The three systems working together provide a very powerful computational tool for the calculation of irreducible Brauer characters for individual finite groups.

1.4 Series of groups

Apart from individual groups, there is also considerable knowledge on some series of simple groups. For the reason of completeness we briefly sketch the known results. Various authors, among them Robinson, Kerber and James have established a theory of modular representations for the symmetric and alternating groups (see for example [75, 84]). Burkhardt did some work for groups of Lie type; in particular he found all decomposition numbers for the simple groups P SL(2, q) [15], all decomposition numbers for the Suzuki groups in odd characteristics [16], and some decomposition numbers for the Suzuki groups and the unitary groups U(3, 2^{f}) in even characteristic [16, 17]. Fong in [36] started the investigation for the small Ree groups ^{2}G_{2}(3^{2m+1}), but he had to leave some problems open. These were later solved by Landrock and Michler [94]. The work on these groups was completed in [53].

The investigation for the groups of Lie type naturally divides into the case where the modular characteristic is the defining characteristic of the group, i.e., the characteristic of the underlying field, and the case where the two characteristics are different. In the defining characteristic case the representation theory of algebraic groups is relevant, in particular the problem of finding the multiplicities of the simple modules in the Weyl modules. An affirmative answer to Lusztig’s conjecture (see [96]) would solve most of the problems for the finite groups of Lie type in this case.

In the non-defining characteristic case, a theory has been evolving starting with a series of papers by Fong and Srinivasan. The highlight of their work is the complete description of all the Brauer trees for the classical groups in [41]. Dipper and James have been developing a representation theory for the general linear groups which has led to the completion of the decomposition numbers for GL_{n}(q), n ≤ 10 [79]. Furthermore, some of the decomposition numbers for G_{2}(q) and Sp_{4}(q) are known by the work of Hiss, Shamash and White. Geck has determined most of the irreducible Brauer characters for U(3, q) and the Steinberg triality groups ^{3}D(4, q^{3}) in [43, 45]. Finally, the Brauer trees for the Ree groups are known completely [53].

1.5 Special defect groups

In general the problem of finding the Brauer character table of a finite group is considerable harder than that of finding the ordinary irreducible characters. This is partly due to the fact that in contrast to the ordinary character theory there is no inner product on the set of Brauer characters which tells the irreducibility of a Brauer character. However, in particular cases, there are deep theoretical results which can be used to good advantage. If a group is p-soluble, the Fong–Swan theorem states that every irreducible Brauer character is liftable to an ordinary character. This leads immediately to an algorithm for the determination of the irreducible Brauer characters.

Brauer has introduced the concepts of blocks, which leads to a subdivision of the problems. The blocks correspond to the primitive idempotents of the centre of the modular group algebra. The decomposition matrix is a matrix direct sum of the decomposition matrices for the block algebras, and so it suffices to consider each block in turn. Brauer has associated to each block a conjugacy class of p-groups, the defect groups of the block. Their isomorphism type measures the difficulty or complexity of finding the Brauer characters for the block. If the defect groups are trivial, the block is called a block of defect zero, having as decomposition matrix a (1 × 1)-matrix with single entry 1. If the defect groups are cyclic, the Brauer–Dade theorem imposes strong restrictions on the decomposition numbers. In this case, the decomposition matrix can be encoded in a certain tree, the Brauer tree. There are other defect groups of a particular type, namely dihedral 2-groups and quaternion groups, where all possible decomposition matrices occuring for blocks with these defect groups are known by work of Brauer, Olsson and Erdmann [29].

Broué and Puig have introduced the notion of nilpotent blocks in [13], generalizing the concept of blocks with cyclic defect groups. Puig [108] has given a remarkable extension of the Brauer–Dade theorem. His results, which are neatly explained in [91], imply the knowledge of the decomposition numbers for nilpotent blocks.

It has been conjectured by Donovan that, given a finite p-group D, there are only finitely many matrices which are decomposition matrices for blocks of finite groups with defect groups isomorphic to D. Scopes [113] has shown the conjecture to be true if we restrict the class of groups to symmetric groups. For cyclic defect groups Donovan’s conjecture is true by the results of Brauer and Dade. Nevertheless, to write down all decomposition matrices belonging to blocks with a given cyclic defect group, one still uses the classification of finite simple groups (c.f. [31]). It is our opinion that finite simple groups and in particular the sporadic groups will play an important role in the description of decomposition matrices for general defect groups, even if Donovan’s conjecture is shown to be true some day. This, we believe, justifies the application of our methods and in particular the usage of MOC.

Background from modular character theory

In this chapter we introduce our notation and recall some basic facts of character theory. We follow the exposition of Goldschmidt [47] and also Chapter 15 of Isaacs’ book [70]. This chapter is intended to give the beginner a quick introduction to those parts of the theory which are essential for the understanding of the following. We give no proofs but references to the appropriate sources. We assume the reader to be familiar with ordinary character theory of finite groups.

Throughout this chapter let G be a finite group, and p a fixed prime number.

2.1 Brauer characters

We use almost the same set-up as Isaacs in Chapter 15 of [70]. Let R denote the ring of algebraic integers over the rationals, and let K be the quotient field of R. Choose a maximal ideal I of R containing p. Finally let F = R/I denote the residue class field of R, a field of characteristic p which is algebraically closed. The canonical epimorphism R → F induces an isomorphism ρ from the set of |G|_{p′}-th roots of unity to F^{∗} (see [70, Lemma (15.1)]), which is used to define Brauer characters as follows.

An element of G of order not divisible by p is called p-regular or a p′-element. An element of order divisible by p is called p-singular. The set of all p-regular elements is denoted by G_{p′}.

Definition 2.1.1 Let X : G → GL_{n}(F) be an F -representation of G. The Brauer character ϑ_{X} of X is a map ϑ_{X} : G_{p′} → K. Let g ∈ G_{p′}, and let ζ_{1}, …, ζ_{n} denote the eigenvalues of X (g). We then define ϑ_{X} (g) = ∑_{i=1}^{n} ρ^{−1}(ζ_{i}).

If we apply the canonical epimorphism to ϑ_{X} (g), we obtain the trace of the matrix X (g). We say that the Brauer character ϑ_{X} is afforded by the representation X. The function complex conjugate to ϑ_{X} is also a Brauer character. It is afforded by the representation dual to X. We sometimes think of a Brauer character as being defined on all of G by letting its values be 0 on p-singular elements. This point of view is taken primarily in this and the next chapter in order to simplify notation. In the implementation of the MOC-system a Brauer character is of course only stored as a function on G_{p′}.

A Brauer character is called irreducible if it corresponds to an irreducible F -representation of G. It is a basic fact that the set of irreducible Brauer characters of G is finite and linearly independent as functions from G to K ([70, Theorem 15.5]).

Note that there are several maximal ideals I inside R containing p. A different choice of I would in general give another Brauer character for the same F -representation. However, all MOC-calculations are independent of the choice of I. The reader is invited to check this assertion, as we go along.

2.2 The decomposition matrix

The rings (K, R, F) are fixed in the following. They form our p-modular system.

The set of ordinary irreducible K-characters of G is denoted by Irr(G), the set of irreducible Brauer characters by IBr(G). Since K is a splitting field for the group ring KG, the number of irreducible K-characters equals the number of conjugacy classes of G.

The set of all ℤ-linear combinations of ordinary characters is a free abelian group with basis Irr(G). It is isomorphic to the Grothendieck group of the category of finitely generated KG-modules (see [20, Proposition (16.10)]) and is therefore denoted by G_{0}(KG). The latter fact is not needed in our book; it is only stated to motivate the notation. Similarly, G_{0}(F G) denotes the set of all ℤ-linear combinations of irreducible Brauer characters. Elements in these groups are called generalized or virtual characters. A function on G which is constant on conjugacy classes is called a class function. Characters are class functions and thus G_{0}(KG) and G_{0}(F G) are subgroups of C(G), the set of K-valued class functions of G. We let C_{p′}(G) denote the set of class functions of G with values in K, vanishing on p-singular elements. Then, by our definition, G_{0}(F G) is contained in C_{p′}(G).

If χ ∈ G_{0}(KG), we define χ̂ by

χ̂(g) = \{ \begin{matrix}χ(g), & if g is p-regular, \\ 0, & otherwise. \end{matrix}  (2.1)

Now comes the important observation. If χ is an ordinary character, χ̂ is a Brauer character. This can be seen as follows. Let X be a Krepresentation of G affording the character χ. Let R̃ denote the localization of R at the maximal ideal I. Then X can be realized over R̃, i.e., there is some representation Y equivalent to X, such that the matrices representing the group elements have all entries in R̃ ([70, Theorem 15.8]). We remark that X can even be realized over R ([68, Satz V.12.5]), but this fact is less elementary. The canonical epimorphism R → F can be extended uniquely to an epimorphism R̃ → F. We thus obtain a representation Ŷ : G → GL_{n}(F) by applying this epimorphism to every entry of a representing matrix. Then the Brauer character afforded by Ŷ obviously is χ̂.

We now consider the decomposition homomorphism:

\begin{matrix}d : G_{0}(KG) & ⟶ & G_{0}(F G) \\ χ & ⟼ & χ̂ \end{matrix}

The matrix of d with respect to the bases Irr(G) and IBr(G) is called the decomposition matrix of G and denoted by D. The entries of D are the non-negative integers d_{χφ} defined by

χ̂ = ∑_{φ∈IBr(G)} d_{χφ} φ.

The purpose of MOC is to support the calculation of decomposition matrices.

It is an important fact, which follows from Brauer’s characterization of characters, that the decomposition homomorphism is surjective ([70, Theorem 15.14]). This means that every irreducible Brauer character is a ℤ-linear combination of {χ̂ | χ ∈ Irr(G)}. The fact that Irr(G) is a basis of C(G) now implies that the set of irreducible Brauer characters spans C_{p′}(G). As IBr(G) is linearly independent, we obtain that |IBr(G)| equals the number of p-regular conjugacy classes of G.

2.3 Projective characters and orthogonality relations

We are now going to introduce the projective characters.

Definition 2.3.1 Let φ ∈ IBr(G). The projective indecomposable character corresponding to φ is the ordinary character Φ_{φ} defined by

Φ_{φ} = ∑_{χ∈Irr(G)} d_{χφ} χ.

A projective indecomposable character will be called a PIM in the following. We denote by IPr(G) the set of all PIMs and by K_{0}(F G) the group of all ℤ-linear combinations of IPr(G). A ℤ-linear combination with non-negative coefficients of PIMs is called a projective character, or a genuine projective character.

It is now most important to observe that a PIM vanishes on p-singular classes ([47, (6.9)(a)]). In particular, K_{0}(F G) ⊆ C_{p′}(G).

This definition of PIMs is suggested by Brauer’s reciprocity law. It can be given without introducing projective modules and some important properties of projective characters can be derived on this more elementary level. Of course, to get deeper results on projective characters one has to give the usual definition via projective modules.

For z ∈ ℂ let z̄ denote the number complex conjugate to z. The usual inner product on C(G) is denoted by ⟨ , ⟩. It is defined by

⟨λ, μ⟩ = \frac{1}{|G|} ∑_{g∈G} λ(g)\overline{μ(g)}.  (2.2)

We recall the orthogonality relations.

Theorem 2.3.2 (Orthogonality relations for Brauer characters): The two sets IPr(G) and IBr(G) are dual bases of C_{p′}(G) with respect to ⟨ , ⟩.

Proof. This follows from the orthogonality relations for ordinary characters and the fact that a PIM vanishes on p-singular classes (see [47, (6.10)]).

It follows from Theorem 2.3.2 that ⟨χ, ψ⟩ ∈ ℤ, if χ ∈ K_{0}(F G) and ψ ∈ G_{0}(F G). Furthermore, ⟨χ, ψ⟩ ∈ ℕ, if χ is a projective character and ψ a Brauer character. The latter observation has the following converse, which will be used in the construction of projective characters.

Proposition 2.3.3 Let Ψ be an ordinary character of G which vanishes on p-singular classes. Then Ψ ∈ K_{0}(F G). If, furthermore,

⟨φ, Ψ⟩ ≥ 0  for all  φ ∈ IBr(G),

then Ψ is a genuine projective character.

Proof. By the orthogonality relations, IPr(G) is a K-basis of C_{p′}(G). Write

Ψ = ∑_{φ∈IBr(G)} a_{φ} Φ_{φ}.

Then a_{φ} = ⟨φ, Ψ⟩, which is an integer since φ is a ℤ-linear combination of restricted ordinary characters.

This proposition is used later on to show that certain constructions yield genuine projective characters. The fundamental principle of the MOCsystem is to produce projective characters to approximate the columns of the decomposition matrix.

In order to simplify calculations based on the orthogonality relations, we introduce some more notation allowing us to write the orthogonality relations as matrix equations. We first choose representatives g_{1} = 1, g_{2}, …, g_{s} for the conjugacy classes of G. We assume that the first s′ of these elements are p-regular, and that g_{s′+1}, …, g_{s} are p-singular. Let C denote the (s × s)-diagonal matrix with entries |C_{G}(g_{1})|^{−1}, |C_{G}(g_{2})|^{−1}, …, |C_{G}(g_{s})|^{−1}. If M = {λ_{1}, …, λ_{m}} is a set of class functions of G, we write

[M] = (λ_{i}(g_{j}))_{1≤i≤m,1≤j≤s}

for the (m × s)-matrix giving the values of the functions in M. For example, [Irr(G)] is the table of ordinary irreducible characters of G. We always assume that any set of class functions is given in a particular order.

If N = {μ_{1}, …, μ_{n}} is another set of class functions, then ⟨M, N⟩ is the matrix (⟨λ_{i}, μ_{j}⟩)_{1≤i≤m,1≤j≤n} of mutual inner products of the elements of M with the elements of N. Finally, \overline{M} := \{λ̄_{1}, …, λ̄_{m}\}, where λ̄_{i} is the class function complex conjugate to λ_{i}.

With these notations we have:

⟨M, N⟩ = [M] C [\overline{N}]^{t}.  (2.3)

Writing E_{ℓ} for the identity matrix of size ℓ, the two orthogonality relations can be written as follows:

[Irr(G)] C [\overline{Irr(G)}]^{t} = E_{s}  (2.4)
[IPr(G)] C [\overline{IBr(G)}]^{t} = E_{s′}.  (2.5)

2.4 Blocks

Let us recall the fundamental notions of Brauer’s block theory. Following Isaacs [70, p. 272], we first introduce the Brauer graph of G. Its vertex set is Irr(G). Two vertices χ and ψ are linked by an edge, if there exists φ ∈ IBr(G) such that d_{χφ} ≠ 0 ≠ d_{ψφ}. The set of characters corresponding to a connected component of the Brauer graph is called a block of G.

If B is a union of blocks, we write Irr(B) and IBr(B) for the sets of irreducible ordinary characters respectively Brauer characters in B. Let B_{1}, …, B_{b} denote the blocks of G. Then

Irr(G) = ⋃_{i=1}^{b} Irr(B_{i}),

a disjoint union. If C(G, B_{i}) denotes the space of class functions spanned by Irr(B_{i}), we obtain

C(G) = ⨁_{i=1}^{b} C(G, B_{i}).

For the set of irreducible Brauer characters we have

IBr(G) = ⋃_{i=1}^{b} IBr(B_{i}),

again a disjoint union. By our definition of projective characters the ordinary constituents of a PIM are all contained in a unique block. In this sense every PIM belongs to a block. We write IPr(B_{i}) for the set of PIMs lying in block B_{i}. Let C_{p′}(G, B_{i}) denote the K-space of class functions spanned by IBr(B_{i}). We thus have

C_{p′}(G) = ⨁_{i=1}^{b} C_{p′}(G, B_{i}).

Therefore, the decomposition matrix splits into a direct sum of block decomposition matrices D_{B}, and we can restrict our attention to the particular blocks, thus greatly simplifying the problem.

The preceding remark of course only makes sense if one can determine the blocks of G from the ordinary character table before knowing the decomposition numbers. This is indeed the case. For χ ∈ Irr(G) let ω_{χ} denote the central character corresponding to χ defined by

ω_{χ}(g) = \frac{|G|χ(g)}{|C_{G}(g)|χ(1)}, g ∈ G.  (2.6)

Then ω_{χ}(g) is an algebraic integer and hence is in R for all g ∈ G (see [70, Theorem 3.7]). Then (see [70, Theorem 15.27]) two characters χ, ψ ∈ Irr(G) are in the same block, if and only if

\frac{|G|χ(g)}{|C_{G}(g)|χ(1)} ≡ \frac{|G|ψ(g)}{|C_{G}(g)|ψ(1)} (mod I) for all g ∈ G_{p′}.  (2.7)

This criterion is independent of the choice of the maximal ideal I of R containing p ([70, Theorem 15.18]).

The easiest blocks to deal with are the so-called blocks of defect 0. They contain exactly one ordinary irreducible character. This is at the same time the unique irreducible Brauer character and the unique PIM in the block. Thus the decomposition matrix of a block of defect 0 is the (1 × 1)-matrix with single entry 1. An irreducible character χ lies in a block of defect 0, if and only if p does not divide |G|/χ(1) ([70, Theorem (15.29)]). In particular, if p does not divide the order of G, the decomposition matrix is the identity matrix.

It is often convenient to know the number of irreducible Brauer characters in a block. Since the decomposition map is surjective, the block decomposition matrices have maximal ℤ-rank and we have:

Remark 2.4.1 Let Îrr(B) = {χ̂ | χ ∈ Irr(B)}. Then the ℤ-rank of [Îrr(B)] equals |IBr(B)|.

More generally, suppose we have a K-linear map

α : C(G) ⟶ C(G),

which leaves Irr(B) and IBr(B) invariant and commutes with the decomposition map d : C(G) → C(G). Such homomorphisms arise, for example, from a group automorphism of G or from the duality operation on the modules of G. We are then interested in the number of characters in B fixed by α. We write ^{α}ϑ for the α-image of a character ϑ and Irr(B)^{α}, respectively IBr(B)^{α} for the set of α-invariant elements. The following result generalizes an observation of Brauer and Feit (see [47, (7.14)(b)]), which was communicated to the authors by Knörr.

Proposition 2.4.2 Let ρ be the character of B defined by:

ρ(g) = ∑_{χ∈Irr(B)} χ(g)^{α}χ(g^{−1}), g ∈ G.

Then

|Irr(B)^{α}| = ⟨1_{G}, ρ⟩  and  |IBr(B)^{α}| = ⟨1_{G}, ρ̂⟩.

Proof. We have ⟨1_{G}, ρ⟩ = ∑_{χ∈Irr(B)}⟨χ,^{α}χ⟩, from which the first equation follows. To prove the second, let D_{B} = (d_{χφ}) denote the decomposition matrix of B. For φ ∈ IBr(G) let Φ_{φ} denote the PIM corresponding to φ. We then have:

⟨1_{G}, ρ̂⟩ = ∑_{χ∈Irr(B)}⟨χ, χ̂^{α}⟩ =

∑_{χ∈Irr(B)} ∑_{φ∈IBr(B)} d_{χφ}⟨χ,^{α}φ⟩ = ∑_{φ∈IBr(B)}⟨Φ_{φ},^{α}φ⟩.

The result follows.

2.5 Generation of characters

We briefly describe the principal methods to produce Brauer characters and projective characters.

2.5.1 Restriction and induction

Suppose H is a subgroup of G. If ϑ ∈ C(G) is a class function of G, the restricted class function ϑ_{H} is defined by:

ϑ_{H}(h) = ϑ(h),  h ∈ H.  (2.8)

If ϑ is an ordinary or a Brauer character, then so is ϑ_{H}. If ϑ ∈ C(H) is a class function of H, the induced class function ϑ^{G} is defined as follows. For g ∈ G choose representatives h_{1}, …, h_{s} for the H-conjugacy classes contained in the G-conjugacy class of g. Then

ϑ^{G}(g) = ∑_{i=1}^{s} \frac{|C_{G}(g)|}{|C_{H}(h_{i})|} ϑ(h_{i}).  (2.9)

The Frobenius reciprocity law asserts that the two operations introduced above are adjoint operations with respect to the inner product ⟨ , ⟩. This means that, given class functions ϑ, ρ of H respectively G, we have

⟨ϑ^{G}, ρ⟩ = ⟨ϑ, ρ_{H}⟩.

Geck has observed that Frobenius reciprocity, together with Proposition 2.3.3, can be used to show that induction takes projective characters to projectives. Let Φ be a projective character of H. Then Φ^{H} is an ordinary character of G; looking at the definition (2.9), we see that it vanishes on the p-singular conjugacy classes. If φ is an irreducible Brauer character of G, then ⟨φ, Φ^{G}⟩ = ⟨φ_{H}, Φ⟩ ≥ 0, since φ_{H} is a genuine Brauer character of H, and Φ is projective. Thus Φ is a genuine projective character of G.

Let ϑ denote a Brauer character of H afforded by some representation X over F. Then ϑ^{G} is a Brauer character of G; indeed, it is the Brauer character afforded by the representation of G induced by X (see [20, §10A, Exercise 10 in §18]). This observation in turn shows that the restriction of a projective character to a subgroup is a projective character.

The following theorem tells us how to find enough projective characters.

Theorem 2.5.1 (Fong [35, Lemma 1]:) Let P be the set of projective characters of G obtained by inducing the PIMs of all maximal subgroups. Then K_{0}(F G) is the ℤ-span of P.

Proof. This follows from Brauer’s induction theorem. See, for example [114, II 4.2,Corollary 2].

2.5.2 Restriction to p-regular classes

If ϑ ∈ C(G) is a class function of G, let ϑ̂ be defined by (2.1). Since the decomposition map is surjective, the set Îrr(G) = {χ̂ | χ ∈ Irr(G)} spans G_{0}(F G) over the integers.

2.5.3 Restriction to blocks

Let B denote a union of p-blocks of G. Let ψ = ∑_{χ∈Irr(G)} m_{ψ,χ}χ ∈ G_{0}(KG). Then the restriction of ψ to B is the generalized character

ψ_{B} = ∑_{χ∈Irr(B)} m_{ψ,χ}χ.  (2.10)

Thus ψ_{B} is just the projection of ψ from C(G) onto C(G, B), the space of class functions spanned by Irr(B).

Similarly, if ϑ is a generalized Brauer character, ϑ_{B} is the projection of ϑ onto C_{p′}(G, B). Thus if ϑ = ∑_{φ∈Irr(G)} d_{ϑ,φ}φ, then

ϑ_{B} = ∑_{φ∈Irr(B)} d_{ϑ,φ}φ.  (2.11)

We remark, that in each case we have to know a basis of the spaces C(G, B) respectively C_{p′}(G, B) in order to calculate the projections, but of course any basis will do.

2.5.4 Tensor products

The product of any two characters is again a character. This is true for ordinary and Brauer characters. For projective characters the following more general statement is true. Namely, if ϑ is a Brauer character and Ψ is a projective character, then the tensor product ϑΨ is a projective character. As Geck has observed, a proof of this fact can be given similar to those in § 2.5.1 by just using Proposition 2.3.3.

This statement for projective characters is extremely important for the purposes of MOC. It is used mainly in the case when Ψ is an ordinary irreducible character of defect 0. These are of course the projective characters which are readily available. The above statement means in particular, that the product of two ordinary irreducible characters is projective, if one of the factors is a defect 0 character. The projectives obtained in this way only depend on the character table of G and not on the set of subgroups of G. If we are able to determine the irreducible Brauer characters of G by using only projectives of this sort, the decomposition numbers are determined by the character table.

2.5.5 Symmetrizations

This paragraph is somewhat less elementary than the preceding ones, since the facts we are going to describe do not seem to follow by just considering characters.

Let χ be an ordinary or modular character of G. The symmetric and skew square of χ are the class functions

χ^{2+}(g) = \frac{1}{2}(χ(g)^{2} + χ(g^{2}))

respectively

χ^{2−}(g) = \frac{1}{2}(χ(g)^{2} − χ(g^{2})), g ∈ G.

The symmetric or skew square of a character is again a character, afforded by the symmetric respectively skew power of the module affording χ (if p ≠ 2). This observation can be generalized to yield, for every positive integer r and every partition λ of r a symmetrized character χ^{λ}.

We now outline this method following James [76]. Let V denote the SG-module affording the character χ. Here, S is one of the fields K or F. Let r denote a positive integer satisfying r ≤ n, where n = Dim_{S}(V). Let V^{⊗r} denote the r-fold tensor product of V on which G acts diagonally. Now V and hence V^{⊗r} is a module for the general linear group GL(V) in its natural action on V and thus every GL(V)-section (a quotient module of a submodule) of V^{⊗r} gives rise to an SG-module by way of restriction. If the characteristic p of S is 0 or larger than r, then V^{⊗r} is a semisimple GL(V)-module. For every partition λ of r there is an irreducible GL(V)-module V_{λ}, the Weyl module corresponding to λ, and V^{⊗r} is a direct sum of the Weyl modules. If p is positive and less than or equal to r, then V^{⊗r} still has a filtration by Weyl modules V_{λ}, but they are no longer direct summands. Furthermore, in general the Weyl modules are not irreducible anymore. Every Weyl module V_{λ} has a unique top composition factor F_{λ}. The other composition factors of V_{λ} are of the form F_{μ} for partitions μ which are strictly smaller than λ in the dominance order of partitions. Thus, if the partitions of r are arranged in reverse lexicographic order, the matrix M_{r,p} giving the multiplicities of the F_{λ} in the V_{μ} is lower unitriangular.

Let ψ^{λ} denote the character of G afforded by the Weyl module V_{λ}. It can be calculated with the help of the character table of the symmetric group S_{r}. Let ρ = (ρ_{1}, ρ_{2}, …, ρ_{r}), ρ_{1} ≥ ρ_{2} ≥ … ≥ ρ_{r} ≥ 0, be a partition of r. Let C(ρ) denote the centralizer in S_{r} of an element with cycle type ρ, and let [λ](ρ) denote the value of the irreducible character of S_{r} corresponding to λ on an element with cycle type ρ. Then, for p-regular g ∈ G,

ψ^{λ}(g) = ∑_{ρ⊢r} \frac{1}{|C(ρ)|}[λ](ρ) ∏_{i=1}^{r} ψ(g^{ρ_{i}}).  (2.12)

If p is non-zero and does not exceed r, than we can improve on the above symmetrization formula, since the Weyl modules are not irreducible in general. Let Σr, p = M_{r,p}^{−1} [Irr(S_{r})]. This matrix is called by James an “extended p-modular character table of S_{r}”. The character φ^{λ} of G afforded by the simple module F_{λ} can then be calculated by the formula (2.12) with [λ] replaced by the corresponding row of Σr, p.

In [76] James has calculated the matrices M_{r,p} and Σ_{r,p} for r ≤ 6 and p = 2, 3. For example, labelling the rows and columns of the following matrices with the partitions of 3 in reverse lexicographic order, we have

M_{3,3} = ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix} )

and

Σ_{3,3} = ( \begin{matrix}1 & −1 & 1 \\ 1 & 1 & −2 \\ 0 & 0 & 3 \end{matrix} ) .

If ψ is a 3-modular character of a group G of degree at least 3, the three 3-modular symmetrizations are the following:

φ^{(1,1,1)}(g) = \frac{1}{6}ψ(g)^{3} − \frac{1}{2}ψ(g^{2})ψ(g) + \frac{1}{3}ψ(g^{3})

φ^{(2,1)}(g) = \frac{1}{6}ψ(g)^{3} + \frac{1}{2}ψ(g^{2})ψ(g) − \frac{2}{3}ψ(g^{3})

φ^{(3)}(g) = ψ(g^{3}).

The formulae of James have been implemented in the CAS system [104]. In his thesis [49], Grabmeier has extended the calculations of James to the case of r ≤ 9 and p = 2, 3.

Computational Concepts

We are now going to introduce the notions of basic sets and atoms and derive some of their most important properties. These notions are fundamental for the algorithms of MOC, in particular those which try to prove the irreducibility of a Brauer character or the indecomposability of a projective character. They allow us to translate some of the problems occuring during the calculation of decomposition matrices into the language of Linear Algebra. Furthermore, the MOC methods generalize all elementary methods for calculating decomposition numbers described in the literature (see [80, Section VI.3] for a collection af these methods). They can be systematically applied and are mechanized so as to be suitable for the treatment with computers.

We also formulate two fundamental computational problems which we believe to be of independent interest.

It turned out that these concepts, originally designed for application to sporadic groups, could also be applied successfully to the study of modular characters of finite groups of Lie type (see [52]). They allow to deal with all groups of a fixed rank and Dynkin type simultaneously. The reason for this is that in a sufficient general situation we can find basic sets which can be described independently of the underlying field (see [46]).

3.1 Basic sets

We continue with the notation introduced in the preceding section. Here, we let B denote a fixed union of p-blocks of G. Let Irr(B) = {χ_{1}, …, χ_{n}}. The subgroups of G_{0}(KG), G_{0}(F G) and K_{0}(F G) generated by the characters in B are denoted by G_{0}(B), G_{0}(B̂) and K_{0}(B) respectively. We write G_{0}(B)^{+} for the set of genuine characters of G_{0}(B), and use an analogous notation for Brauer and projective characters.

Definition 3.1.1 A basic set of Brauer characters BS (basic set of projective characters PS) is a ℤ-basis of G_{0}(B̂) (of K_{0}(B)) consisting of Brauer (projective) characters.

We emphasize the fact that a basic set consists of proper characters, in contrast to the usage in the literature. One of the reasons is the experience that one should never give away the information of a character being proper. We invest a great deal of time to reach the aim of having basic sets at every stage of the calculations. In particular, we introduce an algorithm which for a given set of proper characters determines a ℤ-basis of their span consisting again of proper characters.

The first question we investigate, is the following. Given a set of Brauer characters, how can we decide whether it is a basic set?

Lemma 3.1.2 Let S ⊂ G_{0}(B̂)^{+}. Then S is a basic set if and only if S is linearly independent over ℤ, and every ordinary character of B, restricted to the p-regular classes, is a ℤ-linear combination of elements of S.

Proof. The necessity of the condition is clear. The converse follows from the surjectivity of the decomposition homomorphism.

Example 3.1.3 Here, and in the following examples, we denote irreducible characters by their degrees, which in our examples identifies them uniquely. Consider the principal 7-block B of Conway’s simple group Co_{1}. It contains the following 27 ordinary irreducible characters:

\{ \begin{matrix}\underline{1}, & \underline{276}, & \underline{299}, & \underline{17250}, & \underline{80730}, \\ \underline{94875}, & \underline{822250}, & \underline{871884}, & \underline{1821600}, & \underline{2055625}, \\ \underline{9221850}, & \underline{16347825}, & \underline{21528000}, & \underline{21579129}, & \underline{24667500}, \\ 31574400, & \underline{57544344}, & \underline{66602250}, & \underline{85250880}, & \underline{150732800}, \\ \underline{163478250}, & 191102976, & \underline{207491625}, & 215547904, & 219648000, \\ 299710125, & 326956500 \end{matrix} \} .

If we restrict these to the 7-regular classes, the subset BS of the 21 underlined characters is linearly independent, and the six remaining characters not in BS can be written as ℤ-linear combinations of these according to the following table (we omit the hat on top of the degrees):

\begin{matrix}31574400 & . & . & . & . & . & −1 & . & . & 1 & . & . & −1 & 1 & . & 1 & . & . & . & . & . & . \\ 191102976 & 1 & . & −1 & . & −1 & . & 1 & . & −1 & . & −1 & 1 & . & 1 & . & . & . & . & . & 1 & . \\ 215547904 & . & . & . & −1 & −1 & 1 & 1 & 1 & −1 & . & −1 & 1 & −1 & . & . & . & 1 & . & . & 1 & . \\ 219648000 & 1 & . & . & 1 & . & . & . & . & . & . & . & 1 & . & 1 & −1 & . & −1 & −1 & 1 & . & 1 \\ 299710125 & . & . & . & 1 & . & −1 & −1 & . & 1 & −1 & 1 & . & . & . & . & . & −1 & . & 1 & . & 1 \\ 326956500 & . & −1 & 1 & −1 & . & 1 & . & 1 & −1 & 1 & −1 & 1 & −1 & 1 & −1 & 1 & . & −1 & . & 1 & 1 \end{matrix}

Here, the columns correspond to the 21 characters in increasing order of their degrees. Thus the block contains exactly 21 irreducible Brauer characters and our 21 characters are a basic set.

We have seen how to prove for a given set of Brauer characters that it is a basic set. Now the question arises, of how to do the same for projective characters. By the theorem of Fong cited in § 2.5.1 we know that in principle enough projective characters can be obtained to span the space K_{0}(B).

We assume that we have constructed a set of projective characters and selected from these a maximal linearly independent subset PS. How can we decide whether or not PS is a basic set?

Lemma 3.1.4 Let BS and PS be two sets of Brauer characters respectively projective characters. We assume that each contains exactly |IBr(B)| elements. Let

U = ⟨BS, PS⟩  (3.1)

be the matrix of their mutual scalar products. Then BS and PS are basic sets if and only if U is invertible over ℤ.

Proof. We have

[BS] = U_{1} [IBr(B)]  (3.2)

and

[PS] = U_{2}^{t} [IPr(B)].  (3.3)

with integral matrices U_{1} and U_{2}. From (2.3) and the orthogonality relations we obtain:

U = [BS] C [\overline{PS}]^{t} = U_{1}[IBr(B)] C [\overline{IPr(B)}]^{t}U_{2} = U_{1}U_{2}.

Now BS respectively PS are bases if and only if U_{1} and U_{2} are invertible over ℤ and the assertion follows.

Example 3.1.5 Consider again the principal 7-block of Co_{1}. Let BS be as in Example 3.1.3. After some calculations, which we do not want to comment on right here, (but see Chapter 6), we have found a set PS of projective characters, such that U = ⟨BS, PS⟩ is the matrix given below, where the columns correspond to the projective characters. Now det U = 1, so that PS is indeed a basic set of projectives by Lemma 3.1.4.

\begin{matrix}Ψ : & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\ 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 276 & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 299 & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 17250 & 1 & . & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 80730 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 94875 & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 822250 & 1 & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 871884 & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1821600 & 1 & 1 & . & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\ 2055625 & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 9221850 & 1 & . & . & . & . & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 16347825 & . & . & . & 1 & 2 & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 21528000 & 1 & . & . & . & . & 1 & . & 1 & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 21579129 & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 & . & . & . & . & . & . \\ 24667500 & 1 & . & . & . & 1 & . & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . \\ 57544344 & 2 & . & 1 & 1 & . & . & 1 & . & . & . & 1 & . & . & 1 & . & 1 & . & . & . & . & . \\ 66602250 & . & . & . & 2 & 1 & . & . & . & 1 & 1 & . & . & . & 1 & . & . & 2 & 1 & 1 & . & . \\ 85250880 & 2 & . & 1 & 2 & . & . & . & . & . & 1 & . & . & . & 2 & . & 1 & 1 & . & . & . & . \\ 150732800 & . & . & . & 1 & . & . & . & . & . & 1 & . & . & 1 & 1 & . & 1 & 1 & 1 & . & 1 & 1 \\ 163478250 & 3 & 1 & . & 1 & . & 1 & . & . & . & . & 1 & . & 1 & 1 & . & 1 & 1 & . & . & 1 & 1 \\ 207491625 & 1 & . & . & 1 & . & . & . & . & . & 1 & . & . & 1 & 3 & . & . & 3 & 1 & 1 & 1 & . \end{matrix}

3.2 Atoms

Definition 3.2.1 A ℤ-basis BA (PA) of G_{0}(B̂) (K_{0}(B)) is called a system of atoms of Brauer (projective) characters, if every Brauer (projective) character is a linear combination with non-negative coefficients of elements of BA (PA). A Brauer atom (projective atom) is a generalized Brauer (projective) character, which is a member of a system of atoms.

In contrast to the definition of basic sets, we do not insist that our system of atoms consists of proper characters. The next result shows how to obtain a system of projective atoms from a basic set of Brauer characters. Of course, Brauer characters and projective characters can be interchanged in the formulation of this and many of the following results.

Lemma 3.2.2 Let BS be a basic set of Brauer characters. If PA denotes the basis of K_{0}(B) dual to BS with respect to the bilinear form ⟨ , ⟩, then PA is a system of projective atoms.

Proof. We have

[BS] = U [IBr(B)],

where the entries of U are non-negative integers, and det U = ±1. From

[PA] C [\overline{BS}]^{t} = E_{m}

and the orthogonality relations we obtain

[IPr(B)] = U^{t} [PA],

and thus our assertion.

Given BS, it is not necessary for our purposes, though in principle possible, to know the elements of the dual basis PA explicitly as class functions. Suppose we are given two basic sets BS and PS. The corresponding dual bases of atoms are denoted by PA and BA. Suppose furthermore, that P and B are two sets of projective characters respectively Brauer characters. Then to express the elements of P in terms of the basis PA is equivalent to calculating the inner products of the elements of P with the characters in BS. Namely, let V = ⟨P, BS⟩. Then

[P] = V [PA].

Similarly, if U = ⟨PS, B⟩, then

[B] = U^{t} [BA].  (3.4)

In particular, if we take BS for the set B of Brauer characters in (3.4), we obtain the elements of BA as class functions by inverting the matrix U^{t}.

Before we continue with our exposition, we give an easy but nevertheless very useful application of the notions introduced so far.

Lemma 3.2.3 Suppose PA is a system of projective atoms. If Φ is a projective character contained in PA, then Φ is indecomposable.

Proof. Let PA = {Φ_{1}, …, Φ_{m}} with Φ = Φ_{1}. Furthermore, let Ψ_{1}, …, Ψ_{m} denote the PIMs. We write

Ψ_{i} = ∑_{j=1}^{m} z_{ij}Φ_{j}.

Then all z_{ij} ≥ 0, and their matrix is invertible. Since Φ is projective, we can write

Φ = ∑_{i=1}^{m} x_{i}Ψ_{i}

with x_{i} ≥ 0. Therefore

Φ_{1} = Φ = ∑_{j=1}^{m} (∑_{i=1}^{m} x_{i}z_{ij}) Φ_{j}.

From this it follows that x_{i} = 0 for all i ≠ i_{0} and x_{i_{0}} = 1, i.e. Φ = Ψ_{i_{0}}.

Example 3.2.4 The projective character Ψ_{15} of Example 3.1.5 is an atom, since it has inner product 1 with 21579129, but 0 with all the other characters of BS. Hence it is a PIM.

Of course there is an analogous statement for Brauer characters. The above observation allows to prove indecomposability of projective characters and irreducibility of Brauer characters. It is, however, only a very special case of a much more general procedure, which will be described in Chapter 5.

3.3 The first fundamental problem

We are now ready to formulate the first fundamental problem of our theory. We assume that we are given two basic sets BS and PS. The corresponding dual bases of atoms are denoted by PA and BA. We are interested in all matrices which can possibly be decomposition matrices with respect to the given information. First we calculate the matrix U of the scalar products of the characters of BS with the characters of PS. Writing

[BS] = U_{1} [IBr(B)]  and  [PS] = U_{2}^{t} [IPr(B)],  (3.5)

we have, as in the proof of Lemma 3.1:

U = U_{1} U_{2}.  (3.6)

Notice that by our definition of basic sets, U_{1} and U_{2} have non-negative entries. We write X ≥ 0 to indicate that every entry of the matrix X over the integers is non-negative. Having BS at our hands, we know how to write the restriction of the ordinary characters to the p-regular classes in terms of the characters in BS, say

[Îrr(G)] = V [BS].

Knowing U_{1} is then equivalent to knowing the decomposition matrix:

D_{B} = V U_{1}.

Thus the decomposition matrix is obtained from a solution of

U = U_{1} U_{2},  U_{1}, U_{2} unimodular ,  U_{1}, U_{2} ≥ 0.  (3.7)

Two solutions (U_{1}, U_{2}) and (U′_{1}, U′_{2}) of (3.7) are called equivalent, if there is a permutation matrix X such that U′_{1} = U_{1}X and U′_{2} = X^{t}U_{2}. Equivalent solutions lead to the same set of Brauer characters arranged in a different order. If (3.7) has more than one equivalence class of solutions, then we cannot decide which of them determines the decomposition matrix without further information.

In many cases, however, we have additional conditions which drastically reduce the number of solutions of (3.7). Namely, if we are given a set of Brauer characters B and a set of projective characters P, we can write:

[B] = V [BS] = (V U_{1}) [IBr(B)]

and

[P] = W [PS] = (W U_{2}^{t}) [IPr(B)]

with integer matrices V and W. The solutions U_{1} and U_{2} of (3.7) must therefore satisfy the following conditions:

V U_{1} ≥ 0, and W U_{2}^{t} ≥ 0.  (3.8)

So we have the first fundamental problem:

Fundamental Problem I: Given V ∈ ℤ^{s×m} and W ∈ ℤ^{t×m}, and a unimodular matrix U ∈ ℕ^{m×m}, find all solutions U_{1}, U_{2} ∈ ℤ^{m×m} of

U = U_{1} U_{2}, U_{1}, U_{2} ≥ 0.

satisfying

V U_{1} ≥ 0, and W U_{2}^{t} ≥ 0.

Every solution of the fundamental problem leads to a possible decomposition matrix by equation (3.5): [IBr(B)] = U_{1}^{−1} [BS]. This shows that at the present state of knowledge the fact that the determinant of the Cartan matrix D_{B}^{t} D_{B} is a power of p does not impose any restrictions on the solutions of the fundamental problem (cf. the remarks in [80, p. 273]).

We do not intend to solve this fundamental problem in its full generality. Usually the number of solutions is much too large to be of any practical use. Instead we gradually improve on the two basic sets using the conditions (3.7) and (3.8) leading to a matrix U with smaller entries. How this is done in practise with the help of methods of integral linear programming will be described in Chapter 5.

Example 3.3.1 We continue with our example of the principal 7-block of Co_{1}. Let B be the set of the six ordinary characters which are not contained in BS, restricted to the 7-regular classes of Co_{1}. Then, of course, the matrix V is just the one given in Example 3.1.3.

We also have some projective characters which cannot be expressed in terms of PS of Example (3.1.5) entirely with non-negative coefficients. The corresponding matrix W is as follows.

W = [ \begin{matrix}1 & −1 & −1 & 1 & 1 & . & . & 3 & −2 & 2 & −1 & 3 & 1 & 1 & . & 3 & 4 & 7 & −9 & 2 & −8 \\ 1 & −1 & 1 & . & . & 2 & 6 & 5 & 1 & . & 2 & . & 5 & . & . & 8 & −1 & . & 2 & −4 & 4 \\ . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 & . & −1 & . & . & 1 & 3 & . \\ . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 & . & . & 1 & 1 & −1 & −1 & . \\ 1 & −1 & . & . & . & . & . & . & . & . & −1 & 1 & −1 & . & . & 1 & 1 & 1 & −1 & 1 & −2 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & −1 & 1 & 2 & . & 1 \\ . & . & . & 1 & 2 & . & . & . & −2 & . & . & . & . & . & . & −1 & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & 1 & −1 & −2 & 1 & −1 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & 2 & −2 \\ . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & 1 & . & −1 & . & −1 \\ . & . & . & . & . & . & . & . & . & . & 1 & . & 1 & . & . & 2 & 1 & −1 & −1 & −1 & . \\ . & . & . & . & . & . & . & . & . & . & 2 & . & 2 & . & . & 1 & . & −1 & 1 & −2 & 1 \\ . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & −1 & 1 & 1 & . \\ . & . & . & . & . & . & . & . & . & . & 1 & . & 1 & . & 1 & 1 & 1 & . & −2 & . & −1 \\ . & . & . & . & . & . & . & . & . & . & 2 & . & 2 & . & . & . & 1 & −2 & . & −2 & 1 \\ . & . & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & 1 & . & . & . & −1 & . \end{matrix} ]

With these data, there are only 4 solutions of the fundamental problem I, namely, up to a permutation of the columns, U_{1} is one of the matrices given in Table 3.1. There, a, b ∈ {0, 1}. We do not intend to give a proof of this fact here. An indication of how this is done is given in Chapter 5.

If we are left with a small number of non-equivalent solutions of the fundamental problem, we can distinguish cases. We take each solution in turn and pretend that it leads to the true decomposition matrix. Then we tensor all irreducible Brauer characters with themselves, and write the resulting characters in terms of the irreducibles. If this leads to expressions with negative coefficients, our assumption was wrong and we consider the next solution.

In principle one could extend the fundamental problem by adding the conditions arising from tensor products. Suppose BS = {ϑ_{1}, …, ϑ_{m}} and PS = {Ψ_{1}, …, Ψ_{m}}. Let a_{ijl} = ⟨ϑ_{i}ϑ_{j}, Ψ_{l}⟩. For i = 1, 2, let U_{i}^{−1} = (\underline{u}_{rs}^{i})_{r,s}. Then the solution (U_{1}, U_{2}) which determines the decomposition Table 3.1: Possible decomposition matrices for the principal block of Co_{1}

\begin{matrix}1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 276 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 299 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 17250 & . & . & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 80730 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 94875 & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 822250 & . & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 871884 & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1821600 & . & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\ 2055625 & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 9221850 & . & . & . & . & . & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 16347825 & . & . & . & . & 1 & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 21528000 & . & . & . & . & . & 1 & . & 1 & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 21579129 & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 & . & . & . & . & . & . \\ 24667500 & 1 & . & . & . & 1 & . & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . \\ 57544344 & . & . & 1 & . & . & . & 1 & . & . & . & 1 & . & . & . & . & 1 & . & . & . & . & . \\ 66602250 & . & . & . & 1 & . & . & . & . & 1 & a & . & . & . & . & . & . & 1 & . & . & . & . \\ 85250880 & . & . & 1 & 1 & . & . & . & . & . & 1 & . & . & . & . & . & 1 & . & 1 & . & . & . \\ 150732800 & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & 1 & . & . \\ 163478250 & . & 1 & . & . & . & 1 & . & . & . & . & 1 & . & 1 & b & . & . & . & . & . & 1 & . \\ 207491625 & . & . & . & 1 & . & . & . & . & . & a & . & . & . & . & . & . & 1 & 1 & . & . & 1 \end{matrix}

matrix must satisfy:

∑_{r=1}^{m} ∑_{s=1}^{m} ∑_{t=1}^{m} a_{rst}\underline{u}_{ir}^{1}\underline{u}_{js}^{1}\underline{u}_{tl}^{2} ≥ 0, for all i, j, l,

since the expression on the left hand side equals ⟨φ_{i}φ_{j}, Φ_{l}⟩, where [φ_{i}] respectively [Φ_{i}] is the i-th row of U_{1}^{−1} [BS] respectively U_{2}^{−t}[PS].

In the ideal case, the matrix U in (3.1) is the identity matrix. Then we have solved our problem.

Proposition 3.3.2 Let BS and PS be two basic sets such that the matrix U of their mutual scalar products is the identity matrix. Then BS consists of the irreducible Brauer characters and PS of the PIMs.

Proof. This immediately follows from (3.6), (3.2) and (3.3).

As already mentioned above, MOC tries to reach at this final stage by improving the two basic sets BS and PS step by step by making use of the conditions (3.8).

We finally indicate a proof of a fact mentioned in [80, Remark 6.3.34], namely that the information gained by inducing Brauer characters from a subgroup is the same as that obtained by restricting projective characters to this same subgroup. It is not really necessary to use the language of basic sets to do so, but it can be used to quantify the information.

Let H be a subgroup of G. We assume that the decomposition matrix D_{H} of H is known. Let F denote the G-H-induction-restriction matrix, i.e.,

[Irr(G)_{H}] = F [Irr(H)],

where Irr(G)_{H} denotes the set of restrictions to H of the irreducible characters of G. Then, if Irr(H)^{G} is the set of characters of G induced from the irreducible characters of H, we have

[Irr(H)^{G}] = F^{t} [Irr(G)],

by Frobenius reciprocity. Let BS be a basic set for the Brauer characters of G. Since the decomposition map is surjective, there is an integral matrix Y such that

[BS] = Y [Îrr(G)].

We choose such a Y and fix it in the following. Then

[IBr(G)] = X [Îrr(G)],

with X = U_{1}^{−1} Y. By the surjectivity of the decomposition homomorphism and the definition of the PIMs, this implies that

[Irr(G)] = X^{t} [IPr(G)].

Now [IPr(H)] = D_{H}^{t} [Irr(H)], and so

[IPr(H)^{G}] = D_{H}^{t} [Irr(H)^{G}]

= D_{H}^{t} F^{t} X^{t} [IPr(G)]
= D_{H}^{t} F^{t} X^{t} U_{2}^{−t} [PS]
= D_{H}^{t} F^{t} Y^{t} U^{−t} [PS].

Thus the information gained by inducing the projective characters from H is the following condition for U_{2} coming from (3.8)

(D_{H}^{t} F^{t} Y^{t} U^{−t}) U_{2}^{t} ≥ 0.

On the other hand, restricting the irreducible Brauer characters of G down to H we obtain

[IBr(G)_{H}] = X [Îrr(G)_{H}]
= X F [Irr(H)]
= U_{1}^{−1} Y F D_{H} [IBr(H)]
= U_{2} U^{−1} Y F D_{H} [IBr(H)],

giving exactly the same conditions as above.

3.4 The second fundamental problem

In general, the two matrices of (3.8) have many rows. But if one of these rows is a sum of others, then the row obviously contains no condition at all and can be deleted. As our experience shows, a quick procedure to throw away these obsolete rows is of enormous help. For example, in the course of the calculation of the 7-modular characters of the Conway group 2.Co_{1} we had to produce more than 10000 projective characters in order to find the few which are of any use. This leads us to the formulation of our second fundamental computational problem. Before we do this, we give an ad hoc definition, which is only needed to formulate the problem.

Definition 3.4.1 Let W be a finite subset of ℤ^{1×s}, i.e., a set of rows of integers of length s. A subset W_{0} ⊆ W is called essential, if for all w ∈ W there are non-negative integers n_{v}, v ∈ W_{0}, such that w = ∑_{v∈W_{0}} n_{v}v.

If the zero vector is not in the positive ℤ-span of W, then there is a unique minimal essential subset of W.

Fundamental Problem II: Let W be a finite subset of ℤ^{1×s}. Determine an essential set W_{0} ⊆ W, with |W_{0}| as small as possible.

In general the above problem is slightly more complicated since we do not have a set of vectors but a sequence with some vectors repeated.

3.5 Relations

If we are given a basic set of Brauer characters BS and we know how to write the basic set characters in terms of the irreducible Brauer characters, we can derive the decomposition matrix as remarked at the beginning of Section 3.3. The following notation is convenient in our situation.

Definition 3.5.1 A relation is a linear dependence over ℤ between generalized characters. More specificly: If B is any basis of G_{0}(B̂), and if ϑ ∈ G_{0}(B̂), then the expression of ϑ in terms of B is called a relation with respect to B. A similar notion is used for projective characters.

Example 3.5.2 The matrix in Example 3.1.3 gives the relations of the six ordinary characters not contained in BS in terms of this basic set. The matrix W in Example 3.3.1 gives the relations of some projectives in terms of the basic set of projectives PS of Example 3.1.3.

If we are given a set BS of ordinary irreducible characters, restricted to the p-regular classes, then in order to show that they form a basic set it suffices, as we have seen above, to express every other irreducible ordinary character, restricted to the p-regular classes, as a ℤ-linear combination of the elements of BS. If this can be done, we have at the same time found the relations which are needed to reconstruct the whole decomposition matrix.

Sometimes the relations with respect to a basis are stored rather than the characters themselves. This is for reasons of space but also to simplify and speed up the calculation of inner products. Let BS be a basic set of Brauer characters with dual basis of projective atoms PA. Suppose furthermore, that P and B are two sets of projective characters respectively Brauer characters. Then in order to find the matrix ⟨P, B⟩ of mutual inner products, it suffices to write

[B] = V [BS]  (3.9)

and

[P] = W [PA],  (3.10)

in order to find

⟨P, B⟩ = W V^{t}.  (3.11)

Example 3.5.3 The scalar product of a Brauer character in the set

{31574400, 191102976, 215547904, 219648000, 299710125, 326956500}

of Example 3.1.3 with a projective of PS of Example 3.1.5 is given by matrix multiplication of the rows of matrix in Example 3.1.3 with the columns of the matrix of Example 3.1.5.Thus, for example, the Brauer character 191102976 has inner product 1 − 1 − 1 + 1 − 1 − 1 + 3 = 1 with the projective Ψ_{1} and inner product 1 − 1 = 0 with projective Ψ_{2}.

3.6 Special basic sets

We keep the notation of the previous section. The situation is considerably simpler if we have basic sets of Brauer characters of a special nature.

Definition 3.6.1 A basic set BS of Brauer characters is called special, if it is of the form BS = {φ_{1}, …, φ_{m}}, where the φ_{i} are restrictions of ordinary characters of G to the p-regular conjugacy classes, i.e., φ_{i} = χ̂_{j_{i}} for some χ_{j_{i}} ∈ Irr(B).

Let BS_{0} denote a special basic set and let PA_{0} be the corresponding set of projective atoms. Then write

[Îrr(B)] = S [BS_{0}].  (3.12)

The matrix S thus gives the relations arising from the restrictions of the ordinary characters to the p-regular conjugacy classes with respect to the special basic set BS_{0}. In the present version of MOC we always fix a special basic set and store the matrix S. In particular, MOC will only work if there exists such a special basic set.

It is not clear at all whether there is always a special basic set. In p-soluble groups there is always one by the theorem of Fong and Swan. If the ordinary irreducible characters in a p-soluble group are sorted in increasing order of their degrees, then the first maximal linear independent subsequence of characters consists of the irreducible Brauer characters. In blocks with cyclic defect group there is a special basic set by the results of Brauer and Dade. The authors have checked all blocks for all sporadic simple groups and all possible p, and found a special basic set in all cases. In [46] it is shown that there is a special basic set for groups of Lie type in non-defining characteristics under some additional hypotheses. In searching for a counterexample we would first try a group of Lie type in defining characteristic.

A special basic set has various advantages. First of all, each basic set character contains only modular constituents of a single block, and so the subset BS_{0} ∩ B is a special basic set of B. This is used to find the contribution of a generalized Brauer character to B by (2.11).

The second advantage of a special basic set is the fact that it allows to express a projective character in terms of PA_{0} without much work. Namely, let P denote a set of projective characters expressed in terms of the ordinary irreducible characters. That is to say we know the matrix

Y = ⟨P, Irr(B)⟩.  (3.13)

We then obtain the matrix

Y_{0} = ⟨P, BS_{0}⟩

by just deleting the columns of Y corresponding to characters in Irr(B)\ BS_{0}. Thus

[P] = Y_{0} [PA_{0}].  (3.14)

Provided we have a special basic set of Brauer characters, this observation allows to restrict (in an obvious sense) the decomposition matrix to it, and then give the columns a new interpretation: These are just the PIMs expressed in the dual basis of atoms.

If one tries to calculate the decomposition matrix by constructing projective characters and expressing them in terms of ordinary irreducibles, it suffices to calculate their scalar products with the elements belonging to a special basic set. This is often an enormous saving of space and time.

Example 3.6.2 The basic set BS of Example 3.1.3 is special, and hence the columns of the matrix of Example 3.1.5 give the coefficients in the expression of the corresponding projectives in terms of the basis PA dual to BS.

By comparing (3.12), (3.13) and (3.14) with (3.9), (3.10) and (3.11), we have

Y = Y_{0} S^{t}.

This can be interpreted as finding the matrix Y if a set of projectives is given in terms of PA_{0}. This will be important later on, since in the process of improving the projectives MOC uses their expressions via the matrix Y_{0}.

We finally specialize to the case that P = PS is a basic set of projectives. Then, writing

X = ⟨PS, Irr(B)⟩,  and  X_{0} = ⟨PS, BS_{0}⟩,

we have

X = X_{0} S^{t}.

By the orthogonality relations we also have

[BS_{0}] = X_{0}^{t} [BA].  (3.15)

If B is a set of Brauer characters, [B] = V [BS_{0}], we obtain

[B] = V X_{0}^{t} [BA]

which can also be expressed as

⟨B, PS⟩ = V X_{0}^{t}.

This means that if B is given in terms of BA, i.e., by its inner products with PS, say B = W [BA], we can express B in terms of BS_{0} as follows:

[B] = W X_{0}^{−t} [BS_{0}].

3.7 Triangular decomposition matrices

The solution of the Fundamental Problem I is drastically simplified if the matrix U is a lower triangular matrix. Since U is unimodular, the entries on the main diagonal are all equal to 1. The two matrices U_{1} and U_{2} are then unimodular triangular matrices, too. One obtains U_{1}, say, by iteratively subtracting columns of U from the right (compare also [80, 6.3.22] and the subsequent remarks). Closely related to this special case is the notion of wedge shape of the decomposition matrix.

Definition 3.7.1 The decomposition matrix of B has wedge shape, if the ordinary irreducible characters can be ordered such that the decomposition matrix has the following form:

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

where the entries above the main diagonal are all 0.

In a symmetric group every decomposition matrix has wedge shape (see e.g. [80, Theorem 6.3.60]). The same is true for the general linear and unitary groups in non-defining characteristics, as Dipper [22] and Geck [44] have shown.

We obviously have the following connection.

Lemma 3.7.2 The decomposition matrix has wedge shape, if and only if there is a special basic set of Brauer characters BS_{0} and a basic set PS of projective characters such that the matrix U of (3.1) is a lower triangular matrix.

Data structures

In this chapter we describe the principal data structures underlying the MOC-system. We also present a new and remarkable result due to H. W. Lenstra on certain integral bases of abelian number fields. This result had been suggested by experimental evidence gained in the preparation of the MOC-system.

Based on the theory described in Chapter 3, MOC consists of a collection of FORTRAN-programs and of unformatted FORTRAN-files, which contain characters and information about the characters. The programs call the subroutine RRR to perform reading and writing of files and to do the basic operations on long integers. The files are used for exchanging information between the different programs and for recording the calculations, which have been done. The representation of the various data is the same for the external and the internal format. The programs solve very specific tasks, like tensoring two lists of Brauer characters or multiplying two matrices. MOC lacks a memory management and a language on top of it. Instead more complicated tasks are achieved by concatenating suitable programs via UNIX Bourne-shell scripts. We give an example for such a shell script in the next chapter. We begin by describing the basic data types, which the programs use, and the operations, which are performed on them.

4.1 Long integers

The fundamental data type in MOC is the long integer. A long integer n is represented as a list of coefficients with respect to the basis 10^{4}. If n ≥ 0 then

n = ∑_{i=0}^{k} a_{i}10^{4i}, 0 ≤ a_{i} ≤ 9999, a_{k} ≠ 0.

The list of coefficients a_{k−i+1} is stored in a 1-dimensional FORTRANarray r[ ] and 10000 is added to a_{0}. That is to say, r[i] = a_{k−i+1}, 1 ≤ i ≤ k, and r[k + 1] = a_{0} + 10000. For example, if n = 123456789 then

r[1] = 1, r[2] = 2345, r[3] = 16789.

In case n is negative, we store the coefficients −a_{k−i+1} and add 20000 to −a_{0}. If n = −123456789, then

r[1] = 1, r[2] = 2345, r[3] = 26789.

In order to do the basic operations on long integers, the FORTRAN programs call the arithmetic subroutine RRR, which performs addition, subtraction, multiplication and division of long integers and returns the result to the calling program. The algorithms used by RRR can be found for example in Knuth’s book [88, Chapter 4.3].

4.2 Files

The data stored in memory or on file consists of numbers coded as described above and of separators which are numbers in the range between 30000 and 31000. The separators are used as labels, which indicate the sort of data, following the separator. The separator 30900 for example is followed by class functions evaluated on certain conjugacy classes.

MOC keeps track of the calculation performed for a specific group G and a specific prime p on several files. We shortly mention the four most important ones.

(a) The file G.p contains the following information: A special basic set BS_{0} stored under the label 30900, the matrix of relations S stored under 30550 expressing the ordinary characters in terms of BS_{0}, a basic set of projectives PS in terms of the projective atoms dual to the special basic set, an actual basic set of Brauer characters BS in terms of the Brauer atoms dual to PS. Furthermore it contains information related to the distribution of the characters into blocks. Changing from a representation of characters in a certain basis to a representation in a different basis or writing the characters as class functions is achieved by using the formulae given in Section 3.6.

(b) The file G.p.bras contains a list of Brauer characters stored under the label 30500 expressed in terms of BS_{0}. Whenever new Brauer characters are generated they are put onto this file.

(c) The file G.p.proj contains a list of projective characters stored under the label 30700 expressed in terms of the ordinary irreducible characters. Whenever new projective characters are generated they are put onto this file.

(d) The file G.p.info contains information concerning the calculations already performed by MOC. This can be used to give a conventional proof of the results obtained by MOC. We describe in a later section this important feature in more detail.

4.3 Integers in abelian number fields

The second basic data type deals with character values, which are algebraic integers from abelian number fields. Our representation of character values differs from those used in other systems, such as for example CAS [104]. This is motivated by the fact that Brauer character tables tend to contain more irrationalities than ordinary character tables. One of the important consequences of the new MOC-format for character values is that the basic algorithms involving Brauer characters can be reformulated as algorithms for ℤ-lattices, avoiding explicit calculations in cyclotomic fields. The MOC-format also seems to give a more compact representation of a character table.

Let K be an algebraic number field of degree d over ℚ with ring of integers R. For performing arithmetic in R we choose an integral basis b_{1}, …, b_{d}. The elements of R are represented by the coefficients in this basis. The addition of two elements is now easily performed by adding their coefficient vectors. Next we produce the multiplication matrix of the basis elements, i.e., we store the coefficients of all products b_{i} · b_{j}. The multiplication of two arbitrary elements of R is now achieved by matrix multiplication of the two coefficient vectors with the multiplication matrices of the basis elements. The fields K we mainly deal with have a small degree d over ℚ. Therefore, we decided to store the multiplication tables explicitly, since the cost of storing is relatively small compared to the speedup we gain in performing the multiplication by the method described above.

4.4 Character tables

The character values of finite groups are sums of roots of unity and are therefore algebraic integers of abelian number fields. As described above the character values are stored as coefficient vectors corresponding to certain integral bases, which are to be described below. We shall see that this can be done in such a way that the resulting character table is still a square matrix. The new format of a character table is based upon the following considerations.

We begin by introducing some notation. If K ⊆ L is a Galois extension of fields, we let G(L/K) denote its Galois group. Let n be a positive integer. We write ℚ_{n} for the n-th cyclotomic field, and φ(n) for the degree of ℚ_{n} over ℚ, i.e, φ is the Euler function. If f is an integer coprime to n let ∗f denote the Galois automorphism of ℚ_{n} mapping an n-th root of unity to its f-th power.

Let G be a finite group and x ∈ G. Define ℚ(x) = ℚ(χ(x) | χ ∈ Irr(G)) to be the column field of x. If x has order n then ℚ(x) is contained in ℚ_{n}.

Definition 4.4.1 Two elements of G are called algebraically conjugate if the cyclic subgroups they generate are conjugate in G. The equivalence class of x ∈ G with respect to this relation is called the algebraic conjugacy class of x and denoted by Cl_{a}(x). Note that it is a union of conjugacy classes of G.

Let y be an element of Cl_{a}(x). Then y is conjugate to x^{f} with f relatively prime to n. If χ is a character of G then χ(y) = χ(x)^{∗f}.

Proposition 4.4.2 Let x be an element in G of order n. Suppose the column field ℚ(x) has degree d over ℚ. Then d equals the number of (ordinary) conjugacy classes in the algebraic conjugacy class Cl_{a}(x).

Proof. We put Y = {x^{i} | 1 ≤ i ≤ n, gcd(i, n) = 1} and write S_{Y} for the group of permutations of Y. The normalizer N = N_{G}(⟨x⟩) acts on Y by conjugation. The kernel of the corresponding homomorphism

ρ : N → S_{Y}

is C_{G}(x) and so |ρ(N)| = |N/C_{G}(x)|.

The orbits of N on Y are in one to one correspondence with the G-conjugacy classes intersecting Y non-trivially. The size of the orbits is |N/C_{G}(x)| and the number of orbits is φ(n)/|N/C_{G}(x)|.

Define the monomorphism

ψ : G(ℚ_{n}/ℚ) → S_{Y} ,
∗k ↦ (y ↦ y^{k})

and observe that ψ(G(ℚ_{n}/ℚ(x))) = ρ(N). This follows from the remark preceding the proposition and the fact that two elements a, b ∈ G are conjugate, if and only if χ(a) = χ(b) for all χ ∈ Irr(G). Hence d = |ℚ(x) : ℚ| = φ(n)/|ρ(N)| and the assertion follows.

We are now ready to define the MOC-character table.

Definition 4.4.3 Let x_{1}, …, x_{r} be representatives for the algebraic conjugacy classes of G. Let d_{i} denote the degree of ℚ(x_{i}) over ℚ. For each i, we choose an integral basis b_{1}(x_{i}), …, b_{d_{i}}(x_{i}) of ℚ(x_{i}).

The columns of the MOC-table are indexed by

b_{1}(x_{1}), …, b_{d_{1}}(x_{1}), b_{1}(x_{2}), …, b_{d_{2}}(x_{2}), …, b_{1}(x_{r}), …, b_{d_{r}}(x_{r})

and the entries of χ ∈ Irr(G) at the columns b_{1}(x_{i}), …, b_{d_{i}}(x_{i}) are given by the coefficients of the decomposition of χ(x_{i}) in the basis b_{1}(x_{i}), …, b_{d_{i}}(x_{i}). By the preceding proposition the MOC-character table is a square integral matrix.

Example 4.4.4 Let A_{5} be the alternating group on five letters. The ordinary character table is given by :

      CAS                               MOC
           2   2  2  .  .  .
           3   1  .  1  .  .
           5   1  .  .  1  1
              1a 2a 3a 5a 5b
          2P  1a 1a 3a 5b 5a
          3P  1a 2a 1a 5b 5a
          5P  1a 2a 3a 1a 1a            1  2  3  5  5
           2
      X.1  +   1  1  1  1  1            1  1  1  1  0
      X.2  +   3 -1  .  A *A            3 -1  0  1  1
      X.3  +   3 -1  . *A  A            3 -1  0  0 -1
      X.4  +   4  .  1 -1 -1            4  0  1 -1  0
      X.5  +   5  1 -1  .  .            5  1 -1  0  0

Here, A = (1+\sqrt{5})/2 and ∗A = (1−\sqrt{5})/2. The integral basis of ℚ(5A) in the MOC-table is {1, ϵ + ϵ^{4}}, where ϵ = e^{2πi/5}.

We now indicate how a usual character table can be transformed into a MOC-character table and vice versa.

Definition 4.4.5 Let G(ℚ(x_{i})/ℚ) = {σ_{i,1}, …, σ_{i,d_{i}}}, where the notation is as in Definition 4.4.3. Let A_{i} be the (d_{i} × d_{i})-matrix whose (j, k)-th entry is

σ_{i,j}(b_{k}(x_{i})).

Let d = ∑_{i=1}^{r} d_{i} and let A be the (d × d)-block diagonal matrix whose i-th block is A_{i}.

Note that (det(A_{i}))^{2} equals the discriminant of ℚ(x_{i}) over ℚ; in particular, A is invertible. With a suitable ordering of the classes, we obtain the usual character table from the MOC-character table by a matrix multiplication:

Usual ordinary table = MOC-Table · A.

Note that this observation can be used to calculate the block distribution of the characters by using the MOC-table as follows. Let χ ∈ Irr(G) and let ω_{χ} be the corresponding central character. Let c_{χ}^{i} denote the row of coefficients of ω_{χ}(x_{i}) expressed in the integral basis b_{1}(x_{i}), …, b_{d_{i}}(x_{i}). Then

[ω_{χ}] = [c_{χ}^{1}, …, c_{χ}^{r}] · A.

Since the block matrices A_{i} corresponding to p-regular classes have determinant not divisible by p [105, Lemma I.(10.1), Satz I.(2.11), Korollar III.(2.10)], two characters χ and ψ are in the same p-block of G, if and only if

c_{χ}^{i} ≡ c_{ψ}^{i}(mod p)

for all i corresponding to p-regular classes. We thus have proved the stronger result on block distribution (see [47, (7.10)]), namely that two characters lie in the same p-block if and only if the values of their central characters on p-regular classes differ by elements of pR.

The MOC-system contains a data base for the subfields L of cyclotomic fields up to a certain degree. In this data base a field L is stored by the following information: The minimal n such that L is contained in the cyclotomic field ℚ_{n}, the degree d of L over ℚ, generators for the Galois group L = G(ℚ_{n}/L) and an integral basis of L. Each basis element is a sum over the elements in an orbit of L acting on the n-th roots of unity. Such elements are called orbit sums in the following. It is only necessary to store one representative for each orbit of L whose corresponding sum is a basis element.

Only recently it has been shown by H.W. Lenstra that there always exists such a basis for a given abelian extension over ℚ [95]. We are grateful to him for allowing us to reproduce this previously unpublished result. It is given in the next section.

4.5 Lenstra’s theorem on integral bases

We keep the notation of the preceding section. Furthermore, we put ζ_{m} = exp(2πi/m).

Theorem 4.5.1 (H. W. Lenstra [95]:) Let L/ℚ be a finite abelian extension of the rational numbers. Let f be the smallest positive integer such that L is contained in the f-th cyclotomic field ℚ_{f} (the conductor of L). Finally let H be the subgroup of the Galois group G(ℚ_{f}/ℚ) which fixes L. Then there is an integral basis of L, which consists of orbit sums of H on f-th roots of unity.

Proof. Let p be a prime and k ∈ ℕ. We arrange the set of p^{k}-th roots of unity into the following matrix:

\begin{matrix}1 & ζ_{p} & ζ_{p}^{2} & … & ζ_{p}^{p−1} \\ ζ_{p^{k}} & ζ_{p^{k}}ζ_{p} & ζ_{p^{k}}ζ_{p}^{2} & … & ζ_{p^{k}}ζ_{p}^{p−1} \\ ζ_{p^{k}}^{2} & ζ_{p^{k}}^{2}ζ_{p} & ζ_{p^{k}}^{2}ζ_{p}^{2} & … & ζ_{p^{k}}^{2}ζ_{p}^{p−1} \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ ζ_{p^{k}}^{p^{k−1}−1} & ζ_{p^{k}}^{p^{k−1}−1}ζ_{p} & ζ_{p^{k}}^{p^{k−1}−1}ζ_{p}^{2} & … & ζ_{p^{k}}^{p^{k−1}−1}ζ_{p}^{p−1} \end{matrix}

We introduce an equivalence relation ∼ on T = ⟨ζ_{p^{k}}⟩\{1} as follows: For η, η′ ∈ T we put η ∼ η′ if and only if either η = η′ or else η^{p} = η′^{p} ≠ 1. The equivalence classes are the one element sets {ζ_{p}^{i}}, 1 ≤ i ≤ p − 1 and the p-element sets constituted by the elements of a row different from the first of the above matrix. If C is one of the equivalence classes containing more than one element and if η ∈ C, then

∑_{μ∈C} ℤμ = η ℤ[ζ_{p}],

and thus is a free ℤ-module of rank p − 1. By counting dimensions, we find

ℤ[ζ_{p^{k}}] = ⨁_{C∈T/∼} (∑_{η∈C} ℤη) .  (4.1)

We now generalize these observations. Write

f = ∏_{p prime} p^{k(p)}.

Then

ℤ[ζ_{f}] ≅ ⨂_{p prime} ℤ[ζ_{p^{k(p)}}].  (4.2)

From this and the decomposition (4.1) we obtain a similar direct decomposition of ℤ[ζ_{f}] which we are now going to describe. Let f_{0} be the product of the different prime divisors of f. Put S = {η ∈ ⟨ζ_{f}⟩ | f_{0} divides |η|}. For η ∈ S let d(η) denote the largest squarefree number such that d(η)^{2} | |η|. We introduce an equivalence relation ∼ on S as follows: For η, η′ ∈ S we put η ∼ η′ if and only if η^{d(η)} = η′^{d(η)}. If η ∼ η′, then |η| = |η′| and so our relation is indeed symmetric. From (4.1) and (4.2), we obtain

ℤ[ζ_{f}] = ⨁_{S/∼} (∑_{η∈C} ℤη) .  (4.3)

If C is an equivalence class, η ∈ C and d = d(η), then ∑_{μ∈C} ℤμ = ηℤ[ζ_{d}], and thus the ℤ-rank of ∑_{μ∈C} ℤμ equals φ(d). Thus there is a subset B_{C} ⊂ C which is a ℤ-basis of ∑_{μ∈C} ℤμ, e.g., B_{C} = {ηζ_{d}^{i} | 0 ≤ i < φ(d)}, or B_{C} = {ηζ_{d}^{i} | gcd(i, d) = 1}.

It is clear that H fixes S and permutes the equivalence classes. Suppose first that H acts fixed point freely on the set of equivalence classes of S, i.e., no non-identity element of H fixes an equivalence class. Then, by (4.3) and the subsequent remarks, it is clear that we may choose an integral basis of ℤ[ζ_{f}]^{H}, the set of integers in L, consisting of orbit sums of H on S. Thus we are done in this case.

Suppose there is some 1 ≠ σ ∈ H and η ∈ S such that σ(η) ∼ η. Let d = d(η). Then, by definition, σ(η^{d}) = σ(η)^{d} = η^{d}. The order of η^{d} is divisible by f_{0}, and so σ(ζ_{f_{0}}) = ζ_{f_{0}}. This implies σ ∈ H_{0}, where H_{0} is the intersection of H with the kernel M of the natural map

(ℤ/fℤ)^{∗} → (ℤ/f_{0}ℤ)^{∗}.

The order of M is ∏_{p} p^{k(p)−1}, and the Sylow p-subgroup is cyclic, except if p = 2, and k(2) ≥ 3. Let f′ = f/2^{k(2)} denote the odd part of f. If k(2) ≥ 3, the Sylow 2-subgroup of M is M_{0} × M_{1}, with M_{0} = {∗a | a ≡ 1(mod f′) and a ≡ ±1(mod 2^{k(2)})} and M_{1} = {∗b | b ≡ 1(mod 4f′)}.

The condition that f be the conductor of L is equivalent to saying that for all p dividing f, the group H does not contain the kernel of the natural map

(ℤ/fℤ)^{∗} → (ℤ/\frac{f}{p} ℤ)^{∗}.  (4.4)

This kernel has order p and is generated by ∗(1 + f/p).

So if p is odd or p = 2 and k(2) ≤ 2, then the Sylow p-subgroup of H_{0} is trivial, since otherwise ∗(1 + f/p) would be contained in H, which is not the case by the remarks above. Thus H_{0} is a 2-group, k(2) ≥ 3 and so H_{0} ≤ M_{0} × M_{1}. Let a ∈ ℤ such that ∗a ∈ H_{0} has order 2. Then a ≡ 1(mod f′) and one of the following holds:

a ≡ \{ \begin{matrix}−1 \\ −1 + 2^{k(2)−1} \\ 1 + 2^{k(2)−1} \end{matrix} \} (mod 2^{k(2)}).

By the remarks preceding and following (4.4), H_{0} ∩ M_{1} = 1, and thus the last possibility cannot occur and we must have H_{0} = ⟨a⟩, i.e., H_{0} is a cyclic group of order 2.

So if some non-trivial element σ_{0} of H fixes the equivalence class containing η, then σ_{0} = ∗a. Furthermore, if 2 is the exact power of 2 dividing |η|, then σ_{0}(η) = η, and if 4 is the exact power of 2 dividing |η|, then σ_{0}(η) = −η. On the other hand, if σ(η) = η or σ(η) = −η for σ ∈ H then σ(η) ∼ η. Suppose that 8 divides the order of η. Write d(η) = 2c, c odd. Then η ∼ σ_{0}(η) = η^{a} implies η^{2c} = η^{2ca}, and so η^{2c(a−1)} = 1. However, 2c(a − 1) is not divisible by 8, a contradiction.

Let H_{1} = {∗a ∈ H | a ≡ 1(mod4)}. Then H_{0} ∩ H_{1} = 1, and so H = H_{0} × H_{1}. We now proceed as follows. Since H_{1} acts fixed point freely on the equivalence classes, we can choose an integral basis of ℤ[ζ_{f}]^{H_{1}} consisting of orbit sums of H_{1} on S. If an orbit sum contains an element from an equivalence class fixed by H_{0}, we consider two cases. Either the element has order divisible by 4, then H_{0} acts as −1 on the corresponding ℤH_{1}-module. Otherwise, H_{0} fixes every element in the orbit, in which case we include the corresponding orbit sum, which is now an H-orbit sum, into our basis.

The remaining orbit sums contain only elements of equivalence classes not fixed by H_{0}. The corresponding H-orbit sums are taken to yield the desired integral basis.

Algorithms

In this chapter we describe the principal algorithms utilized by MOC. We first explain two algorithms for dealing with systems of integral linear equations using q-adic approximation. Then we describe the methods of integral linear programming used to attack the two fundamental problems. Furthermore we show how all MOC-computations can be documented and how these documentations are used to give a more conventional proof for the results obtained. Finally we discuss some more advanced methods which can be applied if the MOC-system does not give the complete answer.

5.1 Integral linear equations

Most of the time during a run of MOC we are dealing with the problem of expressing a character as a ℤ-linear combination in terms of a basic set. In other words we have to solve a system of integral linear equations. Dixon [25] independently suggested the following way of solving this problem.

Algorithm 5.1.1 (DEC) PROBLEM: Given m + 1 rows of integers

w, b_{1}, b_{2}, …, b_{m} ∈ ℤ^{1×n},

where b_{1}, b_{2}, …, b_{m} are linearly independent over ℤ. If possible write w as a ℤ-linear combination of these. SOLUTION: Let T be the matrix

T = ( b_{1} ; ⋮ ; b_{m} )

Let q be a (large) prime such that T̄, the reduction modulo q of T, has rank m. Set v := w, j := 0 and w_{0} := 0 (zero vector). Set MAXJ to a reasonable positive integer, representing the maximum number of loops, we are willing to spend our time with.

Step 1. Check whether v̄, the reduction modulo q of v, is in the 𝔽_{q}- span of b̄_{1}, b̄_{2}, …, b̄_{m}. If not, STOP, noting that w is not in the

ℚ-span of b_{1}, b_{2}, …, b_{m}. Otherwise go to Step 2.

Step 2. Define the integers z_{ij}, 1 ≤ i ≤ m by

v̄ = ∑_{i=1}^{m} z̄_{ij}b̄_{i}, where − (q − 1)/2 ≤ z_{ij} ≤ (q − 1)/2,

and set

w_{j+1} := ∑_{s=0}^{j}(z_{1s}, …, z_{ms})q^{s}T.

Then every coefficient of w−w_{j+1} is divisible by q^{j+1}. If w−w_{j+1} = 0, then STOP. Else set

v := (w − w_{j+1})/q^{j+1}

and increase j by 1. If j is larger than MAXJ then STOP, otherwise continue with Step 1.

Remark 5.1.2 Note that w is in the ℚ-span of b_{1}, b_{2}, …, b_{m} if and only if

w = ∑_{j=0}^{∞}(z_{1,j}, …, z_{m,j})q^{j}T  (5.1)

in the q-adic metric, where the integers z_{ij} are obtained by the above algorithm. Furthermore w is in the ℤ-span of b_{1}, b_{2}, …, b_{m} if and only if (5.1) is finite. When the algorithm terminates because j gets larger then MAXJ, this may have three reasons.

(a) w is in the ℤ-span but some coefficient is too large.

(b) w is in the ℚ-span but not in the ℤ-span.

(c) w is not in the ℚ-span.

Our experience tells us that q = 101 and MAXJ = 20 are reasonable choices for the problems we are concerned with.

The first step in the computation of the decomposition numbers is to find a basic set for G_{0}(F G). We start with the restrictions of the ordinary irreducible characters to the p-regular elements which form a generating set for G_{0}(F G). We then have to derive from these a basic set of G_{0}(F G). There are algorithms for constructing bases of ℤ-lattices from generating sets, but in our situation we have the additional restriction that our basis has to consist of proper Brauer characters. This is achieved by an algorithm due to Parker, called FBA, which is based upon Algorithm 5.1.1. We describe FBA in the following general setup.

Algorithm 5.1.3 (FBA) PROBLEM: Let X be a ℤ-lattice with finite basis φ_{1}, …, φ_{m}. Define X^{+} to be the subset of X consisting of all non-negative linear combinations of the basis elements φ_{1}, …, φ_{m}. Suppose that we are given a finite set B = {ϑ_{1}, …, ϑ_{n}} ⊂ X^{+} generating a sublattice Y of X. Derive from B a basis of Y lying in X^{+}.

SOLUTION: Assume inductively that a subset B̃ = {β_{1}, …, β_{k−1}} ⊂ X^{+} has been found with span equal to ⟨ϑ_{1}, …, ϑ_{s}⟩_{ℤ}, s ≤ n, and such that β_{1}, …, β_{k−1} are linearly independent modulo some large prime q.

Step 1. If s = n then STOP and output B̃. Otherwise take the next element ϑ_{s+1} ∈ B and try to express it in terms of β_{1}, …, β_{k−1} using DEC with the prime q. The following cases can occur and are checked in Step 2.

(a) If ϑ_{s+1} ∈ ⟨β_{1}, …, β_{k−1}⟩_{ℤ}, increase s by 1 and go to Step 1.

(b) If β_{1}, …, β_{k−1}, ϑ_{s+1} are linearly independent modulo q, increase s and k by 1, replace B̃ by {β_{1}, …, β_{k−1}, ϑ_{s+1}} and go to Step 1.

(c) If β_{1}, …, β_{k−1}, ϑ_{s+1} are linearly dependent modulo q but linearly independent over ℤ, replace q by a prime ℓ such that β_{1}, …, β_{k−1}, ϑ_{s+1} are linearly independent modulo ℓ. Replace B̃ by {β_{1}, …, β_{k−1}, ϑ_{s+1}}, increase s and k by 1 and go to Step 1.

(d) If β := ϑ_{s+1} ∈ ⟨β_{1}, …, β_{k−1}⟩_{ℚ} but β ∉ ⟨β_{1}, …, β_{k−1}⟩_{ℤ}, we use the first terms of the q-adic expression for the coefficients of β determined by DEC to find the smallest possible natural number m_{1} with

m_{1}β = ∑_{i=1}^{k−1} z_{i}β_{i},

where z_{i} ∈ ℤ. Let m = min{gcd(m_{1}, z_{i}) | 1 ≤ i ≤ k − 1} and suppose m = gcd(m_{1}, z_{j}) > 0. Write m = am_{1} + bz_{j}, a, b ∈ ℤ. For i different from j, let c_{i} be the smallest integer such that c_{i} + bz_{i}m_{1}^{−1} ≥ 0 and define

β̃_{j} = bβ + aβ_{j} + ∑_{i≠j}^{k−1} c_{i}β_{i}.

Then m_{1}β̃_{j} is contained in X^{+} and hence β̃_{j} ∈ X^{+}, too. Replace β_{j} by β̃_{j} in B̃. Increase n by 1 and put ϑ_{n+1} = β_{j}. Do not increase s in this case. Go to Step 1.

Theorem 5.1.4 Algorithm 5.1.3 terminates and finds a ℤ-basis of Y contained in X^{+}.

Proof. For any subset B ⊆ B we define the weight w(B) of B to be the pair (cr(B), d(B)). Here, cr(B) is the co-rank of the ℤ-span of B in Y. The number d(B) is the discriminant of ⟨B⟩_{ℤ}, which is defined in the following way. Choose any ℤ-basis b_{1}, …, b_{s} of ⟨B⟩_{ℤ}, and let M denote the (m × s)-matrix of the coefficients of b_{1}, …, b_{s} expressed in the basis φ_{1}, …, φ_{m} of X. Then d(B) = det MM^{t}. This number is positive and independent of the basis chosen for ⟨B⟩_{ℤ}.

We order the set of weights W (B) = {w(B) | B ⊆ B} lexicographically. Observe that B generates Y if and only if w(B) is minimal in W (B).

In each of the cases occuring in Step 2 of Algorithm 5.1.3, where we change B̃, we decrease its weight. This is clear in cases (b) and (c), since we increase the rank of the generated lattice. In case (d) let S denote the matrix transforming {β_{1}, …, β_{j}, …, β_{k−1}} into {β_{1}, …, β̃_{j}, …, β_{k−1}}. Then S is a diagonal matrix with diagonal entries equal to 1, except that row j of S has the entries

c_{1} + bz_{1}/m_{1}, …, c_{j−1} + bz_{j−1}/m_{1}, a + bz_{j}/m_{1},
c_{j+1} + bz_{j+1}/m_{1}, …, c_{k−1} + bz_{k−1}/m_{1}.

Thus det(S) = a + bz_{j}/m_{1} = m/m_{1} < 1. This implies that the new set B̃ has a smaller discriminant.

Finally, by adding β_{j} to B the minimum of W (B) remains unchanged, since the new enlarged B still generates the same lattice Y. This proves that the algorithm terminates. Since in each step the set B̃ is linearly independent over the integers, the final such set is a ℤ-basis of Y.

5.2 Improving basic sets

Having established the major part of the theory we are now prepared to give some algorithms for dealing with the two fundamental problems stated in Sections 3.3 and 3.4. These algorithms are implemented in the MOC-system and are parts of the program IMPROVE, which uses methods of linear programming. There are some remarks concerning optimization at the end of this section.

Let us fix some notation. Let PS = {Φ_{1}, …, Φ_{s}} be the basic set of projectives and BS = {φ_{1}, …, φ_{s}} the basic set of Brauer characters. We denote by PA = {φ_{1}^{∗}, …, φ_{s}^{∗}} and BA = {Φ_{1}^{∗}, …, Φ_{s}^{∗}} the dual systems of atoms. Let P and B denote additional sets of projective respectively Brauer characters. We denote by U = (⟨φ_{i}, Φ_{j}⟩_{ij}) the matrix of scalar products between BS and PS.

We are now going to restate the first fundamental problem. Recall that we have to find all square integral matrices U_{1} ≥ 0 and U_{2} ≥ 0 so that U = U_{1} · U_{2} subject to the conditions (3.8). We consider the equivalent problem of finding U_{1} ≥ 0 and U_{2} ≥ 0 invertible over ℤ with U_{1}^{−1}·U ·U_{2}^{−1} = E_{s}. Operating by U_{1}^{−1} means changing BS and operating by U_{2}^{−1} means changing PS. We are going to change U according to (3.8) hoping to get E_{s} as result. We have the two following problems to solve:

1. Prove that φ_{i} is irreducible or that Φ_{j} is indecomposable. This is considered in § 5.2.1.

2. Improve PS or BS. We shall show that this can be done in several ways. We present a theorem in § 5.2.2 which allows to subtract irreducible characters from other ones. Then in § 5.2.3 there is an algorithm which computes characters lexicographically smaller. We finally present a criterion in § 5.2.4 for a character to be decomposable or reducible.

5.2.1 Proving indecomposability

In this paragraph we are going to establish a test for proving irreducibility of a Brauer character respectively indecomposability of a projective character. First of all we need the following

Definition 5.2.1 Let

Φ = ∑_{i=1}^{s} n_{i}φ_{i}^{∗} with n_{i} ≥ 0

be a projective character decomposed into PA. A generalized projective character

Φ′ = ∑_{i=1}^{s} n′_{i}φ_{i}^{∗}

is called a part of Φ if 0 ≤ n′_{i} ≤ n_{i} for all i. A similar notion is used for Brauer characters.

The projective indecomposable summands of a projective character Φ are among the parts of Φ. We are able to state the main criterion for indecomposability.

Theorem 5.2.2 (PIM-test:) The projective Φ is indecomposable if for all parts Φ′ of Φ with Φ′ ≠ Φ and Φ′ ≠ 0 there is a Brauer character φ ∈ B with ⟨φ, Φ′⟩ < 0 or ⟨φ, Φ − Φ′⟩ < 0.

Proof. If there is a φ having negative scalar product with Φ′ or with Φ − Φ′ then one of Φ′, Φ − Φ′ is not a genuine projective. If Φ′ is a part of Φ, then so is Φ − Φ′.

Sometimes one can find another test for indecomposability in the literature. It is based on the fact that an ordinary character is a generalized projective character if and only if it is 0 on all p-singular classes. Let Φ = ∑_{i=1}^{n} a_{i}χ_{i} be a projective character expressed as a sum of ordinary characters. Then we call a character Φ′ = ∑_{i=1}^{n} a′_{i}χ_{i} a subsum of Φ if 0 ≤ a′_{i} ≤ a_{i}, Φ′ ≠ 0 and Φ′ ≠ Φ. In the test in question we consider all subsums of Φ and check if one of them is a generalized projective. If there is no such subsum then Φ is indecomposable. Now we are going to compare the two tests.

Proposition 5.2.3 Suppose that BS is a special basic set and that B consists of the restricted ordinary characters. Let Φ be a projective character. Then the PIM-test presented in Theorem 5.2.2 proves that Φ is indecomposable, if and only if there is no subsum of Φ which is a generalized projective.

Proof. Let Ψ be an arbitrary generalized projective character and

Ψ = ∑_{i=1}^{n} a_{i}χ_{i}

its decomposition into the ordinary irreducible characters. Then we have ⟨χ̂_{i}, Ψ⟩ = a_{i}. Let Φ′ and Φ′′ = Φ − Φ′ be parts of Φ. Then Φ′ and Φ′′ are generalized projective characters. Suppose there is no subsum of Φ which is a generalized projective character. Then neither Φ′ nor Φ′′ is a subsum. Therefore at least one of them, say Φ′, has a negative coefficient in its decomposition into ordinary characters. Call this coefficient a_{i}.

But a_{i} is the scalar product of Φ′ with χ̂_{i}, as we have seen above. Hence the PIM-test of Theorem 5.2.2 proves the indecomposability of Φ.

Now suppose that Φ′ is a subsum of Φ and Φ′ is a generalized projective character. Then ⟨χ̂_{i}, Φ′⟩ ≥ 0 and ⟨χ̂_{i}, Φ − Φ′⟩ ≥ 0 for all 1 ≤ i ≤ n. Because of the choice of BS this particular Φ′ is checked by our PIM-test as well. So it can not decide whether Φ is indecomposable or not.

By adding more Brauer characters to B (for example Brauer characters got by induction) our PIM-test may find more PIMs.

Remark 5.2.4 Let Φ = ∑_{i=1}^{s} n_{i}φ_{i}^{∗}, |B| = m and denote by v_{k1}, …, v_{ks} the coefficients of the decomposition of the k-th character in B, 1 ≤ k ≤ m, when decomposed into BS. Then Φ is indecomposable if there are no n′_{1}, …, n′_{s} satisfying the following conditions:

0 ≤ n′_{i} ≤ n_{i} for all 1 ≤ i ≤ s,

∑_{i=1}^{s} n′_{i} ≤ ∑_{i=1}^{s} n_{i} − 1 (that is Φ′ ≠ Φ),

∑_{i=1}^{s} n′_{i} ≥ 1 (that is Φ′ ≠ 0),

∑_{j=1}^{s} −v_{kj}n′_{j} ≤ 0 for all 1 ≤ k ≤ m (that is ⟨φ, Φ′⟩ ≥ 0 for φ ∈ B),

∑_{j=1}^{s} v_{kj}n′_{j} ≤ ∑_{j=1}^{s} v_{kj}⟨φ_{j}, Φ⟩ for all 1 ≤ k ≤ m (that is ⟨φ, Φ − Φ′⟩ ≥ 0).

This system of inequations can be written in matrices as follows. There are no n′_{1}, …, n′_{s} satisfying A · x ≤ b where

A = ( \begin{matrix}1 & & 0 \\ . & . & . \\ 0 & & 1 \\ 1 & … & 1 \\ −1 & … & −1 \\ −v_{11} & … & −v_{1s} \\ ⋮ & & ⋮ \\ −v_{m1} & … & −v_{ms} \\ v_{11} & … & v_{1s} \\ ⋮ & & ⋮ \\ v_{m1} & … & v_{ms} \end{matrix} ) and b = ( ⟨φ_{1}, Φ⟩ ; ⋮ ; ⟨φ_{s}, Φ⟩ ; −1 + ∑_{j=1}^{s}⟨φ_{j}, Φ⟩ ; −1 ; 0 ; ⋮ ; 0 ; ∑_{j=1}^{s} v_{1j}⟨φ_{j}, Φ⟩ ; ⋮ ; ∑_{j=1}^{s} v_{mj}⟨φ_{j}, Φ⟩ )

and x = (n′_{1}, …, n′_{s})^{t}. How this system of inequalities can be solved is discussed in § 5.2.6. For the reader who is familiar with linear programming we mention that this system is dual feasible if we use a dual simplex algorithm and minimize the 0-function. Then, of course, the zero vector is a dual solution. Furthermore we observe that in each column of A the first non-zero entry is 1.

Remark 5.2.5 (i) The number of rows in A depends on |B|. Therefore it is useful to keep |B| as small as possible.

(ii) By interchanging projectives and Brauer characters we can test the irreducibility of a Brauer character.

Example 5.2.6 In this chapter we are going to calculate the 5-modular table of the sporadic simple group Co_{2}. Its ordinary character table can be found in [19] as well as the one of the first maximal subgroup which is isomorphic to U_{6}(2).2. We make use of the sixth maximal subgroup, too. It is isomorphic to (2_{+}^{1+6} × 2^{4}).A_{8}. The 5-modular tables of both maximal subgroups can be constructed by making use only of tensor products.

(i) In U_{6}(2).2 we have six blocks of defect 1 with the following decomposition matrices:

\begin{aligned}1^{+} & 1 & . & . & . \\ 616^{−} & . & 1 & . & . \\ 8064^{−} & . & 1 & 1 & . \\ 11264^{+} & 1 & . & . & 1 \\ 18711_{2}^{−} & . & . & 1 & 1 \\ & Ψ_{4} & Ψ_{3} & Ψ_{2} & Ψ_{1} \end{aligned}  \begin{aligned}1^{−} & 1 & . & . & . \\ 616^{+} & . & 1 & . & . \\ 8064^{+} & . & 1 & 1 & . \\ 11264^{−} & 1 & . & . & 1 \\ 18711_{2}^{+} & . & . & 1 & 1 \\ & Ψ_{8} & Ψ_{7} & Ψ_{6} & Ψ_{5} \end{aligned}

\begin{aligned}22^{+} & 1 & . & . & . \\ 252^{−} & . & 1 & . & . \\ 4928^{−} & . & 1 & 1 & . \\ 32768^{+} & 1 & . & . & 1 \\ 37422^{+} & . & . & 1 & 1 \\ & Ψ_{12} & Ψ_{11} & Ψ_{10} & Ψ_{9} \end{aligned}  \begin{aligned}22^{−} & 1 & . & . & . \\ 252^{+} & . & 1 & . & . \\ 4928^{+} & . & 1 & 1 & . \\ 32768^{−} & 1 & . & . & 1 \\ 37422^{−} & . & . & 1 & 1 \\ & Ψ_{16} & Ψ_{15} & Ψ_{14} & Ψ_{13} \end{aligned}

\begin{aligned}231^{+} & 1 & . & . & . \\ 1386^{−} & . & 1 & . & . \\ 5544^{+} & 1 & . & 1 & . \\ 14784^{−} & . & 1 & . & 1 \\ 18711_{1}^{−} & . & . & 1 & 1 \\ & Ψ_{20} & Ψ_{19} & Ψ_{18} & Ψ_{17} \end{aligned}  \begin{aligned}231^{−} & 1 & . & . & . \\ 1386^{+} & . & 1 & . & . \\ 5544^{−} & 1 & . & 1 & . \\ 14784^{+} & . & 1 & . & 1 \\ 18711_{1}^{+} & . & . & 1 & 1 \\ & Ψ_{24} & Ψ_{23} & Ψ_{22} & Ψ_{21} \end{aligned}

The remaining characters have defect 0. We write them as Ψ_{25}, …, Ψ_{59}, where their order agrees with the one in [19].

(ii) From (2_{+}^{1+6} × 2^{4}).A_{8} we only need one character of defect 0 and the projectives coming from the factor group A_{8}. The ordinary character table of this maximal subgroup was computed by Fischer [33]. We refer the reader to his survey article [34], where he explains his methods. We obtain:

\begin{aligned}1 & 1 & . & . & . \\ 14 & 1 & 1 & . & . \\ 21 & . & . & 1 & . \\ 56 & . & 1 & . & 1 \\ 64 & . & . & 1 & 1 \\ & Φ_{4} & Φ_{3} & Φ_{2} & Φ_{1} \end{aligned}  \begin{aligned}7 & 1 & . \\ 21_{1} & . & 1 \\ 21_{2} & . & 1 \\ 28 & 1 & 1 \\ & Φ_{6} & Φ_{5} \end{aligned}

Let Φ_{7} denote the defect 0 character of degree 15.

We now use MOC to calculate the 5-modular characters of the second Conway group Co_{2}. The Brauer characters are denoted by their degrees. The 60 ordinary characters of Co_{2} split into 26 blocks. The 23 characters

9625_{1}, 9625_{2}, 23000, 31625_{1}, 31625_{2}, 63250, 91125_{1}, 91125_{2}, 221375,

253000, 284625, 442750, 462000, 664125_{1}, 664125_{2}, 664125_{3}, 853875,

1288000, 1771000_{1}, 1771000_{2}, 1992375, 2004750, 2095875

are of defect 0, and therefore each forms a block in its own. The characters 275, 44275, 113850, 398475 and 467775 belong to the second block. It is a block of defect 1 and the decomposition matrix can be found in [56]. It is

\begin{matrix}275 & 1 & . & . & . \\ 44275 & . & 1 & . & . \\ 113850 & 1 & . & 1 & . \\ 398475 & . & 1 & . & 1 \\ 467775 & . & . & 1 & 1 \end{matrix}

In the third block we find the characters 4025, 12650, 177100, 398475 and 558900. The decomposition matrix can be found in [56] as well:

\begin{matrix}4025 & 1 & . & . & . \\ 12650 & . & 1 & . & . \\ 177100 & 1 & . & 1 & . \\ 398475 & . & 1 & . & 1 \\ 558900 & . & . & 1 & 1 \end{matrix}

All remaining characters are in the principal block. We have to find 16 irreducible Brauer characters and PIMs. We induce the projective characters from the two maximal subgroups and call them Ψ_{1}, …, Ψ_{59} and Φ_{1}, …, Φ_{104}. MOC finds 16 linearly independent projective characters lying in the principal block. With Lemma 3.1.4 we check that they form a basic set. They are collected in

PS = {Ψ_{37}, Ψ_{51}, Ψ_{46}, Ψ_{39}, Ψ_{43}, Ψ_{42}, Ψ_{38}, Ψ_{34}, Ψ_{49},
Φ_{6}, Ψ_{11}, Ψ_{32}, Ψ_{31}, Ψ_{20}, Ψ_{8}, Ψ_{4}}.

Next MOC finds a basic set of Brauer characters. It consists of

BS = {1, 23, 253, 1771, 2024, 2277, 7084, 10395_{1}, 31878, 37422, 129536,

184437, 212520, 239085_{1}, 368874, 1291059}.

Furthermore we consider the following three projectives which do not decompose into PS with non-negative coefficients:

Ψ_{37} Ψ_{51} Ψ_{46} Ψ_{39} Ψ_{43} Ψ_{42} Ψ_{38} Ψ_{34} Ψ_{49} Φ_{6} Ψ_{11} Ψ_{32} Ψ_{31} Ψ_{20} Ψ_{8} Ψ_{4}
Φ_{4} −1 . . . . . 1 …. 2 … 1
Φ_{5} −1 1 1 . . 1 . . . . . . . . . .
Φ_{7} −1 . . . . . . . . . 1 … 1 .

We have some more Brauer characters:

φ_{1} φ_{2} φ_{3} φ_{4} φ_{5} φ_{6} φ_{7} φ_{8} φ_{9} φ_{10} φ_{11} φ_{12} φ_{13} φ_{14} φ_{15} φ_{16}
245 916 −1 1 −1 1 . −1 −1 . 1 1 . 1 ….
312 984  . 1 −1 −1 . −1 1 . −1 . 1 . 1 …
637 560  … 1 −1 −1 . . 1 . −1 2 . . 1 .
1 835 008  . −1 . −1 . . 1 . −1 −1 … 1 1 1
2 072 576  … −3 1 . 3 −1 −2 −1 . −1 1 2 1 1

These projectives and Brauer characters define relations (see Section 3.5). With these choices of PS and BS we get the matrix U of scalar products displayed in Table 5.1. By Lemma 3.2.3 we know that Ψ_{37}, Ψ_{46} and Ψ_{39} are PIMs. We are going to show that Ψ_{43} is indecomposable. If we decompose Ψ_{43} into the atoms we get

[Ψ_{43}] = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1) · [PA].

Therefore every part Ψ′ of Ψ_{43} must be of the form

[Ψ′] = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, n′_{12}, 0, 0, 0, n′_{16}) · [PA]

where 0 ≤ n′_{12}, n′_{16} ≤ 1. As we should have Ψ′ ≠ 0 and Ψ′ ≠ Ψ_{43} we get n′_{12} = 0 and n′_{16} = 1 or vice versa. But then we get a negative scalar product with 2072576. So Ψ_{43} is indecomposable. We obtain the same result for Ψ_{42}, Ψ_{38}, Ψ_{49} and Ψ_{32} by using the Brauer characters 637560 and three times 2072576. Now we are going to prove that Ψ_{34} is indecomposable. We have

[Ψ_{34}] = (0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1) · [PA].

Table 5.1: Scalar products in the principal block of Co_{2}

Ψ_{37} Ψ_{51} Ψ_{46} Ψ_{39} Ψ_{43} Ψ_{42} Ψ_{38} Ψ_{34} Ψ_{49} Φ_{6} Ψ_{11} Ψ_{32} Ψ_{31} Ψ_{20} Ψ_{8} Ψ_{4}
1  . . . . . . . . . . . . . . . 1
23  . . . . . . . . . . . . . . 1 .
253  . . . . . . . . . . . . . 1 . .
1771  . . . . . . . . . . . . 1 …
2024  . . . . . . . . . . . 1 … 1
2277  . . . . . . . . . . 1 … 1 .
7084  . . . . . . . . . 1 . . 1 …
10395_{1}  . . . . . . . . 1 . . . . . . .
31878  . . . . . . . 1 . . 1 . . . . .
37422  . . . . . . 1 . . . . . . 1 . .
129536  . . . . . 1 … 1 . . 1 1 . .
184437  …. 1 …. 1 . 1 … 1
212520  … 1 … 1 . . 2 . . . . .
239085_{1}  . . 1 . . . . . . . . . . . . .
368874  . 1 … 1 . . 1 …. 1 1 .
1291059  1 1 . . 1 . 1 1 . . 1 … 1 1

Therefore every part of Ψ_{34} must be some

[Ψ′] = (0, 0, 0, 0, 0, 0, 0, 0, n′_{9}, 0, 0, 0, n′_{13}, 0, 0, n′_{16}) · [PA]

with 0 ≤ n′_{9}, n′_{13}, n′_{16} ≤ 1. Similarly to the preceding proof we have 1 ≤ n′_{9} + n′_{13} + n′_{16} ≤ 2. But for all those parts there is a Brauer character having negative scalar product with Ψ′ or Ψ_{34} − Ψ′ as the following table shows:

n′_{9} n′_{13} n′_{16} Character
0 0 1 1835008
0 1 0  312984
0 1 1  312984
1 0 0  312984
1 0 1  312984
1 1 0 1835008

Therefore, Ψ_{34} is indecomposable. We get similar results for Φ_{6}, Ψ_{11}, Ψ_{31} and Ψ_{20} by applying the five Brauer characters. Up to now, we know that 13 of the 16 projectives belonging to PS are indecomposable. In the next paragraph we are going to apply our algorithms to Ψ_{51}, Ψ_{8} and Ψ_{4}.

5.2.2 Subtracting indecomposables

In this paragraph we are going to present an algorithm for improving basic sets by subtracting irreducible respectively indecomposable characters with the help of relations. We keep the notation introduced at the beginning of this section. To start with, we need some technical definitions.

If one of our projective characters of PS is known to be a PIM, we would like to have upper bounds for the number of times it can possibly be contained in any of the other projectives of the basic set. Such bounds can easily be determined.

Definition 5.2.7 Assume that Φ_{j} is a PIM for j ∈ J, a subset of {1, …, s}. Define non-negative integers (or infinity) m_{ij}, 1 ≤ i, j ≤ s, by:

(i) If i ∉ J, let m_{ij} = ∞.

(ii) If i, j ∈ J, let m_{ij} = δ_{ij}.

(iii) If i ∈ J and j ∉ J, then

m_{ij} = max{n ∈ ℕ_{0} | ⟨φ, Φ_{j} − n · Φ_{i}⟩ ≥ 0 for all φ ∈ B ∪ BS}.

Then obviously m_{ij} denotes how often Φ_{i} can possibly be contained in Φ_{j}. Therefore m_{ij} is called the maximal multiplicity of Φ_{i} in Φ_{j}.

Suppose that Φ ∈ PS is a PIM. If we knew the corresponding irreducible Brauer character φ, we could of course find the multiplicity of Φ in any other projective character Ψ by calculating ⟨φ, Ψ⟩. Of course, such a situation is exceptional. In general, there will be Brauer characters which have positive scalar product with Φ, but none of them is known to be irreducible. However, each of them contains φ as a part (see Definition 5.2.1). Let ϑ be a Brauer character with positive scalar product with Φ. The idea is to consider a set of parts of ϑ containing φ.

If all these special parts, which are called bits and which are introduced in Definition 5.2.9 below, have positive scalar product with Ψ, then Φ is contained in Ψ. We can apply the same idea in a slightly more general situation.

Definition 5.2.8 Let Φ = n_{1}φ_{1}^{∗} + … + n_{s}φ_{s}^{∗}, with n_{i} ≥ 0 for all i, be some generalized projective character. Then Φ is called multiplicity free if n_{i} ∈ {0, 1} for all 1 ≤ i ≤ s.

Definition 5.2.9 Let φ ∈ BS, φ = n_{1}Φ_{1}^{∗} + … + n_{s}Φ_{s}^{∗} with n_{1} = ⟨φ, Φ_{1}⟩ > 0. We are going to define a bit of φ with respect to Φ_{1} if either Φ_{1} is indecomposable or if Φ_{1} is multiplicity free.

(i) Suppose that Φ_{1}, …, Φ_{t} are PIMs for some 1 ≤ t ≤ s. Then φ′ = n′_{1}Φ_{1}^{∗} + … + n′_{s}Φ_{s}^{∗} is called a bit of φ with respect to Φ_{1} if the following conditions are satisfied:

(a) φ′ is a part of φ, i.e., 0 ≤ n′_{i} ≤ n_{i} for all i,

(b) n′_{1} = 1, n′_{2} = … = n′_{t} = 0, n′_{i} ≤ m_{1i} for all i > t,

(c) ⟨φ′, Ψ⟩ ≥ 0 and ⟨φ − φ′, Ψ⟩ ≥ 0 for all Ψ ∈ P.

(ii) If Φ_{1} is multiplicity free and Φ_{t+1}, …, Φ_{s} are indecomposable, then φ′ = n′_{1}Φ_{1}^{∗} + … + n′_{s}Φ_{s}^{∗} is called a bit of φ with respect to Φ_{1} if

(a) φ′ is a part of φ, i.e., 0 ≤ n′_{i} ≤ n_{i} for all i,

(b) n′_{1} = 1 and n′_{i} ≤ m_{i1} for all i > t,

(c) ⟨φ′, Ψ⟩ ≥ 0 and ⟨φ − φ′, Ψ⟩ ≥ 0 for all Ψ ∈ P.

Remark 5.2.10 Let Φ_{1} be a PIM. If |P| = r and v_{k1}, …, v_{ks} are the coefficients of the k-th character in P when decomposed into PS, 1 ≤ k ≤ r, then the conditions that φ is a bit with respect to Φ_{1} can be expressed by the following system of linear inequalities A · x ≤ b with:

A = ( \begin{matrix}1 & & 0 \\ . & . & . \\ 0 & & 1 \\ v_{1,t+1} & … & v_{1,s} \\ ⋮ & & ⋮ \\ v_{r,t+1} & … & v_{r,s} \\ −v_{1,t+1} & … & −v_{1,s} \\ ⋮ & & ⋮ \\ −v_{r,t+1} & … & −v_{r,s} \end{matrix} ) and b = ( min\{n_{t+1}, m_{1,t+1}\} ⋮ min\{n_{s}, m_{1,s}\} −v_{1,1} + ∑_{j=1}^{s} v_{1j}⟨φ, Φ_{j}⟩ ⋮ −v_{r,1} + ∑_{j=1}^{s} v_{rj}⟨φ, Φ_{j}⟩ v_{1,1} ⋮ v_{r,1} )

and x = (n′_{t+1}, …, n′_{s})^{t}. We observe that the number of rows in A and b depends on |P|. A similar remark applies in case Φ_{1} is multiplicity free.

The following two lemmas justify our definition of a bit.

Lemma 5.2.11 If Φ_{1}, …, Φ_{t} are PIMs and φ ∈ BS with ⟨φ, Φ_{1}⟩ > 0 then the irreducible Brauer character β corresponding to Φ_{1} is a bit of φ with respect to Φ_{1}. In particular, the set of bits of φ with respect to Φ_{1} is non-empty.

Proof. As ⟨φ, Φ_{1}⟩ > 0 the irreducible Brauer character β = ∑_{i=1}^{s} n′_{i}Φ_{i}^{∗} corresponding to Φ_{1} is contained in φ. Thus it is a part of φ, hence 0 ≤ n′_{i} ≤ n_{i} for all i. Clearly, n′_{1} = 1. All other PIMs must have scalar product 0 with β, so n′_{i} = 0 for 2 ≤ i ≤ t. If i > t then n′_{i} = ⟨β, Φ_{i}⟩ ≤ m_{1i}. As β is a genuine Brauer character its scalar product with any projective is non-negative, therefore condition (c) is fulfilled. Thus β is a bit.

Lemma 5.2.12 Suppose that Φ_{t+1}, …, Φ_{s} are PIMs and that Φ_{1} ∈ PS is multiplicity free. Let Π_{1}, …, Π_{s} denote all the PIMs. We assume that they are numbered so that Π_{i} = Φ_{i} for t + 1 ≤ i ≤ s. Furthermore, let Φ_{1} = v_{1}Π_{1} + … + v_{s}Π_{s}. Then the following statements hold:

(i) v_{i} ∈ {0, 1} for all i.

(ii) Let 1 ≤ k ≤ s with v_{k} = 1. If φ ∈ BS such that ⟨φ, Π_{k}⟩ > 0, then ⟨φ, Φ_{1}⟩ > 0 and the irreducible Brauer character β corresponding to Π_{k} is a bit of φ with respect to Φ_{1}. In particular, the set of bits of φ with respect to Φ_{1} is non-empty.

Proof.

(i) As Φ_{1} is multiplicity free, its decomposition into PA leads to coefficients in {0, 1}. Now every PIM is a non-negative linear combination of PA. The result follows immediately.

(ii) Let n′_{1}, …, n′_{s} be the coefficients of β when decomposed into BA. Conditions (a) and (c) of Definition 5.2.9 are fulfilled as β is contained in φ and is a genuine Brauer character. We have

⟨β, Φ_{1}⟩ = ∑_{i=1}^{s} v_{i}⟨β, Π_{i}⟩ = v_{k} = 1,

that is n′_{1} = 1. If i > t then

n′_{i} = ⟨β, Φ_{i}⟩ = ⟨β, Π_{i}⟩ = δ_{ik}.

If t < i ≠ k then δ_{ik} = 0 ≤ m_{i1}. If i = k then n′_{k} = 1 = m_{k1} as Φ_{1} is multiplicity free and contains Φ_{k} = Π_{k} exactly once. This completes the proof.

Definition 5.2.13 Let Φ ∈ PS be indecomposable or multiplicity free. For φ ∈ BS with ⟨φ, Φ⟩ > 0 and a projective character Σ ∈ P let

m(Φ, Σ, φ) = min{⟨φ′, Σ⟩ | φ′ is a bit of φ corresponding to Φ}.

Let Φ ∈ PS be indecomposable or multiplicity free. There is at least one φ ∈ BS such that ⟨φ, Φ⟩ > 0. By the preceding lemmas, there is always a bit of φ with respect to Φ and so m(Φ, Σ, φ) is well defined. By our definition of bits, m(Φ, Σ, φ) is a non-negative integer. The set

{m(Φ, Σ, φ) | φ ∈ BS, ⟨φ, Φ⟩ > 0}

is non-empty. It is, of course, finite, since φ has only finitely many parts.

Theorem 5.2.14 Let Φ ∈ PS and Σ ∈ P. If Φ is indecomposable, let

z = max{m(Φ, Σ, φ) | φ ∈ BS, ⟨φ, Φ⟩ > 0}.

If Φ is multiplicity free, let

z = min{m(Φ, Σ, φ) | φ ∈ BS, ⟨φ, Φ⟩ > 0}.

Then Σ − z · Φ is a genuine projective character.

Proof. As remarked above, z is well-defined. Suppose first, that Φ is indecomposable. Let {β_{1}, …, β_{s}} be the irreducible Brauer characters. Without loss of generality let β_{1} be the one corresponding to Φ. For all 1 ≤ i ≤ s with ⟨φ_{i}, Φ⟩ > 0 let z_{i} = m(Φ, Σ, φ_{i}). As

β_{1} ∈ {φ′ | φ′ bit of φ_{i} with respect to Φ}

we have ⟨β_{1}, Σ⟩ ≥ z_{i}. Thus Φ is contained at least z_{i} times in Σ. As z = max z_{i}, also Σ − z · Φ is projective.

Suppose now, that Φ is multiplicity free. Let Φ = v_{1}Π_{1} + … + v_{s}Π_{s} be the decomposition of Φ into the PIMs. Then v_{i} ∈ {0, 1} by Lemma 5.2.12(a). Now let k be such that Π_{k} occurs in Φ, i.e., v_{k} = 1. Let β_{k} be the irreducible Brauer character corresponding to Π_{k}. We shall show that ⟨β_{k}, Σ⟩ ≥ z. This will imply the result since Φ is multiplicity free.

Since BS is a basic set, there is some φ ∈ BS with ⟨φ, Π_{k}⟩ > 0. Then β_{k} is a bit of φ corresponding to Φ by Lemma 5.2.12(b). Hence m(Φ, Σ, φ) ≤ ⟨β_{k}, Σ⟩. Since obviously ⟨φ, Φ⟩ > 0, we have z ≤ m(Φ, Σ, φ).

In § 5.2.6 we shall show how to solve the problem of calculating z. Changing projectives and Brauer characters give analogous results for Brauer characters.

Example 5.2.15 Let us now return to our example Co_{2} mod 5. As mentioned above we still have to find three more PIMs. First we want to show that Ψ_{37} is contained in Ψ_{51}. For this purpose we have to find the irreducible Brauer character corresponding to Ψ_{37}. But this is a bit φ′ of φ = 1291059. Its decomposition into the atoms is

[φ] = (1, n′_{2}, 0, 0, n′_{5}, 0, n′_{7}, n′_{8}, 0, 0, n′_{11}, 0, 0, 0, n′_{15}, n′_{16}) · [BA]

where for all i we have 0 ≤ n′_{i} ≤ 1. As Ψ_{43}, Ψ_{38}, Ψ_{34} and Ψ_{11} are known to be indecomposable we have n′_{5} = n′_{7} = n′_{8} = n′_{11} = 0. So it remains to solve the following system of linear inequalities (remember that ⟨φ′, Ψ_{51}⟩ = n′_{2}):

Minimize n′_{2} subject to

( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \\ −1 & 0 & 0 \\ 0 & −1 & 0 \end{matrix} ) · ( \begin{matrix}n′_{2} \\ n′_{15} \\ n′_{16} \end{matrix} ) ≤ ( \begin{matrix}1 \\ 1 \\ 1 \\ 2 \\ 1 \\ 1 \\ −1 \\ −1 \\ −1 \end{matrix} )

where all n′_{i} ∈ ℕ_{0}. As this minimum equals 1 we find that Ψ_{51} − Ψ_{37} is a genuine projective character. Since this projective is an atom it is indecomposable. Similarly, we show that Ψ_{8} − Ψ_{37} and Ψ_{4} − Ψ_{37} are projective. The two Brauer characters of degree 245916 and 2072576 prove that the resulting projectives are indecomposable. So we have found all PIMs. The resulting decomposition matrix as well as the table of irreducible Brauer characters is printed in the appendix.

5.2.3 Triangular matrix of scalar products

In § 5.2.2 we presented an algorithm for subtracting irreducible or multiplicity free characters. In some cases, it is possible to improve characters without knowing whether they are irreducible or not. For this algorithm we assume that the matrix U of scalar products is lower uni-triangular, i.e., a_{ij} = ⟨φ_{i}, Φ_{j}⟩ = 0 if j > i. Let {Π_{1}, …, Π_{s}} denote the PIMs and {β_{1}, …, β_{s}} the irreducible Brauer characters. We assume that the PIMs and PS are ordered consistently, i.e., that Φ_{j} consists of Π_{j} and a sum of Π_{r}’s with r > j. Here we are going to describe a method which allows to subtract Π_{i} from some Φ_{j_{0}} without knowing what Π_{i} looks like. Suppose we know for some reason that Π_{i} is contained in Φ_{j_{0}} (see Lemma 5.2.17 below). The idea now is to subtract Φ_{i} from Φ_{j_{0}}. The possible errors due to the fact that Φ_{i} might not be indecomposable can easily be corrected by adding some projectives Φ_{l} for l > i (see Theorem 5.2.18). This method produces new characters which are smaller than the older ones in the following sense. If we identify a projective Φ with the column (⟨φ_{1}, Φ⟩, …, ⟨φ_{s}, Φ⟩)^{t}, we can order the set of projectives lexicographically. The new characters obtained are then smaller in this lexicographic ordering.

Lemma 5.2.16 Let Φ_{j} = Π_{j} + ∑_{r=j+1}^{s} b_{rj}Π_{r} be the decomposition of Φ_{j} into the PIMs. Then b_{lj} ≤ a_{lj} = ⟨φ_{l}, Φ_{j}⟩ for all l = j + 1, …, s.

Proof. Let φ_{l} = β_{l} + μ for a certain Brauer character μ. Then a_{lj} = ⟨φ_{l}, Φ_{j}⟩ = ⟨β_{l}, Φ_{j}⟩ + ⟨μ, Φ_{j}⟩ = b_{lj} + ⟨μ, Φ_{j}⟩ ≥ b_{lj}.

Lemma 5.2.17 Let Φ be a projective character, Φ = ∑_{j=1}^{s} v_{j}Φ_{j}. Suppose that v_{i} < 0 for some i. If there exists j_{0} < i with v_{j_{0}} > 0 and

|v_{i}| > ∑_{j=1,j≠j_{0},v_{j}>0}^{i−1} v_{j}a_{ij}

then Π_{i} is contained in Φ_{j_{0}} for a least z times, where

z = ⌈\frac{1}{v_{j_{0}}} (|v_{i}| − ∑_{j=1,j≠j_{0},v_{j}>0}^{i−1} v_{j}a_{ij})⌉.  (5.2)

(Here ⌈x⌉ for x ∈ ℝ denotes the integer y with y − 1 < x ≤ y.)

Proof. Let Λ = ∑_{j=1,j≠i,v_{j}≤0}^{s} v_{j}Φ_{j}. Then

Φ − Λ + |v_{i}|Φ_{i} = ∑_{j=1,v_{j}>0}^{s} v_{j}Φ_{j}

(remember that v_{i} < 0). Therefore |v_{i}|Φ_{i} is contained in the right hand of this equation, hence so is |v_{i}|Π_{i}. By Lemma 5.2.16 Π_{i} is contained in Φ_{j} only for j ≤ i and in this case for at most a_{ij} times. So Π_{i} is contained in v_{j_{0}}Φ_{j_{0}} for at least

|v_{i}| − ∑_{j=1,j≠j_{0},v_{j}>0}^{i−1} v_{j}a_{ij}

times and therefore Π_{i} is contained in Φ_{j_{0}} at least z times.

Theorem 5.2.18 Let the notation be as in Lemma 5.2.16. Suppose that Π_{i} is contained z times in Φ_{j_{0}} for some i > j_{0}. Then

Φ̃_{j_{0}} = Φ_{j_{0}} − zΦ_{i} + z ∑_{l=i+1}^{s} a_{li}Φ_{l}  (5.3)

is a genuine projective character.

Proof. We have

Φ_{j_{0}} − zΦ_{i} + z ∑_{l=i+1}^{s} a_{li}Φ_{l}

= Φ_{j_{0}} − (zΠ_{i} + z ∑_{l=i+1}^{s} b_{li}Π_{l}) + z ∑_{l=i+1}^{s} a_{li}Φ_{l}

= (Φ_{j_{0}} − zΠ_{i}) + z ∑_{l=i+1}^{s} (a_{li} − b_{li})Φ_{l} + z ∑_{l=i+1}^{s} b_{li}(Φ_{l} − Π_{l}),

which by Lemma 5.2.16 is a non-negative linear combination of genuine projectives and therefore is a genuine projective character. The following algorithm derived from Lemma 5.2.17 and Theorem 5.2.18 is implemented in the MOC-system. Suppose that P is a set of projective characters. For Φ ∈ P write Φ = ∑_{j=1}^{s} v_{j}(Φ)Φ_{j}.

Algorithm 5.2.19

For all i in 1, …, s do:
For all j_{0} in 1, …, i − 1 do:

z := 0;

For all Φ ∈ P do:

If v_{j_{0}}(Φ) ≤ 0 or v_{i}(Φ) ≥ 0

z_{Φ} := 0

else

Calculate z_{Φ} according to (5.2); fi z := max{z, z_{Φ}}; od; If z > 0 calculate Φ̃_{j_{0}} according to (5.3);

od;

od.

If, in the above algorithm, z > 0 for some j_{0}, i > j_{0}, then Φ̃_{j_{0}} is lexicographically smaller than Φ_{j_{0}} in the sense described at the beginning of this paragraph.

5.2.4 Proving decomposability

Sometimes one can show that a projective or Brauer character has to be divided into a sum of two or more characters. Let us assume that there is a projective Φ ∈ P which has at least one negative coefficient in its decomposition into the basic set. Let Φ = v_{1}Φ_{1} + … + v_{s}Φ_{s} be this decomposition and assume that v_{i} < 0 for some i. Hence, Φ_{i} is contained in v_{1}Φ_{1} + … + v_{i−1}Φ_{i−1} + v_{i+1}Φ_{i+1} + … + v_{s}Φ_{s}. If Φ_{i} is indecomposable it has to be contained in at least one v_{j}Φ_{j} for j ≠ i. This can be checked by using an algorithm similar to the one used for calculating the m_{ij} (see Definition 5.2.7). If there is no such j then Φ_{i} has to be decomposable.

The easiest and most important case is that Φ_{i} is the sum of two indecomposable projective characters. To check this we find the two lexicographically smallest solutions (n′_{1}, …, n′_{s}) and (n′′_{1}, …, n′′_{s}) fulfilling the system of inequalities of Remark 5.2.4. We put Ψ_{1} = ∑ n′_{j}φ_{j}^{∗} and Ψ_{2} = ∑ n′′_{j}φ_{j}^{∗}. If Ψ_{1} + Ψ_{2} = Φ_{i}, we are done. In two special cases it is now easy to get a better basic set:

(i) Assume that Ψ_{1} ∈ PS. Then {Φ_{1}, …, Φ_{i−1}, Ψ_{2}, Φ_{i+1}, …, Φ_{s}} is a better basic set.

(ii) Assume that the matrix of scalar products is triangular. Let Ψ_{1} = ∑_{k=1}^{s} w_{k}φ_{k}^{∗} and Ψ_{2} = ∑_{k=1}^{s} x_{k}φ_{k}^{∗} be the decompositions into the projective atoms. Then w_{k} = x_{k} = 0 for all k > i. Without loss of generality let w_{i} = 1. As Φ_{i} = Ψ_{1} + Ψ_{2} and U is unitriangular we get x_{i} = 0. In PS we substitute Φ_{i} by Ψ_{1}. Obviously, Ψ_{1} is smaller than Φ_{i}. Let l = max{k | x_{k} > 0}. Then 1 ≤ l ≤ i − 1. Now we substitute Φ_{l} by Ψ_{2} and we get the basic set required.

5.2.5 Solving the second fundamental problem

As we have seen in the last three paragraphs all algorithms depend on the size of P and B. So it becomes clear that these two sets have to be kept small. First of all we remember from Section 3.4 that we only need to consider essential characters. This is made precise below.

Remark 5.2.20 We assume that there is a set E of projectives satisfying the following two conditions:

(i) PS ⊆ E ⊆ P,

(ii) For all Φ ∈ P there exist some v_{1}, …, v_{s}, w_{1}, …, w_{r} ∈ ℕ_{0} such that

Φ = ∑_{i=1}^{s} v_{i}Φ_{i} + ∑_{j=1}^{r} w_{j}Σ_{j}  (5.4)

where E ∩ (P \ PS) = {Σ_{1}, …, Σ_{r}}.

Then E contains “the same information” as P in the following sense. Suppose there is a Brauer character φ and a bit φ′ of φ and a Σ ∈ P \ E with ⟨φ′, Σ⟩ < 0 or ⟨φ − φ′, Σ⟩ < 0. Then we have (without loss of generality ⟨φ′, Σ⟩ < 0) :

Σ = ∑_{i=1}^{s} v_{i}Φ_{i} + ∑_{j=1}^{r} w_{j}Σ_{j}

and therefore

0 > ⟨φ′, Σ⟩ = ∑_{i=1}^{s} v_{i}⟨φ′, Φ_{i}⟩ + ∑_{j=1}^{r} w_{j}⟨φ′, Σ_{j}⟩.

This implies ⟨φ′, Σ_{j}⟩ < 0 for some j. So we do not need Σ.

So again we have to solve a system of linear inequalities over ℕ_{0} which we get by subtracting ∑_{i=1}^{s} v_{i}Φ_{i} on the right hand side of equation (5.4). The problem of integer linear programming is N P-complete. For a proof see [14]. But looking at the proof of Remark 5.2.20 we find that the conditions v_{1}, …, v_{s}, w_{1}, …, w_{r} ≥ 0 are sufficient. So we are able to state the problem again, now over the field of real numbers.

In [83] one can find an polynomial algorithm for solving this problem. Nevertheless, we use an ordinary simplex algorithm for the following reasons: The solution can be constructed explicitly. So, by checking whether it satisfies the inequalities, errors due to truncation are avoided. Finally, it is quite fast. Our task is now:

Find a set E of projectives fulfilling:

(i) PS ⊆ E ⊆ P,

(ii) For all Φ ∈ P there exists some v_{1}, …, v_{s}, w_{1}, …, w_{r} ∈ ℝ_{≥0} with

Φ = ∑_{i=1}^{s} v_{i}Φ_{i} + ∑_{j=1}^{r} w_{j}Σ_{j}

where E ∩ (P \ PS) = {Σ_{1}, …, Σ_{r}}.

Here is our algorithm for solving this problem:

Algorithm 5.2.21 Step 1. Let E := PS, F := P.

Step 2. For all Φ ∈ F \ E do: Try to solve

Φ = ∑_{i=1}^{s} v_{i}Φ_{i} + ∑_{j=1}^{r} w_{j}Σ_{j} for v_{i}, w_{j} ≥ 0,

where E \ PS = {Σ_{1}, …, Σ_{r}}. For this purpose we decompose Φ and Σ_{j} into {Φ_{1}, …, Φ_{s}} and denote by b_{1}, …, b_{s} and a_{j1}, …, a_{js} the resulting coefficients. Then we have to find a solution w = (w_{1}, …, w_{r}), v = (v_{1}, …, v_{s}) and v_{i}, w_{j} ∈ ℝ_{≥0} for the system C · x = b where b = (b_{1}, …, b_{s})^{t}, A = (a_{ij})_{ij}, C = [A, E_{s}] and x = [w^{t}, v^{t}]. We multiply each equation by −1 or +1 so that the right hand side of any equation is non-negative. Let C̄ and b̄ denote the resulting coefficients. Now we introduce some artificial variables y_{1}, …, y_{s} which have to be greater or equal to zero and augment the system as follows:

ā_{11}x_{1} + … + ā_{1,r+s}x_{r+s} + y_{1}  = b̄_{1}
⋮  ⋮  ...  ⋮
ā_{s1}x_{1} + … + ā_{s,r+s}x_{r+s}  + y_{s} = b̄_{s}

Then the system is feasible by setting x_{i} = 0 and y_{i} = b̄_{i}. Now we minimize the function

f = y_{1} + … + y_{s}.

This function counts how many artificial variables we do need to make the system solvable. If f has a minimum equal to 0, the original system is solvable otherwise it is not. If Φ is a non-negative linear combination of elements in E throw it away, i.e. put F := F \ {Φ}.

Step 3. If F ≠ E take a new vector Ψ ∈ F \ E and put E := E ∪ {Ψ}. For example, choose a vector Ψ ∈ F \ E with minimal sum of coefficients. Go to 2.

Example 5.2.22 Let us go back to our example Co_{2} mod 5. One useful method for generating projectives is tensoring characters of defect 0 with Brauer characters. Here we tensor 91125_{1} (it is of defect 0) with 275 and restrict the resulting projective to the principal block. We call this projective Φ. Its decomposition into the first PS we took for Co_{2} mod 5 is

Φ = −Ψ_{37} + Ψ_{51}.

As (using the same notation as in Example 5.2.6) we have

Φ_{5} = Φ + Ψ_{46} + Ψ_{42},

we could have thrown away Φ_{5}. Of course, the final result would have been the same.

5.2.6 An all-integer integer simplex algorithm

As seen in the earlier paragraphs we are given the following problem in linear programming: Let A ∈ ℤ^{m×n}, b ∈ ℤ^{m} and c ∈ ℤ^{n}. Then we have to find min{c^{t}x} subject to x ∈ ℕ_{0}^{n} and Ax ≤ b where we take ≤ in each component. So this problem is similar to that which is solved by a simplex algorithm. The main difference is the fact that we have to find integral solutions x. So we are interested in an algorithm only using integers in order to avoid errors due to truncation. This problem was solved by R.E.Gomory in 1963 [48]. We use this Gomory algorithm to calculate the minimum taken over all bits and to check whether a character is irreducible (in this case we minimize the 0-function). The Gomory algorithm is based on the dual simplex algorithm. Here we are not going to explain the whole theory of duality and linear programming. For details see [18]. Our implementation is based on the book of Burkard [14]. Similarly to Algorithm 5.2.21, Step 2, we first augment the system of inequations by introducing some artificial variables in order to get a system of equations. We are going to explain only how we get integral solutions for the problem. We put

a_{00} = 0,
a_{0j} = c_{j} for j = 1, …, n,
a_{i0} = b_{i} for i = 1, …, m

and write these numbers in the following table:

x_{1} x_{2} … x_{n}
a_{00} a_{01} a_{02} … a_{0n}
x_{n+1} a_{10} a_{11} a_{12} … a_{1n}
⋮  ⋮  ⋮  ⋮  ⋮
x_{n+m} a_{m0} a_{m1} a_{m2} … a_{mn}

This a shortened way of writing

z = min! = ∑_{j=1}^{n} a_{0j}x_{j}

subject to

∑_{j=1}^{n} a_{ij}x_{j} + x_{n+i} = a_{i0} for i = 1, …, m.

As mentioned above, this algorithm is based on the dual simplex algorithm. We assume now that there is a solution for the dual problem such that the first non-zero element in each column a_{j} for 1 ≤ j ≤ n is positive. In the two algorithms presented in §§ 5.2.1 and 5.2.2 these conditions are satisfied as remarked in 5.2.4 and 5.2.10. Suppose now that the last solution the algorithm has found while optimizing has a non-integral coefficient. Then we have to exclude this point from the simplex. So we assure that all vertices of the simplex have integral coefficients. The optimal solution is found if all b_{i} for i = 1, …, m are non-negative. Otherwise one of the b_{i} is negative. From now on we fix this i. We are going to construct a new restriction out of the i-th row. To make the algorithm becoming all-integer the pivot element will become −1. We will repeat this until we get the optimal solution or we get a contradiction so that there is no solution. Now we are going to make the new restriction. Let a_{ij} as defined above. If λ > 0 and a is a real number then a can be written as

a = λ[\frac{a}{λ}] + r_{a} with 0 ≤ r_{a} < λ

where [·] denotes the Gaussian function. Now let a_{i0} < 0 for some 1 ≤ i ≤ m. The corresponding restriction is

a_{i0} = a_{i1}x_{1} + … + a_{in}x_{n} + 1 · x_{n+i}.

Therefore we get

λ·[\frac{a_{i0}}{λ}]+r_{0} = (λ·[\frac{a_{i1}}{λ}]+r_{1})x_{1}+…+(λ·[\frac{a_{in}}{λ}]+r_{n})x_{n}+(λ·[\frac{1}{λ}]+r_{n+i})x_{n+i}

and then

∑_{j=1}^{n} r_{j}x_{j} + r_{n+i}x_{n+i} = r_{0} + λ\{[\frac{a_{i0}}{λ}] − ∑_{j=1}^{n}[\frac{a_{ij}}{λ}]x_{j} − [\frac{1}{λ}]x_{n+i}\}.

Since r_{j} ≥ 0 and x_{j} ≥ 0 for j = 1, …, n, n + i the left hand of this equation is non-negative. Also, the term in brackets is integral. Let

x̄ = [\frac{a_{i0}}{λ}] − ∑_{j=1}^{n}[\frac{a_{ij}}{λ}]x_{j} − [\frac{1}{λ}]x_{n+i}.

If x̄ < 0, then x̄ ≤ −1, hence λx̄ ≤ −λ. As r_{0} < λ we would have r_{0} +λx̄ being negative, a contradiction. In this algorithm we are going to choose λ > 1. Then [\frac{1}{λ}] = 0 and we get the restriction

x̄ = [\frac{a_{i0}}{λ}] − ∑_{j=1}^{n}[\frac{a_{ij}}{λ}]x_{j} ≥ 0.

From this construction we get that every integral point of the given problem satisfies this restriction. We are going to choose a λ such that the pivot element will become −1. Hence the system will remain integral after pivoting. In the dual simplex algorithm the column s containing the pivot element is determined by

\frac{a_{0s}}{[\frac{a_{is}}{λ}]} = max\{\frac{a_{0j}}{[\frac{a_{ij}}{λ}]} | a_{ij} < 0, j = 1, …, n\}.

If there is no such a_{ij} < 0 then the system is not solvable. As we are going to choose λ such that [\frac{a_{is}}{λ}] = −1, we get

−a_{0s} = max\{\frac{a_{0j}}{[\frac{a_{ij}}{λ}]} | a_{ij} < 0, j = 1, …, n\}.

But [\frac{a_{ij}}{λ}] is an integer, less or equal to −1. If we put

μ_{j} = [\frac{a_{ij}}{λ}]

we have

−a_{0s} ≥ \frac{a_{0j}}{μ_{j}} ≥ −a_{0j},

hence s is determined by

−a_{0s} = max{−a_{0j} | a_{ij} < 0, j = 1, …, n}.

This is equivalent to

a_{0s} = min{a_{0j} | a_{ij} < 0, j = 1, …, n}.

If s is not determined uniquely consider the next rows of the table to make a decision. We determine for k = 1, 2 …

a_{ks} = min{a_{kj} | a_{ij} < 0, a_{k−1,j} minimal for a_{ij} < 0}

until the minimum is unique. So we achieve that for all j with a_{ij} < 0 the first element of the vector a_{s} − a_{j} is negative. We are now going to examine how to choose λ best. After pivoting we get a new value

ā_{00} = a_{00} + [\frac{a_{i0}}{λ}]a_{0s}

for the function to be minimized. In order to decrease the value as much as possible we have to choose λ as minimal as possible. But we have to make sure that the first non-zero element ā_{kj} of the columns j = 1, …, n with j ≠ s is

ā_{kj} = a_{kj} + [\frac{a_{ij}}{λ}]a_{ks} > 0.

Let i_{0} be the minimum of i(j) and i(s) where i(j) is first index for which a_{i(j),j} ≠ 0 (and therefore a_{i(j),j} > 0). Similarly, let i(s) be the first index of rows with a_{i(s),s} > 0. Then ā_{i_{0}j} becomes the first non-zero element of column j. If a_{ij} ≥ 0 then we have ā_{i_{0}j} > 0 for all possible λ as we either have a_{i_{0}j} = 0 and a_{i_{0}s} > 0 or a_{i_{0}j} > 0 and a_{i_{0}s} ≥ 0. But if we have a_{ij} < 0 and a_{i_{0}j} > 0 then we get a new restriction for λ resulting from the equation determing ā_{kj}. As 0 ≤ a_{is} ≤ a_{i0} we get

−1 ≥ μ_{j} > −\frac{a_{i_{0}j}}{a_{i_{0}s}} .

As λ has to be chosen as small as possible and a_{ij} < 0 we have to choose μ_{j} as small as possible, too. Therefore we take μ_{j} to be the smallest integer greater than −\frac{a_{i_{0}j}}{a_{i_{0}s}}. Now for j ∈ J = \{j | a_{ij} < 0\} we put:

λ_{j} = \{ \begin{matrix}\frac{a_{ij}}{μ_{j}} if a_{i_{0}s} > 0 \\ 0 otherwise \end{matrix} .

Then

λ = max{λ_{j} | j ∈ J}

satisfies all conditions, i.e. [\frac{a_{is}}{λ}] = −1, the value of the function being optimized decreases as much as possible and after pivoting in all columns the first non-zero element is positive.

Algorithm 5.2.23 Gomory’s all-integer integer algorithm (see [14]) for solving the problem

∑_{j=1}^{n} a_{0j}x_{j} = min!

subject to Ax ≤ a_{0} for x_{j} ∈ ℕ_{0} and 1 ≤ j ≤ n. We assume that the problem is dual feasible. Then:

80 CHAPTER 5. ALGORITHMS

1. If for all i = 1, …, m

a_{i0} ∈ ℕ_{0}

then the optimal solution is found. Stop. Otherwise go to Step 2.

2. Let

r̃:= min{i | a_{i0} ∉ ℕ_{0}, i = 1, …, m}.

3. If a_{r̃j} ≥ 0 for all j = 1, …, n then the problem is not solvable.

Stop. Otherwise let

J := {j | j > 0 and a_{r̃j} < 0}.

4. Determine s to be the index of the pivoting column by

a_{0s} := min{a_{0j} | j ∈ J}.

If s is not determined uniquely repeat for i = 1, 2, …

a_{is} := min{a_{ij} | j ∈ J and a_{i−1,j} minimal }

until the minimum is determined uniquely.

5. For j ∈ J let i(j) be the first index of rows such that a_{i(j),j} ≠ 0.

Determine μ_{j} to be the smallest integer for which

a_{i(j),j} + μ_{j}a_{i(j),s} > 0.

If a_{i(j),s} = 0 put μ_{j} := −∞.

6. For j ∈ J let

λ_{j} := \{ \begin{matrix}0 if μ_{j} = −∞ \\ \frac{a_{r̃j}}{μ_{j}} , otherwise \end{matrix}

and let

λ := max{λ_{j} | j ∈ J}.

If λ = 1, replace λ by 1+ε where ε is some arbitrary little number.

7. For j = 0, 1, …, n put

a_{m+1,j} := ⌈\frac{a_{r̃j}}{λ} ⌉

and add a_{m+1,0}, a_{m+1,1}, …, a_{m+1,n} as (m + 1)st row to the table. Let r := m := m + 1.

8. Put

ā_{rj} := −a_{rj} for j = 0, 1, …, n; j ≠ s,
ā_{ij} := a_{ij} + a_{is}a_{rj}
for i = 0, 1, …, m − 1, j = 0, 1, …, s − 1, s + 1, …, n.

9. Let for i = 0, 1, …, m; j = 0, 1, …, n; j ≠ s

a_{ij} := ā_{ij}

and go to 1.

5.3 Automatic proofs

We once more emphasize the fact that MOC does not provide an algorithm for calculating the irreducible modular characters for a given finite group. No such algorithm is yet known. Rather MOC provides a collection of algorithms for various purposes to support the calculation of some decomposition numbers and irreducible Brauer characters.

Finding the modular characters of a finite group is a kind of problem where a result can be checked with much less effort than was needed to obtain it. We have to produce a very large number of Brauer characters and projective characters to begin with. If the irreducible respectively indecomposable characters happen to be among those of our pool, then we would find them by means of the scalar product (see Proposition 3.3.2).

The methods described in Section 2.5 for constructing characters do not allow to select good candidates beforehand. Usually most of the characters generated are not needed at all. Only after the construction can one recognise those which really contain the information wanted. The other characters constructed during the session can be thrown away. For the final proof, which can usually be given in a conventional form, it suffices, of course, to keep the “good guys”.

Every run of MOC is documented on an info-file corresponding to the pair (G, p). Some additional information, for example output of certain programs is stored on this file, which is called G.p.info. This method of automatic documentation is very much supported by the UNIX operating system. As already mentioned, MOC consists of a collection of FORTRAN programs and UNIX Bourne-shell scripts. The FORTRAN programs serve some well defined purposes such as, for example, matrix multiplication or solving integral linear equations. Transfer of data is via files, and the files are moved by shell scripts according to the needs of the programs.

Let us give an example of such a shell script. In MOC, the actual basic set PS of projective characters is stored on the file G.p under the label 30700 as a matrix X_{0}, expressing PS in terms of the projective atoms PA_{0} corresponding to the special basic set BS_{0}. By equation (3.14) we have X_{0} = ⟨PS, BS_{0}⟩. The following shell script, called bsinba transposes the matrix X_{0} stored under label 30700 on the file G.p. By equation (3.15) this expresses the special basic set BS_{0} in terms of the Brauer atoms BA corresponding to PS. Since the FORTRAN program transp to transpose a matrix expects the matrix under label 30900, we have to merge it first from 30700 to 30900. The shell script get copies the file G.p to the current working directory, which for safety reasons should always be different from the directory containing the data-files. The first argument of the following shell script is the prime p, the second the name of the group G.

     set -x
     # bsinba
     rm -f t1 t2 t3 t4
     get $2.$1 || exit
     cp $2.$1 t1
     cp t1 t2
     merge <<%
     30900
     0
     30700
     %
     rm -f t2
     mv t3 t1
     transp
     mv t2 t3
     rm -f t1 t2

The shell script expressing the Brauer atoms in terms of the special basic set BS_{0} is called bainbs. By equation (3.15) it has to invert the matrix X_{0}^{t}.

set -x
#
# bainbs
#
# Expresses the Brauer atoms corresponding to
# the actual  basic set of projectives, which
# are stored in $2.$1 under the label 30700,
# in terms of the special basic set.
# The result is put onto file t3.
#
# ``bsinba'' is the program inverse to ``bainbs''
#
bsinba $1 $2
mv t3 hilf
#
# ``invers'' calculates the inverse of a matrix
#
invers hilf
rm t1 t2
mv hilf.inv t3

The following shell script calculates the set of Brauer atoms BA as class functions. For this purpose it has to multiply the matrix X_{0}^{−t} with [BS_{0}], the latter being stored on G.p under the label 30900.

set -x
#
# bratoms
#
# This program gives the $1-modular
# Brauer atoms of group $2 corresponding
# to the actual basic set of projective
# characters as class functions.
# The answer is stored in $2.$1.bat.
#

bainbs $1 $2
mv t3 t2
get $2.$1 1
matmul2
mv t3 $2.$1.bat
rm -f $2.$1 t1 t2 t4

We now give an example of an info-file which documents the session for determining the Brauer characters for the Mathieu group M_{11} modulo 5.

Fri Mar 20 11:25:16 MET 1992
/home/euterpe/hiss/mocha/proggy/prp
pst run
fct run
projectives no     1 -    5 obtained by:
defect 0 characters
relations stored under 30550
Brauer characters no     1 -   10 obtained by:
ordinary characters restricted to p-regular classes
Fri Mar 20 11:25:56 MET 1992
/home/euterpe/hiss/mocha/proggy/defzo
projectives no     6 -   55 obtained by:
tensoring ordinaries with defect 0 characters
option:1:all ordinaries tensor all defect 0 characters
Fri Mar 20 11:26:17 MET 1992
/home/euterpe/hiss/mocha/proggy/basepro
input label (i5)
insert depth of search (i3), e.g. 10
input block-number (i3)
number of basic characters found is   4
numbas= 10011 10012 10010 10007

     Fri Mar 20 11:27:21 MET 1992
     /home/euterpe/hiss/mocha/proggy/protest
     projective  nr    11 in block   1 is indecomposable,
       because it is an atom
     projective  nr    12 in block   1 is indecomposable,
       because it is an atom
     projective  nr    10 in block   1 is indecomposable,
       because it is an atom
     projective  nr     7 in block   1 is indecomposable,
       because it is an atom

The info-file serves two principal purposes. It can be used to repeat the calculations for a particular group and prime. This may be necessary if some information is lost by accident or if the files get too large and immovable. In a second run one can usually proceed much faster, because one already knows which of the steps have been redundant.

The second purpose is of course the reconstruction of those characters which are used in the proof. Every character generated is supplied with an identification number. In particular, every irreducible Brauer character or indecomposable projective character has its generating number, which can be used to trace back to its origin with the help of the infofile. We even have programs which do the trace-back automatically, at least if the character was obtained by some standard constructions. This gives the method of automatic proofs. In [56] about two thirds of the proofs were obtained in this fashion. If such a proof is written down in conventional form everybody can reconstruct the characters by the description we have given and thus check the proof by hand. An example of such a proof is given in Chapter 6, where we have collected all the information necessary to determine the decomposition numbers modulo 7 of Conway’s largest group Co_{1}.

5.4 Extensions

There are some features which we would like MOC to have, but which are still not implemented. The first and most important one is the additional information one obtains from duality and group automorphisms. For instance, a real valued ordinary character must be contained in a PIM and its dual with equal multiplicity. This simple observation and analogous statements in the case of group automorphisms can be used to improve the algorithms described in Section 5.2 considerably. This is one of the next things we are planning to add to MOC.

The additional information available in characteristic 2 by Fong’s lemma (see 5.5.3 below) has not been implemented yet.

There are examples such as the Rudvalis group in characteristic 5 which seem to be very hard to settle with the current methods of MOC. Here, no improvement of the characters as described in Section 5.2 is possible, but still there seems to be enough information to rule out most of the cases. Here, it really is desirable to have a program which tries to solve Fundamental Problem I in its full generality. We plan to do some experiments with these factorization problems.

Finally, we would very much like to have a general trace-back program for writing down more complicated proofs. The current versions cannot cope with the numerous possibilities for generating and improving characters.

5.5 Advanced methods

As remarked in the introduction, MOC uses only elementary methods to calculate decomposition numbers. In large examples, however, these methods are not sufficient to find all of the irreducible Brauer characters.

One of the reasons for this is the fact that there may be proper ordinary characters which vanish on all p-singular classes and thus are virtual projective characters, but which are not characters of projective modules. This means that in the decomposition matrix there are two columns, corresponding to PIMs Φ and Ψ, such that one is contained in the other, in other words, Φ − Ψ, say, is a proper ordinary character. This happens for example in the case of the principal 3-block of the second Janko group J_{2}, which has the decomposition matrix displayed in Table 5.2 (for a proof see [55]). In this example Φ_{2} − Φ_{6} is a proper ordinary character. Table 5.3 gives the character table of a basic set of Brauer characters. The table is given in the MOC-character table format. We use the integral bases \{1, (−1 + \sqrt{5})/2\} for the occuring irrationalities.

Let φ denote the irreducible Brauer character corresponding to Φ.

Table 5.2: The 3-modular decomposition of J_{2}

Φ_{1} Φ_{2} Φ_{3} Φ_{4} Φ_{5} Φ_{6} Φ_{7} Φ_{8}
1 1 . . . . . . .
14 1 1 . . . . . .
14 1 . 1 . . . . .
21 … 1 ….
21 …. 1 …
70 . 1 … 1 . .
70 . . 1 … 1 .
160 1 1 1 …. 1
175 … 1 1 . . 1
224 . 1 . . 1 1 . 1
224 . . 1 1 . . 1 1
300 1 2 2 . . 1 1 1
336 2 1 1 1 1 . . 2

The proof that Ψ is not a direct summand of the projective with character Φ is usually achieved by giving a lower bound for the multiplicity of φ in the reduction modulo p of some ordinary character.

In the above example, φ is an irreducible Brauer character of degree 13, and to show that Φ_{2} − Φ_{6} is not projective, is the same as showing that φ is contained in the reduction modulo 3 of the first ordinary character of degree 70. By inspecting the table of Brauer characters 5.3 we see that

169 = 13_{1} ⊗ 13_{1} = 1 − 13_{1} + 21_{1} + 70_{1} + 90,

and thus 13_{1} must be contained in 70_{1}. This example is typical insofar as we have used tensor products of Brauer characters to obtain the bounds. The larger the group and hence the degrees of the irreducible Brauer characters, the weaker gets this method. Sometimes it helps to distinguish cases as indicated in Section 3.3 with the help of Brauer characters obtained by tensor products.

If all this fails, there are more advanced methods available, which make use of structure theory of G-modules, which we are now going to

Table 5.3: Some 3-modular characters of J_{2}

\begin{matrix}1A & 2A & 2A & 4A & 5A & 5A & 5C & 5C & 7A & 8A & 10A & 10A & 10C & 10C \\ 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\ 13 & −3 & 1 & 1 & 2 & 3 & 1 & 1 & −1 & −1 & −1 & 1 & 0 & 1 \\ 13 & −3 & 1 & 1 & −1 & −3 & 0 & −1 & −1 & −1 & −2 & −1 & −1 & −1 \\ 21 & 5 & −3 & 1 & 4 & 1 & 2 & 2 & 0 & −1 & −1 & −1 & 0 & 0 \\ 21 & 5 & −3 & 1 & 3 & −1 & 0 & −2 & 0 & −1 & 0 & 1 & 0 & 0 \\ 70 & −10 & −2 & 2 & 5 & 5 & 0 & 0 & 0 & 0 & 0 & −1 & 0 & 0 \\ 70 & −10 & −2 & 2 & 0 & −5 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 133 & 5 & 1 & −3 & −7 & 0 & −2 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 36 & 4 & 0 & 4 & −4 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & −1 & 0 \\ 90 & 10 & 6 & −2 & 5 & 0 & 0 & 0 & −1 & 0 & 1 & 0 & 0 & 0 \\ 63 & 15 & −1 & 3 & 3 & 0 & −2 & 0 & 0 & 1 & −1 & 0 & 0 & 0 \\ 225 & −15 & 5 & −3 & 0 & 0 & 0 & 0 & 1 & −1 & 0 & 0 & 0 & 0 \\ 189 & −3 & −3 & −3 & 3 & 3 & 2 & 1 & 0 & 1 & 0 & 1 & −1 & −1 \\ 189 & −3 & −3 & −3 & 0 & −3 & 1 & −1 & 0 & 1 & −1 & −1 & 0 & 1 \\ 169 & 9 & 1 & 1 & 13 & 3 & 2 & 1 & 1 & 1 & 2 & −3 & 1 & −1 \end{matrix}

describe. For every such method we also give an example where it can be applied. In fact, all the tricks to be described in the following have come to mind only when we were trying to solve the corresponding problems.

Let S be one of the rings {K, R, F } of our p-modular system for G. We only consider SG-modules X, which are free and finitely generated as S-modules. Let X and Y be two such SG-modules. We set

[X, Y] = rank_{S} Hom_{SG}(X, Y).

The SG-module dual to X is denoted by X^{∗}.

5.5.1 Associativity of the tensor product

Let X, Y and Z denote three SG-modules. Then there is a canonical isomorphism of SG-modules:

(X ⊗ Y) ⊗ Z ≅ X ⊗ (Y ⊗ Z).

This can sometimes be used if the dimensions of the modules involved in the tensor product are small.

5.5.2 Adjointness

Let X, Y and Z denote three SG-modules. We then have (see [93, Corollary II.6.9])

[X ⊗ Y, Z] = [X, Y^{∗} ⊗ Z].

This has been used in the final state of our example of the Conway group modulo 7 (see Section 6.2), and also in the proof for the Janko group J_{2} modulo 5 [55]. This method can be used either to show that some composition factor must occur in a certain tensor product (J_{2} modulo 5) or to show that it cannot occur (Co_{1} modulo 7).

It is worth noting that via tensor products one can transfer information between different blocks with the help of this adjointness property. This is particularly helpful if some of the blocks involved are of defect 1, since for those one can write down all the indecomposables, if the Brauer tree is known. An example of a successful application is provided by the second Janko group J_{2} modulo 3.

5.5.3 Nakayama relations

Let H be a subgroup of G and let X be an SG-module, Y an SH module. Then we have the Nakayama relations (or Frobenius reciprocity) (see [93, Corollary II.1.4])

[X, Y^{G}] = [X_{H}, Y], and [Y^{G}, X] = [Y, X_{H}].

This has been applied for instance in the calculation of the modular characters for the Tits group ([50]).

5.5.4 Self duality

If an F G-module X is self dual, i.e., X ≅ X^{∗}, then its socle series is the dual of its Loewy series (see [93, Lemma I.8.4(i)]). In particular, any self dual composition factor M in the head of X must also occur in the socle. Hence such an M either is twice a composition factor of X or else it is a direct summand. If the latter is not the case, X has a non-trivial endomorphism.

The following lemma, which follows from the above remarks, is often quite useful.

Lemma 5.5.1 Let X be a self dual F G-module with [X, X] = 1. If every composition factor of X is self dual, then X is irreducible. □

5.5.5 Trivial source modules

The methods described above are very powerful in connection with the theory of modules with a trivial source. These are, by definition, indecomposable direct summands of permutation modules over F or R. Every indecomposable F G-module M with trivial source is liftable to a uniquely determined trivial source RG-module X (see [93, Section II.12]). This means that M is isomorphic to the F G-module X ⊗_{R} F G. Furthermore, [M, M] = [X, X].

Any direct summand of the tensor product of two trivial source modules is again a trivial source module. If H is a subgroup of G and Y a trivial source module of H, than every direct summand of Y^{G} is a trivial source module.

If B is a block of G containing only real valued ordinary characters, then B is called strongly real. In a strongly real block every simple F G-module is self dual. This follows from the surjectivity of the decomposition homomorphism: every irreducible Brauer character in such a block is real valued and thus corresponds to a self dual simple module.

Let M be a trivial source F G-module with trivial source lift X. Suppose that the ordinary character of X is irreducible and that M is contained in a strongly real block. Then M is irreducible by Lemma 5.5.1. This applies in particular to permutation modules of doubly transitive permutation representations on n points if p does not divide n. If 1 + χ is the ordinary character of a doubly transitive permutation representation, then χ is irreducible modulo p if χ is contained in a strongly real block and if p does not divide 1 + χ(1). The symplectic group S_{6}(2) has a doubly transitive permutation representation of degree 28, and, since the ordinary character table of S_{6}(2) has only rational entries, the non-trivial constituent of the permutation character remains irreducible modulo 3.

5.5.6 Lattices

An RG-module X, which is free as R-module, is also called an RGlattice. A sublattice Y of X is called pure, if the factor module X/Y again is a lattice. The following result is more or less folclore.

Lemma 5.5.2 Let X be an RG-lattice with ordinary character χ + ψ. Then there exists a pure sublattice Y ≤ X such that the character of Y is χ and the character of X/Y is ψ.

Proof. See [93, Theorem I.17.3].

This has turned out to very powerful in the theory of blocks with a cyclic defect group. But it has also found applications for non-cyclic defect groups, for example in the proof of Theorem 1 of [50].

5.5.7 Fong’s lemma

Lemma 5.5.3 (Fong’s lemma:) Let p = 2. Then every self dual nontrivial simple F G-module has even dimension.

Proof. Such a module carries a non-degenerate G-invariant symplectic form (see [69, Theorem VII.8.13]).

Let χ ∈ Irr(G) be real-valued. Then it easily follows from Fong’s lemma that d_{χ1} ≡ χ(1)(mod 2).

For example, consider the Mathieu group M_{11} modulo 2. There are two characters of defect 0, namely χ_{6} and χ_{7} (in Atlas notation). If we define

Ψ_{1} := χ_{7} ⊗ χ_{6},
Ψ_{2} := χ_{6} ⊗ χ_{6},
Ψ_{3} := χ_{2} ⊗ χ_{6},

we obtain the following basic set of projective characters for the principal 2-block of M_{11}.

Ψ_{1} Ψ_{2} Ψ_{3}
1 1 . .
10 . 1 .
10 . 1 .
10 . 1 .
11 1 1 .
44 1 1 1
45 2 1 1
55 2 2 1

Fong’s lemma shows that Ψ_{3} is contained in Ψ_{1}.

5.5.8 Theorem of Benson and Carlson

Theorem 5.5.4 (Benson-Carlson:) Let M, N be indecomposable F Gmodules such that p | dim_{k} M. Then every indecomposable direct summand of M ⊗ N has dimension divisible by p.

Proof. See [8, Vol. I,Theorem 3.1.9],

This is applied, for example, in [59, p. 111].

5.5.9 Blocks with cyclic defect groups

In a block with a cyclic defect group one can describe all indecomposable F G-modules, once the planar embedded Brauer tree is known (See [1, Chapter V]).

This information is usually easier to obtain than that for a block with a non-cyclic defect group. It is very powerful in connection with the Nakayama relations (where we assume that some block for a subgroup with a cyclic defect group is known) and, if some blocks with a cyclic defect group for G are known, in connection with the adjointness or the theorem of Benson-Carlson.

This has been applied, for example, for some blocks of the Monster group in characteristic 5 (see [56, p. 454f]), and also in the Rudvalis group modulo 3 [54].

5.5.10 Webb’s theorem

Let M be an indecomposable F G-module. The heart of M is the F Gmodule H(M) = rad(M)/soc(M), where rad(M) denotes the intersection of all maximal submodules of M and soc(M) is the sum of all simple submodules. It is not difficult to see that if M is self dual, the same is true for the heart of M. By classifying the possible Auslander-Reiten quivers for a finite-dimensional group algebra, Webb has shown that the heart of a projective indecomposable module of F G has at most 4 indecomposable direct summands. This result has been extended by Bessenrodt.

Theorem 5.5.5 (Webb, Bessenrodt:) Let P be a projective indecomposable module of F G. Then H(P) has at most 3 indecomposable direct summands.

Proof. See [10, Corollary 1.3].

We can apply this in some situations as follows.

Lemma 5.5.6 Let B be a strongly real p-block of G. Let Φ be a PIM of B such that ⟨Φ, Φ⟩ < 4 and

∑_{Ψ∈IPr(B)}⟨Φ, Ψ⟩ ≥ 6.  (5.5)

Then there exists some Θ ∈ IPr(B), Θ ≠ Φ such that ⟨Φ, Θ⟩ ≥ 2.

Proof. Let P denote the projective indecomposable F G-module with character Φ. By assumption (5.5), the heart H(P) of P has at least 4 composition factors. By Bessenrodt’s extension of Webb’s theorem, H(P) has at most 3 indecomposable direct summands, hence it cannot be semi-simple. It follows that H(P) contains some composition factor with multiplicity at least 2. Since ⟨Φ, Φ⟩ < 4, this must be a simple module not isomorphic to P/rad(P) and the assertion follows.

For example, this lemma can be applied in the case of the Rudvalis group modulo 3 [54]. We finally proof a lemma in the same spirit as the one above.

Lemma 5.5.7 Suppose that G has no factor group of order p. Suppose also, that the principal block of G is strongly real. Let Φ denote the character of the projective cover P of the trivial F G-module. If ⟨Φ, Φ⟩ > 2, then there exists some Θ ∈ IPr(B), Θ ≠ Φ such that ⟨Φ, Θ⟩ ≥ 2.

Proof. If the trivial F G-module is in the second Loewy layer of P, then G has some factor group of order p (see [93, Corollary I.10.13]). It follows that H(P) is not semi-simple and that it must contain some non-trivial composition factor with multiplicity at least 2.

Calculating the 7-modular decomposition matrices of the Conway group

The double covering group of the Conway group Co_{1} has nine 7-blocks of defect larger than 0. Seven of these are of defect 1 and their decomposition matrices are given in [56]. The remaining two blocks are of maximal defect 2. Each contains 29 ordinary and 21 irreducible Brauer characters.

6.1 The faithful block of maximal defect

We start with the block of maximal defect containing faithful characters, shortly called the faithful block in the following. The proof for the faithful block is easier than that for the principal block, and the result is used in the proof for the principal block.

We start with the set of projective characters displayed in Table 6.1. The origin of the projective characters is documented in table 6.4. The symbol θ_{i} denotes the ith character of Co_{2} (in ATLAS ordering), and a bar denotes the character times the sign character of 2 × Co_{2}.

The two characters of degree 9 152 000 are complex conjugates of each other. All other characters in the block are real. Since the number of irreducible Brauer characters in the block is 21, there are exactly 19 real valued projective indecomposable characters. Each of Λ_{1}–Λ_{20}, except Λ_{12}, contains one of these. Therefore, Λ_{12} contains a pair of complex conjugate projective indecomposable characters. In particular, the decomposition matrix has wedge shape. From this it immediately follows, that Λ_{2}–Λ_{4}, Λ_{7}, Λ_{9}, Λ_{11} and Λ_{14}–Λ_{20} are projective indecomposable characters.

We now consider the projectives given in Table 6.2. Table 6.4 gives their origin and Table 6.3 their decompositions in terms of the projectives of Table 6.1, the relations.

Using these relations, one can reduce the first projectives to obtain a new set of projectives, given in Table 6.5. For example, Λ_{30} shows, that the PIM Λ_{18} is contained in Λ_{1}, and Λ_{31} shows, that the PIM Λ_{3} is contained in Λ_{1}. This gives Φ_{1} of Table 6.5. All the other new projectives are obtained by similar arguments. Since the two characters of degree 9 152 000 are complex conjugates of each other, Φ_{12} contains a pair of complex conjugate PIMs. This immediately determines the decomposition matrix.

6.2 The principal block

We start with the set of projectives displayed in Table 6.6. The origin of these is documented in Table 6.9. It is not too difficult to check that Λ_{2}, Λ_{4}, Λ_{6}–Λ_{9}, Λ_{12}, Λ_{15} and Λ_{19} are projective indecomposable characters. (For example, this can be done by checking that no subsum of these characters is zero on all 7-singular classes.)

We have displayed further projectives in Table 6.7, their origin in Table 6.9, where the notation Φ_{i,j} stands for the i-th projective indecomposable character in Block j. The numbering of the blocks follows Appendix A.2. The expression of the new projectives in terms of the first set of projectives is given in Table 6.8. Using these relations, one can reduce the basic projectives to obtain a new set of projectives, given in Table 6.10. The arguments are exactly the same as those used for the proof of Block 7.

One immediately checks that all of these, except possibly Φ_{10} and Φ_{14} are characters of projective indecomposable modules. Furthermore, either Φ_{10} is a PIM or Φ_{10} − Φ_{17} is one. Similarly, either Φ_{14} is a PIM or Φ_{14} − Φ_{20} is one.

Thus we are left with four possible decomposition matrices. To decide which of these is correct, we reformulate the problem in terms of Brauer characters. All irreducible Brauer characters are known, except ϕ_{17} and ϕ_{20}. The degree of ϕ_{10} is 2 038 674, that of ϕ_{14} is 10 140 998. If Φ_{10} is a PIM, then the degree of ϕ_{17} is 62 725 301, if Φ_{10} − Φ_{17} is a PIM we have ϕ_{17}(1) = 62 725 301 + 2 038 674 = 64 763 975. Similarly, if Φ_{14} is a PIM, then the degree of ϕ_{20} is 124 375 559 if Φ_{14} − Φ_{20} is a PIM we have ϕ_{20}(1) = 124 375 559 + 10 140 998 = 134 516 557.

Suppose first, that Φ_{10} is a PIM. Consider the irreducible Brauer character of Block 7 of degree 50 207 872. Denoting the irreducible Brauer characters by their degrees, we obtain

24 ⊗ 2 038 674 = 48 928 176,  (6.1)
24 ⊗ 50 207 872 = 62 725 301 + 2 038 674 + ψ,  (6.2)

where ψ is a sum of characters not in the principal block, namely ψ = 52 465 644 + 256 168 549 + 309 429 120 + 522 161 640. Let F denote an algebraically closed field of characteristic 7. For two F G-modules M and N, the dimension of the space of F G-homomorphisms from M to N is denoted by [M, N]. With this notation we have,

[24 ⊗ 2 038 674, 50 207 872] = 0,

by equation (6.1), and so,

[2 038 674, 24 ⊗ 50 207 872] = 0.  (6.3)

Since all modules considered are self-dual, equation (6.2) implies that the irreducible 2 038 674 is in the socle of 24 ⊗ 50 207 872, contradicting equation (6.3). We have proved that Φ_{10} is decomposable.

Now suppose that Φ_{14} is a PIM. This time we consider the irreducible Brauer character of degree 10 039 568 of Block 7. We obtain:

24 ⊗ 10 140 998 = 243 383 952,
24 ⊗ 10 039 568 = 124 375 559 + 10 140 998 + ψ,  (6.4)

where ψ is a sum of characters not in the principal block, namely ψ = 77 700 854 + 25 892 020 + 2 840 201. However, as above,

[10 140 998, 24 ⊗ 10 039 568] = 0,

contradicting equation (6.4). We have shown that Φ_{14} decomposes, and thus completed the proof of the decomposition matrix.

6.3. TABLES 99

6.3 Tables

Here we collect the tables of projective characters which are needed in the previous sections.

Table 6.1: Some projectives in the faithful block

\begin{matrix}Λ : & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ 24 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 2024 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 4576 & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 40480 & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 95680 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 299000 & 1 & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 315744 & 1 & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 789360 & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\ 1937520 & 1 & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 5051904 & . & . & . & 1 & . & . & . & . & 1 & 1 & . & . & . & . & . & . & . & . & . & . \\ 7104240 & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 9152000 & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 9152000 & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 13156000 & . & 1 & . & 1 & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . \\ 17050176 & . & 1 & . & . & . & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . \\ 19734000 & 1 & . & . & . & . & . & . & . & 1 & 2 & . & . & . & 1 & . & . & . & . & . & . \\ 34155000 & 1 & . & 1 & . & . & 1 & 1 & . & . & 2 & . & . & . & . & 1 & . & . & . & . & . \\ 49335000 & . & . & . & . & 2 & 1 & 1 & . & . & . & . & . & . & . & . & 1 & . & . & . & . \\ 50519040 & . & . & . & . & . & . & 1 & . & . & . & . & 1 & 1 & . & . & . & 1 & . & . & . \\ 67358720 & . & . & . & . & . & 1 & . & 1 & . & 3 & . & 2 & . & 1 & 1 & . & . & . & . & . \\ 210496000 & 1 & . & . & . & 1 & 1 & . & . & . & . & 1 & . & 1 & . & . & . & . & 1 & . & . \\ 215547904 & . & . & . & . & . & 1 & . & 3 & . & 1 & . & 6 & 2 & . & . & . & . & . & 1 & . \\ 313524224 & 1 & . & . & . & . & . & . & 2 & . & 4 & . & 4 & . & 1 & 1 & . & . & . & . & 1 \\ 394680000 & . & . & . & . & 1 & 1 & 1 & 1 & . & 2 & . & 5 & 1 & . & 1 & 1 & 1 & . & . & 1 \\ 485760000 & 2 & . & . & . & 2 & 2 & . & 1 & . & . & . & 2 & . & . & . & 1 & . & 1 & . & 1 \\ 517899096 & . & . & . & . & . & 1 & . & 5 & . & 1 & . & 11 & 2 & . & . & . & 1 & . & 1 & 1 \\ 655360000 & 1 & . & . & . & . & 2 & . & 4 & . & 2 & . & 8 & 2 & . & . & . & . & 1 & 1 & 1 \end{matrix}

Table 6.2: More projectives in the faithful block

\begin{matrix}Λ : & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 \\ 24 & . & . & . & . & . & . & . & . & . & 1 & 1 & . \\ 2024 & . & . & . & . & . & . & . & . & . & 1 & . & . \\ 4576 & . & . & . & . & . & . & . & . & . & 2 & . & . \\ 40480 & . & . & . & . & . & 1 & . & . & . & 1 & . & . \\ 95680 & . & . & . & . & 1 & . & . & . & . & 1 & 1 & . \\ 299000 & 2 & . & . & . & . & 3 & . & . & . & 4 & . & . \\ 315744 & . & . & . & . & 1 & . & . & . & . & 2 & . & . \\ 789360 & 2 & . & . & . & . & 4 & . & . & . & 2 & . & 1 \\ 1937520 & 3 & 1 & . & . & . & 7 & . & . & 1 & 5 & 1 & . \\ 5051904 & 3 & 2 & . & 1 & . & 9 & 1 & 1 & 1 & 4 & . & 1 \\ 7104240 & . & . & 2 & . & 2 & . & . & . & . & . & . & . \\ 9152000 & . & . & . & . & . & . & 1 & 1 & 1 & . & . & . \\ 9152000 & . & . & . & . & . & . & 1 & 1 & 1 & . & . & . \\ 13156000 & . & 1 & 1 & 2 & . & 2 & 2 & 2 & . & 1 & . & 1 \\ 17050176 & . & . & 3 & 1 & 1 & . & 1 & 1 & . & 1 & . & . \\ 19734000 & 4 & 5 & . & 3 & . & 14 & 2 & 1 & 2 & 6 & 1 & 2 \\ 34155000 & 2 & 4 & . & 4 & 1 & 7 & . & 1 & 1 & 6 & 1 & 1 \\ 49335000 & . & . & . & . & 2 & . & . & . & . & . & 2 & . \\ 50519040 & . & . & . & . & 1 & . & 2 & 1 & . & . & . & . \\ 67358720 & 3 & 7 & . & 6 & . & 14 & 3 & 3 & 4 & 5 & 1 & 3 \\ 210496000 & . & . & 6 & 1 & 2 & . & 4 & 3 & . & . & 4 & . \\ 215547904 & . & . & 2 & 1 & . & . & 11 & 6 & 1 & . & 1 & 1 \\ 313524224 & 1 & 10 & 1 & 10 & . & 13 & 8 & 7 & 5 & 4 & 6 & 4 \\ 394680000 & . & 6 & 1 & 7 & 1 & 5 & 8 & 7 & 4 & 2 & 7 & 1 \\ 485760000 & . & 2 & 4 & 3 & 1 & 1 & 7 & 5 & 1 & 1 & 11 & . \\ 517899096 & . & 2 & 2 & 3 & . & 2 & 18 & 11 & 4 & . & 5 & 2 \\ 655360000 & . & 3 & 6 & 5 & . & 2 & 19 & 12 & 2 & . & 9 & 2 \end{matrix}

TABLES 101

Table 6.3: Relations in the faithful block

Λ : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Λ_{21} . . . . . 2 . . 3 …. 1 . −2 . −2 −2 .
Λ_{22} . . . . . . . . 1 1 … 2 2 … −1 2
Λ_{23} . . . . . . . . . . 2 . 1 … −1 3 . 1
Λ_{24} . . . . . . . . . 1 . . 1 1 2 . −1 . −2 3
Λ_{25} …. 1 . 1 … 1 …. −1 ….
Λ_{26} … 1 . 3 . 1 7 1 … 5 2 −3 . −3 −7 .
Λ_{27} . . . . . . . . . 1 . 1 1 . −2 . . 3 2 2
Λ_{28} . . . . . . . . . 1 . 1 1 −1 −1 . −1 2 −3 1
Λ_{29} . . . . . . . . 1 . . 1 . 1 1 . −1 . −5 −1
Λ_{30} 1 1 1 . . 2 . . 4 …. 1 2 −2 . −3 −2 .
Λ_{31} 1 . −1 . . . . . . . . . . . 1 2 . 3 1 4
Λ_{32} . . . . . . . 1 . 1 …. −1 … −3 −1

102 DECOMPOSITION MATRICES OF CONWAY’S GROUP

Table 6.4: Origin of projective characters of the faithful block

Char.  Origin
Λ_{1} :  Ind_{2xCo2}(θ̄_{2} + θ̄_{21})
Λ_{2} :  Ind_{2xCo2}(θ̄_{5})
Λ_{3} :  χ_{3} ⊗ χ_{111}
Λ_{4} :  Ind_{2xCo2}(θ̄_{9})
Λ_{5} :  Ind_{2xCo2}(θ̄_{18})
Λ_{6} :  Ind_{2xCo2}(θ̄_{15} + θ̄_{24})
Λ_{7} :  χ_{104} ⊗ χ_{16}
Λ_{8} : Ind_{2xCo2}(θ̄_{22} + θ̄_{23} + θ̄_{24})
Λ_{9} :  χ_{102} ⊗ χ_{36}
Λ_{10} :  χ_{2} ⊗ χ_{141}
Λ_{11} :  χ_{102} ⊗ χ_{39}
Λ_{12} :  χ_{5} ⊗ χ_{118}
Λ_{13} :  χ_{104} ⊗ χ_{33}
Λ_{14} :  χ_{102} ⊗ χ_{49}
Λ_{15} :  χ_{2} ⊗ χ_{127}
Λ_{16} :  χ_{3} ⊗ χ_{118}

Char.  Origin
Λ_{17} :  χ_{2} ⊗ χ_{118}
Λ_{18} :  χ_{2} ⊗ χ_{128}
Λ_{19} :  χ_{2} ⊗ χ_{125}
Λ_{20} :  χ_{102} ⊗ χ_{75}
Λ_{21} :  χ_{2} ⊗ χ_{119}
Λ_{22} :  χ_{2} ⊗ χ_{147}
Λ_{23} :  χ_{2} ⊗ χ_{148}
Λ_{24} :  χ_{2} ⊗ χ_{149}
Λ_{25} :  χ_{3} ⊗ χ_{131}
Λ_{26} :  χ_{4} ⊗ χ_{132}
Λ_{27} :  Ind_{2xCo2}(θ̄_{50})
Λ_{28} :  χ_{3} ⊗ χ_{156}
Λ_{29} :  χ_{4} ⊗ χ_{127}
Λ_{30} :  χ_{7} ⊗ χ_{111}
Λ_{31} : Ind_{2xCo2}(θ̄_{1} + θ̄_{44})
Λ_{32} :  χ_{3} ⊗ χ_{141}

Table 6.5: Refined projectives in the faithful block

\begin{matrix}Φ : & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ 24 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 2024 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 4576 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 40480 & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 95680 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 299000 & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 315744 & . & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 789360 & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\ 1937520 & 1 & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 5051904 & . & . & . & 1 & . & . & . & . & 1 & 1 & . & . & . & . & . & . & . & . & . & . \\ 7104240 & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 9152000 & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 9152000 & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 13156000 & . & 1 & . & 1 & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . \\ 17050176 & . & 1 & . & . & . & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . \\ 19734000 & 1 & . & . & . & . & . & . & . & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . \\ 34155000 & . & . & 1 & . & . & 1 & 1 & . & . & . & . & . & . & . & 1 & . & . & . & . & . \\ 49335000 & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . & . & 1 & . & . & . & . \\ 50519040 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & 1 & . & . & . \\ 67358720 & . & . & . & . & . & 1 & . & 1 & . & . & . & 2 & . & 1 & 1 & . & . & . & . & . \\ 210496000 & . & . & . & . & 1 & . & . & . & . & . & 1 & . & 1 & . & . & . & . & 1 & . & . \\ 215547904 & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 & . & . & . & . & . & 1 & . \\ 313524224 & 1 & . & . & . & . & . & . & 1 & . & 1 & . & 3 & . & 1 & 1 & . & . & . & . & 1 \\ 394680000 & . & . & . & . & . & . & 1 & . & . & . & . & 3 & . & . & 1 & 1 & 1 & . & . & 1 \\ 485760000 & 1 & . & . & . & 1 & . & . & . & . & . & . & 1 & . & . & . & 1 & . & 1 & . & 1 \\ 517899096 & . & . & . & . & . & . & . & 1 & . & . & . & 4 & . & . & . & . & 1 & . & 1 & 1 \\ 655360000 & . & . & . & . & . & . & . & . & . & 1 & . & 2 & 1 & . & . & . & . & 1 & 1 & 1 \end{matrix}

Table 6.6: Basic set of projectives of the principal block

\begin{matrix}Λ : & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\ 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 276 & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 299 & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 17250 & 1 & . & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 80730 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 94875 & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 822250 & 1 & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 871884 & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1821600 & 1 & 1 & . & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\ 2055625 & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 9221850 & 1 & . & . & . & . & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 16347825 & . & . & . & 1 & 2 & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 21528000 & 1 & . & . & . & . & 1 & . & 1 & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 21579129 & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 & . & . & . & . & . & . \\ 24667500 & 1 & . & . & . & 1 & . & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . \\ 31574400 & 2 & . & . & . & . & . & . & 1 & . & . & . & . & 1 & 1 & . & . & . & . & . & . & . \\ 57544344 & 2 & . & 1 & 1 & . & . & 1 & . & . & . & 1 & . & . & 1 & . & 1 & . & . & . & . & . \\ 66602250 & . & . & . & 2 & 1 & . & . & . & 1 & 1 & . & . & . & 1 & . & . & 2 & 1 & 1 & . & . \\ 85250880 & 2 & . & 1 & 2 & . & . & . & . & . & 1 & . & . & . & 2 & . & 1 & 1 & . & . & . & . \\ 150732800 & . & . & . & 1 & . & . & . & . & . & 1 & . & . & 1 & 1 & . & 1 & 1 & 1 & . & 1 & 1 \\ 163478250 & 3 & 1 & . & 1 & . & 1 & . & . & . & . & 1 & . & 1 & 1 & . & 1 & 1 & . & . & 1 & 1 \\ 191102976 & 1 & . & . & 1 & . & . & . & . & . & . & . & 1 & 1 & 2 & 1 & 1 & 1 & . & . & 1 & 1 \\ 207491625 & 1 & . & . & 1 & . & . & . & . & . & 1 & . & . & 1 & 3 & . & . & 3 & 1 & 1 & 1 & . \\ 215547904 & 1 & 1 & . & 2 & 1 & . & . & . & 1 & 1 & . & 1 & . & 2 & . & 1 & 3 & 1 & 1 & 1 & 1 \\ 219648000 & . & . & . & . & . & . & . & . & . & . & . & . & 2 & 1 & 1 & . & 1 & 1 & . & 2 & 1 \\ 299710125 & 2 & . & . & 1 & . & . & . & . & . & . & 1 & . & 2 & 3 & . & 1 & 2 & 1 & . & 2 & 1 \\ 326956500 & 1 & . & . & 1 & . & . & . & . & . & 1 & 1 & . & 1 & 3 & 1 & 1 & 3 & 1 & 1 & 2 & 1 \end{matrix}

Table 6.7: More projectives in the principal block

\begin{matrix}Λ : & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 & 33 & 34 & 35 & 36 & 37 \\ 1 & 1 & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 276 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 299 & . & 2 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 17250 & 1 & 2 & . & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . \\ 80730 & 2 & 1 & . & . & 1 & . & 2 & . & . & . & . & . & . & . & . & . \\ 94875 & . & 2 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 822250 & . & 8 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & 1 \\ 871884 & 4 & 6 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 1821600 & . & 1 & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . \\ 2055625 & 3 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 9221850 & . & 11 & . & . & . & . & . & . & . & . & 1 & 2 & . & 1 & 2 & 1 \\ 16347825 & 4 & 1 & 1 & 1 & 1 & . & 3 & . & . & 1 & . & . & 1 & . & . & . \\ 21528000 & 5 & 13 & . & . & . & . & . & . & . & . & 1 & 2 & . & 1 & 2 & 1 \\ 21579129 & 1 & . & 1 & 1 & . & 1 & . & 1 & 1 & . & . & . & . & 1 & . & . \\ 24667500 & 6 & 1 & 2 & 2 & 2 & 1 & 2 & 1 & . & 1 & . & . & . & . & . & . \\ 31574400 & 7 & 12 & 1 & 1 & 1 & 1 & . & 1 & . & . & 1 & 2 & . & 1 & 2 & 1 \\ 57544344 & 5 & 19 & . & 1 & 2 & 1 & . & 1 & . & . & 3 & 3 & . & 2 & 2 & 2 \\ 66602250 & 10 & 1 & 2 & 3 & 2 & 2 & 2 & . & . & 1 & . & . & 1 & . & . & . \\ 85250880 & 14 & 10 & 1 & 3 & 4 & 1 & 1 & 3 & . & 1 & 3 & 1 & . & 2 & 1 & 1 \\ 150732800 & 13 & 12 & 3 & 2 & 1 & 2 & . & 1 & . & . & 2 & 1 & . & 2 & . & 1 \\ 163478250 & 5 & 18 & 3 & 1 & 1 & 1 & . & 2 & . & . & 4 & 4 & 1 & 3 & 4 & 1 \\ 191102976 & 9 & 13 & 5 & 3 & 2 & 2 & . & 3 & 1 & 1 & 3 & 2 & 1 & 3 & 2 & 1 \\ 207491625 & 20 & 1 & 7 & 5 & 4 & 3 & 1 & 4 & 2 & 2 & 1 & . & 1 & 2 & 1 & . \\ 215547904 & 15 & 8 & 6 & 5 & 4 & 3 & 1 & 2 & . & 2 & 2 & . & 2 & 1 & . & . \\ 219648000 & 10 & 5 & 7 & 1 & . & 2 & . & 2 & 3 & . & . & . & 1 & 3 & . & . \\ 299710125 & 21 & 16 & 8 & 4 & 3 & 3 & . & 5 & 2 & 1 & 4 & 3 & 1 & 5 & 3 & 1 \\ 326956500 & 16 & 11 & 9 & 4 & 3 & 4 & . & 4 & 3 & 1 & 3 & 2 & 2 & 4 & 2 & . \end{matrix}

106 DECOMPOSITION MATRICES OF CONWAY’S GROUP

Table 6.8: Relations in the principal block

Λ : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Λ_{22} 1 −1 −1 1 1 . . 3 −2 2 −1 3 1 1 . 3 4 7 −9 2 −8
Λ_{23} 1 −1 1 . . 2 6 5 1 . 2 . 5 . . 8 −1 . 2 −4 4
Λ_{24} . . . . . . . . . . . 1 . 1 . −1 . . 1 3 .
Λ_{25} . . . . . . . . . . . 1 . 1 . . 1 1 −1 −1 .
Λ_{26} 1 −1 . . . . . . . . −1 1 −1 . . 1 1 1 −1 1 −2
Λ_{27} . . . . . . . . . . . . . 1 . . −1 1 2 . 1
Λ_{28} … 1 2 … −2 . . . . . . −1 . . . . .
Λ_{29} . . . . . . . . . . . . . 1 . . 1 −1 −2 1 −1
Λ_{30} . . . . . . . . . . . . . . 1 …. 2 −2
Λ_{31} . . . . . . . . . . . 1 …. 1 . −1 . −1
Λ_{32} . . . . . . . . . . 1 . 1 . . 2 1 −1 −1 −1 .
Λ_{33} . . . . . . . . . . 2 . 2 . . 1 . −1 1 −2 1
Λ_{34} . . . . . . . . 1 . . . . . . . . −1 1 1 .
Λ_{35} . . . . . . . . . . 1 . 1 . 1 1 1 . −2 . −1
Λ_{36} . . . . . . . . . . 2 . 2 … 1 −2 . −2 1
Λ_{37} . . . . . . 1 . . . . . 1 . . 1 … −1 .

TABLES 107 Table 6.9: Origin of projective characters of the principal block

Char.  Origin
Λ_{1} :  χ_{102} ⊗ Φ_{1,7}
Λ_{2} : Ind_{2xCo2}(θ_{3} + θ_{16})
Λ_{3} :  χ_{102} ⊗ Φ_{1,8}
Λ_{4} :  χ_{102} ⊗ Φ_{7,7}
Λ_{5} :  χ_{2} ⊗ χ_{39}
Λ_{6} :  χ_{102} ⊗ Φ_{4,7}
Λ_{7} :  χ_{102} ⊗ χ_{119}
Λ_{8} :  χ_{2} ⊗ χ_{22}
Λ_{9} :  χ_{2} ⊗ χ_{32}
Λ_{10} :  χ_{6} ⊗ χ_{17}
Λ_{11} :  χ_{102} ⊗ χ_{141}
Λ_{12} :  χ_{4} ⊗ χ_{33}
Λ_{13} :  χ_{2} ⊗ χ_{44}
Λ_{14} :  Ind_{2xCo2}(θ_{43})
Λ_{15} :  χ_{4} ⊗ χ_{26}
Λ_{16} :  χ_{102} ⊗ χ_{149}
Λ_{17} :  χ_{3} ⊗ χ_{73}
Λ_{18} :  χ_{4} ⊗ χ_{32}
Λ_{19} :  (χ_{2} ⊗ χ_{70})/2

Char.  Origin
Λ_{20} :  χ_{102} ⊗ χ_{161}
Λ_{21} :  χ_{2} ⊗ χ_{46}
Λ_{22} :  χ_{16}^{2+}
Λ_{23} :  χ_{112}^{2+}
Λ_{24} :  χ_{7} ⊗ χ_{33}
Λ_{25} :  Ind_{2xCo2}(θ_{45})
Λ_{26} : Ind_{2xCo2}(θ_{1} + θ_{44})
Λ_{27} : Ind_{2xCo2}(θ_{27} + θ_{34})
Λ_{28} : Ind_{2xCo2}(θ_{6} + θ_{17})
Λ_{29} :  χ_{102} ⊗ Φ_{21,7}
Λ_{30} :  χ_{102} ⊗ Φ_{20,7}
Λ_{31} :  χ_{102} ⊗ Φ_{19,7}
Λ_{32} :  χ_{102} ⊗ Φ_{16,7}
Λ_{33} :  χ_{102} ⊗ Φ_{15,7}
Λ_{34} :  χ_{102} ⊗ Φ_{14,7}
Λ_{35} :  χ_{102} ⊗ Φ_{13,7}
Λ_{36} :  χ_{102} ⊗ Φ_{10,7}
Λ_{37} :  χ_{102} ⊗ Φ_{3,8}

Table 6.10: Refined basic set of projectives of the principal block

\begin{matrix}Φ : & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\ 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 276 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 299 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 17250 & . & . & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 80730 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 94875 & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 822250 & . & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 871884 & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1821600 & . & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\ 2055625 & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 9221850 & . & . & . & . & . & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 16347825 & . & . & . & . & 1 & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 21528000 & . & . & . & . & . & 1 & . & 1 & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 21579129 & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 & . & . & . & . & . & . \\ 24667500 & 1 & . & . & . & 1 & . & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . \\ 31574400 & 1 & . & . & . & . & . & . & 1 & . & . & . & . & 1 & 1 & . & . & . & . & . & . & . \\ 57544344 & . & . & 1 & . & . & . & 1 & . & . & . & 1 & . & . & . & . & 1 & . & . & . & . & . \\ 66602250 & . & . & . & 1 & . & . & . & . & 1 & 1 & . & . & . & . & . & . & 1 & . & . & . & . \\ 85250880 & . & . & 1 & 1 & . & . & . & . & . & 1 & . & . & . & . & . & 1 & . & 1 & . & . & . \\ 150732800 & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & 1 & . & . \\ 163478250 & . & 1 & . & . & . & 1 & . & . & . & . & 1 & . & 1 & 1 & . & . & . & . & . & 1 & . \\ 191102976 & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 & 2 & 1 & . & . & . & . & 1 & . \\ 207491625 & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & 1 & 1 & . & . & 1 \\ 215547904 & . & 1 & . & . & . & . & . & . & 1 & 1 & . & 1 & . & 1 & . & . & 1 & . & . & 1 & . \\ 219648000 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & 1 & . & 1 \\ 299710125 & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & 1 & . & 1 & 1 & . & 1 \\ 326956500 & . & . & . & . & . & . & . & . & . & 1 & 1 & . & . & 1 & 1 & . & 1 & . & . & 1 & 1 \end{matrix}

Appendix A

A.1 The 5-decomposition numbers of Co_{2}

\begin{matrix}1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 23 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 253 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1771 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\ 2024 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 2277 & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 7084 & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 10395 & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 10395 & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 31878 & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . \\ 37422 & . & . & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . \\ 129536 & . & . & 1 & 1 & . & . & 1 & . & . & . & 1 & . & . & . & . & . \\ 184437 & 1 & . & . & . & 1 & . & 1 & . & . & . & . & 1 & . & . & . & . \\ 212520 & . & . & . & . & . & 2 & . & . & 1 & . & . & . & 1 & . & . & . \\ 226688 & 1 & . & . & . & 1 & . & . & 1 & . & 1 & . & 1 & . & . & . & . \\ 239085 & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . \\ 239085 & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . \\ 245916 & . & . & . & . & 1 & . & . & . & 1 & 1 & . & 1 & . & . & . & . \\ 312984 & . & . & . & 1 & . & . & 2 & . & . & . & 1 & . & 1 & . & . & . \\ 368874 & . & 1 & 1 & . & . & . & . & 1 & . & . & 1 & . & . & . & 1 & . \\ 430353 & . & . & . & . & . & 1 & 2 & . & . & . & . & . & 1 & 1 & . & . \\ 637560 & 1 & . & . & . & 1 & . & 1 & 1 & 1 & . & . & 2 & . & . & 1 & . \\ 1291059 & . & . & . & . & . & 1 & . & . & 1 & 1 & . & 1 & . & . & . & 1 \\ 1835008 & . & . & . & . & . & . & 1 & 1 & . & . & 1 & 1 & . & 1 & 1 & 1 \\ 1943040 & 1 & . & 1 & . & . & . & 1 & 2 & . & 1 & 2 & 2 & . & . & 1 & 1 \\ 2040192 & . & . & . & . & . & 1 & 2 & . & 1 & . & 1 & 1 & 1 & 1 & 1 & 1 \\ 2072576 & . & 1 & . & . & . & 1 & 2 & . & . & . & 1 & . & 1 & 2 & 1 & 1 \end{matrix}

110 APPENDIX A.

A.2 The 7-decomposition numbers of 2Co_{1}

In this section we give the 7-modular decomposition matrices for the two blocks of maximal defect of 2Co_{1}. The decomposition numbers are followed by tables giving the degrees of the irreducible Brauer characters as well as the prime factorizations of these degrees. The decomposition matrices for the blocks of non-maximal defect can be found in [56].

A.2.1 Decomposition matrix of Block 1

\begin{matrix}Φ : & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\ 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 276 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 299 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 17250 & . & . & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 80730 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 94875 & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 822250 & . & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 871884 & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1821600 & . & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\ 2055625 & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 9221850 & . & . & . & . & . & 1 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 16347825 & . & . & . & . & 1 & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 21528000 & . & . & . & . & . & 1 & . & 1 & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 21579129 & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 & . & . & . & . & . & . \\ 24667500 & 1 & . & . & . & 1 & . & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . \\ 31574400 & 1 & . & . & . & . & . & . & 1 & . & . & . & . & 1 & 1 & . & . & . & . & . & . & . \\ 57544344 & . & . & 1 & . & . & . & 1 & . & . & . & 1 & . & . & . & . & 1 & . & . & . & . & . \\ 66602250 & . & . & . & 1 & . & . & . & . & 1 & . & . & . & . & . & . & . & 1 & . & . & . & . \\ 85250880 & . & . & 1 & 1 & . & . & . & . & . & 1 & . & . & . & . & . & 1 & . & 1 & . & . & . \\ 150732800 & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & 1 & . & . \\ 163478250 & . & 1 & . & . & . & 1 & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & 1 & . \\ 191102976 & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 & 1 & 1 & . & . & . & . & 1 & . \\ 207491625 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 & . & . & 1 \\ 215547904 & . & 1 & . & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . & 1 & . & . & 1 & . \\ 219648000 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & 1 & . & 1 \\ 299710125 & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & 1 & . & 1 & 1 & . & 1 \\ 326956500 & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & 1 & . & 1 & . & . & 1 & 1 \end{matrix}

A.2. THE 7-DECOMPOSITION NUMBERS OF 2CO_{1} 111

A.2.2 Decomposition matrix of Block 7

\begin{matrix}Φ : & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\ 24 & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 2024 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 4576 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 40480 & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 95680 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 299000 & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 315744 & . & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 789360 & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1937520 & 1 & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . \\ 5051904 & . & . & . & 1 & . & . & . & . & 1 & 1 & . & . & . & . & . & . & . & . & . & . & . \\ 7104240 & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . \\ 9152000 & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . \\ 9152000 & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . \\ 13156000 & . & 1 & . & 1 & . & . & . & . & . & 1 & . & . & . & 1 & . & . & . & . & . & . & . \\ 17050176 & . & 1 & . & . & . & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . \\ 19734000 & 1 & . & . & . & . & . & . & . & 1 & 1 & . & . & . & . & 1 & . & . & . & . & . & . \\ 34155000 & . & . & 1 & . & . & 1 & 1 & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . \\ 49335000 & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . \\ 50519040 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . \\ 67358720 & . & . & . & . & . & 1 & . & 1 & . & . & . & 1 & 1 & . & 1 & 1 & . & . & . & . & . \\ 210496000 & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . & 1 & . & . \\ 215547904 & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & 1 & . \\ 313524224 & 1 & . & . & . & . & . & . & 1 & . & 1 & . & 1 & 1 & . & 1 & 1 & . & . & . & . & 1 \\ 394680000 & . & . & . & . & . & . & 1 & . & . & . & . & 1 & 1 & . & . & 1 & 1 & 1 & . & . & 1 \\ 485760000 & 1 & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 & . & 1 \\ 517899096 & . & . & . & . & . & . & . & 1 & . & . & . & 1 & 1 & . & . & . & . & 1 & . & 1 & 1 \\ 655360000 & . & . & . & . & . & . & . & . & . & 1 & . & . & . & 1 & . & . & . & . & 1 & 1 & 1 \end{matrix}

112 APPENDIX A.

A.2.3 Degrees of irreducible Brauer characters

The table on the left hand side lists the degrees of the irreducible Brauer characters of the principal block, the other table those of Block 7.

Char. Degree  Factors
ϕ_{1}  1  1
ϕ_{2}  276  2^{2} 3 23
ϕ_{3}  299  13 23
ϕ_{4}  16951  11 23 67
ϕ_{5}  80729  11 41 179
ϕ_{6}  94599  3^{2} 23 457
ϕ_{7}  821951 13 23 2749
ϕ_{8} 871883 871883
ϕ_{9}  1821324 2^{2} 3 23 6599
ϕ_{10}  2038674 2 3 11 17 23 79
ϕ_{11}  8305300 2^{2} 5^{2} 23^{2} 157
ϕ_{12}  14445772 2^{2} 11 569 577
ϕ_{13}  20561518 2 607 16937
ϕ_{14}  10140998 2 7 227 3191
ϕ_{15}  11438131 11438131
ϕ_{16}  48416794 2 23 37 28447
ϕ_{17}  64763975 5^{2} 23 163 691
ϕ_{18}  34778162 2 23 83 9109
ϕ_{19}  100277332 2^{2} 23 733 1487
ϕ_{20}  134516557 7 19216651
ϕ_{21}  107932537 23 43 109133

Degree  Factors
24  2^{3} 3
2024  2^{3} 11 23
4576  2^{5} 11 13
38456  2^{3}11 19 23
95656  2^{3} 11 1087
294424 2^{3} 13 19 149
311168  2^{7} 11 13 17
494936  2^{3} 13 4759
1937496 2^{3} 3 11 41 179
3075952 2^{4} 11 17477
7008584 2^{3} 11 73 1091
9152000  2^{9} 5^{3} 11 13
9152000  2^{9} 5^{3} 11 13
10039568 2^{4} 7 11 29 281
14720528 2^{4} 283 3251
33544832  2^{7} 262069
48928176 2^{4} 3^{2} 11 17 23 79
50207872 2^{7} 11 13^{2} 211
193352192 2^{9} 11^{2} 3121
205508336 2^{4} 11^{2} 101 1051
243383952 2^{4} 3 7 227 3191

Appendix B

B.1 The 5-modular character table of Co_{2}

;        @      @      @      @    @    @      @    @    @
        42    743     41      1   46   15      3   12    7
305421312000 178240 287680 474560 6560 5520 096576 2880 3728
   p power      A      A      A    A    A      A    B    B
   p' part      A      A      A    A    A      A    A    A
ind       1A     2A     2B     2C   3A   3B     4A   4B   4C
+        1      1      1      1    1    1      1    1    1
+       23     -9      7     -1   -4    5      7   -5    3
+      253     29     13    -11   10   10     29    9    1
+      275     51     35     11    5   14     19   15    7
+     1771    -21    -21     11  -11   16     91   -5   -5
+     2023    231    103     39   -2   25      7   23   23
+     2254   -210    126    -10   13   31     14  -30   10
+     4025   -231    105      1  -25   29    105  -35    5
+     5313    -63    -63     33   21    3     49    1    1
o     9625   -455    105    -15   40   -5     -7    5   29
o     9625   -455    105    -15   40   -5     -7    5   29
+    10395    315    -21    -45   27    0    -21   -9   15
+    12650    554    330     26  -40   59     -6   50    2
+    23000    600    280    120   50    5    184   40    8
+    29624   -168    392    -16   32   14    -56    4  -36
+    31625    265    -55    -55   35   35    377   25   -7
+    31625   1385    505    145   35   35     41   45   53
+    37169   1105    449    -55  -10   71     97   77    5
+    44275  -1869    595    -29   -5   94     35  -85   59
+    63250   -110    210   -110  -65  -20    322  -30    2

o    91125    405   -315     45    0    0    -27  -15    9
o    91125    405   -315     45    0    0    -27  -15    9
+   113575    903   -105    143   40  -23    119   -5    3
+   122199    567   -441    -33  -84   15    343   -5    3
+   173075   -973     35     43    5  -58    483   15    7
+   177100   2828    364    188  -20  -29    -84  -28   20
+   178388  -1932    420     76   53   44    -28  -48  -24
+   221375   4095    735     15 -160  -25    -49  -25   87
+   236004    -92   -188   -108   51  -30     20  -20   -4
+   239085  -2835   -147     45 -108    0   -147   45   45
+   253000   2120   -440    200 -125   10    104  -40   24
+   284625  -3855   1505    -55  -90   45    273 -115    5
+   354200  -4872    840    -32  -40  -58    -56   60   36
+   385825   3457   1105    233   -5  -23    -15    5  -19
+   442750   -770   1470   -130 -185   40   -210   30  -66
+   462000   5040    560   -400   30  120    112   80   16
+   664125  -1155    -35   -275  195  -30    637   85    5
+   664125  -1155    -35    365  195  -30    -35    5  -11
+   664125   2205   1645   -315  195  -30     77  -15  -39
+   853875   7155    435    -45  135    0   -237  -45   51
+  1044912   -720    496    -80  -15  -87   -160   16  -80
+  1288000  -2240  -2240    320  100  100    448    0    0
+  1771000 -12040   1400   -200  205  -20   -168   40   40
+  1771000   1400   -840    -40 -200  115   -168   40  -56
+  1992375   3255  -1225    215  180   45   -441  -25  -25
+  2004750   8910  -1170   -450    0    0   -162  -90   54
+  2095875  -3645  -2205    -45    0    0   -189   75   27

B.1. THE 5-MODULAR CHARACTER TABLE OF CO_{2} 115

@    @    @    @    @    @    @    @    @    @    @    @    @
4
9152 6144 6144 1280 5760 5184 4320 3456  576  288   56  768  768
A    B    B    C   AB   AA   BA   BA   BB   BC    A    A    C
A    A    A    A   AB   AA   BA   BA   BB   BC    A    A    A
4D   4E   4F   4G   6A   6B   6C   6D   6E   6F   7A   8A   8B
1    1    1    1    1    1    1    1    1    1    1    1    1
-1    3   -1   -1    4    0    3   -3    1   -1    2   -1   -3
5    1   -3    1   10    2    2    2   -2   -2    1   -3    3
-5    7    3   -1    5   -3    6    6    2    2    2    3    5
-5   -5    3   -1   21   -3   -6    0    0    2    0    3   -1
7    7    7   -1   -2    6    3    9    1    3    0   -1    3
6   10   -2    2   -3   -3    3   -9    3   -1    0   -2   -4
1    5    1    1   15    3    9   -3   -3    1    0    1   -5
17    1   -7   -3   -3    9   -3    3    3   -3    0    1    1
1   -3   -7    5    0    4   -5   -5    3    3    0    1    3
1   -3   -7    5    0    4   -5   -5    3    3    0    1    3
19   -1    3   -5    3   -9    0    0    0    0    0    3    1
10   18    2    6    0   -4   -1   11    3   -1    1    2    6
24    8    8    0   10    6   15   -3    1    3   -2    8    0
-16   12    0   -8    8   12  -18    6    2    2    0    0    2
-7   -7    1    5   35   -5   -5   -5   -1   -1   -1   -7    1
1    5   25    5   -5   -1    5   11    7    1   -1    1   -1
-23    5   -7   -7  -10   -2    7    7   -1   -1   -1    1    3
-13   11   -5   -1   -5    3    6  -18   -2   -2    0    3   -5
2    2   10  -10   15    7   10    4    0   -2   -2   -6    2

-51    9   -3    5    0    0    0    0    0    0   -1   -3    3
-51    9   -3    5    0    0    0    0    0    0   -1   -3    3
-17    3   -1    7    0   12    3  -15   -3   -1    0   -1   -7
-17    3    7    3  -12    0   -9   -9    3    3    0   -1   -3
-21    7  -21    3    5  -19    2   14    2   -2    0    3    5
-4  -12   -4   12    4  -16   -1  -13   -5   -1    0   -4   -8
12  -24   12    4   -3  -15   12   12  -12    4    0    4    2
-1   -9    7   -5    0    0  -15   15    3   -3    0   -1   -9
36   12   12   -8   -5    7    4   10   -2    0   -1   -4    8
-19   -3   -3    5   12    0    0    0    0    0    0   -3   -3
40    8  -24    0   -5   23  -10    2   -2    2   -1    0    4
-7    5    1    5  -10    6   15   -3    5   -1   -2   -7    3
32  -12  -16   -8    0  -12  -12    6    6    4    0    0   10
 9  -19   25   -7   -5   19    7  -23    1   -1   -1    1   -5
46   14    6   10   15   -5  -20   -8    0   -4    0    6    2
48  -16  -16    0  -10   18    0    0   -4   -4    0    0    0
13    5   -3    5   -5   15    0   -6   -2    4    0   -3   -7
45    5   -3    5   -5   15    0   -6   -2   -4    0   -3    5
37   -7  -11    5   -5   -9    0   18   10    0    0   -3   -5
-77  -13    3   -5   15   27    0    0    0    0    1    3   -1
-32    0   -8   -4    1    9    9    9    1    1    1    8    4
-64    0    0    0  -20   -8  -20    4    4   -4    0    0    0
-40   -8   24    0    5   -7  -10   -4   -4   -2    0    0    4
-8  -24    8    0    0   -4    5  -13    3    5    0   -8    0
39   -9  -17   -5   20  -12   15   -3    5   -1    0   -1    3
-18    6    6   10    0    0    0    0    0    0   -1    6    6
51   27    3   -5    0    0    0    0    0    0   -2    3   -9

@    @    @    @    @    @    @    @    @    @    @    @    @
512  512  256   64   54   11  864  288  288  288   96   96   48
D    D    C    E    A    A   BA   AC   DA   EC   BD   EB   AE
A    A    A    A    A    A   AA   AC   BA   BC   AD   BB   AE
8C   8D   8E   8F   9A  11A  12A  12B  12C  12D  12E  12F  12G
1    1    1    1    1    1    1    1    1    1    1    1    1
3   -1    1    1    2    1   -2    0    1   -3    2    1    0
5    1   -1   -1    1    0    2   -2    2    4    2    0   -2
3   -1    1    1    2    0    1    1   -2    4    1    0    1
7   -1   -1   -1   -2    0    1    1    4   -2    1   -2    1
-1   -1    3   -1   -2   -1   -2    2    1   -1   -2   -1   -2
-2    2    0    0   -2   -1    5    1   -1    1   -3   -3    1
5    1   -1   -1    2   -1   -3   -1   -3   -7    1    1   -1
-3    5    1    1    0    0   -5    1   -5    1   -1    1    1
1   -3   -1   -1    1    0    2   -4   -1   -1   -2   -1    0
1   -3   -1   -1    1    0    2   -4   -1   -1   -2   -1    0
-1    3   -3    1    0    0   -3    3    0    0    1    0   -1
-2   -2   -2    2   -1    0   -6   -4    3   -1   -2   -1    0
0    0    0    0    2   -1    4    2    1    5    0    1    2
-4    0   -2    2    2    1   -2    0   -2    0    2    4    0
5   -3    1    1   -1    0   -1   -1   -1    5   -1    1   -1
1    5    3   -1   -1    0    5   -1   -1   -1    1    3   -1
-3    1   -1   -1   -1    0   -2    2   -5   -1   -2   -1    2
-1   -1    3   -1   -2    0   -1   -1    2    2   -1    2   -1
-2    6    2    2    1    0    7   -1    4   -4   -1    0   -1

1    5   -1   -1    0    1    0    0    0    0    0    0    0
1    5   -1   -1    0    1    0    0    0    0    0    0    0
-5    7   -3   -3   -2    0    2    0    5   -3   -2    1    0
-9   -5    1    1    0    0   10    0   -5   -3   -2    1    0
-9   -5    1    1   -1    1   -3    1    6    4   -3    0    1
4   -4    0    0    1    0    6   -4    3   -1    2   -1    0
8   -4   -2   -2    2    1   -1   -3    8    0    3    0   -3
3    3   -1   -1    2    0   -4    0   -1    3   -4   -1    0
0    0    0   -4   -3   -1   -7   -1    2    2   -3   -2    3
-3   -3   -3    1    0    0    6    0    0    0    2    0    0
0    0    4    0    1    0    5    3    2    0    1   -4   -1
1   -3   -1   -1    0    0   -6    2   -3   -1    2   -1    2
4    0   -2    2    2    0   -2    0   -2    0    2    0    0
5    1   -1   -1    1    0    3   -1   -3   -1    3   -1   -1
-6    2    2   -2    1    0   -3    3    0    0    1    0   -1
0    0    0    0    0    0    4   -2    4   -2    0    2    2
-3    5    1    1    0    0  -11   -1   -2    2    1   -2   -1
5   -3   -3    1    0    0    1    7   -2   -2   -3    2   -1
1   -3   -1   -1    0    0    5    3    2    0    1    0   -1
-5   -5   -1   -1    0    0   -3    3    0    0    1    0   -1
12    4    4    0    0    0   -7    1    5    1    1    1   -3
0    0    0    0    1   -1   -2    0    4    0    2    0    0
0    0    4    0    1    0    3    1    0   -2   -1   -2    1
0    0    0    0    1    0   -6    4    3    1   -2    1    0
-5    3    3   -1    0    0    0   -4   -3   -1    0   -1    0
2    2   -2    2    0    0    0    0    0    0    0    0    0
3   -5   -1   -1    0    1    0    0    0    0    0    0    0

@    @    @    @    @    @    @    @    @    @    @    @
48   56   28   28   32   32   18   23   23   24   24   28
EF   AA   AB   AB    D    C   AB    A    A   CA   BB   AA
BF   AA   AB   AB    A    A   AA    A    A   BA   AB   AA
12H  14A  14B  C**  16A  16B  18A  23A  B**  24A  24B  28A
1    1    1    1    1    1    1    1    1    1    1    1
-1   -2    0    0    1   -1    0    0    0   -1    0    0
0    1   -1   -1    1   -1   -1    0    0    0    0    1
0    2    0    0   -1    1    0   -1   -1    0   -1   -2
0    0    0    0    1    1    0    0    0    0   -1    0
1    0   -2   -2   -1   -1    0   -1   -1   -1    0    0
1    0    0    0   -2    0    0    0    0    1   -1    0
1    0    0    0   -1    1    0    0    0    1    1    0
-1    0    0    0   -1   -1    0    0    0    1    1    0
-1    0    0    0    1   -1    1  b23   **    1    0    0
-1    0    0    0    1   -1    1   **  b23    1    0    0
0    0    0    0    1   -1    0   -1   -1    0    1    0
-1    1    1    1    0    0   -1    0    0   -1    0    1
-1   -2    0    0    0    0    0    0    0   -1    0    2
0    0    0    0    2    0    0    0    0    0    2    0
1   -1    1    1   -1   -1    1    0    0   -1    1   -1
1   -1    1    1    1   -1   -1    0    0    1   -1   -1
-1   -1    1    1   -1    1    1    1    1    1    0   -1
-2    0    0    0    1    1    0    0    0    0    1    0
-2    2    0    0    0    0    1    0    0    0   -1    0

0   -1   i7  -i7   -1    1    0   -1   -1    0    0    1
0   -1  -i7   i7   -1    1    0   -1   -1    0    0    1
-1    0    0    0    1   -1    0    1    1   -1    2    0
1    0    0    0   -1    1    0    0    0   -1    0    0
0    0    0    0    1   -1   -1    0    0    0   -1    0
-1    0    0    0    0    0   -1    0    0   -1   -2    0
0    0    0    0    2    0    0    0    0   -2   -1    0
1    0    0    0    1    1    0    0    0   -1    0    0
0   -1    1    1   -2    2    1    1    1    2   -1   -1
0    0    0    0    1    1    0    0    0    0    0    0
0   -1    1    1    0    0   -1    0    0    0    1   -1
1    2    0    0    1   -1    0    0    0   -1    0    0
2    0    0    0   -2    0    0    0    0    0   -2    0
1   -1   -1   -1   -1    1    1    0    0    1    1   -1
0    0    0    0    0    0    1    0    0    0   -1    0
2    0    0    0    0    0    0   -1   -1    0    0    0
0    0    0    0    1    1    0    0    0    0   -1    0
0    0    0    0    1    1    0    0    0    0   -1    0
-2    0    0    0   -1    1    0    0    0    0    1    0
0    1    1    1   -1   -1    0    0    0    0   -1    1
1    1   -1   -1    2   -2    0   -1   -1   -1    1    1
0    0    0    0    0    0    1    0    0    0    0    0
0    0    0    0    0    0   -1    0    0    0    1    0
-1    0    0    0    0    0   -1    0    0    1    0    0
1    0    0    0    1    1    0    0    0   -1    0    0
0   -1   -1   -1    0    0    0    1    1    0    0   -1
0    2    0    0   -1   -1    0    0    0    0    0    0

Bibliography

[1] J. L. Alperin, Local Representation Theory, Cambridge studies in advanced mathematics 11, Cambridge University Press, Cambridge 1986.

[2] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469–514.

[3] D. J. Benson, Modular Representation Theory: New Trends and Methods, Lecture Notes in Mathematics vol. 1081, Springer-Verlag, 1984.

[4] D. J. Benson, Brauer trees for 12M_{22}, J. Algebra 95 (1985), 398– 408.

[5] D. J. Benson, Some remarks on the decomposition numbers for the symmetric groups, in: The Arcata Conference on Representations of Finite Groups, Proc. Symp. Pure Math. 47, Amer. Math. Soc., 1987.

[6] D. J. Benson, Spin modules for symmetric groups, J. London Math. Soc. 38 (1988), 250–262.

[7] D. J. Benson, Projective modules for the group of twenty-seven lines on a cubic surface, Comm. Algebra 17 (1989), 1017–1068.

[8] D. J. Benson, Representations and Cohomology, I, II, Cambridge studies in advanced mathematics 30, 31, Cambridge University Press, Cambridge 1991.

[9] D. J. Benson and J. F. Carlson, Nilpotent elements in the Green ring, J. Algebra 104 (1986), 329–350.

[10] C. Bessenrodt, The Auslander–Reiten quiver of a modular group algebra revisited, Math. Z. 206 (1991), 25–34.

[11] R. Brauer, Collected Papers, ed. P. Fong and W. J. Wong, The MIT Press, Cambridge, Mass., 1980.

[12] R. Brauer and Nesbitt, On the modular characters of groups, Ann. of Math. 42 (1941), 556–590.

[13] M. Broué and L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117–128.

[14] R. E. Burkard Methoden der ganzzahligen Optimierung, Springer-Verlag, Berlin 1972.

[15] R. Burkhardt, Die Zerlegungsmatrizen der P SL(2, p^{f}), J. Algebra 40 (1976), 75–96.

[16] R. Burkhardt, Über die Zerlegungszahlen der Suzukigruppen, J. Algebra 59 (1979), 421–433.

[17] R. Burkhardt, Über die Zerlegungszahlen der unitären Gruppen P SU(3, 2^{f}), J. Algebra 61 (1979), 548–581.

[18] L. Collatz, W. Wetterling, Optimization Problems, Springer-Verlag, Berlin 1975.

[19] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, London, 1985.

[20] C. W. Curtis and I. Reiner, Methods of Representation Theory, I, II, Wiley, New York, 1981, 1987.

[21] D. I. Deriziotis and G. O. Michler, Character tables and blocks of finite simple triality groups ^{3}D_{4}(q), Trans. Amer. Math. Soc. 303 (1987), 39–70.

[22] R. Dipper, On the decomposition numbers of the finite general linear groups, Trans. Amer. Math. Soc. 290 (1985), 315–343.

[23] R. Dipper, On the decomposition numbers of the finite general linear groups II, Trans. Amer. Math. Soc. 292 (1985), 123–133.

[24] R. Dipper and G. James, Identification of the irreducible modular representations of GL_{n}(q), J. Algebra 104 (1986), 266–288.

[25] J. D. Dixon, Exact solution of linear equations using P -adic expansions, Numer. Math. 40 (1982), 137–141.

[26] J. Du, The Cartan invariants of SL(4, 2^{n}), J. East China Norm. Univ. Natur. Sci. Ed. 4 (1986), 17–25.

[27] K. Erdmann, Algebras and semidihedral defect groups I, Proc. London Math. Soc. 57 (1988), 109–150.

[28] K. Erdmann, Algebras and semidihedral defect groups II, Proc. London Math. Soc. 60 (1990), 123–165.

[29] K. Erdmann, Blocks of Tame Representation Type and Related Algebras, Lecture Notes in Mathematics vol. 1428, Springer-Verlag, Berlin, 1990.

[30] W. Feit, The Representation Theory of Finite Groups, North-Holland Publishing Company, New York, 1982.

[31] W. Feit, Possible Brauer trees, Illinois J. Math. 28 (1984), 43–56.

[32] W. Feit, Blocks with cyclic defect groups for some sporadic groups, in: Representation Theory II, Groups and Orders, Lecture Notes in Mathematics vol. 1187, Springer-Verlag, Berlin, 1986, pp. 25–63.

[33] B. Fischer, Character tables of maximal subgroups of sporadic groups, unpublished.

[34] B. Fischer, Clifford-matrices, in: G. O. Michler and C. M. Ringel (ed.), Representation Theory of Finite Groups and Finite-Dimensional Algebras, Birkhäuser, Basel, 1991, pp. 1–16.

[35] P. Fong, Splitting fields of finite groups, Illinois J. Math. 7 (1963), 515–520.

[36] P. Fong, On decomposition numbers of J_{1} and R(q), in: Symposia Mathematica Vol. XIII, Academic Press, London, 1974, pp. 415– 22.

[37] P. Fong and B. Srinivasan, Blocks with cyclic defect groups in GL(n, q), Bull. Amer. Math. Soc. 3 (1980), 1041–1044.

[38] P. Fong and B. Srinivasan, The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982), 109–153.

[39] P. Fong and B. Srinivasan, Brauer trees in GL(n, q), Math. Z. 187 (1984), 81–88.

[40] P. Fong and B. Srinivasan, The blocks of finite classical groups, J. reine angew. Math. 396 (1989), 122–191.

[41] P. Fong and B. Srinivasan, Brauer trees in classical groups, J. Algebra 131 (1990), 179–225.

[42] G. Frobenius, Gesammelte Abhandlungen, ed. J. P. Serre, Springer-Verlag, Berlin, 1968.

[43] M. Geck, Irreducible Brauer characters of the 3-dimensional special unitary groups in non-defining characteristic, Comm. Algebra 18 (1990), 563–584.

[44] M. Geck, On the decomposition numbers in the finite unitary groups in non-defining characteristic, Math. Z. 207 (1991), 83–89.

[45] M. Geck, Generalized Gelfand–Graev characters for Steinberg’s triality groups and their applications, Comm. Algebra 19 (1991), 3249–3269.

[46] M. Geck and G. Hiss, Basic sets of Brauer characters of finite groups of Lie type, J. reine angew. Math. 418 (1991), 173–188.

[47] D. M. Goldschmidt, Lectures on Character Theory, Mathematics Lecture Series 8, Publish or Perish, Berkeley, 1980.

[48] R. E. Gomory An All-Integer Integer Programming Algorithm, in: J. F. Muth, G. L. Thompson (ed.), Industrial Scheduling, Prentice-Hall, 1963.

[49] J. Grabmeier, Unzerlegbare Moduln mit trivialer Youngquelle und Darstellungstheorie der Schuralgebra, Bayreuther Math. Schriften 20 (1985), 9–152.

[50] G. Hiss, The modular characters of the Tits simple group and its automorphism group, Comm. Algebra 14 (1986), 125–154.

[51] G. Hiss, On the decomposition numbers of G_{2}(q), J. Algebra 120 (1989), 339–360.

[52] G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik, Habilitationsschrift, Aachen, 1990.

[53] G. Hiss, The Brauer trees of the Ree groups, Comm. Algebra 19 (1991), 871–888.

[54] G. Hiss, The 3-modular characters of the Rudvalis group and its covering group, Math. Comput., to appear.

[55] G. Hiss and K. Lux, The Brauer characters of the Hall–Janko group, Comm. Algebra 16 (1988), 357–398.

[56] G. Hiss and K. Lux, Brauer Trees of Sporadic Groups, Oxford University Press, Oxford, 1989.

[57] G. Hiss and K. Lux, appendix by T. Breuer, The 5-modular characters of the sporadic simple Fischer groups F i_{22} and F i_{23}, Comm. Algebra, to appear.

[58] G. Hiss, K. Lux and J. Müller, The 2-modular decomposition matrices of the non-principal blocks of maximal defect of the triple cover of the sporadic simple McLaughlin group, preprint (1992).

[59] G. Hiss, K. Lux and R. Parker, The 5-modular characters of the McLaughlin group and its covering group, Manuscripta Math. 73 (1991), 91–114.

[60] G. Hiss and J. Shamash, 3-blocks and 3-modular characters of G_{2}(q), J. Algebra 131 (1990), 371–387.

[61] G. Hiss and J. Shamash, 2-blocks and 2-modular characters of G_{2}(q), Math. Comput. 59 (1992), 645–672.

[62] G. Hiss and A. Szczepański, On torsion free crystallographic groups, J. Pure Appl. Algebra 74 (1991), 39–56.

[63] G. Hiss and D. L. White, The 5-modular characters of the covering group of the sporadic simple Fischer group F i_{22} and its automorphism group, Comm. Algebra, to appear.

[64] J. E. Humphreys, Modular representations of finite groups of Lie type, in: Finite Simple Groups II, Academic Press, New York, 1980, pp. 259–290.

[65] J. F. Humphreys, The projective characters of the Mathieu group M_{12} and of its automorphism group, Math. Proc. Camb. Phil. Soc. 87 (1980), 401–12.

[66] J. F. Humphreys, The modular characters of the Higman–Sims group, Proc. Roy. Soc. Edinburgh 92A (1982), 319–35.

[67] J. F. Humphreys, Blocks of projective representations of the symmetric groups, J. London Math. Soc. 33 (1986), 441–452.

[68] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.

[69] B. Huppert and N. Blackburn, Finite Groups II, Springer-Verlag, Berlin, 1982.

[70] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976.

[71] G. D. James, The modular characters of the Mathieu groups, J. Algebra 27 (1973), 57–111.

[72] G. D. James, Representations of the symmetric groups over the field of order 2, J. Algebra 38 (1976), 280–308.

[73] G. D. James, On the decomposition matrices of the symmetric groups. I, II, J. Algebra 43 (1976), 42–44, 45–54.

[74] G. D. James, A note on the decomposition matrices of 𝔖_{12} and 𝔖_{13} for the prime 3, J. Algebra 53 (1978), 410–411.

[75] G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics vol. 682, Springer-Verlag, Berlin, 1978.

[76] G. D. James, The decomposition of tensors over fields of prime characteristic, Math. Z. 172 (1980), 161–178.

[77] G. D. James, Representations of General Linear Groups, London Mathematical Society Lecture Notes Series vol. 94, Cambridge University Press, Cambridge, 1984.

[78] G. D. James, The irreducible representations of the finite general linear groups, Proc. London Math. Soc. 52 (1986), 236–268.

[79] G. D. James, The decomposition numbers of GL_{n}(q) for n ≤ 10, Proc. London Math. Soc. 60 (1990), 225–265.

[80] G. D. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. 16 (1981).

[81] Z. Janko, A new finite simple group with abelian Sylow 2subgroup and its characterization, J. Algebra 3 (1966), 147–186.

[82] C. Jansen, K. Lux, R. A. Parker and R. A. Wilson, An Atlas of Modular Character Tables, Cambridge University Press, to appear.

[83] N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), 373–395.

[84] A. Kerber, Representations of Permutation Groups, I, II, Lecture Notes in Mathematics vols. 240, 495, Springer-Verlag, Berlin, 1971, 1975.

[85] F. Khosraviyani, Decomposition numbers of exceptional Weyl groups, I, J. Algebra 91 (1984), 390–409.

[86] F. Khosraviyani and A. O. Morris, Decomposition numbers of exceptional Weyl groups, II, J. Algebra 92 (1985), 525–531.

[87] P. B. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Series vol. 129, Cambridge University Press, Cambridge, 1990.

[88] D. E. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading, 1981.

[89] A. S. Kondrat’ev, Decomposition numbers of the group J_{2}, Algebra and Logic 27 (1988), 333–349.

[90] A. S. Kondrat’ev, Decomposition numbers of the groups Ĵ_{2} and Aut(J_{2}), Algebra and Logic 27 (1988), 429–444.

[91] B. Külshammer, Lectures on Block Theory, London Mathematical Society Lecture Note Series vol. 161, Cambridge University Press, Cambridge, 1991.

[92] P. Landrock, The non-principal 2-blocks of sporadic simple groups, Comm. Algebra 6 (1978), 1865–1891.

[93] P. Landrock, Finite Group Algebras and their Modules, London Mathematical Society Lecture Note Series vol. 84, Cambridge University Press, Cambridge, 1983.

[94] P. Landrock and G. O. Michler, Principal 2-blocks of the simple groups of Ree type, Trans. Amer. Math. Soc. 260 (1980), 83–111.

[95] H. W. Lenstra, Private Communication (1991).

[96] G. Lusztig, Some problems in the representation theory of finite Chevalley groups, in: Proc. Symp. Pure Math. Vol. 37, Amer. Math. Soc., Providence, 1980, pp. 313–317.

[97] G. Lusztig, Characters of Reductive Groups over a Finite Field, Ann. Math. Studies 107, Princeton University Press, 1984.

[98] K. Lux, J. Müller and M. Ringe, Peakword condensation and submodule lattices, preprint (1992).

[99] K. Lux and H. Pahlings, Computational aspects of representation theory of finite groups, in: G. O. Michler and C. M. Ringel (ed.), Representation Theory of Finite Groups and Finite-Dimensional Algebras, Birkhäuser, Basel, 1991, pp. 37–64.

[100] R. P. Martineau, On representations of the Suzuki groups over fields of odd characteristics, J. London Math. Soc. 6 (1972), 153– 160.

[101] B. H. Matzat, Der Kenntnisstand in der konstruktiven Galoistheorie, in: G. O. Michler and C. M. Ringel (ed.), Representation Theory of Finite Groups and Finite-Dimensional Algebras, Birkhäuser, Basel, 1991, pp. 65–98.

[102] G. O. Michler, Ring Theoretical and Computational Methods in Group Representation Theory, Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, Essen, 1989.

[103] G. O. Michler, Contributions to modular representation theory of finite groups, in: G. O. Michler and C. M. Ringel (ed.), Representation Theory of Finite Groups and Finite-Dimensional Algebras, Birkhäuser, Basel, 1991, pp. 99–140.

[104] J. Neubüser, H. Pahlings, and W. Plesken, CAS; Design and use of a system for the handling of characters of finite groups, in: Computational group theory, Academic Press, London, 1984, pp. 195–247.

[105] J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992.

[106] R. A. Parker, The computer calculation of modular characters (the Meat-Axe), in: Computational Group Theory, Academic Press, London, 1984, pp. 267–274.

[107] R. A. Parker and R. W. Wilson, The computer construction of matrix representations of finite groups over finite fields, J. Symbol. Comput. 9 (1990), 583–590.

[108] L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77–116.

[109] M. A. Ronan and S. D. Smith, Computation of 2-modular sheaves and representations for L_{4}(2), A_{7}, 3S_{6} and M_{24}, Comm. Algebra 17 (1989), 1199–1237.

[110] A. J. E. Ryba, Computer condensation of modular representations, J. Symbol. Comput. 9 (1990), 591–600.

[111] M. Schönert et al., GAP — Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, first ed., 1992.

[112] I. Schur, Gesammelte Abhandlungen, ed. A. Brauer und H. Rohrbach, Springer-Verlag, Berlin, 1973.

[113] J. Scopes, Cartan matrices and Morita equivalence classes for blocks of the symmetric groups, J. Algebra 142 (1991), 441–455.

[114] J. P. Serre, Linear Representations of Finite Groups, Springer-Verlag, New-York, 1977.

[115] J. Shamash, Blocks and Brauer trees for groups of type G_{2}(q), in: The Arcata Conference on Representations of Finite Groups, 1987, pp. 283–295.

[116] J. Shamash, Brauer trees for blocks of cyclic defect in the groups G_{2}(q) for primes dividing q^{2}±q+1, J. Algebra 123 (1989), 378–396.

[117] J. Shamash, Blocks and Brauer trees in the groups G_{2}(q) for primes dividing q ± 1, Comm. Algebra 17 (1989), 1901–1949.

[118] J. Shamash, Blocks and Brauer trees for the groups G_{2}(2^{k}), G_{2}(3^{k}), Comm. Algebra 20 (1992), 1375–1387.

[119] I. A. Suleiman, Modular Representations of Finite Simple Groups, PhD-thesis, Birmingham, 1990.

[120] I. A. Suleiman and R. A. Wilson, Computer construction of matrix representations of the covering group of the Higman-Sims group, J. Algebra 148 (1992), 219–224.

[121] I. A. Suleiman and R. A. Wilson, The 3- and 5-modular characters of the covering and the automorphism group of the Higman Sims group, J. Algebra 148 (1992), 225–242.

[122] I. A. Suleiman and R. A. Wilson, The 3- and 5-modular characters of McLaughlin’s group and its automorphism group, Durham Conference Proceedings, 1990, to appear.

[123] J. G. Thackray, Modular Representations of Some Finite Groups, PhD-thesis, Cambridge, 1981.

[124] P. J. Webb, The Auslander–Reiten quiver of a finite group, Math. Z. 179 (1982), 97–121.

[125] D. L. White, The 2-decomposition numbers of Sp(4, q), q odd, J. Algebra 131 (1990), 703–725.

[126] D. L. White, Decomposition numbers of Sp(4, q) for primes dividing q ± 1, J. Algebra 132 (1990), 488–500.

[127] D. L. White, Brauer trees of Sp(4, q), Comm. Algebra 20 (1992), 645–653.

[128] D. L. White, Brauer trees of 2.F_{4}(2), Comm. Algebra 20 (1992), 3353–3368.

[129] D. L. White, Brauer trees of 2.Aut(F_{4}(2)), preprint (1993).

[130] R. A. Wilson, The 2- and 3-modular characters of J_{3}, its covering group and automorphism group J. Symbol. Comput. 10 (1990), 647–656.

[131] R. A. Wilson, The Brauer tree for J_{3} in characteristic 17, J. Symbol. Comput., to appear.

[132] A. J. Woldar, On the 5-decomposition matrix for McLaughlin’s sporadic simple group, Comm. Algebra 14 (1986), 277–291.

[133] J. C. Ye, On the first Cartan invariant of the groups SL_{3}(p^{n}) and SU_{3}(p^{n}), in: Group Theory, Beijing 1984, Lecture Notes in Mathematics vol. 1185, Springer-Verlag, Berlin, 1986, pp. 388–

Index

atom 27 Brauer— 27 projective— 27 system of atoms 27

basic set of Brauer characters 24 of projective characters 24 special— 36

block 14

strongly real— 90 Brauer graph 14

character bit of a— 65 Brauer— 10 generalized— 11 part of a— 56 projective— 12 projective indecomposable—

12 subsum of a— 57 symmetrized— 20 virtual— 11

class function 11

decomposition matrix 11 wedge shape of— 39

essential subset 34 lattice 91

maximal multiplicity 65 multiplicity free 65

orthogonality relations 13

p′-element 10

p-modular system 10 p-regular element 10 p-singular element 10

relation 35 with respect to a basis 35

Weyl module 20 Gerhard Hiss IWR Universität Heidelberg Im Neuenheimer Feld 368 69120 Heidelberg, Germany

Christoph Jansen Lehrstuhl D für Mathematik RWTH Aachen Templergraben 64 52062 Aachen, Germany

Klaus Lux Lehrstuhl D für Mathematik RWTH Aachen Templergraben 64 52062 Aachen, Germany

Richard Parker Perihelion Software Ltd. Shepton Mallet BA4 4PP England