The Representation Theory of Some Groups With Blocks of Defect Group C_{3} × C_{3} in

Characteristic Three

Katrina Hicks Queen’s College

Oxford

A thesis submitted for the degree of Doctor of Philosophy Summer 1994

Abstract

The Representation Theory of Some Groups With Blocks of Defect Group C_{3} × C_{3} in Characteristic Three

Katrina Hicks Queen’s College Oxford

A thesis submitted for the degree of Doctor of Philosophy Summer 1994

This thesis explores the representation theory of both the 3-dimensional projective special unitary groups over q^{2} elements, with q ≡ 2 mod 3 and q ≢ 8 mod 9, and the sporadic simple group Higman–Sims, over an algebraically closed field of characteristic three.

Firstly the Loewy structure and socle structure of the PIMs of the principal block for the 3-dimensional projective special unitary groups described above are completely determined by making use of information about the automorphisms of these groups and applying Auslander–Reiten theory. From this information we are then able to determine a 2-parameter family of algebras to which the basic algebras corresponding to these blocks must belong.

The rest of the thesis deals with the representation theory of the Higman–Sims group, which has two blocks of defect two. By applying induction and restriction techniques to the Higman–Sims group and a subgroup which is the semi-direct product of a projective special unitary group with C_{2}, the Loewy structure and socle structure of the PIMs of these two blocks are completely determined. Using this information we are able to determine a 1-parameter family of algebras to which the basic algebra of the principal block must belong and a 1-parameter family of algebras to which the basic algebra of the non-principal block of defect two must belong.

Acknowledgements

I would like to thank what seems to be half of the algebra department for supervising me. Firstly I need to thank Martin Powell who got me going in this strange new world. Secondly I must thank Dave Benson for throwing me headlong into the world of Representation Theory (I wish he hadn’t aimed at that brick wall though) and for bringing these problems to my attention. I particularly need to thank him for his very timely reappearance, and all the help he has given me in the last panic stricken phase of thesis production. I also need to thank Karin Erdmann for her help and support (and tolerance) and endless notes in my pigeon-hole. Without her guidance over the last year or so this thesis would never have come together.

I would like to thank the graduate students for their support and encouragement. In particular I would like to thank John for being a fellow devotee of science fiction and for all those absolutely necessary 9am post-program analyses, I would like to thank Catherine for being Australian and understanding all those little comments that the British just didn’t get, and I would also like to thank Jane, Martyn, Jonathan, Steven, Stephen and all the other tea-goers for the general discussions and abuse sessions that went on in our offices and in the common room.

Very special thanks to Mike.

I acknowledge the financial support of the Commonwealth Scholarship and Fellowship Plan.

Contents

1 Background 6 1.1 Notation and Standard Theorems . . . . . . . . . . . . . . . . . . 6 1.2 Induction and Restriction . . . . . . . . . . . . . . . . . . . . . . 10 1.3 The functor Ext^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Basic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Quivers and Relations . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.8 Auslander–Reiten Theory . . . . . . . . . . . . . . . . . . . . . . 35

2 The Representation Theory of the Principal Block of U_{3}(q) With Defect Group C_{3} × C_{3} 41 2.1 The Loewy Structure of the PIMs of the Principal Block of U_{3}(q) 42

2.1.1 The Characters of U_{3}(q) . . . . . . . . . . . . . . . . . . . 43 2.1.2 The Loewy Structure of the PIMs . . . . . . . . . . . . . . 44

2.2 The Structure of the Basic Algebra of the Principal Block of U_{3}(q) 53 2.2.1 Finding Generators for the Relations Induced by I …. 54 2.2.2 Determining the Ideal I . . . . . . . . . . . . . . . . . . . 59

3 The Structure of the PIMs in the Non-Principal Blocks of the Higman–Sims Group 68 3.1 The Cartan Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 The PIMs of U_{3}(5) : 2 . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Induction and Restriction Between U_{3}(5) : 2 and HS . . . . . . . 75 3.4 The PIMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 The Structure of the PIMs in the Principal Block of the Higman–

Sims Group 89 4.1 Induction and Restriction Between U_{3}(5) : 2 and HS . . . . . . . 90 4.2 Values of dim_{K}Ext^{1} for the Principal Block . . . . . . . . . . . . . 95 4.3 The PIMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 The Structure of the Basic Algebra of the Second Block of the Higman–Sims Group 116 5.1 Finding Generators for the Relations Induced by I . . . . . . . . . 117 5.2 Determining the Ideal I . . . . . . . . . . . . . . . . . . . . . . . 127

6 The Structure of the Basic Algebra of the Principal Block of the Higman–Sims Group 134 6.1 Finding Generators for the Relations Induced by I . . . . . . . . . 135 6.2 Determining the Ideal I . . . . . . . . . . . . . . . . . . . . . . . 150

A Conjugacy Classes of SU_{3}(q^{2}), Ordinary Character Table for SU_{3}(q^{2}) and Decomposition Matrix for SU_{3}(q^{2}) 161

B Character Tables, Decomposition Matrix, and Cartan Matrix for U_{3}(5) 167

C Character Tables and Decomposition Matrix for U_{3}(5) : 2 170

D Character Tables and Decomposition Matrix for HS 175

E Results on Induction and Restriction Between U_{3}(5) : 2 and HS 181 E.1 The PIMs of U_{3}(5) : 2 . . . . . . . . . . . . . . . . . . . . . . . . . 181 E.2 Calculating the Composition Factors of a Character Using the

Package PARI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 E.3 Structures of the Simple HS-Modules Restricted to U_{3}(5) : 2 … 183 E.4 Structures of Some Simple U_{3}(5) : 2 Modules Induced to HS … 184

F Relations for the Basic Algebra of the Principal Block of HS 186

Introduction

If G is a finite group and K is an algebraically closed field of characteristic zero or characteristic not dividing the order of G, then by Maschke’s theorem, all KG-modules are semisimple. Hence any module is a direct sum of its composition factors and the composition series of a module completely determines its structure. When the characteristic of K does divide the order of G however, the problem of determining the structure of a module is not so straightforward. In this case we can have modules with the same composition factors which are not isomorphic, for we no longer have a specific way of “gluing” the composition factors together. This is the situation in which we will be interested throughout this thesis.

Suppose G is a finite group and K is an algebraically closed field of characteristic dividing the order of G. As a module, KG has, up to isomorphism, a unique decomposition into a direct sum of indecomposable KG-modules, these are the principal indecomposable modules, or PIMs. KG also has a unique decomposition into a direct sum of indecomposable algebras called blocks, and each block, as a module over itself, is a direct sum of certain related PIMs of KG. We therefore study the representation theory of the algebra KG by studying the representation theory of each of the blocks of KG.

It is well known that any block of KG is either of finite, tame or wild representation type, and that the representation type of the block B depends on the structure of the defect group of B. The structure of blocks of finite representation type (cyclic defect group) are well understood, and can be completely described by Brauer trees. A certain amount of progress has also been made with blocks of tame representation type, see for example the five papers by Erdmann listed in the References. Very little can be stated about the general structure of blocks of wild representation type however, and it is examples of this type which we deal with in this thesis.

The first task is to understand something about the structure of the PIMs. To find the composition factors of one of the PIMs it is necessary to set up a system for relating modules for G in characteristic zero to modules for G in characteristic p, i.e., a splitting p-modular system. Given a splitting p-modular system (F, R, k), we can generalise the idea of ordinary characters in characteristic zero to Brauer characters in characteristic p, complex-valued functions defined on the p-regular elements of G. The decomposition matrix D is determined by taking a full set of F -inequivalent irreducible integral representations and finding the decomposition of the corresponding modular representations in terms of composition factors over k. In practice this amounts to restricting each ordinary irreducible character to the p-regular conjugacy classes and finding the unique decomposition into Brauer characters. The relation C = D^{T} D then allows us to determine the Cartan matrix, the matrix which gives the composition factors of the PIMs.

Once we have the composition factors, we consider the structure of the PIMs by finding a decomposition for each into semisimple layers from the top down (the Loewy layers) and from the bottom up (the socle layers). Such decompositions for various group algebras abound in the literature, see for example Benson (1983a) and (1983b) for a discussion of A_{8} and A_{9} in characteristic two, Martin (1993) for a discussion of S_{2p} in characteristic p, and Schneider (1993) for a discussion of M_{12} in characteristic two and three. A large part of this thesis will be concerned with finding these decompositions for blocks of certain group algebras in characteristic three. We then reduce each algebra B to the case where all simple modules are 1-dimensional, the basic algebra to which B is Morita equivalent, and try to determine the overall structure of this basic algebra in terms of a quiver with relations by using the information about the Loewy and socle structure of the PIMs of B. This is the content of the rest of the thesis.

In Chapter 1 we cover the background material which is necessary for understanding the content of the thesis. In Chapter 2 we completely determine the structure of the PIMs of the principal block for the class of 3-dimensional projective special unitary groups over q^{2} elements (q a prime power) with q ≡ 2 mod 3 and q ≢ 8 mod 9. The Cartan matrix can be obtained using information about the decomposition numbers for the corresponding special unitary group, calculated by Geck (1990). We then use information about the automorphisms of the projective special unitary groups and apply Auslander–Reiten theory, to find the Loewy structure and socle structure of the PIMs. From this information we are then able to determine a 2-parameter family of algebras to which the basic algebras corresponding to these blocks must belong.

In Chapters 3 and 4 we use induction and restriction techniques to find the structure of the PIMs for the sporadic simple group Higman–Sims. We first calculate the decomposition matrix and Cartan matrix from the ordinary character table and 3-modular Brauer character table and determine the structure of the blocks of defect zero and defect one. We then induce and restrict characters between the Higman–Sims group and a subgroup which is the semidirect product of a projective special unitary group with C_{2}, obtaining the composition factors of these characters by using the linear system solving aspect of the package PARI. Using the structure of the PIMs for the subgroup, the structure of the restricted modules and the structure of some of the induced modules, we are able to determine completely the structure of the PIMs of the two remaining blocks of the Higman–Sims group.

In Chapters 5 and 6 we use information about the PIMs to obtain the structure of the basic algebra for the two blocks of defect two in terms of a quiver and relations. For each of these blocks we are able to determine a 1-parameter family of algebras to which the corresponding basic algebra must belong.

Appendix A gives a summary of the information needed to calculate the composition factors of the PIMs for the projective special unitary groups of Chapter 2. In Appendices B, C and D, we list relevant character tables, decomposition matrices and Cartan matrices for solving the Higman–Sims group problem. In Appendix E we give a list of all modular structures determined along the way to finding the main results for the Higman–Sims group, as well as a brief summary of how the composition factors of the induced and restricted characters were determined. Appendix F gives a full list of the generating relations found when determining the structure of the basic algebra for the principal block of the Higman–Sims group.

Chapter 1

Background

In what follows we let K denote an algebraically closed field of prime characteristic p and we let A denote a finite dimensional K-algebra with unity. We will often be concerned with the particular case of the group algebra of a finite group G with order divisible by p, which will be denoted KG. All modules are assumed to be left A-modules which are finite dimensional over K. All maps will be written on the left.

1.1 Notation and Standard Theorems

For details see Alperin (1986) and Landrock (1983).

• Rad(M) denotes the radical of M.

• Soc(M) denotes the socle of M.

• The radical series is defined by setting Rad^{0}(M) = M and defining Rad^{i}(M) recursively for i ≥ 1 by setting Rad^{i}(M) = Rad(Rad^{i−1}(M)). The smallest value of i for which Rad^{i}(M) = 0 is called the Loewy length of M and will be denoted by ℓ(M).

• The socle series is defined by setting Soc^{0}(M) = 0, Soc^{1}(M) = Soc(M) and defining Soc^{i}(M) recursively for i > 1 by choosing Soc^{i}(M) to be the module satisfying

\frac{Soc^{i}(M)}{Soc^{i−1}(M)} ≅ Soc (\frac{M}{Soc^{i−1}(M)}) .

The smallest value of i for which Soc^{i}(M) = M is called the socle length of M. It is equal to the Loewy length.

• For 1 ≤ i ≤ ℓ(M), let L_{i}(M) = Rad^{i−1}(M)/Rad^{i}(M) denote the ith Loewy layer of M. The first Loewy layer is called the head of M and is often denoted by Hd(M).

• S_{i}(M) = Soc^{i}(M)/Soc^{i−1}(M) denotes the ith socle layer of M for 1 ≤ i ≤ ℓ(M).

• If we say M has Loewy structure

\begin{matrix}S_{11} & S_{12} & … & S_{1r_{1}} \\ S_{21} & S_{22} & … & S_{2r_{2}} \\ ⋮ & ⋮ & ... & ⋮ \\ S_{ℓ(M)1} & S_{ℓ(M)2} & … & S_{ℓ(M)r_{ℓ(M)}} \end{matrix}

with the S_{ij} simple, this means that L_{i}(M) ≅ S_{i1} ⊕ S_{i2} ⊕ … ⊕ S_{ir_{i}}.

• If we say M has socle structure

\begin{matrix}T_{ℓ(M)1} & T_{ℓ(M)2} & … & T_{ℓ(M)k_{ℓ(M)}} \\ ⋮ & ⋮ & ... & ⋮ \\ T_{21} & T_{22} & … & T_{2k_{2}} \\ T_{11} & T_{12} & … & T_{1k_{1}} \end{matrix}

with the T_{ij} simple, this means that S_{i}(M) ≅ T_{i1} ⊕ T_{i2} ⊕ … ⊕ T_{ik_{i}}.

• If we say M has structure

\begin{matrix}X_{11} & X_{12} & … & X_{1t_{1}} \\ X_{21} & X_{22} & … & X_{2t_{2}} \\ ⋮ & ⋮ & ... & ⋮ \\ X_{n1} & X_{n2} & … & X_{nt_{n}} \end{matrix} ⊕ \begin{matrix}Y_{11} & Y_{12} & … & Y_{1s_{1}} \\ Y_{21} & Y_{22} & … & Y_{2s_{2}} \\ ⋮ & ⋮ & ... & ⋮ \\ Y_{m1} & Y_{m2} & … & Y_{ms_{m}} \end{matrix}

this means that M is a direct sum of a module N with Loewy structure L_{i}(N) ≅ X_{i1}⊕X_{i2}⊕…⊕X_{it_{i}} and socle structure S_{i}(N) ≅ X_{(n−i)1}⊕X_{(n−i)2}⊕ … ⊕ X_{(n−i)t_{n−i}} and a module N′ with Loewy structure L_{i}(N′) ≅ Y_{i1} ⊕ Y_{i2} ⊕ … ⊕ Y_{is_{i}} and socle structure S_{i}(N′) ≅ Y_{(m−i)1} ⊕ Y_{(m−i)2} ⊕ … ⊕ Y_{(m−i)s_{m−i}}.

• For any module M we call Rad(M)/Rad(M) ∩ Soc(M) the heart of M and denote it by Ht(M).

• We call an algebra A symmetric if there is a K-linear map λ : A ⟶ K such that

(i) λ(ab) = λ(ba) for all a, b ∈ A. (ii) ker(λ) contains no non-zero left ideals of A. We call λ the symmetrising form for A. Note that KG is symmetric with symmetrising form λ(∑ a_{g}g) = a_{1}.

• When A = KG, we use M^{∗} to denote the contragredient dual of M, that is, the vector space dual of M with action given by gφ(m) = φ(g^{−1}m), m ∈ M, φ ∈ M^{∗}, and extended linearly. If M ≅ M^{∗} then M is said to be self-dual.

• (M, N) denotes the K-dimension of the vector space Hom_{A}(M, N).

• E (M, N) denotes the K-dimension of the vector space Ext_{A}^{1}(M, N).

• _{A}mod denotes the category of finitely generated A-modules and their homomorphisms.

• We denote the projective cover of M by P_{M} and the injective hull of M by I_{M}.

• Simple modules will frequently be denoted by their dimension. If there is more than one simple module of a given dimension we will use subscripts A, B, … to distinguish them.

We now list some general theorems which will be used later.

Lemma 1.1.1 (see Landrock (1983) Lemma 8.2) Suppose M has Loewy length n. Then Rad^{i}(M) ≤ Soc^{n−i}(M) for all i ≥ 0.

Lemma 1.1.2 (see Landrock (1983) Lemma 8.4) The socle structure of M^{∗} is the dual of the Loewy structure of M, i.e.,

S_{i}(M^{∗}) ≅ L_{i}(M)^{∗}

for all i ≤ ℓ(M) = ℓ(M^{∗}). In particular, if M is self dual and each of the Loewy layers of M is self dual, then S_{i}(M) ≅ L_{i}(M).

In the case where a module M is self dual we see that if S is a composition factor of one of the bottom k Loewy layers of M, then S is a composition factor of one of the first k socle layers of M by Lemma 1.1.1 and therefore by Lemma 1.1.2, S^{∗} is a composition factor of one of the first k Loewy layers of M. Hence for any value of k, the dual of a simple module contained in one of the bottom k Loewy layers of M must also be contained in one of the top k Loewy layers of M. We will use this fact frequently.

Lemma 1.1.3 (Landrock (1980) Lemmas 1 and 2)

(i) Let N be a submodule of M. Then the socle series of N is obtained by intersecting N with the socle series of M.

(ii) M contains a simple module T in S_{i}(M) if and only if M has a submodule of Loewy length i with simple head T.

Corollary 1.1.4 (Landrock (1980) Corollary 1) Let P be a PIM of a block B and T a simple factor of the (ℓ(P) − 1)th socle layer of P. Then

ℓ(P) ≤ ℓ(P_{T}).

Lemma 1.1.5 (see Landrock (1983) Lemmas 8.6) Assume A is a symmetric algebra and let M be an A-module.

(i) Let N be a submodule of M. Then the Loewy series of M/N is the homomorphic image of the Loewy series of M by the canonical map η : M ⟶ M/N.

(ii) M contains a simple module T in L_{i}(M) if and only if M has a factor module of Loewy length i with simple socle T.

We have the following theorem for group algebras.

Theorem 1.1.6 (Webb (1982) Theorem E) Let H_{G} denote the heart of the KGmodule P_{1} and let P be a Sylow p-subgroup of G. If P is not a dihedral 2-group then H_{G} is indecomposable.

1.2 Induction and Restriction

For a discussion of the concepts involved see Alperin (1986) and Curtis and Reiner (1962). In this section we take A to be a group algebra. Let G be a finite group with subgroup H. For any KG-module M we define the restricted KH-module, denoted by M↓_{H}, by restricting the action of G on M to H. Moreover, if we have a KH-module N, then we form the KG-module

N↑^{G} = KG ⊗_{KH} N
≅ ⨁_{g_{i}∈G/H}(g_{i} ⊗ N)

(where G/H denotes a transversal for H in G). The action of g ∈ G is given by

g(g_{i} ⊗ n) = gg_{i} ⊗ n = g_{k}h ⊗ n = g_{k} ⊗ hn

and extended linearly. When we use induction and restriction it will generally be clear which groups we are inducing and restricting between, and therefore we will often drop the subscript H and denote the restricted module by M↓, and drop the superscript G, denoting the induced module by N↑.

We now present some standard results relating to induction and restriction.

Lemma 1.2.1 (see Alperin(1986) Lemma 8.5 parts (2) and (3), and Curtis and Reiner (1981) 32-33)

(i) ((M ⊕ N)↑^{G} ≅ M↑^{G} ⊕ N↑^{G}.

(ii) (M↑^{G})^{∗} ≅ (M^{∗})↑^{G}.

(iii) Induction preserves exact sequences.

Theorem 1.2.2 (Frobenius reciprocity) (see Benson (1991) Prop. 3.3.1) Let M be a KG-module and N a KH-module. Then

(i) Hom_{KH}(N, M↓_{H}) ≅ Hom_{KG}(N ↑^{G}, M).

(ii) Hom_{KH}(M↓_{H}, N) ≅ Hom_{KG}(M, N ↑^{G}).

Theorem 1.2.3 (see Alperin (1986) Theorem 9.2) If H is a subgroup of G containing a Sylow p-subgroup of G then every KG-module is relatively H-projective.

This is equivalent to saying that every surjective KG homomorphism which is split as a KH homomorphism is also split as a KG homomorphism. Hence if

0 ⟶ U ⟶ X ⟶ V ⟶ 0

is a short exact sequence of KG-modules which is split as a short exact sequence of KH-modules, then it is split.

Let e_{1} be a central idempotent of KG and set B_{1} = KGe_{1} and suppose B_{2} = KHe_{2} for some central idempotent e_{2} of KH. Notice that we have defined B_{1} and B_{2} in a manner which allows each of them to be a direct sum of blocks rather than an individual block. This will enable us to deal with situations in which the composition factors of an induced or restricted simple module do not all belong to the same block. We will say that a module belongs to B_{i} (i = 1, 2) if it belongs to one of the block summands of B_{i}. For the rest of this section assume every simple KG-module belonging to B_{1} is semisimple when restricted down to H and every simple module belonging to B_{2} is semisimple when induced up to G. The first three results are all due to Benson (1983a).

For all n we will identify (Rad^{n}(KH))↑ with a subring of KG as follows:

(Rad^{n}(KH))↑ = KG ⊗_{KH} (Rad^{n}(KH)) ≤ KG ⊗_{KH} KH ≅ KG.

Lemma 1.2.4 (Benson (1983a) Lemma 4.1.1)

e_{1}.(Rad(KH))↑ ≤ Rad(KG).

Proof. Let M be a simple KH-module and N a simple KG-module lying in B_{1}

((\frac{P_{M}}{Rad(P_{M})})↑ , N) = (M↑ , N)
= (M, N↓)
= (P_{M}, N↓)
= (P_{M}↑ , N)

(notice that it is the semisimplicity of N↓ which gives the third equality). As this is true for all simple KG-modules N lying in B_{1} and all simple KH-modules M, we have

e_{1}.L_{1} ((\frac{P}{Rad(P)})↑) ≅ e_{1}.L_{1}(P ↑)

1.2 Induction and Restriction 12

for all PIMs P of KH. Taking the direct sum over all direct summands of KH and applying parts (i) and (iii) of Lemma 1.2.1 we have

e_{1}.L_{1} ( \frac{KH↑}{(Rad(KH))↑}) ≅ e_{1}.L_{1}(KH↑)

i.e.

e_{1}.L_{1} ( \frac{KG}{(Rad(KH))↑}) ≅ e_{1}.L_{1}(KG).

Hence

e_{1}.(Rad(KH))↑ ≤ e_{1}.(Rad(KG)) ≤ Rad(KG)

as claimed. □ Lemma 1.2.5 (Benson (1983a) Lemma 4.1.2)

e_{1}.(Rad^{n}(KH))↑ ≤ Rad^{n}(KG)

for all n ≥ 0. Proof. Follows from Lemma 1.2.4 by induction. □ Corollary 1.2.6 For any KH-module M

e_{1}.(Rad^{n}(M))↑ ≤ e_{1}.Rad^{n}(M↑)

for all n ≥ 0. Proof.

e_{1}.(Rad^{n}(M))↑ = e_{1}.((Rad^{n}(KH)).M)↑
= e_{1}.(Rad^{n}(KH))↑ . M↑
≤ e_{1}.(Rad^{n}(KG)). M↑
= e_{1}.Rad^{n}(M↑)

for all values of n, as claimed. □ Lemma 1.2.7 For any KH-module M

Rad^{n} ((e_{2}.M)↑) ≤ (Rad^{n}(e_{2}.M))↑

for all n ≥ 0.

Proof. By part (iii) of Lemma 1.2.1 we have

\frac{(e_{2}.M)↑}{(Rad(e_{2}.M))↑} ≅ (\frac{e_{2}.M}{Rad(e_{2}.M)})↑

which is semisimple, as all simple modules in B_{2} are semisimple when induced up to G. Hence Rad((e_{2}.M)↑) ≤ (Rad(e_{2}.M))↑. Suppose now that Rad^{n}((e_{2}.M)↑) ≤ (Rad^{n}(e_{2}.M))↑. Then

\begin{matrix}(Rad^{n}(e_{2}.M)) & ↑ \\ (Rad^{n+1}(e_{2}.M)) & ↑ \end{matrix} ≅ (\frac{Rad^{n}(e_{2}.M)}{Rad^{n+1}(e_{2}.M)})↑

which is semisimple and hence

Rad((Rad^{n}(e_{2}.M))↑) ≤ (Rad^{n+1}(e_{2}.M))↑ .

Then, by the inductive hypothesis,

Rad(Rad^{n}((e_{2}.M)↑)) ≤ (Rad^{n+1}(e_{2}.M))↑ ,

i.e.

Rad^{n+1}((e_{2}.M)↑) ≤ (Rad^{n+1}(e_{2}.M))↑

and the result is proved. □

Corollary 1.2.8 For any KH-module M

e_{1}.Rad^{n}((e_{2}.M)↑) = e_{1}.(Rad^{n}(e_{2}.M))↑

for all n ≥ 0.

Proof. By Lemma 1.2.7

e_{1}.Rad^{n}((e_{2}.M)↑) ≤ e_{1}.(Rad^{n}(e_{2}.M))↑

and then by Corollary 1.2.6

e_{1}.(Rad^{n}(e_{2}.M))↑ ≤ e_{1}.Rad^{n}((e_{2}.M)↑),

giving the desired equality. □

We have therefore proved the following theorem:

Theorem 1.2.9 Let e_{1} be a central idempotent in KG and e_{2} be a central idempotent in KH. Suppose every simple KG-module belonging to the two-sided ideal B_{1} = KGe_{1} is a semisimple module belonging to the two-sided ideal B_{2} = KHe_{2} when restricted down to H, and every simple KH-module belonging to B_{2} is a semisimple module belonging to B_{1} when induced up to G. Then for any KHmodule M of B_{2}

L_{i}(M↑) = L_{i}(M)↑

for 1 ≤ i ≤ ℓ(M).

1.3 The functor Ext^{n}

For details see Benson (1991) and Hilton and Stammbach (1971). Let M and N be A-modules and let

… ⟶ P_{n} ⟶^{α_{n}} P_{n−1} ⟶ … ⟶ P_{1} ⟶^{α_{1}} P_{0}

be a projective resolution of M, (i.e., an exact sequence with the P_{i} projective and P_{0}/im α_{1} ≅ M). Applying the functor Hom_{A}(−, N) induces a (not necessarily exact) sequence

Hom_{A}(P_{0}, N) ⟶^{δ_{0}} Hom_{A}(P_{1}, N) ⟶^{δ_{1}} …

where δ_{i} is given by f ↦ fα_{i+1}. We then define

Ext_{A}^{n}(M, N) = \{ \begin{matrix}ker δ_{n}/im δ_{n−1} & if n > 0 \\ ker δ_{0} & if n = 0 \end{matrix}

and this is well defined in the sense that any other projective resolution gives rise to an isomorphic vector space for each value of n. Note that Ext_{A}^{0}(M, N) ≅ Hom_{A}(M, N). Also for all values of n > 0, Ext_{A}^{n}(P, N) = 0 for any projective module P and Ext_{A}^{n}(M, I) = 0 for any injective module I.

Lemma 1.3.1 For A-modules N, M, L

(i) Ext_{A}^{n}(N ⊕ M, L) ≅ Ext_{A}^{n}(N, L) ⊕ Ext_{A}^{n}(M, L).

(ii) Ext_{A}^{n}(N, M ⊕ L) ≅ Ext_{A}^{n}(N, M) ⊕ Ext_{A}^{n}(N, L).

1.3 The functor Ext^{n} 15

When A = KG we also have (iii) Ext_{A}^{n}(M^{∗}, N^{∗}) ≅ Ext_{A}^{n}(N, M).

If B is a subalgebra of A, M an A-module and N a B-module, then we have notions of M↓_{B} and N↑^{A} = A ⊗_{B} N which extend the notions of restricted and induced modules for group algebras. We also have a co-induced module N⇑^{A} = Hom_{B}(A, N). Note that if A = KG and B = KH then KG ≅ (KG)^{∗} so that N↑^{G} = KG ⊗_{KH} N ≅ (KG)^{∗} ⊗_{KH} N ≅ Hom_{KH}(KG, N) = N⇑^{G}, i.e., co-induction coincides with induction.

Lemma 1.3.2 (see Benson (1991) Corollary 2.8.4) Suppose B is a subalgebra of A and that A is projective as a B-module. If M is an A-module and N is a B-module, then

(i) Ext_{B}^{n}(N, M↓_{B}) ≅ Ext_{A}^{n}(N↑^{A} , M).

(ii) Ext_{B}^{n}(M↓_{B} , N) ≅ Ext_{A}^{n}(M, N⇑^{A}).

Let φ : P_{M} ⟶ M be an epimorphism of the projective cover P_{M} of M onto M. We define Ω(M) to be the kernel of φ, i.e. the module such that the sequence

0 ⟶ Ω(M) ⟶ P_{M} ⟶^{φ} M ⟶ 0

is exact.

Lemma 1.3.3 (Schanuel) (see Benson (1991) Lemma 1.5.3) Suppose

0 ⟶ M_{1} ⟶ P_{1} ⟶ M ⟶ 0

and

0 ⟶ M_{2} ⟶ P_{2} ⟶ M ⟶ 0

are short exact sequences of modules with the P_{i} projective. Then M_{1} ⊕ P_{2} ≅ M_{2} ⊕ P_{1}.

As a consequence of this lemma we see that given any other epimorphism ψ : P_{M} ⟶ M, ker ψ is isomorphic to Ω(M) and therefore Ω(M) is well defined. We also note that if P is any other projective module and ψ : P ⟶ M is an epimorphism, then ker ψ ≅ Ω(M) ⊕ P′ for some projective direct summand P′ of P. We call Ω the Heller operator.

Given a map f : M ⟶ N, Ω induces a map Ω(f) : Ω(M) ⟶ Ω(N) by lifting as in the following diagram:

0 ⟶ Ω(M) ⟶ P_{M} ⟶ M ⟶ 0
Ω(f) ↓ ↓ f ↓
0 ⟶ Ω(N) ⟶ P_{N} ⟶ N ⟶ 0.

This gives us a well defined map modulo maps that factor through a projective module. We define _{A}\underline{mod} to be the category in which the objects are finitely generated A-modules and two maps of _{A}mod are regarded as being the same in A\underline{mod} if their difference factors through a projective module. Ω is then a well defined functor on this category.

Similarly we can define Ω^{−1}(M) to be the cokernel of the map φ : M ⟶ I_{M} (I_{M} the injective hull of M), i.e. the module such that the sequence

0 ⟶ M ⟶^{φ} I_{M} ⟶ Ω^{−1}(M) ⟶ 0

is exact. If we define _{A}\overline{mod} to be the category in which the objects are finitely generated A-modules and two maps of _{A}mod are regarded as being the same in A\overline{mod} if their difference factors through an injective module, then Ω^{−1} is a well defined functor on _{A}\overline{mod}. In particular, when A is an algebra for which injective modules and projective modules coincide (such as a symmetric algebra), Ω and Ω^{−1} are both functors on _{A}\underline{mod} and they are inverse to each other.

Now take

… ⟶ P_{n} ⟶^{α_{n}} P_{n−1} ⟶ … ⟶ P_{1} ⟶^{α_{1}} P_{0} ⟶^{α_{0}} M ⟶ 0

to be the exact sequence in which P_{0} is the projective cover of M and for each n ≥ 1, P_{n} is the projective cover of ker α_{n−1}. We then define Ω^{n}(M) to be the kernel of α_{n−1} (which is equivalent to making the recursive definition Ω^{n}(M) = Ω(Ω^{n−1}(M))). If we take another projective resolution of M so that we have an exact sequence

… ⟶ P′_{n} ⟶^{β_{n}} P′_{n−1} ⟶ … ⟶ P′_{1} ⟶^{β_{1}} P′_{0} ⟶^{β_{0}} M ⟶ 0

then by Schanuel’s lemma ker β_{n−1} ≅ Ω^{n}(M) ⊕ P for some projective module P. In a similar way we can define Ω^{−n}(M) using injective resolutions so that for any exact sequence

0 ⟶ M ⟶^{β_{0}} I_{0} ⟶^{β_{1}} I_{1} ⟶ … ⟶ I_{n} ⟶^{β_{n+1}} I_{n+1} ⟶ …

with the I_{n} injective, coker β_{n−1} = Ω^{−n}(M) ⊕ I for some injective module I.

If A is a symmetric algebra, we call an A-module M Ω-periodic (or just periodic) if there is an exact sequence

0 ⟶ M ⟶ P_{n−1} ⟶ … ⟶ P_{1} ⟶ P_{0} ⟶ M ⟶ 0

with the P_{i} projective. This is equivalent to saying there exists n > 0 and a projective module P with M ≅ Ω^{n}(M) ⊕ P. We have the following useful theorem for determining when a module over a group algebra is not periodic:

Theorem 1.3.4 (Alperin (1977) Theorem 3) If G has p-rank greater than one, then every periodic KG-module has dimension divisible by p.

Returning to the case of an arbitrary finite dimensional K-algebra we now make use of some of the homological properties of Ext^{n}.

Theorem 1.3.5 (Benson (1991) Proposition 2.5.2) Suppose

0 ⟶ L ⟶^{f} X ⟶^{g} M ⟶ 0

is a short exact sequence of A-modules. Then for any A-module N there is a long exact sequence

0 → Hom_{A}(M, N) →^{g^{∗}} Hom_{A}(X, N) →^{f^{∗}} Hom_{A}(L, N) →^{∂} Ext_{A}^{1}(M, N) → …

… → Ext_{A}^{n}(M, N) →^{g^{∗}} Ext_{A}^{n}(X, N) →^{f^{∗}} Ext_{A}^{n}(L, N) →^{∂} Ext_{A}^{n+1}(M, N) → …

We have a short exact sequence

0 ⟶ Ω(M) ⟶^{f} P_{M} ⟶^{g} M ⟶ 0

and so by the above lemma for any A-module N we have a long exact sequence

0 ⟶ Hom_{A}(M, N) ⟶^{g^{∗}} Hom_{A}(P_{M}, N) ⟶^{f^{∗}} Hom_{A}(Ω(M), N) ⟶^{∂}

Ext_{A}^{1}(M, N) ⟶^{g^{∗}} Ext_{A}^{1}(P_{M}, N) ⟶ …

As Ext_{A}^{1}(P_{M}, N) = 0, the exactness of this sequence tells us that the map

Hom_{A}(Ω(M), N) ⟶^{∂} Ext_{A}^{1}(M, N)

is surjective and therefore

Ext_{A}^{1}(M, N) ≅ \frac{Hom_{A}(Ω(M), N)}{ker ∂ : Hom_{A}(Ω(M), N) ⟶ Ext_{A}^{1}(M, N)}
= \frac{Hom_{A}(Ω(M), N)}{im f^{∗} : Hom_{A}(P_{M}, N) ⟶ Hom_{A}(Ω(M), N)}

and therefore dim_{K}Ext_{A}^{1}(M, N) ≤ dim_{K}Hom_{A}(Ω(M), N).

For us the most important of these spaces Ext_{A}^{n}(M, N) will be the space Ext_{A}^{1}(M, N) which we will denote by Ext^{1}(M, N) when it is clear which algebra we are dealing with. We have the following result relating Ext^{1} directly to Loewy structures:

Lemma 1.3.6 (see Benson (1991) Proposition 2.4.3) Suppose P = Ae and P′ = Ae′ are PIMs for A with P/Rad(P) ≅ S and P′/Rad(P′) ≅ S′. Then

Ext^{1}(S, S′) ≅ Hom_{A}(L_{2}(P), S′).

By Lemma 1.1.5 this tells us that if Ext^{1}(S, S′) > 0 then there exists a module

M with structure \begin{matrix}S \\ S′ \end{matrix} i.e., a non-split short exact sequence

0 ⟶ S′ ⟶ M ⟶ S ⟶ 0.

Any short exact sequence of the form

0 ⟶ N ⟶ M ⟶ L ⟶ 0

is called an extension of N by L, we will usually just say M is an extension of N by L. Two extensions M and M′ will then be called equivalent if the corresponding short exact sequences are isomorphic. If the short exact sequence

0 ⟶ N ⟶ M ⟶ L ⟶ 0

is split, then M ≅ N ⊕ L is called the split extension. The vector space of equivalence classes of extensions of N by L (see Mac Lane (1963) pp 67-72 for a nice description of the additive structure) is isomorphic to Ext^{1}(L, N), giving us a second interpretation of Ext^{1}(L, N) which we will frequently make use of in our calculations. Note that for any modules N and L the split extension always exists and corresponds to the zero element of Ext^{1}(L, N).

Let M be a KG-module with r Loewy layers and let S be a composition factor of L_{i}(M). By Lemma 1.1.5 there is a factor module U of M of Loewy length i with simple socle S. As the Loewy structure of U is a homomorphic image of the Loewy structure of M, the (i − 1)th Loewy layer of U is isomorphic to a submodule of the (i − 1)th Loewy layer of M. As Rad^{i−2}(U) ≤ Soc^{2}(U) this means L_{i−1}(U) ≤ S_{2}(U). Hence, by Lemma 1.1.3, for any T_{j} ∈ L_{i−1}(U) there is a submodule of U of Loewy length 2 with simple head T_{j}. As this module must have structure

\begin{matrix}T_{j} \\ S \end{matrix}

with T_{j} ∈ L_{i−1}(M), this shows that for 1 < i ≤ r any composition factor of L_{i}(M) must be non-trivially extended by some composition factor of L_{i−1}(M). Similarly any composition factor of S_{i}(M), 1 < i ≤ r must non-trivially extend some composition factor of S_{i−1}(M). We will frequently use these facts when determining the structure of modules later on.

1.4 Filtrations

For a more general discussion of filtrations see Siegel (1989).

Suppose we have a module M and we know the Loewy structure of a submodule N of M and the Loewy structure of M/N. How can we make use of this information in determining the Loewy structure of M? If we know further that N is a submodule of Rad^{k}(M) for some k ≥ 0 then we represent this information by the following diagram:

L_{1}(M/N)
L_{2}(M/N)
⋮
L_{k}(M/N)
L_{k+1}(M/N) L_{1}(N)
⋮  L_{2}(N)
⋮

This gives a diagrammatic description of the filtration of M through N. By Lemma 1.1.5 we know that if S_{j} is a composition factor of L_{i}(M/N) then S_{j} is a composition factor of L_{i}(M) so that to determine the Loewy structure of M we need only determine which Loewy layer of M each of the composition factors of N lies in. We already know that the composition factors of N do not lie in L_{1}(M), L_{2}(M), …, L_{k}(M) because N ≤ Rad^{k}(M). We also have the following lemma:

Lemma 1.4.1 (Siegel (1989) Lemma 23) Suppose N ≤ M and L_{i}(M/Rad(N)) ≅ L_{i}(M/N) for i = 1, 2, …, k. Then N ≤ Rad^{k}(M).

Proof. As Rad(N) ≤ N ≤ M

\frac{M/Rad(N)}{N/Rad(N)} ≅ M/N

and therefore M/N is a homomorphic image of M/Rad(N). By Lemma 1.1.5 this means

\frac{M/N}{Rad^{i}(M/N)}

is a homomorphic image of

\frac{M/Rad(N)}{Rad^{i}(M/Rad(N))}

for all appropriate values of i. In particular we have L_{i}(M/Rad(N)) ≅ L_{i}(M/N) for 1 ≤ i ≤ k and therefore for these values of i

\frac{M/Rad(N)}{Rad^{i}(M/Rad(N))} ≅ \frac{M/N}{Rad^{i}(M/N)}

i.e.

\frac{M/Rad(N)}{(Rad^{i}(M) + Rad(N))/Rad(N)} ≅ \frac{M/N}{(Rad^{i}(M) + N)/N}

i.e.

\frac{M}{Rad^{i}(M) + Rad(N)} ≅ \frac{M}{Rad^{i}(M) + N}

so that for 1 ≤ i ≤ k

Rad^{i}(M) + Rad(N) = Rad^{i}(M) + N.  (1.1)

Now N ≤ Rad^{0}(M) = M, and if N ≤ Rad^{j}(M) for j < k, then Rad(N) ≤ Rad^{j+1}(M) for j + 1 ≤ k, so that by equation 1.1, N ≤ Rad^{j+1}(M). Hence by induction we have that for all i ≤ k,

N ≤ Rad^{i}(M).

In particular N ≤ Rad^{k}(M) as required. □

We now construct an example typical of how filtrations will be used later on in order to demonstrate the concepts involved.

Suppose we have simple KG-modules N and N′ and a KG-module M with Loewy structure

\begin{matrix}N′ \\ N & . \end{matrix}

If we consider the module M↓_{H}, then N↓_{H} ≤ M↓_{H} and M↓_{H} / N↓_{H} ≅ N′↓_{H}. Suppose N↓_{H} has Loewy structure

\begin{matrix}C \\ B \\ A \end{matrix}

and N′↓_{H} has Loewy structure

\begin{matrix}A \\ B \end{matrix}

for some simple KH-modules A, B and C. Then we have a filtration for M↓_{H} as follows:

\begin{aligned}A & C \\ B & B \\ & A & . \end{aligned}

If we know that C is not in the head of M and that A cannot extend C nontrivially, then C would have to be in L_{3}(M). By Lemma 1.4.1, this means that N↓_{H} ≤ Rad^{2}(M↓_{H}) and so we actually have filtration

A
B
C
B
A

which immediately forces M↓_{H} to have Loewy structure

A
B
C
B
A .

1.5 Basic Algebras

For a more general discussion of the material in this section see Benson (1991) and Erdmann (1990b) (note that the first author works with left modules while the second author works with right modules). Again we assume that all algebras are finite dimensional over an algebraically closed field K.

Knowing the Loewy and socle structure of the PIMs of an algebra A provides us with a certain degree of information about the structure of A, but to understand the structure of _{A}A we need to completely understand the relationships between the composition factors of A. For an arbitrary algebra such a task seems impossible; however, in this section we will show that to consider the modular structure of A, it suffices to consider the structure of an algebra for which the simple modules are all 1-dimensional. Such an algebra is in some sense structurally less complicated and relatively speaking we can infer a great deal more information about it from its Loewy structure and socle structure. With this in mind we make the following definition:

Definition 1.5.1 We call an algebra A a basic algebra if all the simple A-modules are 1-dimensional.

Let A be an arbitrary finite dimensional algebra over K. As an A-module, A has direct sum decomposition of the form

A = ⨁_{i j} P_{ij} = ⨁_{i j} Ae_{ij}

where the e_{ij} are primitive orthogonal idempotents and the PIMs P_{ij} and P_{kl} are isomorphic if and only if i = k. Now let P = ⨁_{i} P_{i1} and set A_{0} = End_{A}(P)^{op} (i.e. the ring of A-endomorphisms of P with reverse composition). Note that if we set e = ∑_{i} e_{i1} then P = Ae and A_{0} = End_{A}(P)^{op} ≅ eAe. We have:

Lemma 1.5.2 (see Erdmann (1990b) Lemma 1.2.5) Let P be a projective module. Then End_{A}(P)^{op} is basic if and only if P is a direct sum of non-isomorphic indecomposable A-modules.

Hence A_{0} is a basic algebra. The first result we will have need of is the following:

Lemma 1.5.3 (see Erdmann (1990b) Lemma 1.3.3) For an algebra A the following are equivalent:

(i) A is symmetric.

(ii) For any idempotent e of A, the algebra eAe is symmetric.

(iii) A_{0} is symmetric.

We will now show that the categories _{A}mod and _{A_{0}}mod are equivalent.

Definition 1.5.4 Two finite dimensional algebras A and B are said to be Morita equivalent if the categories _{A}mod and _{B}mod are equivalent.

Definition 1.5.5 Let P be a finitely generated projective A-module such that every A-module is a homomorphic image of a direct sum of copies of P. Then P is called a progenerator for _{A}mod.

If we write A ≅ ⨁_{i=1}^{r} n_{i}P_{i1}, then every projective module P ≅ ⨁_{i=1}^{r} m_{i}P_{i1} with m_{i} > 0 for all i is a progenerator for _{A}mod.

Theorem 1.5.6 (see Benson (1991) Theorem 2.2.6) The algebras A and B are Morita equivalent if and only if there is a progenerator P for _{A}mod such that B ≅ End_{A}(P)^{op}.

Hence A is Morita equivalent to A_{0} and so _{A}mod and _{A_{0}}mod are equivalent. In particular there is a one-to-one correspondence between the simple A-modules and the simple A_{0}-modules such that under this correspondence, the Loewy structure and socle structure of a module is preserved. So if we know the Loewy structure and socle structure of the PIMs of one of A and A_{0} then we know the Loewy structure and socle structure of the other.

Lemma 1.5.7 (see Erdmann (1990b) Lemma 1.2.6) Two basic algebras are Morita equivalent if and only if they are isomorphic.

This tells us that A_{0} is unique up to isomorphism and therefore every algebra is Morita equivalent to a unique basic algebra.

1.6 Quivers and Relations

We define a quiver to be a directed graph Q = (Q_{0}, Q_{1}, s, t) where Q_{0} is the set of vertices of Q, where Q_{1} is the set of arrows of Q, and where the maps s, t : Q_{1} ⟶ Q_{0} are defined by taking s(α) to be the vertex at which the arrow α starts, and t(α) to be the vertex at which α terminates. We express this as

t(α) ⟵^{α} s(α).

Note that this definition of Q allows loops and multiple edges.

A non-trivial path in Q is a sequence u = α_{n} … α_{1} where s(α_{i+1}) = t(α_{i}) (i.e. composition of paths is written on the left). This path starts at the vertex s(α_{1}) and terminates at the vertex t(α_{n}). We also define a trivial path for each vertex x denoted by e_{x}. This is the path of length 0 which starts and terminates at x. We then define the path algebra KQ to be the K-vector space with basis the set of paths of Q. Multiplication is given by composition of paths, so that if u and v are paths then

u.v = \{ \begin{matrix}uv & if t(v) = s(u) \\ 0 & otherwise. \end{matrix}

Note that each of the paths e_{x} is a primitive orthogonal idempotent and therefore when Q_{0} is finite

KQ = ⨁_{x∈Q_{0}} KQe_{x}

gives the direct sum decomposition of KQ into PIMs. The PIM P_{x} = KQe_{x} has basis the set of paths starting at x.

For each vertex x of Q we define a simple KQ-module S_{x} by taking the underlying vector space to be K with action given by

αk = 0 ∀α ∈ Q_{1}
e_{y}k = 0 ∀y ∈ Q_{0} with y ≠ x
e_{x}k = k

for all k ∈ K, and extended to elements of KQ in the natural way. These simple modules are distinct and if KQ is finite dimensional these are the only simple KQ-modules.

For each n ≥ 1 we define the two sided ideal KQ^{(n)} to be the span of the set of paths of length ≥ n. This gives us a series

KQ ≥ KQ^{(1)} ≥ KQ^{(2)} ≥ … ≥ KQ^{(n)} ≥ ….

We will often be interested in the algebra KQ/I where I is an ideal contained in KQ^{(2)}. This is equivalent to saying that we will often be interested in KQ subject to a set of relations in paths of length ≥ 2. As such we see that if we identify e_{x} + I with e_{x}, then {e_{x} : x ∈ Q_{0}} is still a set of primitive orthogonal idempotents for KQ/I and when KQ/I is finite dimensional,

KQ/I = ⨁_{x∈Q_{0}} (KQ/I) e_{x}

is the direct sum decomposition of KQ/I into PIMs. We also note that the distinct simple KQ-modules are distinct simple KQ/I-modules.

If KQ/I is finite dimensional then there is a one-to-one correspondence between the PIMs of KQ/I and the simple KQ/I-modules given by

(KQ/I) e_{x} ⟶ \frac{(KQ/I)e_{x}}{Rad((KQ/I)e_{x})}.

The arrows of Q generate the radical of KQ/I and therefore Rad ((KQ/I) e_{x}) is generated by the arrows of Q starting at x (where we identify α + I with α). Hence (KQ/I) e_{x}/Rad((KQ/I)e_{x}) is a 1-dimensional space isomorphic to S_{x} and the S_{x} are a full set of non-isomorphic simple modules for KQ/I. As they are all 1-dimensional, KQ/I is a basic algebra.

Definition 1.6.1 Suppose A is a finite dimensional algebra over an algebraically closed field K. Let S_{1}, …, S_{n} be representatives of the isomorphism classes of simple A-modules and let the projective cover of S_{i} be Ae_{i}. We define a quiver Q with vertices x_{1}, …, x_{n} corresponding to these simple modules and the number of arrows from x_{i} to x_{j} is dim_{K}(e_{j}Rad(A)e_{i}/e_{j}Rad^{2}(A)e_{i}). Note that dim_{K}Ext_{A}^{1}(S_{i}, S_{j}) = dim_{K}(e_{j}Rad(A)e_{i}/e_{j}Rad^{2}(A)e_{j}) (see Benson (1991) Proposition 2.4.3), so the number of arrows from x_{i} to x_{j} is the dimension of Ext_{A}^{1}(S_{i}, S_{j}). We call this quiver the Ext-quiver of A.

Theorem 1.6.2 (Gabriel) (See Benson (1991) Proposition 4.1.7) Any finite dimensional basic algebra over an algebraically closed field K is of the form KQ/I for a unique quiver Q and some ideal I contained in KQ^{(2)}.

Proof. Let A be a finite dimensional algebra over an algebraically closed field K and let Q be the Ext-quiver of A. We prove the theorem by defining a surjective map of algebras from the path algebra KQ onto A which has kernel contained in KQ^{(2)}.

Let A = ⨁ Ae_{i} be a direct sum decomposition of A into PIMs. For each primitive orthogonal idempotent e_{i} of A, there is a primitive orthogonal idempotent e_{x_{i}} of KQ defined to be the trivial path starting and terminating at the vertex x_{i} of the Ext-quiver (the vertex corresponding to the simple module Ae_{i}/Rad(Ae_{i})). We also know that for any vertices x_{i} and x_{j}, the number of arrows from x_{i} to x_{j} is equal to dim_{K}(e_{j}Rad(A)e_{i}/e_{j}Rad^{2}(A)e_{j}), and therefore there is a one-to-one correspondence between the arrows from x_{i} to x_{j} and the basis elements for a basis of the complement of e_{j}Rad^{2}(A)e_{i} in e_{j}Rad(A)e_{i}. We therefore choose a basis for each of these spaces and define a map which sends e_{i} to e_{x_{i}} for all i, and maps the arrows from x_{i} to x_{j} to the corresponding basis elements. By extending linearly, we have a map φ : KQ → A. Note that if the path α_{2}α_{1} = 0 in KQ for arrows α_{1} and α_{2}, then s(α_{2}) ≠ t(α_{1}), i.e., φ(α_{1}) ∈ e_{j}Rad(A)e_{i} and φ(α_{2}) ∈ e_{l}Rad(A)e_{k} with j ≠ k. Therefore φ(α_{2})φ(α_{1}) = e_{l}φ(α_{2})e_{k}e_{j}φ(α_{1})e_{i} = 0 and so φ is a well defined algebra homomorphism.

Given the definition of φ we have

A = Rad^{2}(A) + im (φ).

We now show that if B is a subring of A satisfying A = Rad^{2}(A)+B, then A = B. This will show that φ is surjective and complete the proof of the theorem.

Let B be a subring of A satisfying A = Rad^{2}(A)+B. If x ∈ Rad(A) (which is contained in A) then x = w_{x}+z_{x} with w_{x} in Rad^{2}(A) and z_{x} in B. In fact, as z_{x} = x − w_{x} ∈ Rad(A), we have z_{x} ∈ Rad(A) ∩ B and therefore Rad(A) = Rad^{2}(A) + (Rad(A) ∩ B). Now suppose that Rad^{n}(A) = Rad^{n+1}(A) + (Rad^{n}(A) ∩ B) and consider v ∈ Rad^{n+1}(A). As Rad^{n+1}(A) = Rad(A)Rad^{n}(A) we can express v as x.y with x ∈ Rad(A) and y ∈ Rad^{n}(A). Then x = w_{x}+z_{x} with w_{x} in Rad^{2}(A) and z_{x} in Rad(A) ∩ B and y = w_{y} + z_{y} with w_{y} in Rad^{n+1}(A) and z_{y} in Rad^{n}(A) ∩ B.

We then have

v = (w_{x} + z_{x})(w_{y} + z_{y})
= xw_{y} + w_{x}z_{y} + z_{x}z_{y}
∈ Rad(A)Rad^{n+2}(A) + Rad^{2}(A)(Rad^{n}(A) ∩ B)
+ (Rad(A) ∩ B)(Rad^{n}(A) ∩ B)
⊆ Rad^{n+2}(A) + (Rad^{n+1}(A) ∩ B).

Hence by induction we have shown that for all n ≥ 1,

Rad^{n}(A) = Rad^{n+1}(A) + (Rad^{n}(A) ∩ B).

In particular this means that for all n ≥ 2

Rad^{2}(A) + B = Rad^{n}(A) + B.

As Rad(A) is nilpotent we then have

A = Rad^{2}(A) + B = B

which completes the proof. □

The importance of this theorem is that if we can determine the ideal I for a given finite dimensional basic algebra A, then we have a full structural description of A in terms of the elements of KQ.

Let x_{j} ⟵^{α} x_{i} be an arrow of the Ext-quiver of a basic algebra A. Under the correspondence A ≅ KQ/I, α corresponds to a basis element for the complement of e_{j}Rad^{2}(A)e_{i} in e_{j}Rad(A)e_{i}. Given the isomorphism A ≅ End_{A}(A)^{op} (and e_{j}M ≅ Hom_{A}(Ae_{j}, M)) we can think of α as corresponding to a basis element for the complement of Hom_{A}(Ae_{j}, Rad^{2}(Ae_{i})) in Hom_{A}(Ae_{j}, Rad(Ae_{i})). Composition of paths then corresponds to reverse composition of homomorphisms. The significance of considering paths in this way is that if we understand the action of A on Ae_{j} and Ae_{i}, then an A-module homomorphism Ae_{j} ⟶ Rad(Ae_{i}) is completely determined by the image of e_{j} in Ae_{i}.

Series of extensions within Ae_{i} give us a small amount of information about the way A, acting on the basis element e_{i}, generates Ae_{i}. Let M be a non-zero extension of S_{j} by S_{i}. Then we can find a basis {y_{1}, y_{2}} for M such that Ay_{1} ≅ S_{j} and A(y_{2} + Ay_{1}) ≅ S_{i}. As S_{i} and S_{j} are 1-dimensional this means that y_{1} is a basis for the submodule isomorphic to S_{j} and y_{2} + Ay_{1} is a basis for the quotient space isomorphic to S_{i}. The fact that M is a non-trivial extension means that there is an element a ∈ A such that ay_{2} = λ_{1}y_{1} + λ_{2}y_{2} for some λ_{1}, λ_{2} ∈ K with λ_{1} ≠ 0.

If the composition factor S_{i} of the first Loewy layer of Ae_{i} non-trivially extends a composition factor isomorphic to S_{j} in the second Loewy layer, then any homomorphism in the complement of Hom_{A}(Ae_{j}, Rad^{2}(Ae_{i})) in Hom_{A}(Ae_{j}, Rad(Ae_{i})) must be non-zero. Similarly if we have S_{i_{1}} in L_{1}(P_{S_{i_{1}}}) non-trivially extending S_{i_{2}} in L_{2}(P_{S_{i_{1}}}) non-trivially extending S_{i_{3}} in L_{3}(P_{S_{i_{1}}}) … non-trivially extending S_{i_{r−1}} in L_{r−1}(P_{S_{i_{1}}}) non-trivially extending S_{i_{r}} in L_{r}(P_{S_{i_{1}}}), then a homomorphism

Ae_{i_{r}} ⟶^{f_{1}} Rad(Ae_{i_{r−1}}) ⟶^{f_{2}} … ⟶^{f_{r−1}} Rad^{r−1}(Ae_{i_{1}}) with f_{j} in the complement of

Hom_{A}(Ae_{i_{r+1−j}}, Rad^{2}(Ae_{i_{r−j}})) in Hom_{A}(Ae_{i_{r+1−j}}, Rad(Ae_{i_{r−j}})), can be non-zero. In particular, if Ext^{1}(S_{k}, S_{j}) ≤ 1 for all simple modules of A, then any such homomorphism must be non-zero.

Now suppose we have S_{i_{1}} in L_{1}(P_{S_{i_{1}}}) non-trivially extending S_{i_{2}} in L_{2}(P_{S_{i_{1}}}) non-trivially extending S_{i_{3}} in L_{3}(P_{S_{i_{1}}}) … non-trivially extending S_{i_{r−1}} in L_{r−1}(P_{S_{i_{1}}}) and there is no composition factor isomorphic to S_{i_{r}} in L_{r}(P_{S_{i_{1}}}). Then a homomorphism Ae_{i_{r}} ⟶^{f_{1}} Rad(Ae_{i_{r−1}}) ⟶^{f_{2}} … ⟶^{f_{r−1}} Rad^{r−1}(Ae_{i_{1}}) as above is in Hom_{A}(Ae_{i_{r}}, Rad^{r}(Ae_{i_{1}})) ≅ e_{i_{r}}Rad^{r}(A)e_{i_{1}}. As e_{i_{r}}Rad^{r}(A)e_{i_{1}} is spanned by elements which are a product of at least r of the basis elements, the first being in e_{i_{j}}Rad(A)e_{i_{1}} for some j and the last being in e_{i_{r}}Rad^{r}(A)e_{i_{k}} for some k, the element of e_{i_{r}}Rad^{r}(A)e_{i_{1}} corresponding to the homomorphism is a linear combination of these elements.

Hence we see that by considering the arrows as corresponding to homomorphisms, a relation on series of extensions beginning with S_{i_{1}} and ending with S_{i_{r}} gives rise to a relation on the basis elements corresponding to the arrows, which holds modulo e_{i_{r}}Rad^{r}(A)e_{i_{1}}. Under the correspondence A ≅ KQ/I we see that we then have a relation on the corresponding paths which holds modulo larger paths from x_{i_{1}} to x_{i_{r}}. Therefore we can derive relations on paths defining the basic algebra by considering relationships on non-trivial extensions and then generalising these to include terms in larger paths between appropriate vertices. Note that if all PIMs have r Loewy layers then all paths of length r are zero and all relations involving paths of length r − 1 which hold modulo paths of length greater than r − 1, are relations on KQ that define the basic algebra.

Let A be an arbitrary finite dimensional algebra over an algebraically closed field K. We will now outline the procedure we will use to find generators for the relations that define the basic algebra A_{0}. We will identify the PIMs of A_{0} with the PIMs of A and the simples of A_{0} with the simples of A. The Loewy structure and socle structure of the PIMs of A then gives the Loewy structure and socle structure of the PIMs of A_{0}.

We find the relations by dealing with one PIM at a time, the relations we determine from P_{S_{i}} giving us the relations that start at the vertex corresponding to S_{i} in the Ext-quiver. We begin by determining the extension structure of P_{S_{i}}, a diagram indicating the possible non-trivial extensions between the composition factors of P_{S_{i}}. This diagram will have nodes labelled by the composition factors of P_{S_{i}} such that there is an arrow S_{j} ⟵ S_{k} if S_{k} can extend S_{j} non-trivially in P_{S_{i}} (this is an example of the use of module diagrams, see for example Alperin (1980) and Benson and Carlson (1987)). Note that the fact that we draw an arrow from S_{k} to S_{j} means that it is possible for S_{k} to extend S_{j} non-trivially within P_{S_{i}}, not that it must. The Loewy and socle structure of P_{S_{i}} (along with the values of dim_{K}Ext^{1}(S_{k}, S_{j}) for all simple S_{j} and S_{k}) will be used to determine the possible non-trivial extensions, by making use of the fact that a composition factor of L_{n}(P_{S_{i}}) must be non-trivially extended by a composition factor of L_{n−1}(P_{S_{i}}) for n > 1 and the fact that a composition factor of S_{n}(P_{S_{i}}) must non-trivially extend a composition factor of S_{n−1}(P_{S_{i}}) for n > 1.

We will always have dim_{K}Ext^{1}(S_{k}, S_{j}) ≤ 1 and therefore we associate a nontrivial extension of S_{j} by S_{k} with the arrow x_{j} ⟵^{α} x_{k} in the Ext-quiver. We then determine general relations of linear dependence on series of extensions starting at the S_{i} in the first Loewy layer of P_{S_{i}} and ending with the same composition factor S_{j} of P_{S_{i}}. This frequently involves constructing a basis of non-zero series of extensions (i.e. series of extensions in which each extension is definitely nontrivial) and using these to define the relations of linear dependence. The relations then determine the same relations in the corresponding paths modulo larger paths from x_{i} to x_{j}. We then add in terms in larger paths from x_{i} to x_{j} to give the general form of the relation in these paths.

For each PIM P_{S_{i}} we write down generators for all the relations on paths beginning at the vertex x_{i}. We then take generating relations from all the PIMs and try to reduce the number of generators. This leaves us with a series of relations involving a number of unknown scalar factors, the scalar factors determining the relations of linear dependence. The actual values of these scalars depend on the choice of basis element corresponding to each of the arrows and therefore we rechoose basis elements corresponding to the arrows in order to find a presentation with the least number of unknown scalar factors. As each arrow x_{j} ⟵^{α} x_{k} corresponds to a basis element for the complement of e_{j}Rad^{2}(A)e_{k} in e_{j}Rad(A)e_{k}, we can change our basis elements by adding on an element of e_{j}Rad^{2}(A)e_{k} or by multiplying by a non-zero scalar. We end up with a presentation for the relations defining the basic algebra with as few scalars as possible. If we remove all scalars then we have completely determined the basic algebra, alternatively we have determined a class of algebras to which the basic algebra must belong.

We finish this section by giving an example indicating some of the techniques we have described.

Suppose we have an algebra A with three PIMs, which each have the same Loewy structure and socle structure described as follows:

P_{X} = \begin{aligned}& X \\ Y & Z \\ & X \end{aligned}  P_{Y} = \begin{aligned}& Y \\ Z & X \\ & Y \end{aligned}  P_{Z} = \begin{matrix}Z \\ X & Y \\ Z \end{matrix}

(X, Y, Z the three simple A-modules). By identifying the PIMs of A with the PIMs of A_{0} and the simples of A with the simples of A_{0}, we see that A_{0} has three PIMs with Loewy and socle structure given by the above description, where now X, Y, and Z are 1-dimensional A_{0} modules. The Ext-quiver of A_{0} is

[figure: Y; Z; γ; γ′; X; α; α′; β; β′]

We have dim_{K}Ext^{1}(S_{i}, S_{j}) ≤ 1 for all the simple modules of A_{0}, and so non-trivial series of extensions give rise to non-zero homomorphisms, and hence to non-zero paths. We will therefore associate the arrow x_{j} ⟵ x_{i} with every non-trivial extension of S_{j} by S_{i}.

Consider P_{X}. The X of the head of P_{X} must extend the composition factors of the second Loewy layer non-trivially, and as these are also in the second socle layer, they must both extend the X of the socle non-trivially. Hence the extension structure of P_{X} is

[figure: X; Y; Z; α′; β; X; α; β′]

where each arrow corresponds to a non-trivial extension which must exist.

P_{X} has no composition factor isomorphic to Y in the third Loewy layer and therefore we must have γβ = 0 modulo paths of length greater than two. To see this, let f_{1} be in the complement of Hom(P_{Z}, Rad^{2}(P_{X})) in Hom(P_{Z}, Rad(P_{X})) and let f_{2} be in the complement of Hom(P_{Y} , Rad^{2}(P_{Z})) in Hom(P_{Y} , Rad(P_{Z})) so that P_{Y} ⟶^{f_{2}} Rad(P_{Z}) ⟶^{f_{1}} Rad^{2}(P_{X}). Then f = f_{1}f_{2} : P_{Y} ⟶ Rad^{2}(P_{X}) must lie in Hom(P_{Y} , Rad^{3}(P_{X})). As β and γ correspond to homomorphisms such as f_{1} and f_{2} respectively, this means that γβ = 0 modulo paths of length greater than two. In fact, Rad^{3}(P_{X}) = 0, and so any such map as f above must be zero, i.e., γβ = 0. Using a similar argument we find that γ′α′ = 0.

P_{X} has only one composition factor isomorphic to X in the third Loewy layer, and therefore the paths β′β and αα′ are linearly dependent. To see this, let f_{1} be in the complement of Hom(P_{Z}, Rad^{2}(P_{X})) in Hom(P_{Z}, Rad(P_{X})), let f_{2} be in the complement of Hom(P_{X}, Rad^{2}(P_{Z})) in Hom(P_{X}, Rad(P_{Z})), let g_{1} be in the complement of Hom(P_{Y} , Rad^{2}(P_{X})) in Hom(P_{Y} , Rad(P_{X})) and let g_{2} be in the complement of Hom(P_{X}, Rad^{2}(P_{Y})) in Hom(P_{X}, Rad(P_{Y})). Hence we have

P_{X} ⟶^{f_{2}} Rad(P_{Z}) ⟶^{f_{1}} Rad^{2}(P_{X})

and

P_{X} ⟶^{g_{2}} Rad(P_{Z}) ⟶^{g_{1}} Rad^{2}(P_{Y})

so that f = f_{1}f_{2} and g = g_{1}g_{2} are in Hom(P_{X}, Rad^{2}(P_{X})). Because all the arrows in the diagram of extension structure correspond to non-trivial extensions which must exist, we know that both f and g must be non-zero (using the fact that non-trivial extensions of 1-dimensional modules give a relationship between the basis elements - p. 28). As Rad^{2}(P_{X}) ≅ X, Hom(P_{X}, Rad^{2}(P_{X})) is 1-dimensional and therefore f and g must be linearly dependent, i.e., f = λg for some non-zero scalar λ in K. As the arrows β, β′, α′ and α correspond to maps such as f_{1}, f_{2}, g_{1} and g_{2} respectively, we have β′β = λαα′.

Finally, using similar arguments to these, we see that βαα′ = 0 and α′αα′ = 0 (so that from the above relation also ββ′β = 0 and α′β′β = 0) and therefore the relations

γβ = 0
γ′α′ = 0
β′β = λαα′
βαα′ = 0
α′αα′ = 0

generate all the relations satisfied by paths starting at the vertex X. Hence KQ/I ≅ A where the generators for I starting at the vertex X are

γβ
γ′α′
β′β − λαα′
βαα′
α′αα′

1.7 Automorphisms

Let A = KG be the group algebra and let φ be an automorphism of G. Then φ induces an automorphism of KG by extending linearly. For any submodule M of KG we then have a KG-module φ(M) with action given by

gφ(m) = φ(φ^{−1}(g))φ(m)
= φ(φ^{−1}(g)m).

Now let M be an arbitrary KG-module so that the underlying set of M need not be a subset of KG. We define a KG-module, M^{φ}, by taking the vector space M with action ∗_{φ} given by

g ∗_{φ} m = φ^{−1}(g)m

where the right hand side is action in the KG-module M. This action is well defined in the sense that if the underlying set of M is contained in KG, so that φ(M) is defined, then M^{φ} and φ(M) are isomorphic as KG-modules. We have

Lemma 1.7.1

(i) (M^{φ})^{φ^{−1}} = M.

(ii) (M ⊕ N)^{φ} = M^{φ} ⊕ N^{φ}.

(iii) If N ≤ M then N^{φ} ≤ M^{φ} and M^{φ}/N^{φ} = (M/N)^{φ}.

(iv) M is indecomposable if and only if M^{φ} is indecomposable.

(v) M is simple if and only if M^{φ} is simple.

Notice that by (iii), if N is a maximal submodule of M, then N^{φ} is a maximal submodule of M^{φ}, and if L is a maximal submodule of M^{φ}, then L^{φ^{−1}} is a maximal submodule of M whose image under φ is just L. But if M^{φ} = N^{φ}, then (M^{φ})^{φ^{−1}} = (N^{φ})^{φ^{−1}}, and so φ is a bijection of the set of maximal submodules of M onto the set of maximal submodules of M^{φ}. We therefore have Lemma 1.7.2

L_{i}(M^{φ}) = L_{i}(M)^{φ}.

Proof. We have

Rad(M^{φ}) = ⋂ maximal submodules of M^{φ}
= (⋂ maximal submodules of M)^{φ}
= Rad(M)^{φ}

and then by induction

Rad^{i}(M^{φ}) = Rad^{i}(M)^{φ}

for all values of i. In particular this says that ℓ(M^{φ}) = ℓ(M). We then have

L_{i}(M^{φ}) = \frac{Rad^{i−1}(M^{φ})}{Rad^{i}(M^{φ})}

= \frac{Rad^{i−1}(M)^{φ}}{Rad^{i}(M)^{φ}}
= (\frac{Rad^{i−1}(M)}{Rad^{i}(M)})^{φ}
= L_{i}(M)^{φ}

for all i ≤ ℓ(M). □

Hence we need to understand how to find S^{φ} for each simple module S. For simple groups this can be achieved by using information from the Atlas character tables.

Let φ be an automorphism of the simple group G. Then either φ is an inner automorphism of G or φ induces an outer automorphism φ̄ of G. Let χ_{S} be the irreducible character of S. If φ is an inner automorphism then χ_{S}(g) = χ_{S}(φ^{−1}(g)) for all g ∈ G and so S ≅ S^{φ}. If φ is not an inner automorphism then we look at the fusing and splitting information for the subgroup X = G.⟨φ⟩ of the automorphism group (note that we have identified G with the inner automorphism group of G). We will only ever be interested in automorphisms of prime order, so for the rest of this section we assume that φ is an automorphism of G which is not inner and which has prime order q (and hence G ∩ ⟨φ⟩ = 1).

Consider the induced module S↑^{X}. Because {φ^{i} : 0 ≤ i ≤ q − 1} is a transversal for G in X we have

S↑^{X} ≅ ⨁_{i=0}^{q−1} φ^{i} ⊗ S.

As vector spaces, φ^{i} ⊗ S ≅ S and the action of G on φ^{i} ⊗ S is given by

g(φ^{i} ⊗ s) = φ^{i}(φ^{−i}gφ^{i}) ⊗ s
= φ^{i}φ^{−i}(g) ⊗ s
= φ^{i} ⊗ φ^{−i}(g)s

(here we have used the fact that if ψ is the inner automorphism corresponding to conjugation by g, then φ^{−i}ψφ^{i} is the automorphism corresponding to conjugation by φ^{−i}(g)). Hence φ^{i} ⊗ S ≅ S^{φ^{i}} as KG-modules. Now if φ^{i} ⊗ S ≅ φ^{j} ⊗ S for some i, j, then S, φ⊗S, …, φ^{q−1} ⊗S are all isomorphic KG-modules—this is the splitting case. In particular S ≅ φ⊗S ≅ S^{φ} so that χ_{S}(g) = χ_{S^{φ}}(g) for all g ∈ G.

The other possibility is that none of the φ^{i} ⊗ S are isomorphic—this is the fusing case. In this case we have q non-isomorphic simple modules which are conjugate, and χ_{φ^{i}⊗S}(g) = χ_{S}(φ^{−i}(g)) for all g ∈ G. Hence we can use the character table to determine S^{φ} for each simple module S and given any module M, we can then determine the Loewy structure of the module M^{φ} given the Loewy structure of M.

1.8 Auslander–Reiten Theory

In this section we outline the ideas of almost split sequences and Auslander– Reiten quivers. We then show how knowledge of almost split sequences helps in determining the Auslander–Reiten quiver of a given algebra. For details see Benson (1991).

Definition 1.8.1 A non-split short exact sequence

0 ⟶ N ⟶ E ⟶^{φ} M ⟶ 0

with M and N indecomposable is called an almost split sequence or Auslander– Reiten sequence provided that for every module M′ and any map λ : M′ ⟶ M which is not a split epimorphism, there is a map ψ : M′ ⟶ E such that λ = φψ, i.e. λ factors through E.

If we have an exact sequence

0 ⟶ N ⟶ E ⟶ M ⟶ 0

then we have a commutative diagram with exact rows and columns as follows

0  0  0
↑  ↑  ↑
0 ⟶ N ⟶  E  ⟶ M ⟶ 0
↑  ↑  ↑
0 ⟶ P_{N} ⟶ P_{N} ⊕ P_{M} ⟶ P_{M} ⟶ 0
↑ ↑ ↑
0 ⟶ Ω(N) ⟶ Ω(E) ⊕ P ⟶ Ω(M) ⟶ 0
↑  ↑  ↑
0  0  0

(where P is projective or zero) so that we have an exact sequence

0 ⟶ Ω(N) ⟶ Ω(E) ⊕ P ⟶ Ω(M) ⟶ 0.

Suppose we have a symmetric algebra. Then in the category _{A}mod, the functors Ω and Ω^{−1} are inverse to each other. Therefore if

0 ⟶ N ⟶ E ⟶ M ⟶ 0

is an almost split sequence, so is

0 ⟶ Ω(N) ⟶ Ω(E) ⊕ P ⟶ Ω(M) ⟶ 0.

Hence modulo projectives, Ω maps almost split sequences to almost split sequences.

If A is a symmetric algebra and P is an indecomposable projective module, then there is an almost split sequence

0 ⟶ Rad(P) ⟶ P ⊕ Ht(P) ⟶ P/Soc(P) ⟶ 0

and hence for any simple module S we have an almost split sequence

0 ⟶ Ω(S) ⟶ P_{S} ⊕ Ht(P_{S}) ⟶ Ω^{−1}(S) ⟶ 0.

We will frequently make use of the existence of this sequence.

Theorem 1.8.2 (see Benson (1991) Theorem 4.12.2) Given any finitely generated indecomposable non-projective module M, there exists an almost split sequence ending in M.

Lemma 1.8.3 (see Benson (1991) Lemma 4.12.3) Suppose

0 ⟶ N_{1} ⟶ E_{1} ⟶ M ⟶ 0
0 ⟶ N_{2} ⟶ E_{2} ⟶ M ⟶ 0

are almost split sequences ending at the same module M. Then they are isomorphic as short exact sequences.

Hence given any finitely generated indecomposable non-projective module M, there is (up to isomorphism) a unique indecomposable module N defined by Theorem 1.8.2. We call this module the Auslander–Reiten translate of M and denote it by τ(M). Then

0 ⟶ τ(M) ⟶ E ⟶ M ⟶ 0

is the unique almost split sequence ending at M. The map τ sends indecomposable non-projective modules to indecomposable non-injective modules.

The dual statement of Theorem 1.8.2 is that given any finitely generated indecomposable non-injective module N, there exists an almost split sequence beginning in N and so by the dual of Lemma 1.8.3 there is (up to isomorphism) a unique almost split sequence

0 ⟶ N ⟶ E ⟶ M ⟶ 0.

We set τ^{−1}(N) = M and then τ^{−1} is a map sending indecomposable non-injective modules to indecomposable non-projective modules. The maps τ and τ^{−1} are inverse to each other. For a group algebra τ turns out to be related to an operator we have already met.

Proposition 1.8.4 (see Benson (1991) Proposition 4.12.8) Suppose A is a finite dimensional symmetric algebra. Then for any non-projective indecomposable module M we have τ(M) ≅ Ω^{2}(M).

Before we can define the Auslander–Reiten quiver we need the following definition:

Definition 1.8.5 Suppose M and N are finitely generated A-modules. A map λ : M ⟶ N is said to be irreducible if it is neither a split monomorphism nor a split epimorphism and whenever λ = νμ is a factorisation of λ, either ν is a split epimorphism or μ is a split monomorphism.

Let M = ⨁ M_{i} and N = ⨁ N_{j} with the M_{i} and the N_{j} indecomposable. Denote by Rad(M, N) the space of maps M ⟶ N for which no component M_{i} ⟶ N_{j} is an isomorphism, and denote by Rad^{2}(M, N) the space spanned by homomorphisms of the form νμ with μ ∈ Rad(M, M′) and ν ∈ Rad(M′, N) for some module M′. A map λ is irreducible if and only if λ ∈ Rad(M, N) and λ ∉ Rad^{2}(M, N).

Hence Irr(M, N) = Rad(M, N)/Rad^{2}(M, N) corresponds to the space of irreducible maps from M to N.

Definition 1.8.6 The Auslander–Reiten quiver of an algebra A, denoted by Γ(A), has vertices the isomorphism classes of indecomposable A-modules, and the number of arrows from the isomorphism class containing M to the isomorphism class containing N is the K-dimension of Irr(M, N).

We will often just refer to vertices in the Auslander–Reiten quiver as modules M and N and refer to arrows M → N on the understanding that we are really referring to isomorphism classes.

Definition 1.8.7 The stable Auslander–Reiten quiver of an algebra A, denoted by Γ_{S}(A), is obtained from Γ(A) by removing all τ^{−k}(P) and all τ^{k}(I), k > 0 for P projective and I injective, and all adjacent arrows.

For a symmetric algebra this just means removing the projective modules and adjacent arrows.

We now relate almost split sequences to irreducible maps.

Proposition 1.8.8 (see Benson (1991) Proposition 4.13.3)

(i) Let M be a non-projective indecomposable A-module and let the almost split sequence ending in M be

0 ⟶ τ(M) ⟶ E ⟶^{φ} M ⟶ 0.

Then λ : M′ → M is irreducible if and only if M′ is a direct summand of E and λ = φι where ι is inclusion of M′ into E.

(ii) Let N be a non-injective indecomposable A-module and let the almost split sequence beginning in N be

0 ⟶ N ⟶^{ψ} E ⟶ τ^{−1}(N) ⟶ 0.

Then λ : N → N′ is irreducible if and only if N′ is a direct summand of E and λ = πψ where π is projection of E onto M′.

So for M non-projective, the number of arrows M′ → M is the multiplicity of M′ as a direct summand of the middle term of the almost split sequence ending in M. For N non-injective the number of arrows N → N′ is the multiplicity of N′ as a direct summand of the middle term of the almost split sequence starting in N.

Suppose we have an arrow M′ → M in the stable Auslander–Reiten quiver. Then by part (i) of the proposition there is an almost split sequence

0 ⟶ τ(M) ⟶ E_{M} ⊕ M′ ⟶ M ⟶ 0.

By (ii) therefore, there is also an arrow τ(M) → M′ and so by (i) there is an almost split sequence

0 ⟶ τ(M′) ⟶ E_{M′} ⊕ τ(M) ⟶ M′ ⟶ 0.

Then by (ii) we have an arrow τ(M′) → τ(M) and so we have an almost split sequence

0 ⟶ τ^{2}(M) ⟶ E_{τ(M)} ⊕ τ(M′) ⟶ τ(M) ⟶ 0.

A similar discussion holds for τ^{−1}, and given that τ and τ^{−1} are inverse to each other, we have shown that τ maps the almost split sequence

0 ⟶ τ(M) ⟶ ⨁ E_{i} ⊕ P ⟶ M ⟶ 0

to the almost split sequence

0 ⟶ τ^{2}(M) ⟶ ⨁ τ(E_{i}) ⊕ I ⟶ τ(M) ⟶ 0

where P and I are the projective and injective direct summands of the respective middle terms of the sequences. In terms of the stable Auslander–Reiten quiver this means we have arrows

[figure: τ^{2}(M); τ(E_{1}); τ(E_{2}); ⋮; τ(E_{n}); τ(M); E_{1}; E_{2}; ⋮; E_{n}; M]

so that τ acts on the stable Auslander–Reiten quiver as a shift one to the left.

We saw earlier that when A is symmetric, the Heller operator Ω maps almost split sequences to almost split sequences modulo projectives, and therefore in this case Ω also induces an automorphism of the stable Auslander–Reiten quiver.

Theorem 1.8.9 (Kawata) Let K be an algebraically closed field and let P be a Sylow p-subgroup of G which is not cyclic, dihedral, semi-dihedral, or generalised quaternion. Suppose Θ is a connected component of the stable Auslander–Reiten quiver and Θ contains an indecomposable module whose K-dimension is not divisible by p. Then Θ is isomorphic to ZA_{∞}.

Consider the graph

[figure: x_{1}; x_{2}; x_{3}; …]

which is the Dynkin diagram A_{∞}. From this we define the graph ZA_{∞} by taking the set of vertices to be {(n, x_{i}) : n ∈ Z, x_{i} is a vertex of A_{∞}} and for each arrow x_{i} → x_{i+1} in A_{∞}, defining arrows

(n, x_{i}) → (n, x_{i+1})

and

(n, x_{i}) → (n + 1, x_{i−1}).

Hence the graph ZA_{∞} has the following structure:

[figure: ⋯; ⋮; ⋯; ⋮; ⋮; ⋮; ⋮; ⋯; ⋯; ⋯; ⋯]

where the first row corresponds to {(n, x_{1}) : n ∈ Z}, the second row corresponds to {(n, x_{2}) : n ∈ Z} and so on. In particular notice that there are at most two arrows beginning and ending at any given vertex. We will frequently make use of this fact.

The Representation Theory of the Principal Block of U_{3}(q) With Defect Group C_{3} × C_{3}

Geck (1990) has studied the Brauer characters for SU_{3}(q^{2}), the special unitary group over a field of q^{2} elements, for all odd prime characteristics not dividing q. In particular the 3-modular Brauer characters have been determined and two distinct cases arise depending on whether 3 divides q − 1 or q + 1. When 3 divides q − 1 the Sylow 3-subgroup is cyclic and all blocks have maximal defect or defect zero (i.e. all defect groups are cyclic or trivial). The Brauer trees for all blocks of maximal defect were determined and so the structure of the PIMs is known. In the case when 3 divides q +1 the Sylow 3-subgroups are not cyclic and the decomposition numbers for the principal block were determined up to the presence of a scalar (all other blocks have defect zero or one and so the structure of the PIMs is clear).

Consider now the simple group P SU_{3}(q^{2}), the projective special unitary group over a field of q^{2} elements (q > 2). When 3 divides q − 1 the centre of SU_{3}(q^{2}) is trivial and so P SU_{3}(q^{2}) = SU_{3}(q^{2}) which has already been considered above. We are left with the question then of what happens when 3 divides q + 1. In this case the centre of SU_{3}(q^{2}) is isomorphic to C_{3} and P SU_{3}(q^{2}) has order q^{3}(q + 1)^{2}(q − 1)(q^{2} − q + 1)/3 where the maximal power of 3 dividing q^{2} − q + 1 is 3^{1}. We will deal with the least complicated case, the case when the Sylow 3-subgroup is C_{3} × C_{3}. This occurs when 3 divides q + 1 but 9 does not, i.e. when q ≡ 2 mod 3 but q ≢ 8 mod 9. From now on we will assume q satisfies these relations and adopting Atlas notation (Conway et al. (1985)), we will refer to the projective special unitary group over a field of q^{2} elements as U_{3}(q).

In this chapter we will use information about SU_{3}(q^{2}) determined by Geck to find the Loewy structure of the PIMs of the principal block of U_{3}(q), and then use this information to try and find the relations that determine the basic algebra.

2.1 The Loewy Structure of the PIMs of the

Principal Block of U_{3}(q)

The aim of this section will be to determine the Loewy structure of the PIMs of the principal block of U_{3}(q) over an algebraically closed field, K, of characteristic three. This will rely heavily on information about SU_{3}(q^{2}) determined by Geck (1990) and will make use of some Auslander–Reiten theory. We set out to prove the following theorem:

Theorem 2.1.1 Suppose q is a prime power with q ≡ 2 mod 3 and q ≢ 8 mod 9. The Loewy structures and socle structures of the PIMs in the principal block of U_{3}(q) are as follows:

\begin{aligned}& 1 \\ & D \\ A & B & C \\ & D \\ & 1 \end{aligned}  \begin{aligned}& A \\ & D \\ B & 1 & C \\ & D \\ & A \end{aligned}  \begin{aligned}& B \\ & D \\ C & 1 & A \\ & D \\ & B \end{aligned}

\begin{aligned}& C \\ & D \\ A & 1 & B \\ & D \\ & C \end{aligned}  \begin{aligned}& & D \\ 1 & A & & B & & C \\ D & & D & & D \\ 1 & A & & B & & C \\ & & D \end{aligned}

where 1 is the trivial U_{3}(q)-module, D is a simple U_{3}(q)-module of dimension q^{2} − q over K, and A, B and C are simple U_{3}(q)-modules of dimension (q − 1)(q^{2} − q + 1)/3 over K.

2.1 The Loewy Structure of the PIMs of the Principal Block of U_{3}(q) 43

2.1.1 The Characters of U_{3}(q)

The decomposition matrix for the principal block of SU_{3}(q^{2}) has been determined by Geck (1990) and will be found in appendix A (as will a list of the conjugacy classes of SU_{3}(q^{2}) and the ordinary character table for SU_{3}(q^{2})). Over any field there is a one-to-one correspondence between simple modules of U_{3}(q) and simple modules of SU_{3}(q^{2}) where the central elements all act trivially, and therefore we can deduce the decomposition matrix for the principal block of U_{3}(q). By considering those simple SU_{3}(q^{2})-modules for which the character values on C_{1}^{(0)}, C_{1}^{(1)} and C_{1}^{(2)} (the three central elements) are equal and using the same names, we find that the decomposition matrix for the principal block of U_{3}(q) is:

\begin{matrix}χ_{1} & 1 & 0 & 0 & 0 & 0 \\ χ_{q^{2}−q} & 0 & 1 & 0 & 0 & 0 \\ χ_{(q−1)(q^{2}−q+1)/3}^{(0)} & 0 & 0 & 1 & 0 & 0 \\ χ_{(q−1)(q^{2}−q+1)/3}^{(1)} & 0 & 0 & 0 & 1 & 0 \\ χ_{(q−1)(q^{2}−q+1)/3}^{(2)} & 0 & 0 & 0 & 0 & 1 \\ χ_{q^{3}} & 1 & 2 & 1 & 1 & 1 \end{matrix}

Given this matrix we see that there are five simple modules in the principal block of U_{3}(q), one of dimension 1 which will be denoted by 1, one of dimension q^{2} − q which will be denoted by D, and three of dimension (q − 1)(q^{2} − q + 1)/3 which will be denoted by A, B and C. We then have the following Cartan matrix:

1 D A B C
1 2 2 1 1 1
D 2 5 2 2 2
A 1 2 2 1 1
B 1 2 1 2 1
C 1 2 1 1 2

The simple modules 1 and D are certainly self-dual, given their character values, but as yet we do not know whether all three of the simples A, B and C are self-dual. Given the character values of the corresponding SU_{3}(q^{2})-modules on the conjugacy classes of SU_{3}(q^{2}) we see that these three modules take the same values on all conjugacy classes, except for those classes C_{3}^{(k,l)}. The values of the three modules on these conjugacy classes are as follows:

C_{3}^{(k,0)}  C_{3}^{(k,1)}  C_{3}^{(k,2)}
χ_{(q−1)(q^{2}−q+1)/3}^{(0)} q − (q + 1)/3 −(q + 1)/3 −(q + 1)/3
χ_{(q−1)(q^{2}−q+1)/3}^{(1)} −(q + 1)/3 q − (q + 1)/3 −(q + 1)/3
χ_{(q−1)(q^{2}−q+1)/3}^{(2)} −(q + 1)/3 −(q + 1)/3 q − (q + 1)/3

As these are all integer values, we cannot have two distinct simple modules of dimension (q − 1)(q^{2} − q + 1)/3 which have complex conjugate characters. Therefore these three simple SU_{3}(q^{2})-modules are self-dual and hence the corresponding simple U_{3}(q)-modules are self-dual.

According to Geck (1990) the SU_{3}(q^{2})-modules corresponding to A, B and C are conjugate under an automorphism of SU_{3}(q^{2}) which has order three (conjugation by a diagonal matrix of the general unitary group). Because U_{3}(q) = SU_{3}(q^{2})/Z(SU_{3}(q^{2})) this induces an automorphism of U_{3}(q) of order three which conjugates A, B and C. There is also an automorphism of order two which fixes the SU_{3}(q^{2})-module corresponding to A and interchanges the SU_{3}(q^{2})-modules corresponding to B and C (the map that sends a matrix X to X^{−T}) and this induces an automorphism of U_{3}(q) which fixes A and interchanges B and C. Hence we have a group of automorphisms acting on the principal block which fixes 1 and D and acts like S_{3} on A, B and C.

For any automorphism φ of U_{3}(q) and any PIM P, P^{φ} is also a PIM, and the Loewy structure of P^{φ} is obtained by taking the image of the Loewy structure of P under φ. In particular, Hd(P^{φ}) ≅ Hd(P)^{φ}, and hence for any automorphism φ, P is in the principal block if and only if P^{φ} is in the principal block. The automorphism group above therefore fixes both P_{1} and P_{D} and permutes P_{A}, P_{B} and P_{C} by permuting A, B and C. Therefore we only need to find the Loewy structure of P_{1}, P_{D} and P_{A} to determine the Loewy structure of all of the PIMs in the principal block.

2.1.2 The Loewy Structure of the PIMs

P_{1} has composition factors 2.1 + 2.D + 1.A + 1.B + 1.C. We know that P_{1} is invariant under the automorphism group S_{3}, and therefore the simple modules A, B and C must all lie in the same Loewy layer. It is also self-dual and all the simple composition factors are self-dual, hence, by the discussion following Lemma 1.1.2 the simple modules in the penultimate Loewy layer of P_{1} must be contained in the second Loewy layer of P_{1}. This leaves three possible Loewy structures, one of Loewy length three, one of Loewy length four, and one of Loewy length five. The possible structures of length three and four are

\begin{aligned}& & 1 \\ D & A & B & C & D \\ & & 1 \end{aligned}  and  \begin{aligned}& 1 \\ D & A & B & C \\ & D \\ & 1 \end{aligned}

respectively. In each of these cases, as the composition factors are self-dual, S_{i}(P_{1}) ≅ L_{i}(P_{1}) for all i ≤ ℓ(P_{1}), and hence in both cases the heart of P_{1} is decomposable. However by a theorem of Webb, Theorem 1.1.6, the heart of P_{1} must be indecomposable, a contradiction. Hence P_{1} must have Loewy length five and the Loewy structure of P_{1} must be

1 D

A B C

D

1 .

By duality the socle structure is equal to the Loewy structure.

P_{A} has composition factors 1.1 + 2.D + 2.A + 1.B + 1.C and is fixed by the automorphism of order two interchanging B and C, therefore the simple modules B and C must lie in the same Loewy layer and the Loewy length cannot be greater than six. Furthermore we have

E (A, 1) = E (1^{∗}, A^{∗}) = E (1, A) = 0.

Finally, P_{A} is self-dual, as are all the simple composition factors, and so the simple modules in the last r Loewy layers of P_{A} must also be contained in the first r Loewy layers of P_{A}. Given all this information, the only possible Loewy structures of P_{A} are

\begin{aligned}& A \\ B & D & C \\ & 1 \\ & D \\ & A \end{aligned}  and  \begin{aligned}& A \\ & D \\ B & 1 & C \\ & D \\ & A \end{aligned} .

Suppose the Loewy structure of P_{A} is

A

B D C

1 D

A .

Because all the simple modules are self-dual, by duality we have S_{i}(P_{A}) ≅ L_{i}(P_{A}) for all i ≤ ℓ(P_{A}) and so the heart of P_{A} has structure

( \begin{matrix}D \\ 1 \\ D \end{matrix} ) ⊕ B ⊕ C.

Hence we have an almost split sequence

0 ⟶ Ω(A) ⟶ P_{A} ⊕ ( \begin{matrix}D \\ 1 \\ D \end{matrix} ) ⊕ B ⊕ C ⟶ Ω^{−1}(A) ⟶ 0

and the component of the stable Auslander–Reiten quiver containing Ω(A) has arrows

[figure: Ω(A); ^{( \begin{matrix}D \\ 1 \\ D \end{matrix} )}; B; C; Ω^{−1}(A).]

This tells us that Ω(A) and B are in the same component of the stable Auslander– Reiten quiver. The dimension of B is (q − 1)(q^{2} − q + 1)/3 and (q^{2} − q + 1) is not divisible by 9, therefore the dimension of B is not divisible by 3. By Theorem 1.8.9 the component is isomorphic to ZA_{∞} and so there are at most two arrows starting at ΩA, which contradicts the existence of three indecomposable summands in the heart of P_{A}. Hence the Loewy structure of P_{A} is

\begin{aligned}& A \\ & D \\ B & 1 & C \\ & D \\ & A \end{aligned} ,

which is the same as the socle structure again by duality. By applying the automorphism of order three we find that the Loewy and socle structures of P_{B} and P_{C} are

\begin{aligned}& B \\ & D \\ C & 1 & A \\ & D \\ & B \end{aligned}  and  \begin{aligned}& C \\ & D \\ A & 1 & B \\ & D \\ & C \end{aligned} .

P_{D} has composition factors 2.1 + 5.D + 2.A + 2.B + 2.C. It is invariant under the automorphism group and hence there are two occurrences of the factor A ⊕ B ⊕ C in the Loewy structure of P_{D}. It is also self-dual, as are all the composition factors, so that any factor in the penultimate Loewy layer must also occur in the second Loewy layer. We also have the following values for dim_{K}Ext^{1}:

E (D, 1) = E (1^{∗}, D^{∗}) = E (1, D) = 1
E (D, A) = E (A^{∗}, D^{∗}) = E (A, D) = 1
E (D, B) = E (B^{∗}, D^{∗}) = E (B, D) = 1
E (D, C) = E (C^{∗}, D^{∗}) = E (C, D) = 1.

Given the Loewy structure of P_{1} we know we have a module with Loewy structure

\begin{aligned}& D \\ A & B & C \\ & D \\ & 1 \end{aligned}

which is a quotient of P_{D} of Loewy length length four, with simple socle 1. By Lemma 1.1.5 we have a copy of 1 in the fourth Loewy layer of P_{D}. Similarly we have quotients of P_{D} with the following Loewy structures:

\begin{aligned}& D \\ B & 1 & C \\ & D \\ & A \end{aligned}  \begin{aligned}& D \\ A & 1 & C \\ & D \\ & B \end{aligned}  \begin{aligned}& D \\ A & 1 & B \\ & D \\ & C & & . \end{aligned}

Therefore we have a copy of each of A, B and C in the fourth Loewy layer of P_{D}. Given this information and the fact that any composition factor occurring in the penultimate Loewy layer must also occur in the second Loewy layer, the only possible Loewy structure of Loewy length greater than five is

D
1 A B C D
D
1 A B C
D
D  .

By duality the socle structure of this module is

D
D
1 A B C
D
1 A B C D
D  .

Since Rad^{i}(P_{D}) ≤ Soc^{6−i}(P_{D}) for all i ≤ 6, we must have Rad^{3}(P_{D}) ≤ Soc^{3}(P_{D}).

Let Rad^{3}(P_{D}) be denoted by M. Then the Loewy structure of M is

\begin{aligned}1 & A & B & C \\ & D \\ & D & & . \end{aligned}

The socle series of M is given by the intersection of M with the socle series of Soc^{3}(P_{D}). Furthermore we must have Rad^{i}(M) ≤ Soc^{3−i}(M). This forces the socle structure of M to be

\begin{aligned}1 & A & B & C & D \\ & & D & & . \end{aligned}

But then the socle length and Loewy length of M are different, which is not possible, contradicting the choice of P_{D}. Hence P_{D} must have Loewy length five. Again, using the fact that any composition factor of the penultimate Loewy layer must be a composition factor of the second Loewy layer, the only possibilities for the Loewy structure of P_{D} are as follows:

\begin{aligned}& & D \\ 1 & A & B & C & D \\ & & D \\ 1 & A & B & C & D \\ & & D \end{aligned}  \begin{aligned}& & & D \\ D & 1 & A & B & C & D \\ & & & D \\ & 1 & A & B & C \\ & & & D \end{aligned}

\begin{aligned}& & & D \\ 1 & & A & B & & C & D \\ & & & D & D \\ & 1 & & A & B & & C \\ & & & D \end{aligned}  \begin{aligned}& & D \\ 1 & A & B & C \\ & D & D & D \\ 1 & A & B & C \\ & & D \end{aligned}

Suppose the Loewy structure of P_{D} is

D
1 A B C D
D
1 A B C D
D  .

Then because P_{D} is self-dual and all its composition factors are self-dual, S_{i}(P_{D}) ≅ L_{i}(P_{D}) for all i ≤ ℓ(P_{D}). Hence the Loewy and socle structure of the heart of P_{D} are the same and this is given by

1 A B C D
D
1 A B C D .

The composition factors 1, A, B, C, D of the third Loewy layer of the heart must all be extended non-trivially by the D of the second layer. As there is a composition factor isomorphic to 1 in the third socle layer (i.e., the first Loewy layer), it must extend the D of the second layer non-trivially. Hence we have a submodule with Loewy and socle structure

1 D

1 A B C D .

Because this module has simple head isomorphic to 1 it must be a quotient of P_{1}. But the only quotient of P_{1} with two composition factors isomorphic to 1 is P_{1} itself so this is not possible. Hence this is not the structure of the heart of P_{D} and so P_{D} does not have the above Loewy structure.

If the Loewy structure of P_{D} is

D
D 1 A B C D
D
1 A B C
D

then because P_{D} is self-dual and all its composition factors are self-dual S_{i}(P_{D}) ≅ L_{i}(P_{D}) for all i ≤ ℓ(P_{D}) and so the structure of the heart of P_{D} is

( \begin{matrix}1 & A & B & C \\ & & D \\ 1 & A & B & C \end{matrix} ) ⊕ D ⊕ D.

This means that we have an almost split sequence

0 ⟶ Ω(D) ⟶ P_{D} ⊕ ( \begin{matrix}1 & A & B & C \\ & & D \\ 1 & A & B & C \end{matrix} ) ⊕ D ⊕ D ⟶ Ω^{−1}(D) ⟶ 0

so that the component of the stable Auslander–Reiten quiver containing Ω(D) also contains D. But D has dimension q^{2} − q which is not divisible by 3, so by Theorem 1.8.9 this component is isomorphic to ZA_{∞} and hence there are at most two arrows leaving Ω(D). Hence the heart of P_{D} cannot have three indecomposable summands and so it is not possible for P_{D} to have the above Loewy structure. We will now show that in fact the heart of P_{D} cannot have any indecomposable summands isomorphic to D, leaving only one possibility for the Loewy structure of P_{D}.

Suppose Ht(P_{D}) = X ⊕ D for some indecomposable module X. Then the component of the stable Auslander–Reiten quiver containing Ω(D) has arrows

[figure: Ω(D); X; D; Ω^{−1}(D)]

so that D and Ω(D) are in the same component of the stable Auslander–Reiten quiver. Hence the Heller operator Ω induces a graph automorphism of the component containing D and using the same argument as above we know that this component is isomorphic to ZA_{∞}. Let U be a module corresponding to a vertex in the first row of ZA_{∞}. Then our component has the form

[figure: ⋯; ⋮; ⋯; Ω^{4}(U); ⋮; Ω^{2}(U); ⋮; U; ⋮; Ω^{−2}(U); ⋮; ⋯; Ω^{−4}(U); ⋯]

(because Ω^{2} acts as a shift one to the left for any component of the stable Auslander–Reiten quiver.) As it induces a graph automorphism, Ω must fix the modules in the first row i.e. those vertices with only one arrow starting and ending at them, and so Ω(U) ≅ Ω^{2k}(U) for some integer k > 0. Hence U is Ω-periodic and in fact every module in the first row is Ω-periodic. Now let W be a module in the second row. Then there is some module U in the first row such that this component of the stable Auslander–Reiten quiver has arrows

[figure: U; W; Ω^{−2}(U).]

Because applying Ω to an almost split sequence gives another almost split sequence, the stable Auslander–Reiten quiver also has arrows

[figure: Ω^{2k−1}(U); Ω^{2k−1}(W); Ω^{2k−3}(U)]

so that in fact W ≅ Ω^{2k−1}(W). Hence every module in the second row is Ωperiodic. By applying an inductive argument we find that every module in this component is Ω-periodic, in particular, D is Ω-periodic. But by Alperin’s result on Ω-periodicity, Theorem 1.3.4,, this means that the dimension of D must be divisible by 3, which is not the case. Hence D cannot be a direct summand of the heart of P_{D} as claimed. Therefore, the only possibility left for the Loewy structure of P_{D} is

D
1 A B C
D D D
1 A B C
D  .

By duality, this is the socle structure as well. This completes the proof of Theorem 2.1.1.

2.2 The Structure of the Basic Algebra of the Principal Block of U_{3}(q) 53

2.2 The Structure of the Basic Algebra of the

Principal Block of U_{3}(q)

In this section we will use the information about the Loewy and socle structure of the PIMs of the principal block of U_{3}(q) to determine the structure of the basic algebra of the principal block. We will find that the relations can be determined up to the presence of two scalar factors, so that we have determined a 2-parameter family of algebras to which the basic algebra of the principal block must belong. The relations for this 2-parameter family will be found at the end of the chapter.

We will denote the basic algebra of the principal block of U_{3}(q) by Λ, and denote the primitive orthogonal idempotent in Λ corresponding to P_{S} by e_{S}. The Ext-quiver for Λ is:

[figure: D; 1; B; C; A; α; α′; β; β′; γ; γ′; δ; δ′]

where α corresponds to a basis element for the complement of the space e_{D}Rad^{2}(Λ)e_{A} in e_{D}Rad(Λ)e_{A}, α′ corresponds to a basis element for the complement of e_{A}Rad^{2}(Λ)e_{D} in e_{A}Rad(Λ)e_{D}, and so on. The category of KU_{3}(q)- modules is equivalent to the category of modules for KQ/I where Q is the above Ext-quiver and I is a set of relations on KQ (and K is an algebraically closed field of characteristic three). We will now use our knowledge of the structure of the PIMs of the principal block of U_{3}(q) to calculate the relations on KQ. We then minimise the set of generators and choose basis elements corresponding to the arrows so as to minimise the number of unknown scalar factors involved in the relations. When dealing with elements of the path algebra KQ composition of paths will be written on the left.

2.2.1 Finding Generators for the Relations Induced by I

Given the Loewy and socle structure of P_{1}, the extension structure of P_{1} is as indicated in the following diagram:

[figure: 1; D; δ; B; A; C; α′; β′; γ′; D; α; β; γ; 1; δ′]

From the Loewy and socle structure we know that the composition factor D of the second Loewy layer must non-trivially extend each of the composition factors A, B and C in the third Loewy layer. These factors are also in the third socle layer so each must non-trivially extend the single D in the second socle layer. This tells us that the paths αα′δ, ββ′δ and γγ′δ are all non-zero and span a 1-dimensional space modulo e_{D}Rad^{4}(Λ)e_{1}. But as all paths of length five are zero and there are no paths of length four from 1 to D, this means that αα′δ, ββ′δ and γγ′δ span a 1-dimensional space in KQ/I. As there is no composition factor isomorphic to 1 in the third Loewy layer, we also have δ′δ = 0 modulo e_{1}Rad^{3}(Λ)e_{1}. As the only non-zero paths from 1 to 1 of length greater than two are the paths δ′αα′δ, δ′ββ′δ and δ′γγ′δ which span a 1-dimensional space, we must have δ′δ = k_{1}δ′αα′δ for some scalar k_{1} (possibly equal to zero) in K. We now write down generators for the relations in paths starting at the vertex 1:

δ′δ = k_{1}δ′αα′δ
ββ′δ = λ_{1}αα′δ
γγ′δ = λ_{2}αα′δ

α′αα′δ = 0
β′αα′δ = 0
γ′αα′δ = 0
δδ′αα′δ = 0

where λ_{1} and λ_{2} are non-zero.

Given the Loewy and socle structure of P_{A}, the extension structure of P_{A} is as follows:

[figure: A; D; α; 1; B; C; β′; δ′; γ′; D; β; δ; γ; A; α′]

Using the same type of argument as above, we have generating relations:

α′α = k_{2}α′δδ′α
ββ′α = λ_{3}δδ′α
γγ′α = λ_{4}δδ′α
β′δδ′α = 0
γ′δδ′α = 0
δ′δδ′α = 0
αα′δδ′α = 0

where this time each of ββ′α, δδ′α and γγ′α is non-zero (and therefore λ_{3} and λ_{4} are non-zero).

We have the following extension structures for P_{B} and P_{C}:

[figure: B; D; β; 1; A; C; α′; δ′; γ′; D; α; δ; γ; B; β′; C; D; γ; 1; A; B; α′; δ′; β′; D; α; δ; β; C; γ′]

As a result we have generating relations:

\begin{aligned}β′β & = & k_{3}β′δδ′β \\ αα′β & = & λ_{5}δδ′β \\ γγ′β & = & λ_{6}δδ′β \\ α′δδ′β & = & 0 \\ γ′δδ′β & = & 0 \\ δ′δδ′β & = & 0 \\ ββ′δδ′β & = & 0 \end{aligned}  \begin{aligned}γ′γ & = & k_{4}γ′δδ′γ \\ αα′γ & = & λ_{7}δδ′γ \\ ββ′γ & = & λ_{8}δδ′γ \\ α′δδ′γ & = & 0 \\ β′δδ′γ & = & 0 \\ δ′δδ′γ & = & 0 \\ γγ′δδ′γ & = & 0 \end{aligned}

where this time each of αα′β, δδ′β and γγ′β, αα′γ, δδ′γ and ββ′γ are non-zero (and so λ_{5}, λ_{6}, λ_{7} and λ_{8} are non-zero).

We now determine the relations induced on the paths beginning at the vertex D by considering the extension structure of P_{D} given the generating relations already determined. The set of paths of length one starting at D is the set {δ′, α′, β′, γ′}. The set of paths of length two is {δδ′, αα′, ββ′, γγ′}, and given the non-zero relations determined already, each of these paths must be non-zero. As there are actually only three composition factors isomorphic to D in P_{D} being non-trivially extended by the 1, A, B and C of the second Loewy layer, this tells us that these four non-zero paths must span a 3-dimensional space modulo e_{D}Rad^{3}(Λ)e_{D}. Next, bearing in mind that δ′δ = k_{1}δ′αα′δ, α′α = k_{2}α′δδ′α, β′β = k_{3}β′δδ′β and γ′γ = k_{4}γ′δδ′γ, the set of possible non-zero paths of length three in KQ/I is

{α′δδ′, β′δδ′, γ′δδ′, δ′αα′, β′αα′, γ′αα′, δ′ββ′, α′ββ′, γ′ββ′, δ′γγ′, α′γγ′, β′γγ′}.

As the three composition factors isomorphic to D in the third Loewy layer of P_{D} only non-trivially extend one composition factor isomorphic to A between them, the paths α′δδ′, α′ββ′ and α′γγ′ must span a 1-dimensional space modulo e_{A}Rad^{4}(Λ)e_{D} and hence in KQ/I, and we can make similar statements about the paths beginning with each of β′, γ′ and δ′. Finally, as there is only one composition factor non-trivially extended by the 1, A, B and C in the fourth Loewy layer, all the paths of length four must span a 1-dimensional space and all the paths of length five must be zero. This information is expressed in the following diagram:

[figure: D; δ′; α′; β′; γ′; 1; A; B; C; δ; α; β; γ; D; D; D; D; γ′; β′; α′; γ′; β′; δ′; γ′; α′; δ′; β′; α′; δ′; C; B; A; 1; γ; β; α; δ; D]

We have already seen above that modulo e_{D}Rad^{3}(Λ)e_{D} the four paths of length two span a 3-dimensional space. Hence we have a relation

r_{1}αα′ + r_{2}ββ′ + r_{3}γγ′ + r_{4}δδ′ + (term in paths of length 4) = 0

where not all of r_{1}, r_{2}, r_{3}, r_{4} ∈ K are zero. We have already seen from the structure of P_{1} that the paths δ′αα′, δ′ββ′ and δ′γγ′ are non-zero. Therefore we have relations:

δ′ββ′ = μ_{1}δ′αα′
δ′γγ′ = μ_{2}δ′αα′

with μ_{1} and μ_{2} non-zero. We also have α′δδ′, β′δδ′ and γ′δδ′ non-zero from the structures of P_{A}, P_{B} and P_{C} respectively and hence we can write down the rest of the relations in paths of length three:

α′ββ′ = μ_{3}α′δδ′
α′γγ′ = μ_{4}α′δδ′
β′αα′ = μ_{5}β′δδ′
β′γγ′ = μ_{6}β′δδ′
γ′αα′ = μ_{7}γ′δδ′
γ′ββ′ = μ_{8}α′δδ′.

In fact all these paths are non-zero and so all the μ_{i} are non-zero. We know that the path δδ′αα′ is non-zero, for otherwise there would be a composition factor isomorphic to 1 in the socle of P_{D}, which there is not. Similarly the paths αα′δδ′, ββ′δδ′ and γγ′δδ′ are non-zero and so we can write down the following generators for the relations in paths of length four starting at the vertex D:

\begin{aligned}αα′δδ′ & = & μ_{9}δδ′αα′ \\ ββ′δδ′ & = & μ_{10}δδ′αα′ \\ γγ′δδ′ & = & μ_{11}δδ′αα′ \end{aligned}

where the μ_{i} are non-zero. The relation involving the paths of length two can now be expressed as follows:

r_{1}αα′ + r_{2}ββ′ + r_{3}γγ′ + r_{4}δδ′ = k_{5}δδ′αα′.

Finally the generators for the relations in paths of length five are:

\begin{aligned}α′δδ′αα′ & = & 0 \\ β′δδ′αα′ & = & 0 \\ γ′δδ′αα′ & = & 0. \end{aligned}

2.2.2 Determining the Ideal I

Consider the relations

δ′δ = k_{1}δ′αα′δ
α′α = k_{2}α′δδ′α
β′β = k_{3}β′δδ′β
γ′γ = k_{4}γ′δδ′γ
k_{5}δδ′αα′ = r_{1}αα′ + r_{2}ββ′ + r_{3}γγ′ + r_{4}δδ′.

Each arrow x_{j} ⟵ x_{i} corresponds to a basis element for the complement of e_{j}Rad^{2}(Λ)e_{i} in e_{j}Rad(Λ)e_{i} and the above are the relations that depend on the component of each of the basis elements in Rad^{2}(Λ) (all the other generating relations determined depend only on the choice of basis elements modulo Rad^{2}(Λ)). Hence if we rechoose the basis elements in e_{j}Rad(Λ)e_{i} by adding elements of e_{j}Rad^{2}(Λ)e_{i}, then only these relations will be affected.

Let δ′_{0} be the choice of basis element for the complement of e_{1}Rad^{2}(Λ)e_{D} in e_{1}Rad(Λ)e_{D}, associated with the arrow δ′ and let α_{0} and α′_{0} be the choices of basis elements associated with the arrows α and α′ respectively. Set δ′_{1} = δ′_{0}(1−k_{1}α_{0}α′_{0}) and let this now be our choice of basis element corresponding to the arrow δ′. The relations are then

δ′δ = 0
α′α = k_{2}α′δδ′α
β′β = k_{3}β′δδ′β
γ′γ = k_{4}γ′δδ′γ
k′_{5}δδ′αα′ = r_{1}αα′ + r_{2}ββ′ + r_{3}γγ′ + r_{4}δδ′.

Let α′_{0} be the choice of basis element for the complement of e_{A}Rad^{2}(Λ)e_{D} in e_{A}Rad(Λ)e_{D}, associated with the arrow α′ and let δ_{0} and δ′_{0} be the current choices of basis elements associated with the arrows δ and δ′ respectively. Set α′_{1} = α′_{0}(1−k_{2}δ_{0}δ′_{0}) and let this now be our choice of basis element corresponding to the arrow α′. The relations in α′ are then

α′α = 0
k′′_{5}δδ′αα′ = r_{1}αα′ + r_{2}ββ′ + r_{3}γγ′ + r_{4}δδ′.

Similarly let β′_{0} and γ′_{0} be the choices of basis elements corresponding to the arrows β′ and γ′. Set β′_{1} = β′_{0}(1 − k_{3}δ_{0}δ′_{0}) and γ′_{1} = γ′_{0}(1 − k_{4}δ_{0}δ′_{0}). Then if we take β′_{1} and γ′_{1} as our new choices of basis elements corresponding to the arrows β′ and γ′, the relations are

δ′δ = 0
α′α = 0
β′β = 0
γ′γ = 0
kδδ′αα′ = r_{1}αα′ + r_{2}ββ′ + r_{3}γγ′ + r_{4}δδ′

for some scalar k ∈ K.

We have now shown that the ideal determining the basic algebra is generated by the relation

r_{1}αα′ + r_{2}ββ′ + r_{3}γγ′ + r_{4}δδ′ = kδδ′αα′,

where not all of μ_{1}, μ_{2}, μ_{3} and μ_{4} are zero, and the relations

\begin{aligned}δ′δ & = & 0 \\ α′α & = & 0 \end{aligned}  \begin{aligned}β′β & = & 0 \\ γ′γ & = & 0 \end{aligned}

\begin{aligned}α′αα′δ & = & 0 \\ β′αα′δ & = & 0 \\ γ′αα′δ & = & 0 \\ β′δδ′α & = & 0 \\ γ′δδ′α & = & 0 \\ δ′δδ′α & = & 0 \\ α′δδ′β & = & 0 \end{aligned}  \begin{aligned}γ′δδ′β & = & 0 \\ δ′δδ′β & = & 0 \\ α′δδ′γ & = & 0 \\ β′δδ′γ & = & 0 \\ δ′δδ′γ & = & 0 \\ δδ′αα′δ & = & 0 \\ αα′δδ′α & = & 0 \end{aligned}  \begin{aligned}ββ′δδ′β & = & 0 \\ γγ′δδ′γ & = & 0 \\ α′δδ′αα′ & = & 0 \\ β′δδ′αα′ & = & 0 \\ γ′δδ′αα′ & = & 0 \end{aligned}

\begin{aligned}ββ′δ & = & λ_{1}αα′δ \\ γγ′δ & = & λ_{2}αα′δ \\ ββ′α & = & λ_{3}δδ′α \\ γγ′α & = & λ_{4}δδ′α \\ αα′β & = & λ_{5}δδ′β \\ γγ′β & = & λ_{6}δδ′β \\ αα′γ & = & λ_{7}δδ′γ \\ ββ′γ & = & λ_{8}δδ′γ \end{aligned}  \begin{aligned}δ′ββ′ & = & μ_{1}δ′αα′ \\ δ′γγ′ & = & μ_{2}δ′αα′ \\ α′ββ′ & = & μ_{3}α′δδ′ \\ α′γγ′ & = & μ_{4}α′δδ′ \\ β′αα′ & = & μ_{5}β′δδ′ \\ β′γγ′ & = & μ_{6}β′δδ′ \\ γ′αα′ & = & μ_{7}γ′δδ′ \\ γ′ββ′ & = & μ_{8}γ′δδ′ \end{aligned}

\begin{aligned}αα′δδ′ & = & μ_{9}δδ′αα′ \\ ββ′δδ′ & = & μ_{10}δδ′αα′ \\ γγ′δδ′ & = & μ_{11}δδ′αα′ \end{aligned}

where all the λ_{i} and μ_{i} are non-zero.

All the relations in paths of length four and five are an immediate consequence of the relations in paths of length two and three, so that we may consider these as generated by the relations in paths of length two and three. As an example we have

β′αα′δ = λ_{1}^{−1}β′ββ′δ
= λ_{1}^{−1}(0)β′δ
= 0.

Also

αα′δδ′ = μ_{3}^{−1}αα′ββ′
= μ_{3}^{−1}λ_{5}δδ′ββ′
= μ_{3}^{−1}λ_{5}μ_{1}δδ′αα′.

Hence (μ_{9} − μ_{3}^{−1}λ_{5}μ_{1})δδ′αα′ = 0 and since δδ′αα′ is non-zero this means that μ_{9} = μ_{3}^{−1}λ_{5}μ_{1}. Therefore, a set of generators for the relations defining the basic algebra is:

\begin{aligned}δ′δ & = & 0 \\ α′α & = & 0 \end{aligned}  \begin{aligned}β′β & = & 0 \\ γ′γ & = & 0 \end{aligned}
r_{1}αα′ + r_{2}ββ′ + r_{3}γγ′ + r_{4}δδ′ = kδδ′αα′

\begin{aligned}ββ′δ & = & λ_{1}αα′δ \\ γγ′δ & = & λ_{2}αα′δ \\ ββ′α & = & λ_{3}δδ′α \\ γγ′α & = & λ_{4}δδ′α \\ αα′β & = & λ_{5}δδ′β \\ γγ′β & = & λ_{6}δδ′β \\ αα′γ & = & λ_{7}δδ′γ \\ ββ′γ & = & λ_{8}δδ′γ \end{aligned}  \begin{aligned}δ′ββ′ & = & μ_{1}δ′αα′ \\ δ′γγ′ & = & μ_{2}δ′αα′ \\ α′ββ′ & = & μ_{3}α′δδ′ \\ α′γγ′ & = & μ_{4}α′δδ′ \\ β′αα′ & = & μ_{5}β′δδ′ \\ β′γγ′ & = & μ_{6}β′δδ′ \\ γ′αα′ & = & μ_{7}γ′δδ′ \\ γ′ββ′ & = & μ_{8}γ′δδ′. \end{aligned}

Given the generating relations in paths of length three, we see that

δ′ββ′δ = μ_{1}δ′αα′δ

and

δ′ββ′δ = λ_{1}δ′αα′δ

and as δ′αα′δ is non-zero this means that λ_{1} = μ_{1}. Applying similar arguments we find that the generating relations in the scalars λ_{i} and μ_{i} are:

\begin{aligned}ββ′δ & = & λ_{1}αα′δ \\ γγ′δ & = & λ_{2}αα′δ \\ ββ′α & = & λ_{3}δδ′α \\ γγ′α & = & λ_{4}δδ′α \\ αα′β & = & λ_{5}δδ′β \\ γγ′β & = & λ_{6}δδ′β \\ αα′γ & = & λ_{7}δδ′γ \\ ββ′γ & = & λ_{8}δδ′γ \end{aligned}  \begin{aligned}δ′ββ′ & = & λ_{1}δ′αα′ \\ δ′γγ′ & = & λ_{2}δ′αα′ \\ α′ββ′ & = & λ_{3}α′δδ′ \\ α′γγ′ & = & λ_{4}α′δδ′ \\ β′αα′ & = & λ_{5}β′δδ′ \\ β′γγ′ & = & λ_{6}β′δδ′ \\ γ′αα′ & = & λ_{7}γ′δδ′ \\ γ′ββ′ & = & λ_{8}γ′δδ′. \end{aligned}

As each of our paths α, β, γ, δ, α′ and so on correspond to bases of 1dimensional spaces, we can rechoose our basis elements by replacing a given basis element by a non-zero scalar multiple of itself. Let β_{0} be the basis element corresponding to the arrow β. We set β_{1} = λ_{1}^{−1}β_{0} and take this as the new basis element corresponding to the arrow β. Similarly, let γ_{0} be the basis element corresponding to the arrow γ and take γ_{1} = λ_{2}^{−1}γ_{0} to be the new basis element corresponding to the arrow γ. These choices of basis elements have no effect on the zero relations, but the relations in the scalars are now:

r_{1}αα′ + r′_{2}ββ′ + r′_{3}γγ′ + r_{4}δδ′ = kδδ′αα′.

\begin{aligned}ββ′δ & = & αα′δ \\ γγ′δ & = & αα′δ \\ ββ′α & = & λ′_{3}δδ′α \\ γγ′α & = & λ′_{4}δδ′α \\ αα′β & = & λ_{5}δδ′β \\ γγ′β & = & λ′_{6}δδ′β \\ αα′γ & = & λ_{7}δδ′γ \\ ββ′γ & = & λ′_{8}δδ′γ \end{aligned}  \begin{aligned}δ′ββ′ & = & δ′αα′ \\ δ′γγ′ & = & δ′αα′ \\ α′ββ′ & = & λ′_{3}α′δδ′ \\ α′γγ′ & = & λ′_{4}α′δδ′ \\ β′αα′ & = & λ_{5}β′δδ′ \\ β′γγ′ & = & λ′_{6}β′δδ′ \\ γ′αα′ & = & λ_{7}γ′δδ′ \\ γ′ββ′ & = & λ′_{8}γ′δδ′. \end{aligned}

where r′_{2} = r_{2}λ_{1}, r′_{3} = r_{3}λ_{2}, λ′_{3} = λ_{3}λ_{1}^{−1}, λ′_{4} = λ_{4}λ_{2}^{−1}, λ′_{6} = λ_{6}λ_{2}^{−1} and λ′_{8} = λ_{8}λ_{1}^{−1}. We now rechoose basis elements by setting

δ_{1} = λ′_{3}δ_{0}.

The above relations are then

s_{1}αα′ + s_{2}ββ′ + s_{3}γγ′ + s_{4}δδ′ = k′δδ′αα′.

\begin{aligned}ββ′δ & = & αα′δ \\ γγ′δ & = & αα′δ \\ ββ′α & = & δδ′α \\ γγ′α & = & t_{1}δδ′α \\ αα′β & = & t_{2}δδ′β \\ γγ′β & = & t_{3}δδ′β \\ αα′γ & = & t_{4}δδ′γ \\ ββ′γ & = & t_{5}δδ′γ \end{aligned}  \begin{aligned}δ′ββ′ & = & δ′αα′ \\ δ′γγ′ & = & δ′αα′ \\ α′ββ′ & = & α′δδ′ \\ α′γγ′ & = & t_{1}α′δδ′ \\ β′αα′ & = & t_{2}β′δδ′ \\ β′γγ′ & = & t_{3}β′δδ′ \\ γ′αα′ & = & t_{4}γ′δδ′ \\ γ′ββ′ & = & t_{5}γ′δδ′. \end{aligned}

where s_{1} = r_{1}, s_{2} = r′_{2}, s_{3} = r′_{3}, s_{4} = r_{4}(λ′_{3})^{−1}, k′ = k(λ′_{3})^{−1}, t_{1} = λ′_{4}λ_{3}^{−1}, t_{2} = λ_{5}(λ′_{3})^{−1}, t_{3} = λ′_{6}(λ′_{3})^{−1}, t_{4} = λ_{7}(λ′_{3})^{−1} and t_{5} = λ′_{8}(λ′_{3})^{−1}.

Now we have that

δ′αα′ = δ′ββ′

so

δδ′αα′ = δδ′ββ′
= t_{2}^{−1}αα′ββ′
= t_{2}^{−1}αα′δδ′.

Now KG is a symmetric algebra, so the corresponding basic algebra is also symmetric, and hence the basic algebra for the principal block is symmetric. Let φ be the symmetrising form

φ : KQ/I ⟶ K.

By the properties of symmetrising forms we have

φ(δδ′αα′) = φ(αα′δδ′).

But φ is a linear transformation and so we also have from above

φ(δδ′αα′) = t_{2}^{−1}φ(αα′δδ′).

Hence either φ(αα′δδ′) = 0 or t_{2} = 1. But αα′δδ′ is non-zero and spans a 1-dimensional left ideal while ker(φ) has no non-zero left ideals, therefore we cannot have φ(αα′δδ′) = 0. Hence t_{2} = 1.

Similarly we have

δδ′αα′ = δδ′γγ′
= t_{4}^{−1}αα′γγ′
= t_{4}^{−1}t_{1}αα′δδ′.

Using the fact that KQ/I is a symmetric algebra we then have t_{4} = t_{1}. Finally we have

δδ′αα′ = δδ′ββ′
= t_{3}^{−1}γγ′ββ′
= t_{3}^{−1}t_{5}γγ′δδ′
= t_{3}^{−1}t_{5}αα′δδ′.

Using the fact that KQ/I is a symmetric algebra we then have t_{5} = t_{3}. Hence generators for the relations defining the basic algebra are:

\begin{aligned}δ′δ & = & 0 \\ α′α & = & 0 \end{aligned}  \begin{aligned}β′β & = & 0 \\ γ′γ & = & 0 \end{aligned}
s_{1}αα′ + s_{2}ββ′ + s_{3}γγ′ + s_{4}δδ′ = k′δδ′αα′.

\begin{aligned}ββ′δ & = & αα′δ \\ γγ′δ & = & αα′δ \\ ββ′α & = & δδ′α \\ γγ′α & = & t_{1}δδ′α \\ αα′β & = & δδ′β \\ γγ′β & = & t_{3}δδ′β \\ αα′γ & = & t_{1}δδ′γ \\ ββ′γ & = & t_{3}δδ′γ \end{aligned}  \begin{aligned}δ′ββ′ & = & δ′αα′ \\ δ′γγ′ & = & δ′αα′ \\ α′ββ′ & = & α′δδ′ \\ α′γγ′ & = & t_{1}α′δδ′ \\ β′αα′ & = & β′δδ′ \\ β′γγ′ & = & t_{3}β′δδ′ \\ γ′αα′ & = & t_{1}γ′δδ′ \\ γ′ββ′ & = & t_{3}γ′δδ′. \end{aligned}

Now consider the relation

s_{1}αα′ + s_{2}ββ′ + s_{3}γγ′ + s_{4}δδ′ = k′δδ′αα′.

2.2 The Structure of the Basic Algebra of the Principal Block of U_{3}(q) 65 Bearing in mind that δ′δ = 0 we then have that

s_{1}δ′αα′ + s_{2}δ′ββ′ + s_{3}δ′γγ′ = 0

i.e.,

s_{1}δ′αα′ + s_{2}δ′αα′ + s_{3}δ′αα′ = 0

so that

(s_{1} + s_{2} + s_{3})δ′αα′ = 0.

As δ′αα′ ≠ 0 we have

s_{1} + s_{2} + s_{3} = 0.  (2.1)

By multiplying our relation on the left by each of α′, β′ and γ′ respectively we also have the following equations:

s_{2} + s_{3}t_{1} + s_{4} = 0  (2.2)
s_{1} + s_{3}t_{3} + s_{4} = 0  (2.3)
s_{1}t_{1} + s_{2}t_{3} + s_{4} = 0.  (2.4)

By equation 2.1

s_{1} = −s_{2} − s_{3}.

By equation 2.2 we have

s_{4} = −s_{2} − s_{3}t_{1}.

Substituting for s_{1} and s_{4} in equation 2.3 we have

−s_{2} − s_{3} + s_{3}t_{3} − s_{2} − s_{3}t_{1} = 0

i.e.,

−2s_{2} + s_{3}(−1 + t_{3} − t_{1}) = 0.

So, remembering that we are working in a field of characteristic three, we have

s_{2} = s_{3}(1 − t_{3} + t_{1}).

We then have

s_{1} = s_{3}(−1 + t_{3} − t_{1}) − s_{3}
= s_{3}(1 + t_{3} − t_{1})

and

s_{4} = s_{3}(−1 + t_{3} − t_{1}) − s_{3}t_{1}
= s_{3}(−1 + t_{3} + t_{1}).

This tells us that s_{3} is not zero, for if s_{3} = 0 then s_{2} = s_{4} = s_{1} = 0 which is not possible. Hence we can divide our original equation through by s_{3} and the relation becomes

(1 + t_{3} − t_{1})αα′ + (1 − t_{3} + t_{1})ββ′ + γγ′ + (−1 + t_{3} + t_{1})δδ′ = k′′δδ′αα′

for some scalar k′′.

Now, given equation 2.4, we have

(1 + t_{3} − t_{1})t_{1} + (1 − t_{3} + t_{1})t_{3} + (−1 + t_{3} + t_{1}) = 0

i.e.,

−1 − t_{1} − t_{3} − t_{1} − t_{1}t_{3} − t_{1}^{2} − t_{3}^{2} = 0

or equivalently

t_{3}^{2} + (t_{1} + 1)t_{3} + (t_{1}^{2} + t_{1} + 1) = 0.

Hence we have a quadratic in t_{3} and using the quadratic formula this means

t_{3} = \frac{−(t_{1} + 1) ± \sqrt{(t_{1} + 1)^{2} − 4(t_{1}^{2} + t_{1} + 1)}}{2}
= (t_{1} + 1) ± \sqrt{t_{1}}
= (\sqrt{t_{1}} ± 1)^{2}
= (−\sqrt{t_{1}} ∓ 1)^{2}.

Hence there exists a non-zero scalar λ ∈ K with λ^{2} = t_{1} and (λ − 1)^{2} = t_{3}.

It does not appear to be possible to reduce the number of scalars any further and therefore the relations

\begin{aligned}α′α & = & 0 \\ β′β & = & 0 \end{aligned}  \begin{aligned}γ′γ & = & 0 \\ δ′δ & = & 0 \end{aligned}
(λ − 1)αα′ + (−λ)ββ′ + γγ′ + λ(1 − λ)δδ′ = k′′δδ′αα′

\begin{aligned}ββ′δ & = & αα′δ \\ γγ′δ & = & αα′δ \\ ββ′α & = & δδ′α \\ γγ′α & = & λ^{2}δδ′α \\ αα′β & = & δδ′β \\ γγ′β & = & (λ − 1)^{2}δδ′β \\ αα′γ & = & λ^{2}δδ′γ \\ ββ′γ & = & (λ − 1)^{2}δδ′γ \end{aligned}  \begin{aligned}δ′ββ′ & = & δ′αα′ \\ δ′γγ′ & = & δ′αα′ \\ α′ββ′ & = & α′δδ′ \\ α′γγ′ & = & λ^{2}α′δδ′ \\ β′αα′ & = & β′δδ′ \\ β′γγ′ & = & (λ − 1)^{2}β′δδ′ \\ γ′αα′ & = & λ^{2}γ′δδ′ \\ γ′ββ′ & = & (λ − 1)^{2}γ′δδ′ \end{aligned}

determine a 2-parameter family of algebras to which the basic algebra of the principal block must belong.

The Structure of the PIMs in the Non-Principal Blocks of the Higman–Sims Group

In this chapter we will determine the Loewy structure and socle structure of the PIMs in the non-principal blocks of the Higman–Sims group over an algebraically closed field of characteristic three (the more difficult case of the principal block will be dealt with in the next chapter). We will see that among the non-principal blocks there are three blocks of defect zero, one block of defect one and one block of defect two. The Loewy structure of the defect zero and defect one blocks will be obvious and we will essentially only have to deal with the block of defect two. In order to determine the structure of the PIMs of this block we will need to calculate the Loewy structure of the PIMs of a subgroup of Higman–Sims closely related to one of the unitary groups of Chapter 2 and then make use of induction and restriction techniques. We prove the following theorem: Theorem 3.0.1 The Loewy structures and socle structures of the PIMs of the non-principal block of Higman–Sims of defect two are as follows:

\begin{matrix}49_{a} \\ 77 \\ 49_{b} & 770_{b} \\ 77 \\ 49_{a} \end{matrix}  \begin{matrix}49_{b} \\ 77 \\ 49_{a} & 770_{a} \\ 77 \\ 49_{b} \end{matrix}  \begin{matrix}154_{b} \\ 770_{b} \\ 77 & 154_{c} \\ 770_{a} \\ 154_{b} \end{matrix}

\begin{matrix}154_{c} \\ 770_{a} \\ 77 & 154_{b} \\ 770_{b} \\ 154_{c} \end{matrix}  \begin{aligned}& 770_{a} \\ 77 & 154_{b} \\ 49_{b} & 770_{a} & 770_{b} \\ 77 & 154_{c} \\ & 770_{a} \end{aligned}  \begin{aligned}& & 770_{b} \\ & 77 & 154_{c} \\ 49_{a} & 770_{b} & 770_{a} \\ & 77 & 154_{b} \\ & & 770_{b} \end{aligned} ; \begin{aligned}& 77 \\ 770_{a} & 49_{a} & 49_{b} & 770_{b} \\ 154_{b} & 77 & 77 & 77 & 154_{c} \\ 770_{a} & 49_{a} & 49_{b} & 770_{b} \\ & 77 \end{aligned}

In accordance with Atlas notation (Conway et al. (1985)), we will frequently denote the Higman–Sims group by HS. The ordinary character table and 3modular Brauer character table for HS will be found in Appendix D.

3.1 The Cartan Matrix

The decomposition matrix for HS has been determined using the ordinary and 3-modular Brauer character tables taken from Conway et al. (1985) and Parker respectively and may be found in Appendix D. Given this decomposition matrix we find that the Cartan matrix for the non-principal blocks of HS is

49_{a} 49_{b} 77 154_{b} 154_{c} 770_{a} 770_{b} 231 825 693 1386 2520
49_{a} 2 1 2 0  0  0  1
49_{b} 1 2 2 0  0  1  0
77 2 2 5 1  1  2  2
154_{b} 0 0 1 2  1  1  1
154_{c} 0 0 1 1  2  1  1
770_{a} 0 1 2 1  1  3  1
770_{b} 1 0 2 1  1  1  3
231  2 1
825  1 2
693  1
1386  1
2520  1

From the Cartan matrix the Loewy structures of the PIMs of all but one of the non-principal blocks are immediate. Determining the structure of the PIMs in the block of defect two is far more difficult however and we will have to resort to inducing and restricting between HS and one of its subgroups. From now on we will refer to this block of defect two as the second block of Higman–Sims (the first being the principal block).

HS contains two classes of maximal subgroup isomorphic to U_{3}(5) : 2, the semi-direct product of U_{3}(5) with C_{2}, corresponding to a point stabiliser and a quadric stabiliser in the Higman lattice respectively, and these are conjugate under the duality which interchanges points and quadrics (Conway et al (1985)). These groups have index 176 which is not divisible by three and so information about the structure of the PIMs will be very useful in determining the structure of the PIMs of HS. With the aid of the material of Chapter 2 we will now determine the structure of the PIMs of U_{3}(5) : 2 and the values of dim_{K}Ext^{1}(S, T) for all simple U_{3}(5) : 2-modules S and T.

3.2 The PIMs of U_{3}(5) : 2

The group U_{3}(5) : 2 contains a normal subgroup isomorphic to U_{3}(5) of index 2 and we will find this group very useful in calculating the Loewy structures of the PIMs of U_{3}(5) : 2. The character tables, decomposition matrix and Cartan matrix for U_{3}(5) may be found in Appendix B. Given the Cartan matrix for U_{3}(5) and the fact that 5 ≡ 2 mod 3 and 5 ≢ 8 mod 9, from Theorem 2.1.1, the Loewy and socle structure of the PIMs of U_{3}(5) are:

\begin{aligned}& 1 \\ & 20 \\ 28_{a} & 28_{b} & 28_{c} \\ & 20 \\ & 1 \end{aligned}  \begin{aligned}& 28_{a} \\ & 20 \\ 28_{b} & 1 & 28_{c} \\ & 20 \\ & 28_{a} \end{aligned}  \begin{aligned}& 28_{b} \\ & 20 \\ 28_{c} & 1 & 28_{a} \\ & 20 \\ & 28_{b} \end{aligned}

\begin{aligned}& 28_{c} \\ & 20 \\ 28_{a} & 1 & 28_{b} \\ & 20 \\ & 28_{c} \end{aligned}  \begin{aligned}& 20 \\ 1 & 28_{a} & 28_{b} & 28_{c} \\ 20 & 20 & 20 \\ 1 & 28_{a} & 28_{b} & 28_{c} \\ & 20 \end{aligned}  21 84 21  84 21 84

126_{a}  126_{b}  126_{c}  144_{a}  144_{b}

3.2 The PIMs of U_{3}(5) : 2 71

We may regard U_{3}(5) : 2 as being both the semi-direct product of U_{3}(5) with C_{2} and a subgroup of the automorphism group of U_{3}(5) (which, using the notation of Conway et al., has structure U_{3}(5).S_{3}, as the outer automorphism group of U_{3}(5) is isomorphic to S_{3}). Given this second interpretation we can use the Atlas of Finite Groups (Conway et al. (1985)) and the Modular Atlas (Parker) to calculate character tables for U_{3}(5) : 2. The automorphism group of U_{3}(5) contains three extensions of U_{3}(5) by elements of order two and these are all isomorphic. We choose the extension in which 28_{a} splits, and 28_{b} and 28_{c} fuse. The fusing and splitting information for this group can be found in tables B.1 and B.2 of the Appendix. Notice that the simple U_{3}(5)-module 28_{a} splits in U_{3}(5) : 2 into a direct sum of two simple U_{3}(5) : 2-modules of dimension 28, we will call these modules 28_{A} and 28_{B}. This notation should not be too confusing as it will be clear when we are dealing with U_{3}(5)-modules and when we are dealing with U_{3}(5) : 2-modules.

Given the fusing and splitting information for this group, we can determine the character tables and decomposition matrix for U_{3}(5) : 2, these will be found in Appendix C. From the decomposition matrix we see that there are four blocks of defect zero and that the Cartan matrix for the rest of the PIMs of U_{3}(5) : 2 is

1_{A} 1_{B} 20_{A} 20_{B} 28_{A} 28_{B} 56 21_{A} 84_{A} 21_{B} 84_{B}
1_{A} 2 0 1 1 1 0 1
1_{B} 0 2 1 1 0 1 1
20_{A} 1 1 3 2 1 1 2
20_{B} 1 1 2 3 1 1 2
28_{A} 1 0 1 1 2 0 1
28_{B} 0 1 1 1 0 2 1
56 1 1 2 2 1 1 3
21_{A}  2 1
84_{A}  1 2
21_{B}  2 1
84_{B}  1 2

The Loewy and socle structure of the PIMs in all but the principal block of U_{3}(5) : 2 follow immediately, so we only need deal with the principal block of U_{3}(5) : 2.

Consider the structure of the following modules induced from U_{3}(5) to U_{3}(5) : 2:

\begin{aligned}1↑ & = & 1_{A} ⊕ 1_{B} \\ 20↑ & = & 20_{A} ⊕ 20_{B} \\ 28_{a}↑ & = & 28_{A} ⊕ 28_{B} \end{aligned}  \begin{aligned}28_{b}↑ & = & 56 \\ 28_{c}↑ & = & 56 \end{aligned}

and the following modules restricted from U_{3}(5) : 2 to U_{3}(5):

\begin{aligned}1_{A}↓ & = & 1 \\ 1_{B}↓ & = & 1 \\ 20_{A}↓ & = & 20 \\ 20_{B}↓ & = & 20 \end{aligned}  \begin{aligned}28_{A}↓ & = & 28_{a} \\ 28_{B}↓ & = & 28_{a} \\ 56↓ & = & 28_{b} ⊕ 28_{c}. \end{aligned}

We see that for all simple modules S in the principal block of U_{3}(5), S↑ is a semisimple module belonging to the principal block of U_{3}(5) : 2, and that for all simple modules T of the principal block of U_{3}(5) : 2, T ↓ is a semisimple module belonging to the principal block of U_{3}(5). Hence by Theorem 1.2.9

L_{i}(M↑) = L_{i}(M)↑

for all modules M in the principal block of U_{3}(5).

Given the structure of the induced modules and the fact that L_{i}(P_{1}↑) = L_{i}(P_{1})↑, the Loewy structure of P_{1}↑ = P_{1_{A}} ⊕ P_{1_{B}} is

1_{A} 1_{B}
20_{A} 20_{B}
56 28_{A} 28_{B} 56
20_{A} 20_{B}
1_{A} 1_{B} .

From this we see that either E (1_{A}, 20_{A}) = 1 or E (1_{A}, 20_{B}) = 1 (but not both). As 20_{A} and 20_{B} differ only on the conjugacy classes of elements of order 20 and these two conjugacy classes differ only in the values taken by the two characters of dimension 20, we can choose the module 20_{A} to be the module non-trivially extended by 1_{A}, and the conjugacy class 20A to be the class on which 20_{A} takes the value i5. Then we have E (1_{A}, 20_{A}) = 1 and E (1_{A}, 20_{B}) = 0. As the composition factors of P_{1_{A}} are 2.1_{A} + 20_{A} + 20_{B} + 28_{A} + 56 and the composition factors of P_{1_{B}} are 2.1_{B} + 20_{A} + 20_{B} + 28_{B} + 56, this forces the Loewy structures of P_{1_{A}} and P_{1_{B}} to be

\begin{matrix}1_{A} \\ 20_{A} \\ 56 & 28_{A} \\ 20_{B} \\ 1_{A} \end{matrix}  and  \begin{matrix}1_{B} \\ 20_{B} \\ 56 & 28_{B} \\ 20_{A} \\ 1_{B} \end{matrix}

respectively. For any module M we have S_{i}(M^{∗}) ≅ L_{i}(M)^{∗} for all i ≤ ℓ(M) and as 20_{A} and 20_{B} are mutually dual and the other composition factors are all self-dual, the socle structure is equal to the Loewy structure in each of these cases.

If we induce P_{28_{a}} of U_{3}(5) to U_{3}(5) : 2 in the same way, the Loewy structure of P_{28_{A}} ⊕ P_{28_{B}} is

28_{A} 28_{B}
20_{A} 20_{B}
56 1_{A} 1_{B} 56
20_{A} 20_{B}
28_{A} 28_{B} .

This time we have either E (28_{A}, 20_{A}) = 1 or E (28_{A}, 20_{B}) = 1. But from the socle structure of P_{1_{A}} we see that E (28_{A}, 20_{B}) ≥ 1 and therefore E (28_{A}, 20_{B}) = 1. Given the composition factors of P_{28_{a}} and P_{28_{b}}, this forces the Loewy structures of P_{28_{A}} and P_{28_{B}} to be

\begin{matrix}28_{A} \\ 20_{B} \\ 56 & 1_{A} \\ 20_{A} \\ 28_{A} \end{matrix}  and  \begin{matrix}28_{B} \\ 20_{A} \\ 56 & 1_{B} \\ 20_{B} \\ 28_{B} \end{matrix}

respectively. Again as each of these modules is self-dual and S_{i}(M^{∗}) ≅ L_{i}(M)^{∗} for all i ≤ ℓ(M), the socle structure is equal to the Loewy structure in each of these cases.

3.2 The PIMs of U_{3}(5) : 2 74

Similarly, P_{20_{A}} ⊕ P_{20_{B}} has Loewy structure

20_{A} 20_{B}
1_{A} 1_{B} 28_{A} 28_{B} 56 56
20_{A} 20_{B} 20_{A} 20_{B} 20_{A} 20_{B}
1_{A} 1_{B} 28_{A} 28_{B} 56 56
20_{A} 20_{B}  .

Now, by Lemma 1.3.1 (iii),

E (20_{A}, 1_{A}) = E (1_{A}^{∗}, 20_{A}^{∗})
= E (1_{A}, 20_{B})
= 0.

Similarly, we have the following values:

E (20_{B}, 1_{B}) = 0
E (20_{A}, 1_{B}) = 1
E (20_{B}, 1_{A}) = 1.

We also have:

\begin{aligned}E (20_{A}, 28_{A}) & = & 1 \\ E (20_{B}, 28_{B}) & = & 1 \end{aligned}  \begin{aligned}E (20_{A}, 28_{B}) & = & 0 \\ E (20_{B}, 28_{A}) & = & 0. \end{aligned}

Furthermore, for both 20_{A} and 20_{B},

E (20_{i}, 56) = E (20_{i}, 28_{b}↑)
= E (20, 28_{b})
= 1

by Lemma 1.3.2. Given all this information the Loewy structures of P_{20_{A}} and P_{20_{B}} are

\begin{aligned}& 20_{A} \\ 1_{B} & 56 & 28_{A} \\ 20_{B} & 20_{A} & 20_{B} \\ 1_{A} & 56 & 28_{B} \\ & 20_{A} \end{aligned}  and  \begin{aligned}& 20_{B} \\ 1_{A} & 56 & 28_{B} \\ 20_{A} & 20_{B} & 20_{A} \\ 1_{B} & 56 & 28_{A} \\ & 20_{B} \end{aligned}

respectively. As S_{i}(M^{∗}) ≅ L_{i}(M)^{∗} for all i ≤ ℓ(M) and 20_{A} and 20_{B} are mutually dual, the socle structure is equal to the Loewy structure in both cases.

Finally, by inducing P_{28_{b}}, we have

P_{56} = \begin{aligned}& 56 \\ 20_{A} & 20_{B} \\ 1_{A} & 1_{B} & 28_{A} & 28_{B} & 56 \\ 20_{A} & 20_{B} \\ & 56 \end{aligned}

which is both the Loewy structure and the socle structure of P_{56}.

The table of values of dim_{K}Ext^{1}(S, T) for the simple U_{3}(5) : 2-modules of the non-trivial blocks is then:

S\T 1_{A} 1_{B} 20_{A} 20_{B} 28_{A} 28_{B} 56 21_{A} 84_{A} 21_{B} 84_{B}
1_{A} 0 0 1 0 0 0 0
1_{B} 0 0 0 1 0 0 0
20_{A} 0 1 0 0 1 0 1
20_{B} 1 0 0 0 0 1 1
28_{A} 0 0 0 1 0 0 0
28_{B} 0 0 1 0 0 0 0
56 0 0 1 1 0 0 0
21_{A}  0 1
84_{A}  1 0
21_{B}  0 1
84_{B}  1 0

3.3 Induction and Restriction Between U_{3}(5) : 2

and HS

3.3 Induction and Restriction Between U_{3}(5) : 2 and HS 76

Before we can compare characters of HS and U_{3}(5) : 2 we need to know which conjugacy classes of HS the conjugacy classes of U_{3}(5) : 2 belong to. By looking at the restrictions to U_{3}(5) : 2 of some of the smaller characters of HS, we will see that by an appropriate renaming of conjugacy classes of HS and simple HSmodules, there is a maximal subgroup of HS isomorphic to U_{3}(5) : 2 in which the conjugacy classes lie in the following conjugacy classes of HS

\begin{matrix}U_{3}(5) : 2 & HS \\ 1A & 1A \\ 2A & 2A \\ 2B & 2B \\ 3A & 3A \\ 4A & 4C \\ 4B & 4A \\ 5A & 5A \\ 5B & 5B \\ 5C & 5C \\ 6A & 6B \end{matrix}  \begin{matrix}U_{3}(5) : 2 & HS \\ 6B & 6A \\ 7A & 7A \\ 8A & 8B \\ 8B & 8C \\ 10A & 10A \\ 10B & 10B \\ 12B & 12A \\ 20A & 20A \\ B^{∗∗} & B^{∗∗} \end{matrix}

By restricting characters of HS to U_{3}(5) : 2, we can determine which conjugacy classes of HS all the conjugacy classes of U_{3}(5) : 2 lie in, except those with elements of order 8 and those with elements of order 20. For the conjugacy classes of elements of order 8 we find that we do not have enough information to tell whether the conjugacy class 8A of U_{3}(5) : 2 lies in the conjugacy class 8B or 8C of HS (or whether the conjugacy class 8B of U_{3}(5) : 2 lies in the conjugacy class 8C or 8B of HS). But the conjugacy classes 8B and 8C of HS differ only in the values taken on them by the HS characters 154_{b} and 154_{c}, and these characters differ only in the values they take on the conjugacy classes 8B and 8C. Therefore, by relabelling if necessary, we choose the conjugacy class 8B of HS to be the conjugacy class which 8A of U_{3}(5) : 2 lies in (and then 8B of U_{3}(5) : 2 lies in 8C of HS), and we choose 154_{b} to be the simple HS-module of dimension 154 which takes character value 2 on the conjugacy class 8B (and −2 on 8C).

Similarly, it is not possible to determine whether the conjugacy class 20A of U_{3}(5) : 2 lies in the conjugacy class 20A of HS or the conjugacy class B^{∗∗} of HS. By relabelling if necessary, we can choose 20A to be the conjugacy class of HS which the conjugacy class 20A of U_{3}(5) : 2 lies in (and then B^{∗∗} of U_{3}(5) : 2 lies in B^{∗∗} of HS) and choose the characters 49_{a} and 770_{a} to be the characters of degrees 49 and 770 respectively, which take the value i5 on 20A (and −i5 on B^{∗∗}).

In order to find the Loewy structure of the restricted modules, we will need to make use of information about the structure of some of the HS-modules induced from the simples of U_{3}(5) : 2. We calculate the characters of these modules using the formula

χ(g, M↑^{G}) = ∑ ; c.c of h ∈ H with h ∼ g in G[C_{G}(h) : C_{H}(h)]χ(h, M)

where the sum is taken over those representatives of the conjugacy classes of H which are conjugate to g in G (see Isaacs (1976) p. 64). The induced modules then have composition factors as follows:

1_{A}↑ ∼ 1 + 77 + 49_{a} + 49_{b}
1_{B}↑ ∼ 154_{b} + 22
20_{A}↑ ∼ 770_{a} + 1253 + 1176 + 321
20_{B}↑ ∼ 770_{b} + 1253 + 1176 + 321
21_{A}↑ ∼ 2520 + 693 + 231 + 2.77 + 49_{a} + 49_{b}
21_{B}↑ ∼ 825 + 770_{a} + 770_{b} + 748 + 154_{b} + 2.154_{c} + 77 + 2.22
28_{A}↑ ∼ 2520 + 1386 + 693 + 154_{a} + 77 + 49_{a} + 49_{b}
28_{B}↑ ∼ 1176 + 2.748 + 770_{a} + 770_{b} + 2.154_{b} + 2.154_{c} + 77 + 22 + 1
56↑ ∼ 2520 + 2.1253 + 1176 + 2.825 + 2.748 + 231 + 154_{a}
+ 77 + 2.22 + 2.1.

The simple modules of the second block of HS restricted down to the subgroup U_{3}(5) : 2 have the following composition factors:

\begin{aligned}49_{a}↓ & ∼ & 28_{A} + 20_{A} + 1_{A} \\ 49_{b}↓ & ∼ & 28_{A} + 20_{B} + 1_{A} \\ 77↓ & ∼ & 56 + 21_{A} \\ 154_{b}↓ & ∼ & 56 + 2.28_{B} + 20_{A} + 20_{B} + 2.1_{B} \\ 154_{c}↓ & ∼ & 84_{B} + 28_{B} + 2.21_{B} \\ 770_{a}↓ & ∼ & 288 + 252 + 126_{A} + 84_{A} + 20_{A} \\ 770_{b}↓ & ∼ & 288 + 252 + 126_{A} + 84_{A} + 20_{B}. \end{aligned}

Now we determine the structures of the simple modules of the second block of HS restricted down to U_{3}(5) : 2.

49_{a}↓ has composition factors 1_{A}+20_{A}+28_{A}. As 20_{A}↑ has composition factors 770_{a} + 321 + 1176 + 1253, Frobenius reciprocity tells us that

(49_{a}↓ , 20_{A}) = (20_{A}, 49_{a}↓) = 0.

Therefore 20_{A} is extended non-trivially by some other composition factor of 49_{a}↓ and extends another composition factor of 49_{a}↓ non-trivially. As E (1_{A}, 20_{A}) = 1 and E (28_{A}, 20_{A}) = 0, the structure of 49_{a}↓ must be

49_{a}↓ = \begin{matrix}1_{A} \\ 20_{A} \\ 28_{A} \end{matrix} .

By duality,

49_{b}↓ = \begin{matrix}28_{A} \\ 20_{B} \\ 1_{A} \end{matrix} .

As 56 and 21_{A} are in different blocks we have

77↓ = 56 ⊕ 21_{A}.

To find the structure of 154_{b}↓, which has composition factors 56 + 2.28_{B} + 20_{A} + 20_{B} + 2.1_{B}, we note firstly that as the composition factors of 1_{B}↑ are in different blocks,

1_{B}↑ = 154_{b} ⊕ 22

and therefore

(1_{B}, 154_{b}↓) = (154_{b}↓ , 1_{B}) = 1.

We also have

(20_{i}, 154_{b}↓) = (154_{b}↓ , 20_{i}) = 0

for both 20_{A} and 20_{B}, as neither 20_{A}↑ nor 20_{B}↑ has a composition factor isomorphic to 154_{b}. Similarly,

(56, 154_{b}↓) = (154_{b}↓ , 56) = 0.

Hence 56 is extended non-trivially by one of 20_{A} and 20_{B}, and non-trivially extends the other, while the 20s themselves cannot lie in the head or the socle.

Given the values of dim_{K}Ext^{1} for the simple U_{3}(5) : 2-modules, the only possibilities for the structure of 154_{b}↓ are

1_{B} ⊕ \begin{matrix}28_{B} \\ 20_{A} \\ 1_{B} & 56 \\ 20_{B} \\ 28_{B} \end{matrix} , 28_{B} ⊕ \begin{matrix}1_{B} \\ 20_{B} \\ 28_{B} & 56 \\ 20_{A} \\ 1_{B} \end{matrix} , or 28_{B} ⊕ 28_{B} ⊕ \begin{matrix}1_{B} \\ 20_{B} \\ 56 \\ 20_{A} \\ 1_{B} \end{matrix} .

Notice that for the third choice to be possible we would have a module 1_{B} 20_{B} 56 20_{A} 1_{B}

which must be a quotient of P_{1_{B}}. But the only quotient of P_{1_{B}} with two composition factors isomorphic to 1_{B} is P_{1_{B}}, hence this is not possible. Therefore we have

154_{b}↓ = 1_{B} ⊕ \begin{matrix}28_{B} \\ 20_{A} \\ 1_{B} & 56 \\ 20_{B} \\ 28_{B} \end{matrix}  or  154_{b}↓ = 28_{B} ⊕ \begin{matrix}1_{B} \\ 20_{B} \\ 28_{B} & 56 \\ 20_{A} \\ 1_{B} \end{matrix} .

Given the values of dim_{K}Ext^{1} for the simple U_{3}(5) : 2-modules and the fact that 154_{c} and therefore 154_{c}↓ are self-dual, the only possible structures for 154_{c}↓ are

28_{B} ⊕ 21_{B} ⊕ 84_{B} ⊕ 21_{B}  or  28_{B} ⊕ \begin{matrix}21_{B} \\ 84_{B} \\ 21_{B} \end{matrix} .

Consider the structure of that part of 21_{B}↑ in the second block. Given that none of the modules 77↓, 154_{b}↓, 770_{a}↓ and 770_{b}↓ have a composition factor isomorphic to 21_{B}, we have:

\begin{aligned}(21_{B}↑ , 77) & = & (77, 21_{B}↑) & = & 0 \\ (21_{B}↑ , 154_{b}) & = & (154_{b}, 21_{B}↑) & = & 0 \\ (21_{B}↑ , 770_{i}) & = & (770_{i}, 21_{B}↑) & = & 0 \end{aligned}

for both 770_{a} and 770_{b}. Hence for that part of the module lying in the second block, the head must be 154_{c} and the socle must be 154_{c}, forcing

(21_{B}, 154_{c}↓) = (154_{c}↓ , 21_{B}) = 1.

Therefore

154_{c}↓ = 28_{B} ⊕ \begin{matrix}21_{B} \\ 84_{B} \\ 21_{B} \end{matrix} .

Finally, as 288, 252, 126_{A}, 84_{A} and 20_{i} are all in different blocks for both 20_{A} and 20_{B}, we have

770_{a}↓ = 288 ⊕ 252 ⊕ 126_{A} ⊕ 84_{A} ⊕ 20_{A}

and

770_{b}↓ = 288 ⊕ 252 ⊕ 126_{A} ⊕ 84_{A} ⊕ 20_{B}.

3.4 The PIMs

Before dealing with the structure of the PIMs, let us determine the structure of certain induced modules.

The composition factors of 1_{A}↑ are 1+49_{a}+49_{b}+77. By Frobenius reciprocity we have:

\begin{aligned}(1_{A}↑ , 49_{a}) & = & 0 \\ (1_{A}↑ , 49_{b}) & = & 1 \\ (1_{A}↑ , 77) & = & 0 \end{aligned}  \begin{aligned}(49_{a}, 1_{A}↑) & = & 1 \\ (49_{b}, 1_{A}↑) & = & 0 \\ (77, 1_{A}↑) & = & 0. \end{aligned}

As 1 is in a different block from the other composition factors we know that 1 is a direct summand of 1_{A}↑ and therefore the structure of 1_{A}↑ is

1_{A}↑ = 1 ⊕ \begin{matrix}49_{b} \\ 77 \\ 49_{a} \end{matrix} .

In particular the right hand summand has simple socle isomorphic to 49_{a} and so must be a submodule of P_{49_{a}}.

Now consider the structure of the second block component of the module 28_{A}↑. By Frobenius reciprocity we have:

\begin{aligned}(28_{A}↑ , 77) & = & 0 \\ (28_{A}↑ , 49_{a}) & = & 1 \\ (28_{A}↑ , 49_{b}) & = & 0 \end{aligned}  \begin{aligned}(77, 28_{A}↑) & = & 0 \\ (49_{a}, 28_{A}↑) & = & 0 \\ (49_{b}, 28_{A}↑) & = & 1 \end{aligned}

and so the component of 28_{A}↑ in the second block of HS is

\begin{matrix}49_{a} \\ 77 \\ 49_{b} \end{matrix}

Note that this module must be a quotient of P_{49_{a}}.

P_{49_{a}} has composition factors 2.49_{a} + 49_{b} + 2.77 + 770_{b}. The existence of a quotient of P_{49_{a}} of the form

\begin{matrix}49_{a} \\ 77 \\ 49_{b} \end{matrix}

means that we have a composition factor isomorphic to 77 in the second Loewy layer of P_{49_{a}} and a composition factor isomorphic to 49_{b} in the third Loewy layer P_{49_{a}}, by Lemma 1.1.5. The module

\begin{matrix}49_{b} \\ 77 \\ 49_{a} \end{matrix}

must then be a submodule of Rad^{2}(P_{49_{a}}) by Lemma 1.4.1. Now P_{49_{a}} has only six composition factors and so must have Loewy length less than or equal to six. But the only possible Loewy structures of Loewy length six are

\begin{matrix}49_{a} \\ 77 \\ 49_{b} \\ 77 \\ 770_{b} \\ 49_{a} \end{matrix}  and  \begin{matrix}49_{a} \\ 77 \\ 49_{b} \\ 770_{b} \\ 77 \\ 49_{a} \end{matrix}

neither of which has a submodule

\begin{matrix}49_{b} \\ 77 \\ 49_{a} . \end{matrix}

Hence the Loewy length must be five. This immediately forces the Loewy structure of P_{49_{a}} to be

\begin{aligned}& 49_{a} \\ 77 & 770_{b} \\ & 49_{b} \\ & 77 \\ & 49_{a} \end{aligned} ,  \begin{matrix}49_{a} \\ 77 \\ 49_{b} & 770_{b} \\ 77 \\ 49_{a} \end{matrix} ,  or  \begin{aligned}& 49_{a} \\ & 77 \\ & 49_{b} \\ 77 & 770_{b} \\ & 49_{a} \end{aligned} .

But 49_{b} cannot extend 770_{b} non-trivially, as 770_{b} is not a composition factor of P_{49_{b}} and so the only possibilities are in fact

\begin{aligned}& 49_{a} \\ 77 & 770_{b} \\ & 49_{b} \\ & 77 \\ & 49_{a} \end{aligned}  and  \begin{matrix}49_{a} \\ 77 \\ 49_{b} & 770_{b} \\ 77 \\ 49_{a} \end{matrix} .

Suppose we have a module

M = \begin{matrix}49_{a} \\ 770_{b} \end{matrix} .

Then M↓ has composition factors 288 + 252 + 126_{A} + 84_{A} + 20_{B} + 28_{A} + 20_{A} + 1_{A} and given the structure of the modules on pages 80–81, we have by Frobenius reciprocity,

(1_{A}, M↓) = 0 and (M↓ , 1_{A}) = 1

while

(28_{A}, M↓) = 0 and (M↓ , 28_{A}) = 0.

Now 20_{A}↑ has composition factors 770_{a} + 321 + 1176 + 1253 and therefore

(20_{A}, M↓) = 0 and (M↓ , 20_{A}) = 0.

20_{B}↑ has composition factors 770_{b} + 321 + 1176 + 1253 and as 770_{b} is the only composition factor of 20_{B}↑ in the second block, it is a direct summand of 20_{B}↑. Therefore by Frobenius reciprocity we have

(20_{B}, M↓) = 1 and (M↓ , 20_{B}) = 0.

Hence the structure of M↓ is

288 ⊕ 252 ⊕ 126_{a} ⊕ 84_{A} ⊕ N

where N has Loewy structure

\begin{matrix}1_{A} \\ 20_{A} \\ 28_{A} \\ 20_{B} . \end{matrix}

This module must be a quotient of P_{1_{A}}, so there exists a homomorphism φ : P_{1_{A}} ⟶ N, and then ker φ is a submodule of P_{1_{A}} with composition factors 1_{A} and 56. As 56 and 1_{A} do not non-trivially extend each other ker φ must have structure 1_{A} ⊕56. But then 56 would be in the socle of P_{1_{A}} which is not possible. Hence there is no such module as M and 49_{a} cannot extend 770_{b} non-trivially. Therefore the Loewy structure of P_{49_{a}} is

49_{a} 77

49_{b} 770_{b}

77

49_{a} .

The HS-modules 49_{a} and 49_{b} are conjugate under an outer automorphism of HS which has order two (Conway et al. (1985)). Hence there is an automorphism of order two which swaps the PIMs P_{49_{a}} and P_{49_{b}} such that the Loewy structure of P_{49_{b}} is the image of the Loewy structure of P_{49_{a}} under the automorphism. As this automorphism swaps 49_{a} and 49_{b}, 154_{b} and 154_{c}, and 770_{a} and 770_{b}, while fixing the simple module 77, the Loewy structure of P_{49_{b}} is

49_{b} 77

49_{a} 770_{a}

77

49_{b} .

Note that by duality the socle structures of P_{49_{a}} and P_{49_{b}} are the same as their Loewy structures.

Next we consider P_{154_{c}} which has composition factors 77 + 154_{b} + 2.154_{c} + 770_{a} + 770_{b}. Any extension of 77↓ by 154_{c}↓ has composition factors 56 + 21_{A} + 28_{B} + 21_{B} + 84_{B} + 21_{B}, and so given the values of dim_{K}Ext^{1} for U_{3}(5) : 2 and the fact that there is a submodule isomorphic to 77↓ with quotient isomorphic to 154_{c}↓, the only possibility for the structure of such an extension is

56 ⊕ 21_{A} ⊕ 28_{B} ⊕ \begin{matrix}21_{B} \\ 84_{B} \\ 21_{B} \end{matrix} .

Therefore E (154_{c}↓ , 77↓) = 0 and hence by the discussion following Lemma 1.2.3, we have E (154_{c}, 77) = 0. Similarly if we consider extensions of 154_{b}↓ by 154_{c}↓ we see that

\begin{matrix}21_{B} \\ 84_{B} \\ 21_{B} \end{matrix}

is in a different block from the rest of the composition factors and 28_{B} cannot extend 1_{B}, P_{28_{B}}, 28_{B} or P_{1_{B}} non-trivially. Hence for either possible structure of 154_{b}↓, the only possible extension of 154_{b}↓ by 154_{c}↓ is split, so that E (154_{c}, 154_{b}) = 0. Finally we see that the only possibility for an extension of 770_{b}↓ by 154_{c}↓ is

288 ⊕ 252 ⊕ 126_{A} ⊕ 84_{A} ⊕ 20_{B} ⊕ 28_{B} ⊕ \begin{matrix}21_{B} \\ 84_{B} \\ 21_{B} \end{matrix}

as 28_{B} cannot extend 20_{B} non-trivially. Hence E (154_{c}, 770_{b}) = 0. As P_{154_{c}} is self-dual, this forces the Loewy structure to be

154_{c}
770_{a}
77 154_{b}
770_{b}
154_{c} .

The socle structure is then the same by duality. By taking the image of P_{154_{c}} under the outer automorphism of HS of order two, we see that the Loewy structure of P_{154_{b}} is then

154_{b}
770_{b}
77 154_{c}
770_{a}
154_{b} .

The socle structure is the same by duality.

P_{770_{a}} has composition factors 49_{b} + 2.77 + 154_{b} + 154_{c} + 3.770_{a} + 770_{b}. Given the structure of P_{154_{c}} we have a module with Loewy and socle structure

\begin{matrix}770_{a} \\ 77 & 154_{b} \\ 770_{b} \\ 154_{c} \end{matrix}

which is a quotient of P_{770_{a}}. Given the structure of P_{49_{b}} we have a module with Loewy and socle structure

\begin{matrix}770_{a} \\ 77 \\ 49_{b} \end{matrix}

which is also a quotient of P_{770_{a}}. Hence we know that 770_{a} extends 154_{b} and 77, that the third Loewy layer contains composition factors isomorphic to 770_{b} and 49_{b}, and that the fourth Loewy layer contains a composition factor isomorphic to 154_{c}. As 154_{c} is not in the socle of P_{770_{a}}, this means that P_{770_{a}} must have Loewy length strictly greater than four.

Any extension of 770_{a}↓ by 770_{a}↓ must have the form

288 ⊕ 288 ⊕ 252 ⊕ 252 ⊕ 126_{A} ⊕ 126_{A} ⊕ 84_{A} ⊕ 84_{A} ⊕ 20_{A} ⊕ 20_{A}

and so E (770_{a}, 770_{a}) = 0. Therefore the only possible Loewy structure of Loewy length seven is

770_{a}

77 154_{b}
49_{b} 770_{b}

154_{c} 770_{a} 77

770_{a} .

But then Rad^{3}(P_{770_{a}}) is a quotient of P_{154_{c}} with two composition factors isomorphic to 770_{a}, which is not possible. Now suppose P_{770_{a}} has Loewy length six. Then since 770_{a} does not extend itself non-trivially, the fifth Loewy layer must be isomorphic to 77. Given the structure of P_{49_{b}} we have E (49_{b}, 770_{a}) = 0. Furthermore, any extension of 770_{a}↓ by 770_{b}↓ must have the form

288 ⊕ 288 ⊕ 252 ⊕ 252 ⊕ 126_{A} ⊕ 126_{A} ⊕ 84_{A} ⊕ 84_{A} ⊕ 20_{A} ⊕ 20_{B}

which is the trivial extension, so that E (770_{b}, 770_{a}) = 0. Hence 770_{a} does not lie in the fourth Loewy layer of P_{770_{a}}, and Rad^{3}(P_{770_{a}}) must therefore have Loewy structure

\begin{matrix}154_{c} \\ 77 \\ 770_{a} \end{matrix}

which contradicts the fact that E (154_{c}, 77) = 0. Hence P_{770_{a}} must have Loewy length five.

Because P_{770_{a}} has Loewy length five, and as 770_{a} cannot extend itself nontrivially, the third factor isomorphic to 770_{a} must lie in the third Loewy layer of P_{770_{a}}. As 49_{b} does not lie in the socle of P_{770_{a}}, it must non-trivially extend some composition factor of Rad^{3}(P_{770_{a}}). But the only factor 49_{b} can extend nontrivially is 77, and therefore the second composition factor isomorphic to 77 must be in the fourth Loewy layer of P_{770_{a}}. Hence the Loewy structure of P_{770_{a}} is

770_{a}
77 154_{b}
49_{b} 770_{a} 770_{b}
77 154_{c}
770_{a}  .

By taking the image of P_{770_{a}} under the outer automorphism of HS of order two, we then see that the Loewy structure of P_{770_{b}} is

770_{b}
77 154_{c}
49_{a} 770_{b} 770_{a}
77 154_{b}
770_{b}  .

The socle structures of both modules are the same as their Loewy structures by duality.

Finally, we consider P_{77}, which has composition factors 2.49_{a} + 2.49_{b} + 5.77 + 154_{b} + 154_{c} + 2.770_{a} + 2.770_{b}. By using the property that E (A, B) = E (B^{∗}, A^{∗}) we have the following values for dim_{K}Ext^{1}:

\begin{aligned}E (77, 49_{a}) & = & 1 \\ E (77, 49_{b}) & = & 1 \\ E (77, 154_{b}) & = & 0 \end{aligned}  \begin{aligned}E (77, 154_{c}) & = & 0 \\ E (77, 770_{a}) & = & 1 \\ E (77, 770_{b}) & = & 1. \end{aligned}

If we consider the number of ways in which 77↓ can extend itself, we see that the only possible module is the split extension

21_{A} ⊕ 56 ⊕ 21_{A} ⊕ 56

and therefore E (77, 77) = 0.

We now know that the first two Loewy layers of P_{77} are

( \begin{matrix}& 77 \\ 770_{a} & 49_{a} & 49_{b} & 770_{b} \end{matrix} )

and therefore, by duality, any factor of the penultimate Loewy layer must be one of 49_{a}, 49_{b}, 770_{a}, or 770_{b}. As P_{77} is self-dual, the penultimate socle layer of P_{77} is isomorphic to the dual of the penultimate Loewy layer of P_{77}, and so again contains one of the factors 49_{a}, 49_{b}, 770_{a}, or 770_{b}. As all of P_{49_{a}}, P_{49_{b}}, P_{770_{a}}, and P_{770_{b}} have Loewy length five, P_{77} must have Loewy length less than or equal to five by Corollary 1.1.4. In fact, as none of 49_{a}, 49_{b}, 770_{a} or 770_{b} can extend any of the factors in the second Loewy layer non-trivially, P_{77} must have five Loewy layers exactly and each of the remaining composition factors isomorphic to 49_{a}, 49_{b}, 770_{a}, and 770_{b} must lie in the fourth Loewy layer. Hence the Loewy structure of P_{77} must be

77

770_{a} 49_{a} 49_{b} 770_{b}
154_{b} 77 77 77 154_{c}
770_{a} 49_{a} 49_{b} 770_{b}
77  .

This completes the proof of Theorem 3.0.1.

The Structure of the PIMs in the Principal Block of the Higman–Sims Group

In this chapter we will determine the Loewy and socle structure of the PIMs in the principal block of the Higman–Sims group over an algebraically closed field of characteristic three. From the decomposition matrix (see Appendix D) we find that the Cartan matrix for the principal block is

1 22 154_{a} 321 748 1176 1253
1 4 1 1  0 2  2  2
22 1 4 1  1 2  1  2
154_{a} 1 1 3  1 0  0  2
321 0 1 1  2 0  1  2
748 2 2 0  0 3  2  1
1176 2 1 0  1 2  3  2
1253 2 2 2  2 1  2  4

To find the Loewy structures and socle structures of the PIMs in this block, we will once again rely heavily on induction and restriction techniques. We first calculate the values of dim_{K}Ext^{1} for the simple modules in the principal block by using information about a maximal subgroup of HS isomorphic to U_{3}(5) : 2 and then use these values to determine the Loewy and socle structures. We will prove the following theorem:

Theorem 4.0.1 The Loewy structures and socle structures of the PIMs of the principal block of the Higman–Sims group are:

\begin{matrix}1 \\ 1253 748 \\ 154_{a} 1 1176 22 1 1176 \\ 1253 748 \\ 1 \end{matrix}  \begin{matrix}22 \\ 1253 748 \\ 154_{a} 321 22 1 1176 22 \\ 1253 748 \\ 22 \end{matrix}

\begin{matrix}154_{a} \\ 1253 \\ 321 154_{a} 1 22 \\ 1253 \\ 154_{a} \end{matrix}  \begin{matrix}321 \\ 1253 \\ 154_{a} 1176 22 \\ 1253 \\ 321 \end{matrix}  748 1 1176 22 1253 748 1 1176 22 748

1176 1253 748 321 1 22 1176 1 1253 748 1176  \begin{matrix}1253 \\ 321 154_{a} 1 1176 22 \\ 1253 1253 748 \\ 321 154_{a} 1 1176 22 \\ 1253 \end{matrix}

4.1 Induction and Restriction Between U_{3}(5) : 2

and HS

We consider the same class of subgroup as in the preceding chapter, and hence the conjugacy class information is the same and the composition factors of the induced characters are the same. The simple modules of the principal block of HS restricted down to this subgroup have the following composition factors:

\begin{aligned}1↓ & ∼ & 1_{A} \\ 22↓ & ∼ & 21_{B} + 1_{B} \\ 154_{a}↓ & ∼ & 126_{A} + 28_{A} \\ 321↓ & ∼ & 252 + 28_{B} + 20_{A} + 20_{B} + 1_{A} \\ 748↓ & ∼ & 288 + 126_{A} + 126_{B} + 2.56 + 2.28_{B} + 20_{A} + 20_{B} \\ 1176↓ & ∼ & 288 + 2.252 + 126_{B} + 2.84_{B} + 28_{A} + 21_{B} + 20_{A} + 20_{B} + 1_{B} \\ 1253↓ & ∼ & 2.288 + 252 + 84_{B} + 2.84_{A} + 2.56 + 21_{A} + 20_{A} + 20_{B}. \end{aligned}

We now determine the structure of these modules. Note that we will frequently make use of the information about the composition factors of the induced simple U_{3}(5) : 2-modules, given on page 77, and the values of dim_{K}Ext^{1} for the simple U_{3}(5) : 2-modules, given on page 75.

We certainly have

1↓ = 1_{A}.

As 21_{B} and 1_{B} are in different blocks we have

22↓ = 21_{B} ⊕ 1_{B}.

As 126_{A} and 28_{A} are in different blocks we have

154_{a}↓ = 126_{A} ⊕ 28_{A}.

The module 321↓ has composition factors 252 + 28_{B} + 20_{A} + 20_{B} + 1_{A} and as neither 28_{B}↑ nor 1_{A}↑ has a composition factor isomorphic to 321, by Frobenius reciprocity we have

(321↓ , 28_{B}) = (28_{B}, 321↓) = 0

and

(321↓ , 1_{A}) = (1_{A}, 321↓) = 0.

Given the values of dim_{K}Ext^{1} for U_{3}(5) : 2 and the fact that 321↓ is self-dual, we must have

321↓ = 252 ⊕ \begin{matrix}20_{B} \\ 1_{A} & 28_{B} \\ 20_{A} \end{matrix} .

We now consider the module 748↓ which has composition factors 288+126_{A}+ 126_{B} + 2.56 + 2.28_{B} + 20_{A} + 20_{B}. By Frobenius reciprocity,

(20_{i}, 748↓) = (748↓ , 20_{i}) = 0

for both 20_{A} and 20_{B}, as neither has a composition factor isomorphic to 748. To find (28_{B}, 748↓) and (748↓ , 28_{B}), consider the principal block component of 28_{B}↑, which has composition factors 1176 + 2.748 + 22 + 1. By Frobenius reciprocity we have the following values:

\begin{aligned}(28_{B}↑ , 1) & = & 0 \\ (28_{B}↑ , 22) & = & 0 \\ (28_{B}↑ , 1176) & = & 0 \end{aligned}  \begin{aligned}(1, 28_{B}↑) & = & 0 \\ (22, 28_{B}↑) & = & 0 \\ (1176, 28_{B}↑) & = & 0. \end{aligned}

As 28_{B} and therefore 28_{B}↑ are self-dual, the Loewy structure of the principal block component of 28_{B}↑ is

\begin{aligned}& 748 \\ 1 & 1176 & 22 \\ & 748. \end{aligned}

Hence

(28_{B}↑ , 748) = (748, 28_{B}↑) = 1

and therefore

(28_{B}, 748↓) = (748↓ , 28_{B}) = 1.

As 748 and therefore 748↓ are self-dual, 748↓ = 288 ⊕ 126_{A} ⊕ 126_{B} ⊕ M where M has one of the structures

28_{B} ⊕ \begin{matrix}56 \\ 20_{B} \\ 28_{B} \\ 20_{A} \\ 56 \end{matrix} ,  56 ⊕ \begin{matrix}28_{B} \\ 20_{A} \\ 56 \\ 20_{B} \\ 28_{B} \end{matrix} ,  or  \begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} .

But the first two possibilities involve modules which are not quotients of projectives, so we must have

748↓ = 288 ⊕ 126_{A} ⊕ 126_{B} ⊕ \begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} .

The module 1176↓ has composition factors 288 + 2.252 + 126_{B} + (2.84_{B} + 21_{B}) + (28_{A} + 20_{A} + 20_{B} + 1_{B}), and by Frobenius reciprocity we have:

\begin{aligned}(21_{B}, 1176↓) & = & 0 \\ (28_{A}, 1176↓) & = & 0 \\ (1_{B}, 1176↓) & = & 0 \end{aligned}  \begin{aligned}(1176↓ , 21_{B}) & = & 0 \\ (1176↓ , 28_{A}) & = & 0 \\ (1176↓ , 1_{B}) & = & 0. \end{aligned}

As 1176↓ is self-dual, we must have

1176↓ = 288 ⊕ 252 ⊕ 252 ⊕ 126_{B} ⊕ \begin{matrix}84_{B} \\ 21_{B} \\ 84_{B} \end{matrix} ⊕ \begin{matrix}20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} \end{matrix} .

We now determine the structure of the module 1253↓. The composition factors of 1253↓ are 2.288 + 252 + 84_{B} + (2.84_{A} + 21_{a}) + (2.56 + 20_{A} + 20_{B}), and by Frobenius reciprocity,

(21_{A}, 1253↓) = (1253↓ , 21_{A}) = 0.

Now 20_{A}↑ has composition factors 770_{a} + (321 + 1176 + 1253) and we know that

(321, 20_{A}↑) = 0  and  (20_{A}↑ , 321) = 1

and

(1176, 20_{A}↑) = 1 and (20_{A}↑ , 1176) = 0.

Therefore we have the following possibilities for 20_{A}↑:

\begin{matrix}770_{a} ⊕ 1253 ⊕ {321 \atop 1176} & 770_{a} ⊕ ( \begin{matrix}1253 & 321 \\ 1176 \end{matrix} ) \\ 770_{a} ⊕ ( \begin{matrix}321 \\ 1253 & 1176 \end{matrix} ) & 770_{a} ⊕ 321 1253 1176 . \end{matrix}

Notice that in three of these cases we must have 321 extending 1176 non-trivially. We now show that 321 cannot extend 1176 non-trivially, hence eliminating the first three cases. Lemma 4.1.1

E (321, 1176) = 0.

Proof. Suppose we have a module

M = {321 \atop 1176} .

Then M↓ = 252 ⊕ 288 ⊕ 252 ⊕ 252 ⊕ 126_{B} ⊕ \begin{matrix}84_{B} \\ 21_{B} \\ 84_{B} \end{matrix} ⊕ N where N has

filtration

\begin{matrix}20_{B} \\ 1_{A} & 28_{B} \\ 20_{A} \end{matrix}  \begin{matrix}20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} & . \end{matrix}

Therefore Hd(N) = 20_{A} ⊕ 20_{B} or Hd(N) = 20_{B} and Soc(N) = 20_{A} ⊕ 20_{B} or Soc(N) = 20_{B}. Suppose N has three Loewy layers. Then N has Loewy structure

20_{B} 20_{A}
1_{A} 28_{B} 28_{A} 1_{B}
20_{A} 20_{B} .

Given the values of dim_{K}Ext^{1} for the simple U_{3}(5) : 2-modules, N must have structure

\begin{matrix}20_{B} \\ 1_{A} & 28_{B} \\ 20_{A} \end{matrix} ⊕ \begin{matrix}20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} \end{matrix}

which means that M↓ is just 321↓ ⊕ 1176↓. By Lemma 1.4.1, and given the values of dim_{K}Ext^{1} for U_{3}(5) : 2, the only other possible Loewy structures of N are

\begin{matrix}20_{B} & 20_{A} \\ 1_{A} & 28_{B} & 28_{A} \\ 20_{A} \\ 1_{B} \\ 20_{B} \end{matrix} ,  \begin{matrix}20_{B} & 20_{A} \\ 1_{A} & 28_{B} & 1_{B} \\ 20_{A} \\ 28_{A} \\ 20_{B} \end{matrix}  and  \begin{matrix}20_{B} \\ 1_{A} & 28_{B} \\ 20_{A} & 20_{A} \\ 1_{B} & 28_{A} \\ 20_{B} \end{matrix} .

For the first two cases N has socle isomorphic to 20_{B}, but there is no submodule of P_{20_{B}} with composition factors 1_{A} + 1_{B} + 2.20_{A} + 2.20_{B} + 28_{A} + 28_{B}, so these Loewy structures are not possible. For the third case N has head isomorphic to 20_{B} and is a quotient of P_{20_{B}}. But there is no quotient of P_{20_{B}} with these composition factors and so this Loewy structure is also not possible. Hence the only extension of 1176↓ by 321↓ is the split extension and so by the discussion following Lemma 1.2.3, we must have

E (321, 1176) = 0. □

As 321 cannot extend 1176 non-trivially we must have

20_{A}↑ = 770_{a} ⊕ 321 1253 1176

and by duality

20_{B}↑ = 770_{b} ⊕ 1176 1253 321

so that

(20_{A}, 1253↓) = (1253↓ , 20_{A}) = 0

and

(20_{B}, 1253↓) = (1253↓ , 20_{B}) = 0.

Given the values of dim_{K}Ext^{1} for U_{3}(5) : 2 and the fact that 1253↓ is self-dual this means we must have

1253↓ = 288 ⊕ 288 ⊕ 252 ⊕ 84_{B} ⊕ \begin{matrix}84_{A} \\ 21_{A} \\ 84_{A} \end{matrix} ⊕ \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 \end{matrix} .

4.2 Values of dim_{K}Ext^{1} for the Principal Block

Before looking at the values of dim_{K}Ext^{1}, let us consider the structure of some modules in the principal block which we already know about. Firstly, given the structures of 20_{A}↑ and 20_{B}↑, we know that we have modules

321 1253 1176  and  1176 1253 321 .

Given the structure of 28_{B}↑ we know that we have a module

\begin{aligned}& 748 \\ 1 & 1176 & 22 \\ & 748 & & . \end{aligned}

Finally, let us consider the principal block component of 21_{B}↑, which has composition factors 748 + 22 + 22. Now

(22, 21_{B}↑) = (21_{B}↑ , 22) = 1

4.2 Values of dim_{K}Ext^{1} for the Principal Block 96

and

(748, 21_{B}↑) = (21_{B}↑ , 748) = 0

by Frobenius reciprocity. As 21_{B}↑ is self-dual, this forces the principal block component to have structure

22 748 22

and so P_{22} must have such a submodule and quotient.

We now determine dim_{K}Ext^{1}(S, T) for the simple modules of the principal block using the structures of the restricted modules determined in Section 4.1 and the values of dim_{K}Ext^{1}(S, T) for the simple U_{3}(5) : 2-modules (p. 75). As the simple modules are all self-dual, we know dim_{K}Ext^{1}(T, S) once we have found dim_{K}Ext^{1}(S, T) and we therefore need only determine the values of dim_{K}Ext^{1}(S, T) with dim S ≤ dim T.

dim_{K}Ext^{1}(1, T)

Recall from Chapter 3 (p. 80) that

1_{A}↑ = 1 ⊕ \begin{matrix}49_{b} \\ 77 \\ 49_{a} \end{matrix}

and therefore for any module M in the principal block we have, by Frobenius reciprocity,

E (1, M) = E (1_{A}↑ , M)
= E (1_{A}, M↓).

As 1_{A} does not extend 1_{A} non-trivially, we have

E (1, 1) = 0.

Furthermore, as 1_{A} does not extend 21_{B}, 1_{B}, 126_{A}, or 28_{A} non-trivially, we also have

E (1, 22) = 0

and

E (1, 154_{a}) = 0.

Given the Cartan matrix we know that 1 cannot extend 321 non-trivially, i.e.,

E (1, 321) = 0.

Given that there is a module

748 1 1176 22 748

we must have

E (1, 748) ≥ 1.

But E (1, 748) = E (1_{A}, 748↓), where 748↓ is

288 ⊕ 126_{A} ⊕ 126_{B} ⊕ \begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix}

and the only possible way 1_{A} can extend 748↓ non-trivially is if 1_{A} extends the composition factor of 748↓ isomorphic to 20_{A} non-trivially. As E (1_{A}, 20_{A}) = 1, we must have E (1_{A}, 748↓) ≤ 1 and so E (1, 748) ≤ 1. Hence

E (1, 748) = 1.

Now

1176↓ = 288 ⊕ 252 ⊕ 252 ⊕ 126_{B} ⊕ \begin{matrix}84_{B} \\ 21_{B} \\ 84_{B} \end{matrix} ⊕ \begin{matrix}20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} \end{matrix}

and the only possible way 1_{A} can extend 1176↓ non-trivially is if 1_{A} extends

\begin{matrix}20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} \end{matrix}

non-trivially. But given the values of dim_{K}Ext^{1} for U_{3}(5) : 2, the only possible non-trivial extension is

\begin{matrix}1_{A} \\ 20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} \end{matrix}

which is not a quotient of P_{1_{A}} (as P_{1_{A}} does not have a composition factor isomorphic to 1_{B}). Therefore

E (1, 1176) = E (1_{A}, 1176↓) = 0.

Given the structure of 1253↓, the only possible way 1_{A} can extend 1253↓ non-trivially is if 1_{A} extends

\begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 \end{matrix}

non-trivially. As 20_{A} is the only composition factor that 1_{A} can non-trivially extend, and E (1_{A}, 20_{A}) = 1, we must have E (1, 1253) ≤ 1. Suppose that E (1, 1253) = 0 and consider the structure of P_{1}, which has composition factors 4.1 + 22 + 154_{a} + 2.748 + 2.1176 + 2.1253. As E (1, S) = E (S, 1) = 0 for all simple modules other than 748, it must be the case that P_{1} has more than four Loewy layers. Furthermore, the first two Loewy layers must be

{1 \atop 748}

and the first two socle layers must be

{748 \atop 1}

so that the heart of P_{1} has simple socle and simple head isomorphic to 748. But then the heart of P_{1} has two composition factors isomorphic to 1 which are not in the head or the socle of Ht(P_{1}), and each of these factors must be extended nontrivially by a composition factor in the Loewy layer above it. As the 748 of the head is the only simple module of Ht(P_{1}) which can extend 1 non-trivially (the other composition factor isomorphic to 748 is in the socle of Ht(P_{1}) and therefore cannot extend any composition factor of Ht(P_{1}) non-trivially), we must have two composition factors isomorphic to 1 in L_{2}(Ht(P_{1})). Therefore E (748, 1) = 2. As we know that E (748, 1) = E (1, 748) = 1 this is a contradiction and therefore we must have

E (1, 1253) = 1.

dim_{K}Ext^{1}(22, T) with dim T ≥ 22

The only possible extension of 22↓ by 22↓ is

21_{B} ⊕ 1_{B} ⊕ 21_{B} ⊕ 1_{B}.

Hence

E (22, 22) = 0.

As neither 21_{B} nor 1_{B} extend 126_{A} or 28_{A} non-trivially,

E (22, 154_{a}) = 0.

Given the values of dim_{K}Ext^{1} for U_{3}(5) : 2, the only possible non-trivial extension of 321↓ by 22↓ is

21_{B} ⊕ 252 ⊕ \begin{matrix}1_{B} \\ 20_{B} \\ 1_{A} & 28_{B} \\ 20_{A} \end{matrix} .

But

\begin{matrix}1_{B} \\ 20_{B} \\ 1_{A} & 28_{B} \\ 20_{A} \end{matrix}

is not a quotient of P_{1_{B}} (as P_{1_{B}} has no composition factor isomorphic to 1_{A}), and therefore

E (22, 321) = 0.

We already know (from p. 96), that there is at least one non-trivial extension of 748 by 22 because we have a module

22 748 22

and therefore E (22, 748) ≥ 1. Suppose there is a module

M = ( \begin{matrix}& 22 \\ 748 & & 748 \end{matrix} ) .

Then M↓ = 288 ⊕ 288 ⊕ 126_{A} ⊕ 126_{A} ⊕ 126_{B} ⊕ 126_{B} ⊕ N where N has filtration

1_{B} 21_{B}  \begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} ⊕ \begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 . \end{matrix}

By Frobenius reciprocity,

(21_{B}, M↓) = (21_{B}↑ , M) = ( \begin{matrix}22 \\ 748 \\ 22 \end{matrix} , ( \begin{matrix}& 22 \\ 748 & & 748 \end{matrix} )) = 0

as M↓ has no submodule isomorphic to 22, no submodule with composition factors 22 and 748, and no submodule with composition factors 22, 748 and 22. This means that 21_{B} must extend some composition factor of 748↓ ⊕ 748↓ nontrivially, which is not possible given the values of dim_{K}Ext^{1} for U_{3}(5) : 2. Hence there can be no such module as M and we must have

E (22, 748) = 1.

Suppose there exists a module

M = {22 \atop 1176} .

Then M↓ = 288 ⊕ 252 ⊕ 252 ⊕ 126_{B} ⊕ \begin{matrix}84_{B} \\ 21_{B} \\ 84_{B} \end{matrix} ⊕ N where N has filtration

21_{B} 1_{B}  \begin{matrix}20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} & . \end{matrix}

Given the structure of 20_{A}↑ (p. 94), by Frobenius reciprocity we have,

(M↓ , 20_{A}) = (M, 20_{A}↑) = 0

and therefore we must have 21_{B} or 1_{B} extending 20_{A} non-trivially. But neither of these simple modules can actually extend 20_{A} non-trivially, hence such a module as M does not exist and we must have

E (22, 1176) = 0.

The composition factors of P_{22} are 1+4.22+154_{a}+321+2.748+1176+2.1253, and because P_{22} has only two composition factors isomorphic to 1253, we know that E (22, 1253) ≤ 2. Suppose E (22, 1253) = 0 and consider the structure of P_{22}. We have E (22, S) = E (S, 22) = 0 for all simple modules other than 748, and therefore we know that the heart of P_{22} has simple socle and simple head isomorphic to 748. But then the heart of P_{22} has two composition factors isomorphic to 22 which must be non-trivially extended by the 748 of the head, and therefore E (748, 22) = 2. Because E (748, 22) = E (22, 748) = 1 this is a contradiction and therefore we must have

E (22, 1253) ≥ 1.

Now suppose that E (22, 1253) = 2. Then P_{22} has two composition factors isomorphic to 1253 in the second Loewy layer, and by duality, in the second socle layer, and so the heart of P_{22} has structure 1253 ⊕ 1253 ⊕ N, for some module N with simple socle and simple head isomorphic to 748. Just as above, N must have two composition factors isomorphic to 22 which are non-trivially extended by the 748 of the head of N, contradicting the fact that E (748, 22) = 1. Hence we must have

E (22, 1253) = 1.

dim_{K}Ext^{1}(154_{a}, T) with dim T ≥ 154

Any extension of 154_{a}↓ by 154_{a}↓ must have the form

126_{A} ⊕ 28_{A} ⊕ 126_{A} ⊕ 28_{A}

which is the split extension and therefore

E (154_{a}, 154_{a}) = 0.

Given the values of dim_{K}Ext^{1} for U_{3}(5) : 2, the only possible non-trivial extension of 321↓ by 154_{a}↓ must have the form:

252 ⊕ 126_{A} ⊕ \begin{matrix}28_{A} \\ 20_{B} \\ 1_{A} & 28_{B} \\ 20_{A} \end{matrix} .

But there is no quotient of P_{28_{A}} with a composition factor isomorphic to 28_{B} and therefore

E (154_{a}, 321) = 0.

Given the Cartan matrix we know

E (154_{a}, 748) = 0

and

E (154_{a}, 1176) = 0.

Finally, since no other simple module in P_{154_{a}} extends or is extended by 154_{a} non-trivially, it must be the case that P_{154_{a}} has more than three Loewy layers. As P_{154_{a}} is self-dual and all the simple modules are self-dual, whatever simples are in the penultimate Loewy layer must also be in the second Loewy layer. But if E (154_{a}, 1253) = 2, we cannot have 1253 in the penultimate Loewy layer, and no other simple module extends 154_{a} non-trivially. Hence

E (154_{a}, 1253) = 1.

dim_{K}Ext^{1}(321, T) with dim T ≥ 321

As there are only two composition factors isomorphic to 321 in P_{321} we must have

E (321, 321) = 0.

From the Cartan matrix

E (321, 748) = 0.

We have already seen that

E (321, 1176) = 0.

We already have a module

321 1253 1176

and therefore E (321, 1253) ≥ 1. Now as 321 does not extend any of the other simples in P_{321} non-trivially, we know that P_{321} must have more than three Loewy layers. Furthermore we know that the second Loewy layer must contain a single composition factor isomorphic to 1253 and that the penultimate Loewy layer must contain a single composition factor isomorphic to 1253 (as there are only two composition factors isomorphic to 1253 in P_{321}). Hence

E (321, 1253) = 1.

dim_{K}Ext^{1}(748, T) with dim T ≥ 748

Suppose we have a module

M = {748 \atop 748} .

Then M↓ = 288 ⊕ 288 ⊕ 126_{A} ⊕ 126_{A} ⊕ 126_{B} ⊕ 126_{B} ⊕ N, where N has filtration

\begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} \begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix}

and hence the head of N has at least one composition factor isomorphic to 28_{B} and at least one composition factor isomorphic to 56. By Frobenius reciprocity,

(M↓ , 28_{B}) = (M, 28_{B}↑) = ( \begin{matrix}748 \\ 748 \end{matrix} , \begin{matrix}& 748 \\ 1 & 1176 & 22 \\ & 748 \end{matrix} ) = 1

and given that 56↑ has principal block composition factors 2.1 + 2.22 + 154_{a} + 2.748 + 1176 + 2.1253, we know that

1 ≤ (M↓ , 56) ≤ 2.

Lemma 4.2.1

(M↓ , 56) = 1.

Proof. Suppose for a contradiction that (M↓ , 56) = 2 and therefore (M, 56↑) = 2. By Frobenius reciprocity we also have:

\begin{aligned}(1, 56↑) & = & 0 \\ (22, 56↑) & = & 0 \\ (154_{a}, 56↑) & = & 0 \\ (748, 56↑) & = & 1 \\ (1176, 56↑) & = & 0 \\ (1253, 56↑) & = & 1 \end{aligned}  \begin{aligned}(56↑ , 1) & = 0 \\ (56↑ , 22) & = 0 \\ (56↑ , 154_{a}) & = 0 \\ (56↑ , 748) & = 1 \\ (56↑ , 1176) & = 0 \\ (56↑ , 1253) & = 1 \end{aligned}

which tells us that both the socle and head of the principal block component of 56↑ are isomorphic to the module 748 ⊕ 1253. Note that 56↑ has a composition factor isomorphic to 154_{a} which is not in the socle or the head of 56↑ and which must therefore extend some composition factor of 56↑ non-trivially and be nontrivially extended by some composition factor of 56↑. The only simple module which can non-trivially extend or be extended by 154_{a} non-trivially is 1253 and so we must have two composition factors isomorphic to 1253 in the indecomposable summand of 56↑ with composition factor 154_{a}. Hence 56↑ does not have an indecomposable summand isomorphic to 1253.

In order to have (M, 56↑) = 2 we must have

( \begin{matrix}748 \\ 748 \end{matrix} , 56↑) = 2

and therefore as well as having a single composition factor isomorphic to 748 in the socle, we know that 56↑ must also have a submodule isomorphic to M. Let ι be the inclusion of M into 56↑. As M is self-dual and 56↑ is self-dual, we must also have

(56↑ , \begin{matrix}748 \\ 748 \end{matrix} ) = 2

and therefore as well as having a single composition factor isomorphic to 748 in the head, we know that 56↑ must also have a factor module isomorphic to M. Let π be the projection of 56↑ onto M. Because 56↑ only has two composition factors isomorphic to 748 we see that ker π ∩ im ι = 0 and therefore 56↑ has a direct summand isomorphic to M. The other summand of 56↑ then has simple head isomorphic to 1253 and simple socle isomorphic to 1253. We know that there are two composition factors isomorphic to 1 in 56↑ which must be in this second summand of 56↑. We also know that they are not in the head and the only simple modules that can extend 1 non-trivially in this summand is 1253. Since the 1253 of the socle of 56↑ cannot extend a composition factor isomorphic to 1 non-trivially, the composition factor 1253 of the head must non-trivially extend both of the composition factors isomorphic to 1. Therefore we have E (1253, 1) = 2, a contradiction. Hence we must actually have (M, 56↑) = 1 and therefore (M↓ , 56) = 1. □

Given the values of dim_{K}Ext^{1} for U_{3}(5) : 2 and the fact that

(M↓ , 28_{B}) = 1

and

(M↓ , 56) = 1,

by Lemma 1.4.1, N must actually have filtration

\begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} \begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix}

and so the Loewy structure of N is

56 28_{B}
20_{B} 20_{A}
28_{B} 56 56 28_{B}
20_{B} 20_{A}
28_{B} 56 .

But then N is a quotient of P_{56} ⊕ P_{28_{B}} and L_{3}(P_{56} ⊕ P_{28_{B}}) has only one composition factor isomorphic to 28_{B}, so L_{3}(N) can only have one composition factor isomorphic to 28_{B} at most, a contradiction. Hence no such module as M can exist and we must have

E (748, 748) = 0.

We now consider E (748, 1176). We already know we have a module

748 1 1176 22 748

and so E (748, 1176) ≥ 1. As there are only two composition factors isomorphic to 1176 in P_{748} we also have E (748, 1176) ≤ 2. Suppose E (748, 1176) = 2. Then by duality, E (1176, 748) = 2. We consider the structure of P_{1176}, which has composition factors 2.1 + 22 + 321 + 2.748 + 3.1176 + 2.1253. Because E (1176, 748) = 2, both composition factors isomorphic to 748 are in the second Loewy layer, so by duality are also in the second socle layer, and therefore the heart of P_{1176} has structure 748 ⊕ 748 ⊕ N for some module N. Hence we have arrows

[figure: Ω(1176); 748; 748; N; Ω^{−1}(1176)]

in the component of the stable Auslander–Reiten quiver containing Ω(1176). But 748 is also in this component and 748 is not divisible by 3, so by Theorem 1.8.9 this component is isomorphic to ZA_{∞}. Therefore there are at most two arrows starting at Ω(1176), contradicting the assumption that E (748, 1176) = 2. Hence we must have E (1176, 748) = 1 and therefore

E (748, 1176) = 1.

Suppose we have a module

M = {748 \atop 1253} .

Then M↓ = 288 ⊕ 288 ⊕ 288 ⊕ 252 ⊕ 126_{A} ⊕ 126_{B} ⊕ 84_{B} ⊕ \begin{matrix}84_{A} \\ 21_{A} \\ 84_{A} \end{matrix} ⊕ N, where N

has filtration

\begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 & . \end{matrix}

By Frobenius reciprocity,

(28_{B}, M↓) = (28_{B}↑ , M) = ( \begin{matrix}& 748 \\ 1 & 1176 & 22 \\ & 748 \end{matrix} , \begin{matrix}748 \\ 1253 \end{matrix} ) = 0

and hence the only possibility for the socle of N is that it contains one or two composition factors isomorphic to 56. Lemma 4.2.2

(56, M↓) = 1.

Proof. Suppose for a contradiction that (56, M↓) = 2. Then (56↑ , M) = 2, that is

(56↑ , \begin{matrix}748 \\ 1253 \end{matrix} ) = 2.

We saw earlier that 56↑ has first block composition factors 2.1 + 2.22 + 154_{a} + 2.748 + 1176 + 2.1253 and both the socle and head of 56↑ are isomorphic to 748 ⊕ 1253. For the above to be true therefore, it must be the case that 56↑ has a factor module isomorphic to

{748 \atop 1253} .

Hence not only do we have a composition factor isomorphic to 1253 in the head of 56↑, we also have a composition factor isomorphic to 1253 in the second Loewy layer of 56↑. But there is a composition factor isomorphic to 154_{a} in 56↑ which is not a composition factor of the socle or the head. Since the only simple module 154_{a} can non-trivially extend is 1253, it must non-trivially extend the 1253 of the second Loewy layer. But then we would have a composition factor isomorphic to 154_{a} in the head of 56↑, a contradiction. This means that there can be no factor module of the form

{748 \atop 1253}

and hence we must actually have (56↑ , M) = 1 and therefore (56, M↓) = 1. □

We now see that N has simple socle isomorphic to 56 and therefore N is a submodule of P_{56}. Given the composition factors of N this is not possible, and so no such module as M can exist. Hence we must have

E (748, 1253) = 0.

dim_{K}Ext^{1}(1176, T) with dim T ≥ 1176

Suppose there is a module

M = {1176 \atop 1176} .

Then M↓ = 288 ⊕ 288 ⊕ 252 ⊕ 252 ⊕ 252 ⊕ 252 ⊕ 126_{B} ⊕ 126_{B} ⊕ \begin{matrix}84_{B} \\ 21_{B} \\ 84_{B} \end{matrix}

⊕ \begin{matrix}84_{B} \\ 21_{B} \\ 84_{B} \end{matrix} ⊕ N where N has filtration

\begin{matrix}20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} \end{matrix}  \begin{matrix}20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} & . \end{matrix}

Given the structure of 20_{A}↑ (p. 94), by Frobenius reciprocity we have,

(M↓ , 20_{A}) = (M, 20_{A}↑) = 1

and therefore the 20_{A} of the submodule must be extended non-trivially. But none of 20_{A}, 28_{A}, 1_{B}, or 20_{B} can extend 20_{A} non-trivially, so this is not possible. Hence E (1176↓ , 1176↓) = 0 and therefore

E (1176, 1176) = 0.

Given that we have a module

M = 1176 1253 321

(see p. 95) we know that E (1176, 1253) ≥ 1, and as there are precisely two copies of 1253 in P_{1176}, we must have E (1176, 1253) ≤ 2. Suppose E (1176, 1253) = 2. Then both composition factors isomorphic to 1253 are in the second Loewy layer and by duality are also in the second socle layer. Hence the heart of P_{1176} has structure 1253 ⊕ 1253 ⊕ N, for some module N, which contradicts the existence of the module M above (for M is a quotient of P_{1176} and therefore Rad(M) must be a quotient of the heart of P_{1176}). Therefore we cannot have both composition factors isomorphic to 1253 in the second Loewy layer, which means that

E (1176, 1253) = 1.

4.2 Values of dim_{K}Ext^{1} for the Principal Block 109

dim_{K}Ext^{1}(1253, T) with dim T ≥ 1253

We now determine whether or not 1253 extends 1253. Let M = 1253↓, that is

M = 288 ⊕ 288 ⊕ 252 ⊕ 84_{B} ⊕ \begin{matrix}84_{A} \\ 21_{A} \\ 84_{A} \end{matrix} ⊕ \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 \end{matrix} .

Then P_{M}, the projective cover of M, is

288 ⊕ 288 ⊕ 252 ⊕ \begin{matrix}84_{B} \\ 21_{B} \\ 84_{B} \end{matrix} ⊕ \begin{matrix}84_{A} \\ 21_{A} \\ 84_{A} \end{matrix} ⊕ \begin{aligned}& 56 \\ 20_{A} & 20_{B} \\ 1_{A} & 1_{B} & 28_{A} & 28_{B} & 56 \\ 20_{A} & 20_{B} \\ & 56 \end{aligned}

so that Ω(M) is

\begin{matrix}21_{B} \\ 84_{B} \end{matrix} ⊕\begin{aligned}1_{A} & 1_{B} & 28_{A} & 28_{B} \\ 20_{A} & 20_{B} \\ & 56 & & . \end{aligned}

This tells us that

(Ω(M), M) = ( \begin{matrix}21_{B} \\ 84_{B} \end{matrix} ⊕\begin{aligned}1_{A} & 1_{B} & 28_{A} & 28_{B} \\ 20_{A} & 20_{B} \\ & 56 \end{aligned} , 84_{B} ⊕ \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 \end{matrix} )
= ( \begin{matrix}21_{B} \\ 84_{B} \end{matrix} , 84_{B}) + ( \begin{matrix}1_{A} & 1_{B} & 28_{A} & 28_{B} \\ 20_{A} & 20_{B} \\ & 56 \end{matrix} , \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 \end{matrix} )
= 0 + 0
= 0.

By the discussion following Theorem 1.3.5, this means that E (1253↓ , 1253↓) = 0 and therefore

E (1253, 1253) = 0.

Given all these calculations, we now have the values of dim_{K}Ext^{1} for the principal block of Higman–Sims. The table of values of dim_{K}Ext^{1} is as follows:

1 22 154_{a} 321 748 1176 1253
1 0 0 0  0 1  0  1
22 0 0 0  0 1  0  1
154_{a} 0 0 0  0 0  0  1
321 0 0 0  0 0  0  1
748 1 1 0  0 0  1  0
1176 0 0 0  0 1  0  1
1253 1 1 1  1 0  1  0

Note that if we call the simple modules 1, 22, 154_{a}, 321 and 1176 type A modules and 748 and 1253 type B modules, then we have no type A modules extending each other non-trivially and no type B modules extending each other non-trivially. This will be very useful when we try to determine the Loewy structure of the PIMs of the principal block.

4.3 The PIMs

P_{321} has composition factors 22 + 154_{a} + 2.321 + 1176 + 2.1253. Given that only 1253 extends 321 non-trivially or can be extended by 321 non-trivially, and that 22, 154_{a}, and 1176 are type A modules (so do not extend each other non-trivially), the Loewy structure of P_{321} must be

321 1253 154_{a} 1176 22 1253

321  .

Because all the simple modules in the first block are self-dual, the socle structure is the same as the Loewy structure by duality.

P_{154_{a}} has composition factors 1 + 22 + 3.154_{a} + 321 + 2.1253. Again only 1253 extends 154_{a} non-trivially or can be extended by 154_{a} non-trivially, and 1, 22, 154_{a} and 321 are type A modules. Hence the Loewy structure of P_{154_{a}} must be 154_{a} 1253 321 154_{a} 1 22 1253

154_{a}  .

Again, because all the simple modules in the first block are self-dual, the socle structure is the same as the Loewy structure by duality.

P_{748} has composition factors 2.1+2.22+3.748+2.1176+1253. We know that each of 1, 22 and 1176 occurs once in the second Loewy layer. We also know, since 1253 does not extend 748 non-trivially and cannot be extended non-trivially by 748, that P_{748} has at least five Loewy layers. If P_{748} has more then five Loewy layers, then since the layers must alternate type B, type A and there are only four type B modules, P_{748} must have seven Loewy layers. As P_{748} is self-dual and all its composition factors are self-dual, any composition factor occurring in the third last Loewy layer must also occur in the third Loewy layer. But this is not possible and so P_{748} must have five Loewy layers. Hence the Loewy structure (and by duality, the socle structure) of P_{748} must be

748 1 1176 22 1253 748 1 1176 22

748  .

P_{1} has composition factors 4.1 + 22 + 154_{a} + 2.748 + 2.1176 + 2.1253. The first two Loewy layers of P_{1} are

(\begin{matrix}1 \\ 1253 748 \end{matrix} ) .

The number of possible Loewy layers then depends on whether we have one or both of 748 and 1253 in the penultimate Loewy layer, and since the Loewy layers must alternate type A, type B, this means that there are either five or seven Loewy layers. Suppose we have seven Loewy layers. If the penultimate Loewy layer is isomorphic to 748, then we can describe the Loewy structure of P_{1} as follows:

1 1253 748 type A 1253 type A 748

1  .

By duality, we can describe the socle structure as follows:

1 748 type A 1253 type A 1253 748

1  .

But we then see that the heart of P_{1} has an indecomposable summand isomorphic to 1253, which contradicts the fact that the heart of P_{1} must be indecomposable, by Theorem 1.1.6. In exactly the same way, if we assume that P_{1} has seven Loewy layers with the penultimate Loewy layer isomorphic to 1253, then the heart of P_{1} will have an indecomposable summand isomorphic to 748, a contradiction. Hence P_{1} must have five Loewy layers and this forces the Loewy structure to be

1 1253 748 154_{a} 1 1176 22 1 1176 1253 748

1  .

The socle structure is then the same by duality.

P_{1176} has composition factors 2.1 + 22 + 321 + 2.748 + 3.1176 + 2.1253. The first two Loewy layers of P_{1176} are

( \begin{matrix}1176 \\ 1253 748 \end{matrix} ) .

As with P_{1}, there are either seven or five Loewy layers and the number depends on whether we have one or both of 748 and 1253 in the penultimate Loewy layer. If we have seven Loewy layers with the penultimate Loewy layer isomorphic to 748, then the Loewy structure of P_{1176} takes the form:

1176 1253 748 type A 1253 type A 748

1176 .

Whatever type A simples are in the fifth Loewy layer must also be in the third layer and this means that the composition factor 321 must be in the third layer, because it only occurs once. As P_{1176} is self-dual and there is a composition factor isomorphic to 1253 in the second Loewy layer, there is also a composition factor isomorphic to 1253 in the second socle layer and therefore 1253 is a direct summand of the heart of P_{1176}. But this tells us that the 748 in the second Loewy layer must non-trivially extend 321 and as E (748, 321) = 0, this is not possible and so no such module exists.

Now suppose that 1253 is in the penultimate Loewy layer instead. Then the Loewy structure of P_{1176} takes the form:

1176 1253 748 type A 748 type A 1253

1176 .

In fact, since the only type A composition factor that occurs twice in P_{1176} (other than 1176) is 1, we must actually have the following Loewy structure:

1176 1253 748 321 1 22 1176 748 1 1253

1176  .

But then we have a composition factor isomorphic to 1 in the fifth Loewy layer of P_{1176} and hence, by Lemma 1.1.5, there is a factor module of P_{1176} of Loewy length five with simple socle isomorphic to 1 and simple head isomorphic to 1176. But there can be no such submodule of P_{1} (because it only has Loewy length five) and hence P_{1176} must have Loewy length five. This forces the Loewy structure to be

1176 1253 748 321 1 22 1176 1 1253 748 1176

and the socle structure is the same by duality.

P_{22} has composition factors 1 + 4.22 + 154_{a} + 321 + 2.748 + 1176 + 2.1253 and the first two Loewy layers of P_{22} are

(\begin{matrix}22 \\ 1253 748 \end{matrix} ) .

Following the same argument as before, the only possible Loewy structure for P_{22} with seven Loewy layers is

22 1253 748 154_{a} 321 22 1 1176 748 22 1253

22  .

In this case the heart of P_{22} has structure 748 ⊕ N where N has simple head and simple socle isomorphic to 1253. We know that we have a module

22 748 22

which is the structure of the principal block component of 21_{B}↑, and this module is a submodule of P_{22}. As it is a proper submodule of P_{22}, it must also be a submodule of the unique maximal submodule Rad(P_{22}), and therefore the module

{22 \atop 748}

must be a submodule of Ht(P_{22}). But this is not possible if P_{22} has the above structure. Therefore P_{22} has five Loewy layers and the Loewy structure must be

22 1253 748 154_{a} 321 22 1 1176 22 1253 748

22  .

The socle structure is once again the same by duality.

P_{1253} has composition factors 2.1+2.22+2.154_{a}+2.321+748+2.1176+4.1253. As E (1253, 1253) = 0, we know that 1253 is not in the penultimate socle layer of P_{1253}. As all the other PIMs in the principal block have five Loewy layers, P_{1253} must therefore have five or fewer Loewy layers by Corollary 1.1.4. Given the fact that the Loewy structure must alternate type B, type A and not all of the type A modules belong to the second Loewy layer, this immediately tells us that the Loewy length is five and forces the Loewy structure of P_{1253} to be

1253 321 154_{a} 1 1176 22 1253 1253 748 321 154_{a} 1 1176 22

1253  .

By duality the socle structure is then equal to the Loewy structure. This completes the proof of Theorem 4.0.1.

The Structure of the Basic Algebra of the Second Block of the Higman–Sims Group

Now that we have determined the Loewy structure and socle structure of the PIMs of the second block of the Higman–Sims group, we are ready to determine the structure of the basic algebra of the second block (once again, we will deal with the more difficult case of the principal block in the next chapter). Using our knowledge of the Loewy and socle structure of the PIMs we will be able to determine a 1-parameter family of algebras to which the basic algebra of the second block must belong. The relations defining this 1-parameter family of algebras will be found at the end of the chapter. Note that composition of paths will be written on the left.

Let A denote the basic algebra of the second block of HS, and let e_{S} denote the primitive orthogonal idempotent in A corresponding to P_{S}. By Theorem 3.0.1 (p. 68), the Ext-quiver for A is:

[figure: 77; 770_{a}; 49_{a}; a; c; ā; c̄; 770_{b}; 49_{b}; b; b̄; d̄; d; 154_{b}; w; x; 154_{c}; z; y]

where a corresponds to a basis element for the complement of e_{77}Rad^{2}(A)e_{49_{a}} in e_{77}Rad(A)e_{49_{a}}, ā corresponds to a basis element for the complement of e_{49_{a}}Rad^{2}(A)e_{77} in e_{49_{a}}Rad(A)e_{77}, and so on. We now consider the extension structure of P_{S} for each simple module S of the second block and use this information to find generators for the relations induced by I (where A ≅ KQ/I). We then minimise the set of generators and choose basis elements corresponding to the arrows so as to minimise the number of unknown scalar factors involved in the relations.

5.1 Finding Generators for the Relations In-

duced by I

Given the Loewy structure and socle structure of P_{49_{a}} (see Theorem 3.0.1), the extension structure of P_{49_{a}} must be:

[figure: 49_{a}; 77; a; 49_{b}; 770_{b}; b̄; d̄; 77; b; d; 49_{a}; ā]

5.1 Finding Generators for the Relations Induced by I 118

From the Loewy and socle structure, we know that the composition factor 77 of the second Loewy layer must non-trivially extend the composition factor 770_{b}, and that this in turn must non-trivially extend the composition factor 77 of the fourth Loewy layer. In terms of paths in KQ/I this tells us that dd̄a is non-zero. A similar argument tells us that bb̄a is non-zero, and hence dd̄a and bb̄a are nonzero paths spanning a 1-dimensional space modulo e_{77}Rad^{4}(A)e_{49_{a}}. As all paths of length five are zero and there are no paths of length four from 49_{a} to 77, this means that dd̄a and bb̄a span a 1-dimensional space in KQ/I. Note that from the diagram we also see that āa = 0 modulo e_{49_{a}}Rad^{3}(A)e_{49_{a}} (because there is no composition factor isomorphic to 49_{a} in the third Loewy layer of P_{49_{a}}). As the only non-zero paths from 49_{a} to 49_{a} of length greater than two are the paths ābb̄a and ādd̄a, which are linearly dependent, we must have āa = k_{1}ābb̄a for some scalar k_{1} (possibly equal to zero) in K. We can now write down generators for the relations in paths starting at the vertex 49_{a}:

āa = k_{1}ābb̄a
c̄a = 0
yd̄a = 0
dd̄a = λ_{1}bb̄a
b̄bb̄a = 0
c̄bb̄a = 0
d̄bb̄a = 0
aābb̄a = 0

where λ_{1} is non-zero.

Given the Loewy structure and socle structure of P_{49_{b}} the extension structure of P_{49_{b}} must be:

[figure: 49_{b}; 77; b; 49_{a}; 770_{a}; ā; c̄; 77; a; c; 49_{b}; b̄]

Using the same type of argument as above, we have generating relations:

b̄b = k_{2}b̄aāb

d̄b = 0
wc̄b = 0
cc̄b = λ_{2}aāb
āaāb = 0
c̄aāb = 0
d̄aāb = 0
bb̄aāb = 0

where λ_{2} is non-zero.

The extension structures of P_{154_{b}} and P_{154_{c}} are as follows:

[figure: 154_{b}; 770_{b}; x; 77; 154_{c}; d; y; 770_{a}; c̄; z; 154_{b}; w; 154_{c}; 770_{a}; z; 77; 154_{b}; c; w; 770_{b}; d̄; x; 154_{c}; y]

and so we have generating relations:

\begin{aligned}ādx & = & 0 \\ b̄dx & = & 0 \\ d̄dx & = & 0 \\ c̄dx & = & λ_{3}zyx \\ czyx & = & 0 \\ xwzyx & = & 0 \end{aligned}  \begin{aligned}ācz & = & 0 \\ b̄cz & = & 0 \\ c̄cz & = & 0 \\ d̄cz & = & λ_{4}xwz \\ dxwz & = & 0 \\ zyxwz & = & 0 \end{aligned}

where λ_{3} and λ_{4} are non-zero.

Now we turn to P_{770_{a}}. Given the Loewy structure and socle structure of P_{770_{a}} and the values of dim_{K}Ext^{1} for the simple modules in the second block (which can be easily obtained using Theorem 3.0.1), we know firstly that both the composition factors 77 and 154_{b} of the second Loewy layer are non-trivially extended by the 770_{a} of the head of P_{770_{a}}, and secondly that both the composition factors 77 and 154_{c} of the fourth Loewy layer, which is also the second socle layer, must non-trivially extend the socle. We also know that the composition factor 154_{b} of the second Loewy layer, which is also the fourth socle layer, can only non-trivially extend the composition factor 770_{b} of the third Loewy layer, (the third socle layer) and that this factor must non-trivially extend the factor 154_{c} of the fourth Loewy layer. Furthermore, the composition factors 770_{a} and 49_{b} of the third Loewy layer must be non-trivially extended by the 77 of the second Loewy layer (because they cannot be extended non-trivially by 154_{b}) and they must both non-trivially extend the 77 of the fourth Loewy layer, or equivalently, the second socle layer (because they cannot extend 154_{c} non-trivially). Finally, because E (77, 770_{b}) = 1 and E (770_{b}, 77) = 1, we could have the 77 of the second Loewy layer non-trivially extending the 770_{b} of the third Loewy layer, which could non-trivially extend the 77 of the fourth Loewy layer, also the second socle layer. Therefore we can represent the extension structure of P_{770_{a}} as follows:

[figure: 770_{a}; 77; 154_{b}; c; w; 770_{a}; 49_{b}; 770_{b}; b̄; c̄; d̄; x; 77; 154_{c}; b; c; d; y; 770_{a}; c̄; z]

where the paths d̄c and dd̄c may be zero, modulo e_{770_{b}}Rad^{3}(A)e_{770_{a}} and e_{77}Rad^{4}(A)e_{770_{a}} respectively, depending on whether the 77 of the second Loewy layer non-trivially extends 770_{b}, and whether 770_{b} non-trivially extends the 77 of the fourth Loewy layer respectively. As there are actually no non-zero paths from 770_{a} to 770_{b} of length greater than two, d̄c = 0 modulo e_{770_{b}}Rad^{3}(A)e_{770_{a}} if and only if d̄c = 0. But we have already seen that the path d̄cz is non-zero in KQ/I given the discussion of the relations on the paths starting at 154_{c}, so certainly d̄c is non-zero. As the rest of the paths shown are non-zero, we have generating relations:

\begin{aligned}āc & = & 0 \\ wc̄c & = & 0 \\ ābb̄c & = & 0 \\ b̄bb̄c & = & 0 \\ d̄bb̄c & = & 0 \\ wzyxw & = & 0 \\ czyxw & = & 0 \end{aligned}  \begin{aligned}d̄c & = & λ_{5}xw \\ cc̄c & = & λ_{6}bb̄c \\ dd̄c & = & λ_{7}bb̄c \\ c̄bb̄c & = & λ_{8}zyxw \end{aligned}

where λ_{5}, λ_{6} and λ_{8} are non-zero.

Similarly, using the Loewy and socle structure of P_{770_{b}} and the values of dim_{K}Ext^{1}, we can determine the extension structure of P_{770_{b}} up to the existence of non-trivial extensions between 770_{a} and the 77s. The extension structure can then be represented as follows:

[figure: 770_{b}; 77; 154_{c}; d; y; 770_{b}; 49_{a}; 770_{a}; ā; d̄; c̄; z; 77; 154_{b}; a; d; c; w; 770_{b}; d̄; x]

We have that c̄d is non-zero as there is a non-zero path c̄dx in KQ/I, and therefore generators for the relations in paths starting at 770_{b} are:

\begin{aligned}b̄d & = & 0 \\ yd̄d & = & 0 \\ āaād & = & 0 \\ b̄aād & = & 0 \\ c̄aād & = & 0 \\ yxwzy & = & 0 \\ dxwzy & = & 0 \end{aligned}  \begin{aligned}c̄d & = & λ_{9}zy \\ cc̄d & = & λ_{10}aād \\ dd̄d & = & λ_{11}aād \\ d̄aād & = & λ_{12}xwzy \end{aligned}

where λ_{9}, λ_{11} and λ_{12} are non-zero. Now, we have that d̄c = λ_{5}xw and that c̄d = λ_{9}zy and so c̄dd̄c = λ_{9}λ_{5}zyxw and d̄cc̄d = λ_{5}λ_{9}xwzy where each of λ_{5}, λ_{9} and the paths zyxw and xwzy are non-zero. Hence c̄dd̄c and d̄cc̄d are both non-zero paths. In particular, dd̄c and cc̄d are both non-zero paths and so λ_{7} and λ_{11} are non-zero.

We now determine generators for the relations we have so far. The relations we already have are:

\begin{aligned}āa & = & k_{1}ābb̄a \\ c̄a & = & 0 \\ b̄b & = & k_{2}b̄aāb \\ d̄b & = & 0 \\ āc & = & 0 \\ b̄d & = & 0 \\ b̄bb̄a & = & 0 \\ c̄bb̄a & = & 0 \\ d̄bb̄a & = & 0 \\ āaāb & = & 0 \\ c̄aāb & = & 0 \\ d̄aāb & = & 0 \\ czyx & = & 0 \\ dxwz & = & 0 \end{aligned}  \begin{aligned}yd̄a & = & 0 \\ wc̄b & = & 0 \\ ādx & = & 0 \\ b̄dx & = & 0 \\ d̄dx & = & 0 \\ ācz & = & 0 \\ ābb̄c & = & 0 \\ b̄bb̄c & = & 0 \\ d̄bb̄c & = & 0 \\ āaād & = & 0 \\ b̄aād & = & 0 \\ c̄aād & = & 0 \\ aābb̄a & = & 0 \\ bb̄aāb & = & 0 \end{aligned}  \begin{aligned}b̄cz & = & 0 \\ c̄cz & = & 0 \\ wc̄c & = & 0 \\ yd̄d & = & 0 \\ xwzyx & = & 0 \\ zyxwz & = & 0 \\ wzyxw & = & 0 \\ czyxw & = & 0 \\ yxwzy & = & 0 \\ dxwzy & = & 0 \end{aligned}

\begin{aligned}d̄c & = & λ_{5}xw \\ c̄d & = & λ_{9}zy \\ dd̄a & = & λ_{1}bb̄a \\ cc̄b & = & λ_{2}aāb \\ c̄dx & = & λ_{3}zyx \end{aligned}  \begin{aligned}d̄cz & = & λ_{4}xwz \\ cc̄c & = & λ_{6}bb̄c \\ dd̄c & = & λ_{7}bb̄c \\ cc̄d & = & λ_{10}aād \\ dd̄d & = & λ_{11}aād \end{aligned}

\begin{aligned}c̄bb̄c & = & λ_{8}zyxw \\ d̄aād & = & λ_{12}xwzy \end{aligned}

where the λ_{i} are all non-zero. It is now easy to shows that all the relations in paths of length four and all the relations in paths of length five can be generated by the relations involving paths of length two and three, for example

c̄bb̄a = λ_{1}^{−1}c̄dd̄a
= λ_{1}^{−1}λ_{9}zyd̄a
= λ_{1}^{−1}λ_{9}z0
= 0.

Also

c̄bb̄c = λ_{7}^{−1}c̄dd̄c
= λ_{7}^{−1}λ_{9}λ_{5}zyxw

so that λ_{8} is actually λ_{7}^{−1}λ_{9}λ_{5}, and c̄bb̄c = λ_{8}zyxw is a direct consequence of the relations c̄d = λ_{9}zy and d̄c = λ_{5}xw. We also have that

c̄dx = λ_{9}zyx

and

d̄cz = λ_{5}xwz

so that λ_{3} = λ_{9} and λ_{4} = λ_{5}, and therefore c̄dx = λ_{3}zyx is a direct consequence of the relation c̄d = λ_{9}zy, and d̄cz = λ_{4}xwz is a direct consequence of the relation d̄c = λ_{5}xw. Finally,

b̄dx = 0x = 0

and

ācz = 0z = 0

so that these two relations are generated by the relations involving paths of length two. Hence the generating relations so far are:

\begin{aligned}āa & = & k_{1}ābb̄a \\ c̄a & = & 0 \\ b̄b & = & k_{2}b̄aāb \\ d̄b & = & 0 \\ āc & = & 0 \\ b̄d & = & 0 \end{aligned}  \begin{aligned}yd̄a & = & 0 \\ wc̄b & = & 0 \\ ādx & = & 0 \\ d̄dx & = & 0 \end{aligned}  \begin{aligned}b̄cz & = & 0 \\ c̄cz & = & 0 \\ wc̄c & = & 0 \\ yd̄d & = & 0 \end{aligned}

\begin{aligned}d̄c & = & λ_{5}xw \\ c̄d & = & λ_{9}zy \\ dd̄a & = & λ_{1}bb̄a \\ cc̄b & = & λ_{2}aāb \end{aligned}  \begin{aligned}cc̄c & = & λ_{6}bb̄c \\ dd̄c & = & λ_{7}bb̄c \\ cc̄d & = & λ_{10}aād \\ dd̄d & = & λ_{11}aād. \end{aligned}

We now determine the rest of the relations induced by I by considering the extension structure of P_{77}, given the generating relations already determined. The set of paths of length one is {ā, b̄, c̄, d̄} and then the set of paths of length two is {aā, bb̄, cc̄, wc̄, dd̄, yd̄}. Now there are actually only three composition factors isomorphic to 77 in the third Loewy layer of P_{77} and therefore the four nonzero paths aā, bb̄, cc̄, and dd̄ span a 3-dimensional space modulo e_{77}Rad^{3}(A)e_{77}. Bearing in mind that āa = k_{1}ābb̄a, that b̄b = k_{2}b̄aāb, that c̄a = d̄b = 0 and that āc = b̄d = 0, the set of possible non-zero paths of length three is

{b̄aā, d̄aā, ābb̄, c̄bb̄, b̄cc̄, c̄cc̄, d̄cc̄, xwc̄, ādd̄, c̄dd̄, d̄dd̄, zyd̄}.

Furthermore, as there is only one composition factor isomorphic to each of 49_{a}, 49_{b}, 770_{a} and 770_{b} in the fourth Loewy layer, we know that all the paths beginning with ā span a 1-dimensional space in KQ/I, that all the paths beginning with b̄ span a 1-dimensional space in KQ/I, that all the paths beginning with c̄ and z span a 1-dimensional space in KQ/I and that all the paths beginning with d̄and x span a 1-dimensional space in KQ/I. As there is only one composition factor in the fifth Loewy layer of P_{77}, all the paths of length four must span a single 1-dimensional space in KQ/I and all the paths of length five must be zero. We represent the above information in the following diagram:

[figure: 77; c̄; ā; b̄; d̄; 770_{a}; 49_{a}; 49_{b}; 770_{b}; w; c; a; b; d; y; 154_{b}; 77; 77; 77; 77; 154_{c}; x; d̄; b̄; c̄; d̄; b̄; ā; c̄; d̄; ā; c̄; z; 770_{b}; 49_{b}; 49_{a}; 770_{a}; d; b; a; c; 77]

From the above discussion, we have already seen that we must have a relation

μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + μ_{4}dd̄ + (term in paths of length 4) = 0.

We know the path ābb̄a is non-zero from the extension structure of P_{49_{a}}, so in particular, ābb̄ is non-zero. Indeed we can show in a similar manner that all the paths of length three are non-zero. Finally, we know that each of aābb̄, bb̄aā, cc̄bb̄ and dd̄aā are non-zero, for otherwise we would have that 49_{a}, 49_{b}, 770_{a} and 770_{b} respectively, do not non-trivially extend the 77 of the socle and so lie in the socle themselves, which is not true. Hence we also have the following set of generating relations:

\begin{aligned}ādd̄ & = & μ_{5}ābb̄ \\ b̄cc̄ & = & μ_{6}b̄aā \\ c̄cc̄ & = & μ_{7}c̄bb̄ \\ c̄dd̄ & = & μ_{8}c̄bb̄ \\ d̄cc̄ & = & μ_{9}d̄aā \\ d̄dd̄ & = & μ_{10}d̄aā \end{aligned}  \begin{aligned}zyd̄ & = & μ_{11}c̄bb̄ \\ xwc̄ & = & μ_{12}d̄aā \\ bb̄aā & = & μ_{13}aābb̄ \\ cc̄bb̄ & = & μ_{14}aābb̄ \\ dd̄aā & = & μ_{15}aābb̄ \end{aligned}

\begin{aligned}wc̄bb̄ & = & 0 \\ yd̄aā & = & 0 \\ āaābb̄ & = & 0 \end{aligned}  \begin{aligned}b̄aābb̄ & = & 0 \\ c̄aābb̄ & = & 0 \\ d̄aābb̄ & = & 0 \end{aligned}

where all of the μ_{i} are non-zero. Our relation in paths of length two can therefore be written as

μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + μ_{4}dd̄= k_{3}aābb̄.

Again we can show that the relations in paths of length four and five are a direct result of all the relations involving paths of length two and three. For example

bb̄aā = λ_{1}^{−1}dd̄aā
= λ_{1}^{−1}μ_{10}^{−1}dd̄dd̄
= λ_{1}^{−1}μ_{10}^{−1}λ_{11}aādd̄
= λ_{1}^{−1}μ_{10}^{−1}λ_{11}μ_{5}aābb̄
= μ_{13}aābb̄

and

d̄aābb̄ = μ_{5}^{−1}d̄aādd̄

= μ_{5}^{−1}μ_{10}^{−1}d̄dd̄dd̄
= μ_{5}^{−1}d̄dd̄aā
= μ_{5}^{−1}λ_{1}d̄bb̄aā
= μ_{5}^{−1}λ_{1}0b̄aā
= 0.

We also have that

zyd̄ = λ_{9}^{−1}c̄dd̄
= λ_{9}^{−1}μ_{8}c̄bb̄
= μ_{11}c̄bb̄

and

xwc̄ = λ_{5}^{−1}d̄cc̄
= λ_{5}^{−1}μ_{9}d̄aā
= μ_{12}d̄aā.

Hence we can now write down the generating relations induced by I:

\begin{aligned}āa & = & k_{1}ābb̄a \\ c̄a & = & 0 \\ b̄b & = & k_{2}b̄aāb \\ d̄b & = & 0 \\ āc & = & 0 \\ b̄d & = & 0 \end{aligned}  \begin{aligned}yd̄a & = & 0 \\ wc̄b & = & 0 \\ ādx & = & 0 \\ d̄dx & = & 0 \end{aligned}  \begin{aligned}b̄cz & = & 0 \\ c̄cz & = & 0 \\ wc̄c & = & 0 \\ yd̄d & = & 0 \end{aligned}

μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + μ_{4}dd̄= k_{3}aābb̄

\begin{aligned}d̄c & = & λ_{5}xw \\ dd̄a & = & λ_{1}bb̄a \\ cc̄b & = & λ_{2}aāb \\ cc̄c & = & λ_{6}bb̄c \\ dd̄c & = & λ_{7}bb̄c \\ cc̄d & = & λ_{10}aād \\ dd̄d & = & λ_{11}aād \end{aligned}  \begin{aligned}c̄d & = & λ_{9}zy \\ ādd̄ & = & μ_{5}ābb̄ \\ b̄cc̄ & = & μ_{6}b̄aā \\ c̄cc̄ & = & μ_{7}c̄bb̄ \\ c̄dd̄ & = & μ_{8}c̄bb̄ \\ d̄cc̄ & = & μ_{9}d̄aā \\ d̄dd̄ & = & μ_{10}d̄aā. \end{aligned}

5.2 Determining the Ideal I 127

5.2 Determining the Ideal I

We now simplify the relations in the k_{i}, μ_{i} and λ_{i}.

Given the relations

dd̄a = λ_{1}bb̄a

and

ādd̄= μ_{5}ābb̄

we have that

ādd̄a = λ_{1}ābb̄a

and

ādd̄a = μ_{5}ābb̄a

and as ābb̄a is not equal to zero, λ_{1} = μ_{5}. By applying similar arguments we see that the relations in the λ_{i} are:

\begin{aligned}d̄c & = & λ_{5}xw \\ dd̄a & = & λ_{1}bb̄a \\ cc̄b & = & λ_{2}aāb \\ cc̄c & = & λ_{6}bb̄c \\ dd̄c & = & λ_{7}bb̄c \\ cc̄d & = & λ_{10}aād \\ dd̄d & = & λ_{11}aād \end{aligned}  \begin{aligned}c̄d & = & λ_{9}zy \\ ādd̄ & = & λ_{1}ābb̄ \\ b̄cc̄ & = & λ_{2}b̄aā \\ c̄cc̄ & = & λ_{6}c̄bb̄ \\ c̄dd̄ & = & λ_{7}c̄bb̄ \\ d̄cc̄ & = & λ_{10}d̄aā \\ d̄dd̄ & = & λ_{11}d̄aā. \end{aligned}

Choosing Basis Elements

Consider the relations

āa = k_{1}ābb̄a
b̄b = k_{2}b̄aāb
k_{3}aābb̄ = μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + μ_{4}dd̄.

Recall that each arrow x_{j} ⟵ x_{i} corresponds to a basis element for the complement of e_{j}Rad^{2}(A)e_{i} in e_{j}Rad(A)e_{i} and the above are the relations that depend on the component of each of the basis elements in Rad^{2}(A) (all the other generating relations determined depend only on the choice of basis elements modulo Rad^{2}(A)). Hence if we rechoose the basis elements in e_{j}Rad(A)e_{i} by adding elements of e_{j}Rad^{2}(A)e_{i}, then only these relations will be affected.

Let ā_{0} be the choice of basis element for the complement of e_{49_{a}}Rad^{2}(A)e_{77} in e_{49_{a}}Rad(A)e_{77}, associated with the arrow ā and let b_{0} and b̄_{0} be the choices of basis elements associated with the arrows b and b̄ respectively. Set ā_{1} = ā_{0}(1 − k_{1}b_{0}b̄_{0}) and let this now be our choice of basis element corresponding to the arrow ā. Then the relations in ā are

āa = 0
b̄b = k_{2}b̄aāb
k′_{3}aābb̄ = μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + μ_{4}dd̄.

Similarly we have b̄_{0} as the choice of basis element corresponding to the arrow b̄ and let a_{0} and ā_{0} be the current choices of basis elements associated with the arrows a and ā respectively. Set b̄_{1} = b̄_{0}(1 − k_{2}a_{0}ā_{0}). Then if we take b̄_{1} as our new choice of basis element corresponding to the arrow b̄, the relations in b̄ may be written

b̄b = 0
k′′_{3}aābb̄ = μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + μ_{4}dd̄.

Now consider the relation

μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + μ_{4}dd̄= k′′_{3}aābb̄.

Lemma 5.2.1

μ_{4} ≠ 0.

Proof. Suppose μ_{4} = 0, then the relation is

μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ = k′′_{3}aābb̄.

We then have

μ_{1}aāa + μ_{2}bb̄a + μ_{3}cc̄a = k′′_{3}aābb̄a.

But given that c̄a = 0, aābb̄a = 0, and with our new choice of basis element corresponding to ā, āa = 0, this can be written as

μ_{2}bb̄a = 0.

As the path bb̄a is non-zero, we must have μ_{2} = 0. Hence our relation is

μ_{1}aā + μ_{3}cc̄ = k′′_{3}aābb̄.

We then have

μ_{1}aāc + μ_{3}cc̄c = k′′_{3}aābb̄c

i.e.

μ_{3}cc̄c = 0.

As cc̄c ≠ 0, we must have μ_{3} = 0. Hence our relation is

μ_{1}aā = k′′_{3}aābb̄.

But then

μ_{1}aāb = k′′_{3}aābb̄b = 0

which means that μ_{1} = 0. Hence μ_{4} = 0 ⇒ μ_{1} = μ_{2} = μ_{3} = 0, a contradiction. Therefore μ_{4} ≠ 0. □

Given that μ_{4} ≠ 0 and given the fact that aādd̄ = λ_{1}aābb̄ with λ_{1} ≠ 0, our relation can be expressed as

μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + (1 − μ_{4}^{−1}k′′_{3}λ_{1}^{−1}aā)μ_{4}dd̄= 0.

Set d_{1} = (1 − μ_{4}^{−1}k′′_{3}λ_{1}^{−1}a_{0}ā_{0})d_{0}, and take this as our new choice of basis element corresponding to the arrow d. The relation in d is then

μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + μ_{4}dd̄= 0.

Hence with these new choices of basis elements corresponding to the arrows, the generating relations are

\begin{aligned}āa & = & 0 \\ c̄a & = & 0 \\ b̄b & = & 0 \\ d̄b & = & 0 \\ āc & = & 0 \\ b̄d & = & 0 \end{aligned}  \begin{aligned}yd̄a & = & 0 \\ wc̄b & = & 0 \\ ādx & = & 0 \\ d̄dx & = & 0 \end{aligned}  \begin{aligned}b̄cz & = & 0 \\ c̄cz & = & 0 \\ wc̄c & = & 0 \\ yd̄d & = & 0 \end{aligned}

μ_{1}aā + μ_{2}bb̄ + μ_{3}cc̄ + μ_{4}dd̄= 0

\begin{aligned}d̄c & = & λ_{5}xw \\ dd̄a & = & λ_{1}bb̄a \\ cc̄b & = & λ_{2}aāb \\ cc̄c & = & λ_{6}bb̄c \\ dd̄c & = & λ_{7}bb̄c \\ cc̄d & = & λ_{10}aād \\ dd̄d & = & λ_{11}aād \end{aligned}  \begin{aligned}c̄d & = & λ_{9}zy \\ ādd̄ & = & λ_{1}ābb̄ \\ b̄cc̄ & = & λ_{2}b̄aā \\ c̄cc̄ & = & λ_{6}c̄bb̄ \\ c̄dd̄ & = & λ_{7}c̄bb̄ \\ d̄cc̄ & = & λ_{10}d̄aā \\ d̄dd̄ & = & λ_{11}d̄aā. \end{aligned}

As each of our arrows a, b, c, x, y and so on correspond to bases of 1dimensional spaces, we can rechoose our basis elements by replacing a given basis element by a non-zero scalar multiple of itself. Let c_{0} and d_{0} be the basis elements corresponding to the arrows c and d respectively. We choose new basis elements c_{1} and d_{1} as follows:

c_{1} = λ_{6}^{−1}c_{0}
d_{1} = λ_{1}^{−1}d_{0}.

These choices of basis elements have no effect on the zero relations, but the relations in the scalars λ_{i} and μ_{i} are now

μ_{1}aā + μ_{2}bb̄ + μ′_{3}cc̄ + μ′_{4}dd̄= 0

\begin{aligned}d̄c & = & λ′_{5}xw \\ dd̄a & = & bb̄a \\ cc̄b & = & λ′_{2}aāb \\ cc̄c & = & bb̄c \\ dd̄c & = & λ′_{7}bb̄c \\ cc̄d & = & λ′_{10}aād \\ dd̄d & = & λ′_{11}aād \end{aligned}  \begin{aligned}c̄d & = & λ′_{9}zy \\ ādd̄ & = & ābb̄ \\ b̄cc̄ & = & λ′_{2}b̄aā \\ c̄cc̄ & = & c̄bb̄ \\ c̄dd̄ & = & λ′_{7}c̄bb̄ \\ d̄cc̄ & = & λ′_{10}d̄aā \\ d̄dd̄ & = & λ′_{11}d̄aā \end{aligned}

where μ′_{3} = μ_{3}λ_{6}, μ′_{4} = μ_{4}λ_{1}, λ′_{5} = λ_{5}λ_{6}^{−1}, λ′_{9} = λ_{9}λ_{1}^{−1}, λ′_{2} = λ_{2}λ_{6}^{−1}, λ′_{7} = λ_{7}λ_{1}^{−1}, λ′_{10} = λ_{10}λ_{6}^{−1} and λ′_{11} = λ_{11}λ_{1}^{−1}. Now rechoose basis elements as follows:

a_{1} = λ′_{2}a_{0}
x_{1} = λ′_{5}x_{0}
z_{1} = λ′_{9}z_{0}.

5.2 Determining the Ideal I 131 Then the relations are

α_{1}aā + α_{2}bb̄ + α_{3}cc̄ + α_{4}dd̄= 0
\begin{aligned}d̄c & = & xw \\ dd̄a & = & bb̄a \\ cc̄b & = & aāb \\ cc̄c & = & bb̄c \\ dd̄c & = & β_{1}bb̄c \\ cc̄d & = & β_{2}aād \\ dd̄d & = & β_{3}aād \end{aligned}  \begin{aligned}c̄d & = & zy \\ ādd̄ & = & ābb̄ \\ b̄cc̄ & = & b̄aā \\ c̄cc̄ & = & c̄bb̄ \\ c̄dd̄ & = & β_{1}c̄bb̄ \\ d̄cc̄ & = & β_{2}d̄aā \\ d̄dd̄ & = & β_{3}d̄aā \end{aligned}

for appropriate choices of scalars α_{i} and β_{i}.

Solving for Scalars

Given the relations in paths of length three we have the following two relations

dd̄aā = bb̄aā
= bb̄cc̄

and

dd̄aā = β_{2}^{−1}dd̄cc̄
= β_{2}^{−1}β_{1}bb̄cc̄.

Hence β_{2}^{−1}β_{1} = 1, i.e. β_{1} = β_{2}, and we have

\begin{aligned}d̄c & = & xw \\ dd̄a & = & bb̄a \\ cc̄b & = & aāb \\ cc̄c & = & bb̄c \\ dd̄c & = & β_{1}bb̄c \\ cc̄d & = & β_{1}aād \\ dd̄d & = & β_{3}aād \end{aligned}  \begin{aligned}c̄d & = & zy \\ ādd̄ & = & ābb̄ \\ b̄cc̄ & = & b̄aā \\ c̄cc̄ & = & c̄bb̄ \\ c̄dd̄ & = & β_{1}c̄bb̄ \\ d̄cc̄ & = & β_{1}d̄aā \\ d̄dd̄ & = & β_{3}d̄aā \end{aligned}

Turning to the relation

α_{1}aā + α_{2}bb̄ + α_{3}cc̄ + α_{4}dd̄= 0

we have that

0 = α_{2}bb̄a + α_{4}dd̄a
= α_{2}bb̄a + α_{4}bb̄a

and

0 = α_{1}aāb + α_{3}cc̄b
= α_{1}aāb + α_{3}aāb

and as each of bb̄a and aāb are non-zero, this tells us that α_{3} = −α_{1} and α_{4} = −α_{2}. Therefore we have

α_{1}aā + α_{2}bb̄ − α_{1}cc̄ − α_{2}dd̄= 0.

As a result

0 = α_{2}bb̄c − α_{1}cc̄c − α_{2}dd̄c
= α_{2}bb̄c − α_{1}bb̄c − α_{2}β_{1}bb̄c

and hence

α_{2} − α_{1} − α_{2}β_{1} = 0

i.e.,

α_{2}(1 − β_{1}) = α_{1}.

In particular, if α_{2} = 0 then also α_{1} = 0 which is not possible, therefore α_{2} is non-zero. If we divide the relation through by α_{2} we have

(1 − β_{1})aā + bb̄ − (1 − β_{1})cc̄ − dd̄= 0.

We now have

0 = (1 − β_{1})aād − (1 − β_{1})cc̄d − dd̄d
= (1 − β_{1})aād − β_{1}(1 − β_{1})aād − β_{3}aād

so that

β_{3} = (1 − β_{1}) − β_{1}(1 − β_{1})
= (1 − β_{1})^{2}.

We therefore have relations

(1 − β_{1})aā + bb̄ − (1 − β_{1})cc̄ − dd̄= 0

\begin{aligned}d̄c & = & xw \\ dd̄a & = & bb̄a \\ cc̄b & = & aāb \\ cc̄c & = & bb̄c \\ dd̄c & = & β_{1}bb̄c \\ cc̄d & = & β_{1}aād \\ dd̄d & = & (1 − β_{1})^{2}aād \end{aligned}  \begin{aligned}c̄d & = & zy \\ ādd̄ & = & ābb̄ \\ b̄cc̄ & = & b̄aā \\ c̄cc̄ & = & c̄bb̄ \\ c̄dd̄ & = & β_{1}c̄bb̄ \\ d̄cc̄ & = & β_{1}d̄aā \\ d̄dd̄ & = & (1 − β_{1})^{2}d̄aā. \end{aligned}

It does not appear to be possible to reduce the number of scalars any further and therefore the relations

\begin{aligned}āa & = & 0 \\ c̄a & = & 0 \\ b̄b & = & 0 \\ d̄b & = & 0 \\ āc & = & 0 \\ b̄d & = & 0 \end{aligned}  \begin{aligned}yd̄a & = & 0 \\ wc̄b & = & 0 \\ ādx & = & 0 \\ d̄dx & = & 0 \end{aligned}  \begin{aligned}b̄cz & = & 0 \\ c̄cz & = & 0 \\ wc̄c & = & 0 \\ yd̄d & = & 0 \end{aligned}

(1 − β_{1})aā + bb̄ − (1 − β_{1})cc̄ − dd̄= 0

\begin{aligned}d̄c & = & xw \\ dd̄a & = & bb̄a \\ cc̄b & = & aāb \\ cc̄c & = & bb̄c \\ dd̄c & = & β_{1}bb̄c \\ cc̄d & = & β_{1}aād \\ dd̄d & = & (1 − β_{1})^{2}aād \end{aligned}  \begin{aligned}c̄d & = & zy \\ ādd̄ & = & ābb̄ \\ b̄cc̄ & = & b̄aā \\ c̄cc̄ & = & c̄bb̄ \\ c̄dd̄ & = & β_{1}c̄bb̄ \\ d̄cc̄ & = & β_{1}d̄aā \\ d̄dd̄ & = & (1 − β_{1})^{2}d̄aā \end{aligned}

determine a 1-parameter family of algebras to which the basic algebra of the second block must belong.

The Structure of the Basic Algebra of the Principal Block of the Higman–Sims Group

We will now use the information we have about the Loewy structure and socle structure of the PIMs to determine the structure of the basic algebra of the principal block of HS. We will find that the relations can be determined up to the presence of a single scalar, so that we have determined a 1-parameter family of algebras to which the basic algebra of the principal block must belong. The relations for this 1-parameter family will be found at the end of the chapter.

We will denote the basic algebra of the principal block of HS by A, and denote the primitive orthogonal idempotent in A corresponding to P_{S} by e_{S}. By Theorem 4.0.1 (p. 90), the Ext-quiver for A is:

[figure: 748; x; y; z; x̄; ȳ; z̄; 321; 154_{a}; 1; 1176; 22; 1253; ā; b̄; c̄; d̄; ē; a; b; c; d; e]

where a corresponds to a basis element for the complement of e_{1253}Rad^{2}(A)e_{321} in e_{1253}Rad(A)e_{321}, ā corresponds to a basis element for the complement of e_{321}Rad^{2}(A)e_{1253} in e_{321}Rad(A)e_{1253}, and so on. We now consider the extension structure of P_{S} for each simple module S of the principal block and use this information to find generators for the relations induced by I (where I is the ideal of KQ such that A ≅ KQ/I). We then minimise the set of generators and choose basis elements corresponding to the arrows so as to minimise the number of unknown scalar factors involved in the relations.

6.1 Finding Generators for the Relations In-

duced by I

Part I: P_{321}, P_{154_{a}}, P_{1}, P_{22} and P_{1176}

Given the Loewy structure and socle structure of P_{321} (see Theorem 4.0.1), the extension structure of P_{321} must be as indicated in the following diagram:

[figure: 321; 1253; a; 1176; 154_{a}; 22; b̄; d̄; ē; 1253; b; d; e; 321; ā]

From the Loewy and socle structure, we know that the composition factor 1253 of the second Loewy layer must non-trivially extend the composition factor 154_{a} of the third Loewy layer, and that this in turn must non-trivially extend the composition factor 1253 of the fourth Loewy layer. In terms of paths, this means that bb̄a is non-zero. A similar argument tells us that dd̄a and eēa are non-zero, and hence eēa, dd̄a and bb̄a are non-zero paths spanning a 1-dimensional space modulo e_{1253}Rad^{4}(A)e_{321}. But as all paths of length five are zero and there are no paths of length four from 321 to 1253, this means that eēa, dd̄a and bb̄a span a 1-dimensional space in KQ/I. We also have āa = 0 modulo e_{321}Rad^{3}(A)e_{321}. As the only non-zero paths from 321 to 321 of length greater than two are the paths ābb̄a, ādd̄a and āeēa which span a 1-dimensional space, we must have āa = k_{1}ābb̄a for some scalar k_{1} (possibly equal to zero) in K. We can now write down generators for the relations in paths starting at the vertex 321:

\begin{aligned}āa & = & k_{1}ābb̄a \\ c̄a & = & 0 \\ yd̄a & = & 0 \\ zēa & = & 0 \\ dd̄a & = & λ_{1}bb̄a \\ eēa & = & λ_{2}bb̄a \end{aligned}  \begin{aligned}b̄bb̄a & = & 0 \\ c̄bb̄a & = & 0 \\ d̄bb̄a & = & 0 \\ ēbb̄a & = & 0 \\ aābb̄a & = & 0 \end{aligned}

where λ_{1} and λ_{2} are non-zero.

Given the Loewy structure and socle structure of P_{154_{a}}, the extension structure of P_{154_{a}} must be:

[figure: 154_{a}; 1253; b; 154_{a}; 321; 1; 22; ā; b̄; c̄; ē; 1253; a; b; c; e; 154_{a}; b̄]

and using the same type of argument as above, we have generating relations:

\begin{aligned}d̄b & = & 0 \\ xc̄b & = & 0 \\ zēb & = & 0 \\ bb̄b & = & λ_{3}aāb \\ cc̄b & = & λ_{4}aāb \\ eēb & = & λ_{5}aāb \end{aligned}  \begin{aligned}āaāb & = & 0 \\ c̄aāb & = & 0 \\ d̄aāb & = & 0 \\ ēaāb & = & 0 \\ bb̄aāb & = & 0 \end{aligned}

where the λ_{i} are non-zero.

Given the Loewy and socle structure of P_{1}, the extension structure of P_{1} must be:

[figure: 1; c; x; 1253; 748; b̄; c̄; d̄; ē; z̄; x̄; ȳ; 154_{a}; 1; 1176; 22; 1; 1176; b; c; d; e; c; d; x; y; z; x; y; 1253; 748; c̄; x̄; 1]

We know that there is only one composition factor isomorphic to 22 in the third Loewy layer of P_{1} and as there are no non-zero paths of length ≥ 3 from 1 to 22, the paths ēc and z̄x span a 1-dimensional space in KQ/I. Although at least one of the paths ēc and z̄x must be non-zero, at present we have no way of knowing which it is, so we are not yet able to choose a basis for this 1-dimensional space. Lemma 6.1.1

z̄x ≠ 0.

Proof. Consider the structure of the principal block component of 28_{B}↑, determined in Chapter 4 (p. 92), which has the form

\begin{aligned}& 748 \\ 1 & 1176 & 22 \\ & 748 & . \end{aligned}

Given the existence of such a module, we know that P_{748} must have both a submodule and a factor module with structure

\begin{aligned}& 748 \\ 1 & 1176 & 22 \\ & 748 & . \end{aligned}

Now consider the extension structure of P_{748}, which, given the Loewy structure and socle structure of P_{748} must be:

[figure: 748; 1176; 1; 22; ȳ; x̄; z̄; 1253; 748; c; x; d; y; e; z; 1176; 1; 22; c̄; x̄; d̄; ȳ; ē; z̄; y; x; z; 748]

The existence of the factor module

748 1 1176 22 748

means that the paths xx̄, yȳ and zz̄, are non-zero. The fact that we have a submodule with structure

748 1 1176 22 748

means that also the paths xx̄xx̄, yȳxx̄, zz̄xx̄, xx̄yȳ, yȳyȳ, zz̄yȳ, xx̄zz̄, yȳzz̄ and zz̄zz̄ are all non-zero. In particular, as zz̄xx̄ is non-zero, the path z̄x must also be non-zero. □

Returning to P_{1} we have found a basis for the 1-dimensional space spanned by the paths ēc and z̄x. We also then know that the path xx̄x is non-zero (as xx̄xx̄ is non-zero) and so we may take xx̄x to be the basis of the 1-dimensional space corresponding to the composition factor 748 of the fourth Loewy layer of P_{1}. Finally, we know that the factor 154_{a} of the third Loewy layer must be extended non-trivially by the 1253 of the second Loewy layer, and must non-trivially extend the 1253 of the fourth Loewy layer, so that the path bb̄c is non-zero. Hence we have generating relations:

\begin{aligned}āc & = & 0 \\ ēc & = & r_{1}z̄x \\ xc̄c & = & r_{2}xx̄x \\ yd̄c & = & r_{3}xx̄x \\ zz̄x & = & r_{4}xx̄x \\ yȳx & = & r_{5}xx̄x \\ cc̄c & = & r_{6}bb̄c \\ dd̄c & = & r_{7}bb̄c \\ ez̄x & = & r_{8}bb̄c \\ cx̄x & = & r_{9}bb̄c \\ dȳx & = & r_{10}bb̄c \end{aligned}  \begin{aligned}ābb̄c & = & 0 \\ b̄bb̄c & = & 0 \\ d̄bb̄c & = & 0 \\ ēbb̄c & = & 0 \\ ȳxx̄x & = & 0 \\ z̄xx̄x & = & 0 \\ c̄bb̄c & = & r_{11}x̄xx̄x \\ cx̄xx̄x & = & 0 \\ xx̄xx̄x & = & 0. \end{aligned}

Given the Loewy structure and socle structure of P_{1176}, the extension structure of P_{1176} must be:

[figure: 1176; d; y; 1253; 748; ā; c̄; ē; d̄; z̄; ȳ; x̄; 321; 1; 22; 1176; 1; a; c; e; d; c; x; z; y; x; 1253; 748; d̄; ȳ; 1176]

As the paths zz̄yȳ, yȳyȳ and xx̄yȳ are non-zero, the paths z̄y, ȳy and xx̄y are non-zero. Furthermore, as the only factor that 321 can non-trivially extend or be extended non-trivially by is 1253, we must have the composition factor 1253 of the second Loewy layer non-trivially extending the factor 321 of the third Loewy layer, which in turn must non-trivially extend the factor 1253 of the fourth Loewy layer. Hence the path aād is non-zero. Generators for the relations in paths starting at the vertex 1176 are therefore:

\begin{aligned}b̄d & = & 0 \\ ēd & = & s_{1}z̄y \\ d̄d & = & s_{2}ȳy + k_{2}ȳxx̄y \\ xc̄d & = & s_{3}xx̄y \\ zz̄y & = & s_{4}xx̄y \\ yȳy & = & s_{5}xx̄y \\ cc̄d & = & s_{6}aād \\ ez̄y & = & s_{7}aād \\ dȳy & = & s_{8}aād \\ cx̄y & = & s_{9}aād \end{aligned}  \begin{aligned}āaād & = & 0 \\ b̄aād & = & 0 \\ c̄aād & = & 0 \\ ēaād & = & 0 \\ x̄xx̄y & = & 0 \\ z̄xx̄y & = & 0 \\ d̄aād & = & s_{10}ȳxx̄y \\ dȳxx̄y & = & 0 \\ yȳxx̄y & = & 0. \end{aligned}

Given the Loewy and socle structure of P_{22}, the extension structure of P_{22} must be:

[figure: 22; e; z; 1253; 748; b̄; ā; ē; c̄; d̄; x̄; ȳ; z̄; 154_{a}; 321; 22; 1; 1176; 22; b; a; e; c; d; e; z; x; y; z; 1253; 748; ē; z̄; 22]

Using the same types of arguments as before, the paths x̄z, ȳz xx̄z, bb̄e and aāe are non-zero. Hence generators for the relations in paths starting at the vertex 22 are:

\begin{aligned}c̄e & = & t_{1}x̄z \\ d̄e & = & t_{2}ȳz \end{aligned}  \begin{aligned}āaāe & = & 0 \\ b̄aāe & = & 0 \end{aligned}

\begin{aligned}zēe & = & t_{3}xx̄z \\ yȳz & = & t_{4}xx̄z \\ zz̄z & = & t_{5}xx̄z \\ bb̄e & = & t_{6}aāe \\ eēe & = & t_{7}aāe \\ cx̄z & = & t_{8}aāe \\ dȳz & = & t_{9}aāe \\ ez̄z & = & t_{10}aāe \end{aligned}  \begin{aligned}c̄aāe & = & 0 \\ d̄aāe & = & 0 \\ x̄xx̄z & = & 0 \\ ȳxx̄z & = & 0 \\ ēaāe & = & t_{11}z̄xx̄z \\ ez̄xx̄z & = & 0 \\ zz̄xx̄z & = & 0 \end{aligned}

where t_{6} is non-zero.

Part II: P_{748} and P_{1253}

We have already seen from page 138 that the extension structure of P_{748} is:

[figure: 748; 1176; 1; 22; ȳ; x̄; z̄; 1253; 748; c; x; d; y; e; z; 1176; 1; 22; c̄; x̄; d̄; ȳ; ē; z̄; y; x; z; 748]

We know that xx̄ is a basis for the space spanned by xx̄, yȳ and zz̄ modulo e_{748}Rad^{3}(A)e_{748}, but we do not have a basis for the space spanned by cx̄, dȳ and ez̄. We will now show that the path dȳ is non-zero, giving us a basis for this space. Then using the fact that the paths x̄xx̄, ȳxx̄, z̄xx̄ and xx̄xx̄ are non-zero, we will be able to write down generators for the relations in paths starting at the vertex 748.

Lemma 6.1.2

dȳ ≠ 0.

Proof. Suppose dȳ is zero. Then P_{748} must have a quotient with one of the following structures:

748 1 22 1253  748 1 1253  \begin{matrix}748 \\ 22 \\ 1253 . \end{matrix}

Suppose we have a module

M = \begin{matrix}748 \\ 1 \\ 1253 . \end{matrix}

Then, given the structure of the restricted modules (Section 4.1), M↓ = 288 ⊕

126_{A} ⊕ 126_{B} ⊕ 288 ⊕ 288 ⊕ 252 ⊕ 84_{B} ⊕ \begin{matrix}84_{A} \\ 21_{A} \\ 84_{A} \end{matrix} ⊕ N where N has filtration

\begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} 1_{A}  \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 & . \end{matrix}

By Frobenius Reciprocity we have

(1_{A}, M↓) = (1_{A}↑ , M) = (1 ⊕ \begin{matrix}49_{b} \\ 77 \\ 49_{a} \end{matrix} , \begin{matrix}748 \\ 1 \\ 1253 \end{matrix} ) = 0

and that

(28_{B}, M↓) = (28_{B}↑ , M) = ( \begin{matrix}& 748 \\ 1 & 1176 & 22 \\ & 748 \end{matrix} , \begin{matrix}748 \\ 1 \\ 1253 \end{matrix} ) = 0

and hence the only possibility for the socle of N is that it contains one or two composition factors isomorphic to 56. Suppose that (56, M↓) = 2. Then (56↑ , M) = 2. We have already seen that 56↑ has first block composition factors 2.1+2.22+154_{a} +2.748+1176+2.1253, where both the socle and head of 56↑ are isomorphic to 748 ⊕ 1253 (Lemma 4.2.1, p. 103). In order to have (56↑ , M) = 2 we must have

(56↑ , \begin{matrix}748 \\ 1 \\ 1253 . \end{matrix}) = 2.

Hence as well as having a single composition factor isomorphic to 1253 in the head, 56↑ must also have a factor module isomorphic to

{1 \atop 1253}

or a factor module isomorphic to

\begin{matrix}748 \\ 1 \\ 1253 . \end{matrix}

The first case is not possible as there is no composition factor isomorphic to 1 in the head of 56↑. The second case is also not possible; there is a composition factor isomorphic to 154_{a} in 56↑ which is not contained in the socle or the head and which therefore must extend the 1253 of the socle non-trivially. Hence, any factor module of 56↑ having the 1253 of the socle as a composition factor must also have the 154_{a} as a composition factor, and so there can be no factor module of the form

\begin{matrix}748 \\ 1 \\ 1253 . \end{matrix}

Hence we must actually have (56↑ , M) = 1 and therefore (56, M↓) = 1. This tells us that N has simple socle isomorphic to 56 and so N is a submodule of P_{56}. Given the composition factors of N this is not possible, and hence M does not exist.

Now suppose we have a module

M = \begin{matrix}748 \\ 22 \\ 1253 . \end{matrix}

6.1 Finding Generators for the Relations Induced by I 144 Then M↓ = 288 ⊕ 126_{A} ⊕ 126_{B} ⊕ 288 ⊕ 288 ⊕ 252 ⊕ \begin{matrix}84_{A} \\ 21_{A} \\ 84_{A} \end{matrix} ⊕ N where N has filtration

\begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} 21_{B} ⊕ 1_{B} 84_{B} ⊕ \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 \end{matrix} .

By Frobenius Reciprocity we have

(M↓ , 21_{B}) = (M, 21_{B}↑) = ( \begin{matrix}748 \\ 22 \\ 1253 \end{matrix} , \begin{matrix}22 \\ 748 \\ 22 \end{matrix} ) = 1

and

(21_{B}, M↓) = (21_{B}↑ , M) = ( \begin{matrix}22 \\ 748 \\ 22 \end{matrix} , \begin{matrix}748 \\ 22 \\ 1253 \end{matrix} ) = 0.

Hence N = \begin{matrix}21_{B} \\ 84_{B} \end{matrix} ⊕ U where U has filtration

\begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} 1_{B}  \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 & . \end{matrix}

Now, by Frobenius Reciprocity,

(1_{B}, M↓) = (1_{B}↑ , M) = (154_{b} ⊕ 22, \begin{matrix}748 \\ 22 \\ 1253 \end{matrix} ) = 0

and

(28_{B}, M↓) = (28_{B}↑ , M) = ( \begin{matrix}& 748 \\ 1 & 1176 & 22 \\ & 748 \end{matrix} , \begin{matrix}748 \\ 22 \\ 1253 \end{matrix} ) = 0

and hence the only possibility for the socle of U is that it contains at most two composition factors isomorphic to 56. Suppose that (56, M↓) = 2. Then (56↑ , M) = 2 and so we must have

(56↑ , \begin{matrix}748 \\ 22 \\ 1253 \end{matrix} ) = 2.

Therefore, as well as having a single composition factor isomorphic to 1253 in the head, 56↑ must also have a factor module isomorphic to

{22 \atop 1253}

or a factor module isomorphic to

\begin{matrix}748 \\ 22 \\ 1253 . \end{matrix}

The first case is not possible as there is no composition factor isomorphic to 22 in the head of 56↑. The second case is also not possible as again, any factor module of 56↑ having the 1253 of the socle as a composition factor must also have the 154_{a} as a composition factor. Hence we must actually have (56↑ , M) = 1 and so (56, M↓) = 1. This tells us that U has simple socle isomorphic to 56 and therefore U is a submodule of P_{56}. Given the composition factors of U, this is not possible and so M does not exist.

Finally, suppose we have a module

M = \begin{matrix}748 \\ 1 & 22 \\ 1253 & & . \end{matrix}

Using a similar argument to above, we find that M↓ = 288⊕126_{A} ⊕126_{B} ⊕288⊕

288 ⊕ 252 ⊕ \begin{matrix}84_{A} \\ 21_{A} \\ 84_{A} \end{matrix} ⊕ \begin{matrix}21_{B} \\ 84_{B} \end{matrix} ⊕ N where N has filtration

\begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} 1_{A} ⊕ 1_{B}  \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 & . \end{matrix}

Again N can be shown to be a submodule of P_{56}, which is not possible, and so M does not exist. □

We have now shown that dȳ is non-zero, and so we can write out generators for the relations in paths starting at the vertex 748. They are:

\begin{aligned}yȳ & = & α_{1}xx̄ + k_{3}xx̄xx̄ \\ zz̄ & = & α_{2}xx̄ + k_{4}xx̄xx̄ \\ cx̄ & = & α_{3}dȳ \\ ez̄ & = & α_{4}dȳ \\ ādȳ & = & 0 \\ b̄dȳ & = & 0 \\ c̄dȳ & = & α_{5}x̄xx̄ \\ d̄dȳ & = & α_{6}ȳxx̄ \\ ēdȳ & = & α_{7}z̄xx̄ \end{aligned}  \begin{aligned}cx̄xx̄ & = & 0 \\ dȳxx̄ & = & 0 \\ ez̄xx̄ & = & 0 \\ yȳxx̄ & = & α_{8}xx̄xx̄ \\ zz̄xx̄ & = & α_{9}xx̄xx̄ \\ x̄xx̄xx̄ & = & 0 \\ ȳxx̄xx̄ & = & 0 \\ z̄xx̄xx̄ & = & 0 \end{aligned}

where α_{1}, α_{2}, α_{3} and α_{4} are non-zero.

We now determine the relations induced on the paths beginning at the vertex 1253 by considering the extension structure of P_{1253} given the generating relations already determined. The set of paths of length one is {ā, b̄, c̄, d̄, ē} and then the set of paths of length two is {aā, bb̄, cc̄, xc̄, dd̄, yd̄, eē, zē}. There is only one composition factor isomorphic to 748 in the third Loewy layer of P_{1253} and as all the paths from 1253 to 748 of length greater than two are equal to zero in KQ/I, the three paths xc̄, yd̄and zē span a 1-dimensional space in KQ/I. Furthermore, there are actually only two composition factors isomorphic to 1253 in the third Loewy layer of P_{1253}, and therefore the five paths aā, bb̄, cc̄, dd̄ and eē span a 2dimensional space modulo e_{1253}Rad^{3}(A)e_{1253}. Bearing in mind that āa = k_{1}ābb̄a, that āc = c̄a = 0, and that d̄b = b̄d = 0, the set of paths of length three which could be non-zero is:

{b̄aā, d̄aā, ēaā, ābb̄, b̄bb̄, c̄bb̄, ēbb̄, b̄cc̄, c̄cc̄, d̄cc̄, ēcc̄, ādd̄, c̄dd̄, d̄dd̄, ēdd̄,

āeē, b̄eē, c̄eē, d̄eē, ēeē, x̄xc̄, ȳxc̄, z̄xc̄, x̄yd̄, ȳyd̄, z̄yd̄, x̄zē ȳzē z̄zē}.

Furthermore, as there is only one composition factor isomorphic to each of 321, 154_{a}, 1, 1176 and 22 in the fourth Loewy layer, we know that all the paths beginning with ā span a 1-dimensional space in KQ/I, that all the paths beginning with b̄ span a 1-dimensional space in KQ/I, that all the paths beginning with c̄ and x̄ span a 1-dimensional space in KQ/I, that all the paths beginning with d̄ and ȳ span a 1-dimensional space in KQ/I and that all the paths beginning with ē and z̄ span a 1-dimensional space in KQ/I. As there is only one composition factor in the fifth Loewy layer of P_{1253}, all the paths of length four must span a single 1-dimensional space in KQ/I and all the paths of length five must be zero. We represent this information in the following diagram:

[figure: 1253; 321; 154_{a}; 1; 1176; 22; ā; b̄; c̄; d̄; ē; a; b; c; d; e; x; y; z; 1253; 1253; 1253; 1253; 1253; 748; b̄; d̄; ē; ā; b̄; c̄; ē; b̄; c̄; d̄; ē; ā; c̄; d̄; ē; ā; b̄; c̄; d̄; ē; x̄; ȳ; z̄; 321; 154_{a}; 1; 1176; 22; a; b; c; d; e; 1253]

We saw earlier from Lemma 6.1.2 that there is a module with three Loewy layers which has head isomorphic to 748, socle isomorphic to 1253, and has a composition factor isomorphic to 1176 in the second Loewy layer (i.e., dȳ ≠ 0). If we take the dual of this module we have a module with three Loewy layers which has head isomorphic to 1253, socle isomorphic to 748, and a composition factor isomorphic to 1176 in the second Loewy layer. This is a quotient of P_{1253} and therefore the path yd̄ is non-zero and we can take this as the basis for the 1-dimensional space spanned by xc̄, yd̄ and zē. Each of the paths aā, bb̄, cc̄, dd̄ and eē is non-zero (given the structure of P_{321} and P_{154_{a}}), so we choose our first basis element for the 2-dimensional space spanned by these paths to be aā. Suppose bb̄ and aā span a 1-dimensional space modulo e_{1253}Rad^{3}(A)e_{1253}. Then

bb̄ = λaā + (term in paths of length 4).

We then have bb̄c = λaāc = 0. But we know that bb̄c is non-zero, so this is not possible. Hence aā and bb̄ span a 2-dimensional space, and so we take them to be our basis elements. We then have

cc̄ = γ_{1}aā + γ_{2}bb̄ + (term in paths of length 4)

for some scalars γ_{1} and γ_{2}. But this says that

0 = cc̄a = γ_{1}aāa + γ_{2}bb̄a = γ_{2}bb̄a

and since bb̄a ≠ 0 we must have γ_{2} = 0. Hence

cc̄ = γ_{1}aā + (term in paths of length 4)

for some non-zero scalar γ_{1}. Now

dd̄= γ_{2}aā + γ_{3}bb̄ + (term in paths of length 4)

for some scalars γ_{2} and γ_{3}. In fact γ_{3} is non-zero, for if γ_{3} = 0 then

dd̄= γ_{2}aā + (term in paths of length 4)

for some non-zero scalar γ_{2}, which means

dd̄a = γ_{2}aāa = 0

and dd̄a is actually non-zero, a contradiction. Finally, we have

eē = γ_{4}aā + γ_{5}bb̄ + (term in paths of length 4)

for some scalars γ_{4} and γ_{5}. This time we can see that γ_{5} is non-zero since if γ_{5} were zero we would have

eē = γ_{4}aā + (term in paths of length 4)

for some non-zero scalar γ_{4}, which means

eēa = γ_{4}aāa = 0

and just as above, eēa is non-zero so this is not possible. Before writing out generators for the relations in paths starting at the vertex 1253, we notice that given the comments we have already made about the other projective modules, the following paths are non-zero; ābb̄, ādd̄, āeē, b̄aā, b̄bb̄, b̄cc̄, b̄eē, c̄bb̄, d̄aā, ēaā and ēbb̄. In particular this means that we have basis elements ābb̄, b̄aā, c̄bb̄, d̄aā and ēaā for the five 1-dimensional spaces corresponding to each of the type A simple modules in the fourth Loewy layer of P_{1253}. Finally, we also know that each of the simple modules in the fourth Loewy layer extends the socle non-trivially, and so the paths aābb̄, bb̄aā, cc̄bb̄, dd̄aā and eēaā are non-zero. Generators for the relations induced by I can now be seen to be:

\begin{aligned}ādd̄ & = & β_{3}ābb̄ \\ āeē & = & β_{4}ābb̄ \\ b̄bb̄ & = & β_{5}b̄aā \\ b̄cc̄ & = & β_{6}b̄aā \\ b̄eē & = & β_{7}b̄aā \\ c̄cc̄ & = & β_{8}c̄bb̄ \\ c̄dd̄ & = & β_{9}c̄bb̄ \\ c̄eē & = & β_{10}c̄bb̄ \\ x̄yd̄ & = & β_{11}c̄bb̄ \\ d̄cc̄ & = & β_{12}d̄aā \\ d̄dd̄ & = & β_{13}d̄aā \\ d̄eē & = & β_{14}d̄aā \\ ȳyd̄ & = & β_{15}d̄aā \\ ēbb̄ & = & β_{16}ēaā \\ ēcc̄ & = & β_{17}ēaā \\ ēdd̄ & = & β_{18}ēaā \\ ēeē & = & β_{19}ēaā \\ z̄yd̄ & = & β_{20}ēaā \end{aligned}  \begin{aligned}xc̄ & = & β_{1}yd̄ \\ zē & = & β_{2}yd̄ \\ cc̄ & = & γ_{1}aā + k_{5}aābb̄ \\ dd̄ & = & γ_{2}aā + γ_{3}bb̄ + k_{6}aābb̄ \\ eē & = & γ_{4}aā + γ_{5}bb̄ + k_{7}aābb̄ \\ bb̄aā & = & β_{21}aābb̄ \\ cc̄bb̄ & = & β_{22}aābb̄ \\ dd̄aā & = & β_{23}aābb̄ \\ eēaā & = & β_{24}aābb̄ \\ āaābb̄ & = & 0 \\ b̄aābb̄ & = & 0 \\ c̄aābb̄ & = & 0 \\ d̄aābb̄ & = & 0 \\ ēaābb̄ & = & 0 \end{aligned}

where β_{3}, β_{4}, β_{5}, β_{6}, β_{7}, β_{16}, β_{21}, β_{22}, β_{23}, β_{24}, γ_{1}, γ_{3} and γ_{5} are non-zero.

For a list of all the generating relations that we have now determined, see Appendix F.

6.2 Determining the Ideal I

Choosing Basis Elements I

Consider the relations

\begin{aligned}āa & = & k_{1}ābb̄a \\ d̄d & = & s_{2}ȳy + k_{2}ȳxx̄y \\ yȳ & = & α_{1}xx̄ + k_{3}xx̄xx̄ \\ zz̄ & = & α_{2}xx̄ + k_{4}xx̄xx̄ \end{aligned}  \begin{aligned}cc̄ & = & γ_{1}aā + k_{5}aābb̄ \\ dd̄ & = & γ_{2}aā + γ_{3}bb̄ + k_{6}aābb̄ \\ eē & = & γ_{4}aā + γ_{5}bb̄ + k_{7}aābb̄. \end{aligned}

Let ā_{0} be the choice of basis element for the complement of e_{321}Rad^{2}(A)e_{1253} in e_{321}Rad(A)e_{1253}, associated with the arrow ā, and let b_{0} and b̄_{0} be the choices of basis elements corresponding to the arrows b and b̄ respectively. Set ā_{1} = ā_{0}(1 − k_{1}b_{0}b̄_{0}) and let this now be our choice of basis element corresponding to the arrow ā. Then the relations in ā are

āa = 0
cc̄ = γ_{1}aā + k′_{5}aābb̄
dd̄ = γ_{2}aā + γ_{3}bb̄ + k′_{6}aābb̄
eē = γ_{4}aā + γ_{5}bb̄ + k′_{7}aābb̄.

Now

cc̄ = γ_{1}aā(1 + γ_{1}^{−1}k′_{5}bb̄)

as γ_{1} ≠ 0. Let c̄_{0} be the choice of basis element corresponding to the arrow c̄ and let b_{0} and b̄_{0} be the choices of basis elements corresponding to the arrows b and b̄ respectively. As u = 1 + γ_{1}^{−1}k′_{5}b_{0}b̄_{0} is a unit we can set c̄_{1} = c̄_{0}u^{−1}. Then if we take c̄_{1} as our new choice of basis element for the arrow c̄, the relations in c̄ are

cc̄ = γ_{1}aā.

Now eēaā = β_{24}aābb̄, with β_{24} ≠ 0, and therefore

eē = γ_{4}aā + γ_{5}bb̄ + k′_{7}β_{24}^{−1}eēaā.

Set ē_{1} = ē_{0}(1−k′_{7}β_{24}^{−1}a_{0}ā_{0}), where ē_{0} is the choice of basis element corresponding to the arrow ē, and a_{0} and ā_{0} are the current choices of basis elements corresponding to the arrows a and ā. Take ē_{1} as the new choice of basis element corresponding to ē, then we have

eē = γ_{4}aā + γ_{5}bb̄.

In the same way we take d̄_{1} = d̄_{0}(1 − k′_{6}β_{23}^{−1}a_{0}ā_{0}) as the choice of basis element corresponding to d̄, then the relations in d̄ may be written

dd̄= γ_{2}aā + γ_{3}bb̄

and

d̄d = s_{2}ȳy + k′_{2}ȳxx̄y.

As s_{2} is non-zero we can rechoose the basis element corresponding to the arrow ȳ by taking ȳ_{1} = ȳ_{0}(1 + s_{2}^{−1}k′_{2}x_{0}x̄_{0}). The relations in ȳ are then

d̄d = s_{2}ȳy

and

yȳ = α_{1}xx̄ + k′_{3}xx̄xx̄.

Set x̄_{1} = x̄_{0}(1 + α_{1}^{−1}k′_{3}x_{0}x̄_{0}), then the relations in x̄ will be

yȳ = α_{1}xx̄

and

zz̄ = α_{2}xx̄ + k′_{4}xx̄xx̄.

Let x_{0} and x̄_{0} now be the current basis elements corresponding to the arrows x and x̄ respectively. As v = 1 + α_{2}^{−1}k′_{4}x_{0}x̄_{0} is a unit we can set z̄_{1} = z̄_{0}v^{−1}. With this as our choice of basis element corresponding to z̄ our relation in z̄ is

zz̄ = α_{2}xx̄.

Hence with these new choices of basis elements corresponding to the arrows, the relations involving paths of length two are:

\begin{aligned}āa & = & 0 \\ c̄a & = & 0 \\ d̄b & = & 0 \end{aligned}  \begin{aligned}d̄e & = & t_{2}ȳz \\ yȳ & = & α_{1}xx̄ \\ zz̄ & = & α_{2}xx̄ \end{aligned}

\begin{aligned}āc & = & 0 \\ b̄d & = & 0 \\ ēc & = & r_{1}z̄x \\ ēd & = & s_{1}z̄y \\ d̄d & = & s_{2}ȳy \\ c̄e & = & t_{1}x̄z \end{aligned}  \begin{aligned}cx̄ & = & α_{3}dȳ \\ ez̄ & = & α_{4}dȳ \\ xc̄ & = & β_{1}yd̄ \\ zē & = & β_{2}yd̄ \\ cc̄ & = & γ_{1}aā \\ dd̄ & = & γ_{2}aā + γ_{3}bb̄ \\ eē & = & γ_{4}aā + γ_{5}bb̄. \end{aligned}

Reducing the Number of Generators

We will now show that the scalars r_{i}, s_{i}, t_{i}, α_{i}, β_{i} and γ_{i} are non-zero, and hence that these relations generate all the relations induced by I.

Firstly we remark that we know the scalars α_{1} and α_{2} are non-zero. Secondly, we saw earlier from Lemma 6.1.2 that there is a module M with three Loewy layers which has head isomorphic to 748, socle isomorphic to 1253, and has a composition factor isomorphic to 1176 in the second Loewy layer. The dual of M, M^{∗}, has three Loewy layers with head isomorphic to 1253, socle isomorphic to 748, and a composition factor isomorphic to 1176 in the second Loewy layer. Hence not only is M a quotient of P_{748}, we also have M^{∗} a submodule of P_{748} and this means that the path yd̄dȳ is non-zero. In particular, d̄d is non-zero and s_{2} is non-zero.

Given

cc̄ = γ_{1}aā

with γ_{1} ≠ 0, we have

cc̄e = γ_{1}aāe

We also have

cc̄e = t_{1}cx̄z
= t_{1}α_{3}dȳz
= t_{1}α_{3}t_{9}aāe

and as aāe ≠ 0, we must have

γ_{1} = t_{1}α_{3}t_{9}.

Since γ_{1} ≠ 0, this means that t_{1} and α_{3} are non-zero.

Further,

ēcc̄ = γ_{1}ēaā.

Also

ēcc̄ = r_{1}z̄xc̄
= r_{1}β_{1}z̄yd̄
= r_{1}β_{1}β_{20}ēaā

and since ēaā ≠ 0, we must have

γ_{1} = r_{1}β_{1}β_{20}.

As γ_{1} ≠ 0, it must be the case that r_{1} and β_{1} are non-zero.

Given that

dd̄= γ_{2}aā + γ_{3}bb̄,

we have

0 = γ_{2}aāb + γ_{3}bb̄b
= γ_{2}aāb + γ_{3}λ_{3}aāb.

As aāb ≠ 0

γ_{2} = −γ_{3}λ_{3}

and because γ_{3} and λ_{3} are non-zero, we know that γ_{2} is non-zero.

Given that

eē = γ_{4}aā + γ_{5}bb̄

we have

eēc = γ_{5}bb̄c.

Also

eēc = r_{1}ez̄x
= r_{1}α_{4}dȳx
= r_{1}α_{4}r_{10}bb̄c.

As bb̄c ≠ 0, we must have

γ_{5} = r_{1}α_{4}r_{10}

and because γ_{5} is non-zero we know that α_{4} must be non-zero.

As ez̄ ≠ 0 there is a module M with three Loewy layers which has head isomorphic to 748, socle isomorphic to 1253 and has a composition factor isomorphic to 22 in the second Loewy layer. Then M^{∗} has head isomorphic to 1253, socle isomorphic to 748 and has a composition factor isomorphic to 22 in the second Loewy layer. As this is a quotient of P_{1253}, we know that zē ≠ 0. Hence β_{2} is non-zero.

We now have that

d̄ez̄ = α_{4}d̄dȳ.

But d̄dȳ and α_{4} are non-zero, so d̄ez̄ is also non-zero. In particular, d̄e is nonzero, and therefore t_{2} is non-zero. We then have a quotient of P_{22} with three Loewy layers, which has head isomorphic to 22, socle isomorphic to 1176 and a composition factor isomorphic to 1253 in the second Loewy layer. Taking the dual of this module, we see that we have a module with three Loewy layers with head isomorphic to 1176, socle isomorphic to 22 and a composition factor isomorphic to 1253 in its second Loewy layer. This module is a quotient of P_{1176}, and therefore d̄eēd ≠ 0. In particular ēd ≠ 0, and so s_{1} is non-zero. Furthermore d̄eēd = γ_{4}d̄aād and as d̄aād ≠ 0 we have γ_{4} ≠ 0.

We have now shown that all the relations in paths of length two involving scalars are non-zero. It is now simple but time-consuming to show that these relations, along with the zero relations in paths of length two, generate all the relations we have determined. We give a few sample calculations:

xc̄c = β_{1}yd̄c
= β_{1}β_{2}^{−1}zēc
= β_{1}β_{2}^{−1}r_{1}zz̄x
= β_{1}β_{2}^{−1}r_{1}α_{2}xx̄x.

As xx̄x ≠ 0 we have

r_{2} = β_{1}β_{2}^{−1}r_{1}α_{2}.
bb̄e = γ_{3}^{−1}(dd̄e − γ_{2}aāe)
= γ_{3}^{−1}(t_{2}dȳz − γ_{2}aāe)
= γ_{3}^{−1}(t_{2}α_{3}^{−1}cx̄z − γ_{2}aāe)

= γ_{3}^{−1}(t_{2}α_{3}^{−1}t_{1}^{−1}cc̄e − γ_{2}aāe)
= γ_{3}^{−1}(t_{2}α_{3}^{−1}t_{1}^{−1}γ_{1}aāe − γ_{2}aāe)
= (γ_{3}^{−1}t_{2}α_{3}^{−1}t_{1}^{−1}γ_{1} − γ_{3}^{−1}γ_{2})aāe.

As aāe ≠ 0 we have

t_{6} = γ_{3}^{−1}t_{2}α_{3}^{−1}t_{1}^{−1}γ_{1} − γ_{3}^{−1}γ_{2}.
x̄xx̄z = t_{1}^{−1}x̄xc̄e
= t_{1}^{−1}β_{1}x̄yd̄e
= t_{1}^{−1}β_{1}β_{2}^{−1}x̄zēe
= t_{1}^{−2}β_{1}β_{2}^{−1}c̄eēe
= t_{1}^{−2}β_{1}β_{2}^{−1}(γ_{4}c̄aāe + γ_{5}c̄bb̄e)
= t_{1}^{−2}β_{1}β_{2}^{−1}(γ_{4}c̄aāe + γ_{5}t_{6}c̄aāe)
= 0.

Finally,

zz̄xx̄z = α_{2}x(x̄xx̄z)
= 0.

Choosing Basis Elements II

As each of our paths a, b, c, d, e and so on correspond to bases of 1-dimensional spaces, we can rechoose our basis elements by replacing a given basis element by a non-zero scalar multiple of itself. Let y_{0} be the basis element corresponding to the arrow y. We set y_{1} = α_{1}^{−1}y_{0} and take this as the new basis element corresponding to the arrow y. Similarly let z_{0} be the basis element corresponding to the arrow z and take z_{1} = α_{2}^{−1}z_{0} to be the new basis element corresponding to the arrow z. These choices of basis elements have no effect on the zero relations, but the relations in the scalars may now be written:

\begin{aligned}ēc & = & r_{1}z̄x \\ ēd & = & s′_{1}z̄y \\ d̄d & = & s′_{2}ȳy \end{aligned}  \begin{aligned}cx̄ & = & α_{3}dȳ \\ ez̄ & = & α_{4}dȳ \\ xc̄ & = & β′_{1}yd̄ \end{aligned}

\begin{aligned}c̄e & = & t′_{1}x̄z \\ d̄e & = & t′_{2}ȳz \\ yȳ & = & xx̄ \\ zz̄ & = & xx̄ \end{aligned}  \begin{aligned}zē & = & β′_{2}yd̄ \\ cc̄ & = & γ_{1}aā \\ dd̄ & = & γ_{2}aā + γ_{3}bb̄ \\ eē & = & γ_{4}aā + γ_{5}bb̄. \end{aligned}

We now rechoose our basis elements again, this time as follows:

c_{1} = α_{3}^{−1}c_{0}
e_{1} = α_{4}^{−1}e_{0}.

With these choices of basis elements the relations in the scalars are:

\begin{aligned}ēc & = & r′_{1}z̄x \\ ēd & = & s′_{1}z̄y \\ d̄d & = & s′_{2}ȳy \\ c̄e & = & t′′_{1}x̄z \\ d̄e & = & t′′_{2}ȳz \\ yȳ & = & xx̄ \\ zz̄ & = & xx̄ \end{aligned}  \begin{aligned}cx̄ & = & dȳ \\ ez̄ & = & dȳ \\ xc̄ & = & β′_{1}yd̄ \\ zē & = & β′_{2}yd̄ \\ cc̄ & = & γ′_{1}aā \\ dd̄ & = & γ_{2}aā + γ_{3}bb̄ \\ eē & = & γ′_{4}aā + γ′_{5}bb̄. \end{aligned}

Now take

c̄_{1} = (t′′_{1})^{−1}c̄_{0}
ē_{1} = (r′_{1})^{−1}ē_{0}.

These relations are then:

\begin{aligned}ēc & = & z̄x \\ ēd & = & s′′_{1}z̄y \\ d̄d & = & s′_{2}ȳy \\ c̄e & = & x̄z \\ d̄e & = & t′′_{2}ȳz \\ yȳ & = & xx̄ \\ zz̄ & = & xx̄ \end{aligned}  \begin{aligned}cx̄ & = & dȳ \\ ez̄ & = & dȳ \\ xc̄ & = & β′′_{1}yd̄ \\ zē & = & β′′_{2}yd̄ \\ cc̄ & = & γ′′_{1}aā \\ dd̄ & = & γ_{2}aā + γ_{3}bb̄ \\ eē & = & γ′′_{4}aā + γ′′_{5}bb̄. \end{aligned}

Now take

d̄_{1} = (s′_{2})^{−1}d̄_{0}
a_{1} = γ′′_{1}a_{0}.

6.2 Determining the Ideal I 157 These relations are then:

\begin{aligned}ēc & = & z̄x \\ ēd & = & s′′_{1}z̄y \\ d̄d & = & ȳy \\ c̄e & = & x̄z \\ d̄e & = & t′′′_{2}ȳz \\ yȳ & = & xx̄ \\ zz̄ & = & xx̄ \end{aligned}  \begin{aligned}cx̄ & = & dȳ \\ ez̄ & = & dȳ \\ xc̄ & = & β′′′_{1}yd̄ \\ zē & = & β′′′_{2}yd̄ \\ cc̄ & = & aā \\ dd̄ & = & γ′_{2}aā + γ′_{3}bb̄ \\ eē & = & γ′′′_{4}aā + γ′′_{5}bb̄. \end{aligned}

Finally, take

b_{1} = γ′_{3}b_{0}.

These relations are then:

\begin{aligned}ēc & = & z̄x \\ ēd & = & μ_{1}z̄y \\ d̄d & = & ȳy \\ c̄e & = & x̄z \\ d̄e & = & μ_{2}ȳz \\ yȳ & = & xx̄ \\ zz̄ & = & xx̄ \end{aligned}  \begin{aligned}cx̄ & = & dȳ \\ ez̄ & = & dȳ \\ xc̄ & = & μ_{3}yd̄ \\ zē & = & μ_{4}yd̄ \\ cc̄ & = & aā \\ dd̄ & = & μ_{5}aā + bb̄ \\ eē & = & μ_{6}aā + μ_{7}bb̄ \end{aligned}

for appropriate choices of non-zero scalars μ_{i}.

Solving for Scalars

Now we calculate relations between the paths of length four.

d̄aād = μ_{5}^{−1}d̄dd̄d
= μ_{5}^{−1}d̄dȳy
= μ_{5}^{−1}d̄ez̄y
= μ_{2}μ_{5}^{−1}ȳzz̄y
= μ_{2}μ_{5}^{−1}ȳyȳy
= μ_{2}μ_{5}^{−1}d̄dd̄d
= μ_{2}d̄aād.

Since d̄aād ≠ 0, μ_{2} = 1. Also

d̄aād = μ_{5}^{−1}d̄dd̄d
= μ_{5}^{−1}ȳyd̄d
= μ_{4}^{−1}μ_{5}^{−1}ȳzēd
= μ_{1}μ_{4}^{−1}μ_{5}^{−1}ȳzz̄y
= μ_{1}μ_{4}^{−1}μ_{5}^{−1}d̄dd̄d
= μ_{1}μ_{4}^{−1}d̄aād.

Hence μ_{1}μ_{4}^{−1} = 1, i.e., μ_{4} = μ_{1}. Also

d̄aād = μ_{5}^{−1}d̄dd̄d
= μ_{5}^{−1}d̄dȳy
= μ_{5}^{−1}d̄ez̄y
= μ_{5}^{−1}μ_{1}^{−1}d̄eēd
= μ_{5}^{−1}μ_{1}^{−1}μ_{6}d̄aād.

Hence μ_{5}^{−1}μ_{1}^{−1}μ_{6} = 1, i.e., μ_{6} = μ_{5}μ_{1}.

We also have

c̄eēc = x̄zēc
= μ_{4}x̄yd̄c
= μ_{4}μ_{3}^{−1}x̄xc̄c

and

c̄eēc = c̄ez̄x
= c̄dȳx
= c̄cx̄x.

Hence we have

c̄cx̄x = μ_{4}μ_{3}^{−1}x̄xc̄c

where x̄xc̄c ≠ 0 (because x̄xc̄c = x̄xx̄x which is non-zero). But we are dealing with a symmetric algebra, so there is a symmetrising form φ such that

φ(x̄xc̄c) = φ(c̄cx̄x) = μ_{4}μ_{3}^{−1}φ(x̄xc̄c).

As φ(x̄xc̄c) ≠ 0 (because ker(φ) contains no non-zero left ideals) we must have μ_{4}μ_{3}^{−1} = 1, i.e., μ_{3} = μ_{4}.

Finally,

xx̄xx̄ = xx̄zz̄
= xc̄ez̄
= μ_{3}yd̄ez̄
= μ_{3}yd̄dȳ
= μ_{3}yȳyȳ
= μ_{3}xx̄xx̄.

Hence μ_{3} = 1, and so we also have μ_{4} = μ_{1} = 1, and μ_{6} = μ_{5}.

Given these values for the scalars μ_{1} up to μ_{6}, we now have

aābb̄ = μ_{7}^{−1}aāeē

and

aābb̄ = aādd̄
= μ_{5}^{−1}eēdd̄
= μ_{5}^{−1}(μ_{5}eēaā + eēbb̄)
= eēaā + μ_{5}^{−1}eēbb̄
= eēaā + μ_{5}^{−1}(eēdd̄− μ_{5}eēaā)
= eēaā + μ_{5}^{−1}(ez̄yd̄− μ_{5}eēaā)
= eēaā + μ_{5}^{−1}(ez̄xc̄ − μ_{5}eēaā)
= eēaā + μ_{5}^{−1}(eēcc̄ − μ_{5}eēaā)
= eēaā + μ_{5}^{−1}(eēaā − μ_{5}eēaā)
= μ_{5}^{−1}eēaā.

Hence aāeē = μ_{7}μ_{5}^{−1}eēaā.

By symmetry we must have

φ(eēaā) = φ(aāeē) = μ_{7}μ_{5}^{−1}φ(eēaā).

As eēaā ≠ 0 and φ(eēaā) ≠ 0 we must have μ_{7} = μ_{5}. The relations in the scalars therefore become:

\begin{aligned}ēc & = & z̄x \\ ēd & = & z̄y \\ d̄d & = & ȳy \\ c̄e & = & x̄z \\ d̄e & = & ȳz \\ yȳ & = & xx̄ \\ zz̄ & = & xx̄ \end{aligned}  \begin{aligned}cx̄ & = & dȳ \\ ez̄ & = & dȳ \\ xc̄ & = & yd̄ \\ zē & = & yd̄ \\ cc̄ & = & aā \\ dd̄ & = & μ_{5}aā + bb̄ \\ eē & = & μ_{5}aā + μ_{5}bb̄. \end{aligned}

It does not appear to be possible to reduce the number of scalars any further and therefore the relations

\begin{aligned}āa & = & 0 \\ c̄a & = & 0 \\ d̄b & = & 0 \\ āc & = & 0 \\ b̄d & = & 0 \end{aligned}  \begin{aligned}ēc & = & z̄x \\ ēd & = & z̄y \\ d̄d & = & ȳy \\ c̄e & = & x̄z \\ d̄e & = & ȳz \\ yȳ & = & xx̄ \\ zz̄ & = & xx̄ \end{aligned}  \begin{aligned}cx̄ & = & dȳ \\ ez̄ & = & dȳ \\ xc̄ & = & yd̄ \\ zē & = & yd̄ \\ cc̄ & = & aā \\ dd̄ & = & μ_{5}aā + bb̄ \\ eē & = & μ_{5}aā + μ_{5}bb̄ \end{aligned}

determine a 1-parameter family of algebras to which the basic algebra of the principal block must belong.

Appendix A

Conjugacy Classes of SU_{3}(q^{2}), Ordinary Character Table for SU_{3}(q^{2}) and Decomposition Matrix for SU_{3}(q^{2})

In this appendix we present information about the conjugacy classes of SU_{3}(q^{2}) and the ordinary character table for SU_{3}(q^{2}) for all values of q. We also present the decomposition matrix for the principal block of SU_{3}(q^{2}) over a field of characteristic three when 3 divides q + 1. This information has all been presented by Geck (1990). As SU_{3}(q^{2}) has order q^{3}(q + 1)^{2}(q − 1)(q^{2} − q + 1)/d where d = gcd(3, q + 1), the information for our special case can be deduced from the given information by setting d = 3 and q ≡ 2 mod 3 with q ≢ 8 mod 9. Table A.1 gives the q^{2} + q + 1 + d^{2} conjugacy classes of SU_{3}(q^{2}) and the canonical form for each conjugacy class, Table A.2 gives the ordinary character table for SU_{3}(q^{2}) and Table A.3 gives the decomposition matrix for the principal block of SU_{3}(q^{2}).

In what follows, σ is a generator for the multiplicative group of the field F_{q^{2}}, then ρ = σ^{q−1}, ω = ρ^{(q+1)/d} and τ is an element of order q^{2} − q + 1 in some extension field of F_{q^{2}}.

Appendix A 162

Table A.1 Conjugacy Classes of SU_{3}(q^{2})

name canonical form

parameters

C_{1}^{(k)}  ( \begin{matrix}ω^{k} \\ & ω^{k} \\ & & ω^{k} \end{matrix} )  0 ≤ k ≤ d − 1
C_{2}^{(k)}  ( \begin{matrix}ω^{k} \\ 1 & ω^{k} \\ & & ω^{k} \end{matrix} )  0 ≤ k ≤ d − 1
C_{3}^{(k,l)}  ( \begin{matrix}ω^{k} \\ 1 & ω^{k} \\ & 1 & ω^{k} \end{matrix} )  0 ≤ k, l ≤ d − 1
C_{4}^{(k)}  ( \begin{matrix}ρ^{k} \\ & ρ^{k} \\ & & ρ^{−2k} \end{matrix} )  \begin{matrix}0 ≤ k ≤ q \\ \frac{q+1}{d} |̸k \end{matrix}
C_{5}^{(k)}  ( \begin{matrix}ρ^{k} \\ 1 & ρ^{k} \\ & & ρ^{−2k} \end{matrix} )  \begin{matrix}0 ≤ k ≤ q \\ \frac{q+1}{d} |̸k \end{matrix}
C_{6}^{(k,l,m)}  ( \begin{matrix}ρ^{k} \\ & ρ^{i} \\ & & ρ^{m} \end{matrix} )  \begin{matrix}0 ≤ k, l, m ≤ q \\ k < l < m \\ k + l + m ≡ 0 (mod q + 1) \end{matrix}

C_{7}^{(k)} ( \begin{matrix}σ^{k} \\ & ρ^{k} \\ & & σ^{−qk} \end{matrix} )

0 ≤ k ≤ q^{2} − 2
k ≢ 0 (mod q − 1);
if k_{1} ≡ −kq (mod q^{2} − 1),
then C_{7}^{(k)} = C_{7}^{(k_{1})}

C_{8}^{(k)} ( \begin{matrix}τ^{k} \\ & τ^{−qk} \\ & & τ^{q^{2}k} \end{matrix} )

0 ≤ k ≤ q^{2} − q
(q^{2} − q + 1)/d |̸ k;
if k_{1} ≡ −qk (mod q^{2} − q + 1)
or k_{2} ≡ kq^{2} (mod q^{2} − q + 1),
then C_{8}^{(k)} = C_{8}^{(k_{1})} = C_{8}^{(k_{2})}

Appendix A 163

In the following table, ∑_{(k,l,m)} denotes summation over all permutations of (k, l, m). Furthermore we have δ = exp\frac{2πi}{q^{2}−1}, ϵ = δ^{q−1}, β = ϵ^{(q+1)/3}, γ = exp\frac{2πi}{q^{2}−q+1} and δ_{ij} is the Kronecker delta.

Table A.2: Ordinary Character Table of SU_{3}(q^{2})

χ_{1} χ_{q^{2}−q} χ_{q^{3}}  χ_{q^{2}−q+1}^{(u)}  χ_{q(q^{2}−q+1)}^{(u)}

parameters  1 ≤ u ≤ q  1 ≤ u ≤ q

C_{1}^{(k)}  1 q^{2} − q q^{3} (q^{2} − q + 1)β^{uk}  q(q^{2} − q + 1)β^{uk}

C_{2}^{(k)}  1 −q  0  (1 − q)β^{uk}  qβ^{uk}

C_{3}^{(k,l)}  1  0  0  β^{uk}  0

C_{4}^{(k)}  1 1 − q q (1 − q)ϵ^{uk} + ϵ^{−2uk} (q − 1)ϵ^{uk} + qϵ^{−2uk}

C_{5}^{(k)}  1  1  0  ϵ^{uk} + ϵ^{−2uk}  −ϵ^{uk}

C_{6}^{(k,l,m)}  1  2  −1 ϵ^{uk} + ϵ^{ul} + ϵ^{um}  −ϵ^{uk} − ϵ^{ul} − ϵ^{um}

C_{7}^{(k)}  1  0  1  ϵ^{uk}  ϵ^{uk}

C_{8}^{(k)}  1 −1 −1  0  0

(continued)

Appendix A 164

Table A.2 cont.

χ_{(q−1)(q^{2}−q+1)}^{(u,v)}  χ_{q^{3}+1}^{(u)}  χ_{(q+1)^{2}(q−1)}^{(u)}

parameters \begin{matrix}1 ≤ u ≤ (q + 1)/3 \\ u < v < 2(q + 1)/3 \end{matrix}

1 ≤ u ≤ q^{2} − 1
(q − 1) |̸ u;
if u_{1} ≡ −uq
(mod q^{2} − 1), then
χ^{(u)} = χ^{(u_{1})}

0 ≤ u ≤ q^{2} − q
(q^{2} − q + 1)/d |̸ u;
if u_{1} ≡ −uq
or u_{2} ≡ uq^{2}
(mod q^{2} − q + 1), then
χ^{(u)} = χ^{(u_{1})} = χ^{(u_{2})}
C_{1}^{(k)}  (q − 1)(q^{2} − q + 1)β^{(u+v)k}  (q^{3} + 1)β^{uk}  (q + 1)^{2}(q − 1)β^{uk}
C_{2}^{(k)}  (2q − 1)β^{(u+v)k}  β^{uk}  −(q + 1)β^{uk}
C_{3}^{(k,l)}  −β^{(u+v)k}  β^{uk}  −β^{uk}
C_{4}^{(k)}  \begin{matrix}(q − 1)(ϵ^{(u+v)k} \\ +ϵ^{(v−2u)k} + ϵ^{(u−2v)k}) \end{matrix}  (q + 1)ϵ^{uk}  0
C_{5}^{(k)}  \begin{matrix}−ϵ^{(u+v)k} \\ −ϵ^{(u−2v)k} − ϵ^{(v−2u)k} \end{matrix}  ϵ^{uk}  0
C_{6}^{(k,l,m)}  − ∑_{(k,l,m)} ϵ^{(uk+vl)}  0  0
C_{7}^{(k)}  0  δ^{uk} + δ^{−ukq}  0
C_{8}^{(k)}  0  0  −γ^{uk} − γ^{ukq} − γ^{ukq^{2}}

Appendix A 165

If d = 3 we have in addition the following characters:

χ_{(q−1)(q^{2}−q+1)/3}^{(u)}  χ_{(q+1)^{2}(q−1)/3}^{(u,v)}
parameters  0 ≤ u ≤ 2  0 ≤ u ≤ 2, v = 1, 2
C_{1}^{(k)}  \frac{(q−1)(q^{2}−q+1)}{3}  \frac{(q+1)^{2}(q−1)}{3}β^{vk}
C_{2}^{(k)}  \frac{2q−1}{3}  −\frac{q+1}{3}β^{vk}
C_{3}^{(k,l)}  qδ_{lu} − \frac{q+1}{3}  (qδ_{lu} − \frac{q+1}{3})β^{vk}
C_{4}^{(k)}  q − 1  0
C_{5}^{(k)}  −1  0
C_{6}^{(k,l,m)}  −β^{k−l} − β^{l−k}  0
C_{7}^{(k)}  0  0
C_{8}^{(k)}  0  −β^{vk}

Appendix A 166 Table A.3: 3-Modular Decomposition Matrix for the Principal Block of SU_{3}(q^{2})

character number
χ_{1}  1 0 0 0 0
χ_{q^{2}−q}  0 1 0 0 0
χ_{(q−1)(q^{2}−q+1)/3}^{(0)}  0 α 1 0 0
χ_{(q−1)(q^{2}−q+1)/3}^{(1)}  0 α 0 1 0
χ_{(q−1)(q^{2}−q+1)/3}^{(2)}  0 α 0 0 1
χ_{q^{3}}  1 3α + 2 1 1 1
χ_{q^{2}−q+1}^{(u)}, \frac{q+1}{3^{a}}| u  1 1 0 0 0 3^{a} − 1
χ_{q(q^{2}−q+1)}^{(u)}, \frac{q+1}{3^{a}}| u  1 3α + 1 1 1 1 3^{a} − 1
χ_{(q−1)(q^{2}−q+1)}^{(v,w)}, \frac{q+1}{3^{a}}| v, w 0 3α 1 1 1 \frac{1}{2}3^{a}(3^{a} − 1)
χ_{(q+1)^{2}(q−1)/3}^{(0,v)}, v = 1, 2 0 α + 1 1 0 0  2
χ_{(q+1)^{2}(q−1)/3}^{(1,v)}, v = 1, 2 0 α + 1 0 1 0  2
χ_{(q+1)^{2}(q−1)/3}^{(2,v)}, v = 1, 2 0 α + 1 0 0 1  2

where a is chosen such that 3^{a} is the largest power of 3 dividing q + 1 and

0 ≤ α ≤ \frac{1}{2}(3^{a} − 3).

In particular, α = 0 if a = 1.

Appendix B

Character Tables, Decomposition Matrix, and Cartan Matrix for U_{3}(5)

Table B.1 gives the irreducible ordinary character table for U_{3}(5) taken from Conway et al. (1985), Table B.2 gives the 3-modular Brauer character table for U_{3}(5) taken from Parker, Table B.3 gives the decomposition matrix for U_{3}(5), and Table B.4 gives the Cartan matrix for U_{3}(5).

In both of the character tables we have made use of the Atlas convention for expressing algebraic numbers. The following convention has been used

i2 = i\sqrt{2}

and

−b7 = \frac{1 − i\sqrt{7}}{2}
∗∗ = \frac{1 + i\sqrt{7}}{2}.

Appendix B 168

Table B.1: Ordinary Character Table for U_{3}(5)

1A 2A 3A 4A 5A 5B 5C 5D 6A 7A B ∗ ∗ 8A B ∗ ∗ 10A U_{3}(5).2
χ_{1} 1 1 1 1 1 1 1 1 1 1  1  1  1  1  :
χ_{2} 20 −4 2 0 −5 0 0 0 2 −1 −1  0  0  1  :
χ_{3} 21 5 3 1 −4 1 1 1 −1 0  0  −1 −1  0  :
χ_{4} 28 4 1 0 3 3 −2 −2 1 0  0  0  0  −1  :
χ_{5} 28 4 1 0 3 −2 3 −2 1 0  0  0  0  −1  ·|
χ_{6} 28 4 1 0 3 −2 −2 3 1 0  0  0  0  −1  ·|
χ_{7} 84 −4 3 0 9 −1 −1 −1 −1 0  0  0  0  1  :
χ_{8} 105 1 −3 1 5 0 0 0 1 0  0  −1 −1  1  :
χ_{9} 125 5 −1 1 0 0 0 0 −1 −1 −1  1  1  0  :
χ_{10} 126 6 0 −2 1 1 1 1 0 0  0  0  0  1  :
χ_{11} 126 −6 0 0 1 1 1 1 0 0  0  i2 −i2 −1  ·|
χ_{12} 126 −6 0 0 1 1 1 1 0 0  0 −i2 i2 −1  ·|
χ_{13} 144 0 0 0 −6 −1 −1 −1 0 −b7 ∗∗  0  0  0  ·|
χ_{14} 144 0 0 0 −6 −1 −1 −1 0 ∗∗ −b7 0  0  0  ·|

Table B.2: Modular Character Table for U_{3}(5) mod 3

1A 2A 4A 5A 5B 5C 5D 7A B ∗ ∗ 8A B ∗ ∗ 10A U_{3}(5).2
1  1 1 1 1 1 1 1  1  1  1  1  1  :
20 20 −4 0 −5 0 0 0 −1 −1  0  0  1  :
21 21 5 1 −4 1 1 1  0  0  −1 −1  0  :
28_{a} 28 4 0 3 3 −2 −2 0  0  0  0  −1  :
28_{b} 28 4 0 3 −2 3 −2 0  0  0  0  −1  ·|
28_{c} 28 4 0 3 −2 −2 3  0  0  0  0  −1  ·|
84 84 −4 0 9 −1 −1 −1 0  0  0  0  1  :
126_{a} 126 6 −2 1 1 1 1  0  0  0  0  1  :
126_{b} 126 −6 0 1 1 1 1  0  0  i2 −i2 −1  ·|
126_{c} 126 −6 0 1 1 1 1  0  0 −i2 i2 −1  ·|
144_{a} 144 0 0 −6 −1 −1 −1 −b7 ∗∗  0  0  0  ·|
144_{b} 144 0 0 −6 −1 −1 −1 ∗∗ −b7 0  0  0  ·|

Appendix B 169

Table B.3: Decomposition Matrix for U_{3}(5)

1 20 21 28_{a} 28_{b} 28_{c} 84 126_{a} 126_{b} 126_{c} 144_{a} 144_{b}
χ_{1} 1
χ_{2}  1
χ_{3}  1
χ_{4}  1
χ_{5}  1
χ_{6}  1
χ_{7}  1
χ_{8} 1 1
χ_{9} 1 2  1 1 1
χ_{10}  1
χ_{11}  1
χ_{12}  1
χ_{13}  1
χ_{14}  1

Table B.4: Cartan Matrix for U_{3}(5)

1 20 28_{a} 28_{b} 28_{c} 21 84 126_{a} 126_{b} 126_{c} 144_{a} 144_{b}
1 2 2 1 1 1
20 2 5 2 2 2
28_{a} 1 2 2 1 1
28_{b} 1 2 1 2 1
28_{c} 1 2 1 1 2
21  2 1
84  1 2
126_{a}  1
126_{b}  1
126_{c}  1
144_{a}  1
144_{b}  1

Appendix C

Character Tables and Decomposition Matrix for U_{3}(5) : 2

Table C.1 gives the ordinary irreducible character table for U_{3}(5) : 2 and Table C.2 gives the 3-modular Brauer character table for U_{3}(5) : 2. Both tables are calculated from the information presented with the character tables of U_{3}(5) found in the Atlas of Finite Groups (Conway et al. (1985)), and the Modular Atlas (Parker). We make use of the following notation when using algebraic integers:

i5 = i\sqrt{5}

Finally, the decomposition matrix for U_{3}(5) : 2 is presented in Table C.3.

Appendix C 171 Table C.1: Ordinary Character Table for U_{3}(5) : 2

1A 2A 2B 3A 4A 4B 5A 5B 5C 6A 6B
χ_{1} 1 1 1 1 1 1 1 1 1 1 1
χ_{2} 1 1 −1 1 1 −1 1 1 1 1 −1
χ_{3} 20 −4 0 2 0 0 −5 0 0 2 0
χ_{4} 20 −4 0 2 0 0 −5 0 0 2 0
χ_{5} 21 5 1 3 1 5 −4 1 1 −1 1
χ_{6} 21 5 −1 3 1 −5 −4 1 1 −1 −1
χ_{7} 28 4 4 1 0 4 3 3 −2 1 1
χ_{8} 28 4 −4 1 0 −4 3 3 −2 1 −1
χ_{9} 56 8 0 2 0 0 6 −4 1 2 0
χ_{10} 84 −4 4 3 0 −4 9 −1 −1 −1 1
χ_{11} 84 −4 −4 3 0 4 9 −1 −1 −1 −1
χ_{12} 105 1 5 −3 1 1 5 0 0 1 −1
χ_{13} 105 1 −5 −3 1 −1 5 0 0 1 1
χ_{14} 125 5 5 −1 1 5 0 0 0 −1 −1
χ_{15} 125 5 −5 −1 1 −5 0 0 0 −1 1
χ_{16} 126 6 6 0 −2 −6 1 1 1 0 0
χ_{17} 126 6 −6 0 −2 6 1 1 1 0 0
χ_{18} 252 −12 0 0 0 0 2 2 2 0 0
χ_{19} 288 0 0 0 0 0 −12 −2 −2 0 0

(continued)

Appendix C 172

Table C.1 cont.

7A 8A 8B 10A 10B 12A 20A B^{∗∗}
χ_{1} 1 1 1 1 1 1 1 1
χ_{2} 1 1 −1 1 −1 −1 −1 −1
χ_{3} −1 0 0 1  0  0 i5 −i5
χ_{4} −1 0 0 1  0  0 −i5 i5
χ_{5} 0 −1 −1 0  1 −1 0  0
χ_{6} 0 −1 1 0 −1 1  0  0
χ_{7} 0 0 0 −1 −1 1 −1 −1
χ_{8} 0 0 0 −1 1 −1 1  1
χ_{9} 0 0 0 −2 0 0 0 0
χ_{10} 0 0 0 1 −1 −1 1  1
χ_{11} 0 0 0 1  1  1 −1 −1
χ_{12} 0 −1 −1 1  0  1  1  1
χ_{13} 0 −1 1 1  0 −1 −1 −1
χ_{14} −1 1 1 0  0 −1 0  0
χ_{15} −1 1 −1 0  0  1  0  0
χ_{16} 0 0 0 1  1  0 −1 −1
χ_{17} 0 0 0 1 −1 0  1  1
χ_{18} 0 0 0 −2 0  0  0  0
χ_{19} 1 0 0 0  0  0  0  0

Appendix C 173

Table C.2: Modular Character Table for U_{3}(5) : 2 mod 3

1A 2A 2B 4A 4B 5A 5B 5C 7A 8A 8B 10A 10B 20A B^{∗∗}
φ_{1} 1  1  1 1 1  1  1 1 1 1 1  1  1  1  1
φ_{2} 1  1 −1 1 −1 1  1 1 1 1 −1 1 −1 −1 −1
φ_{3} 20 −4 0 0 0 −5 0 0 −1 0 0  1  0  i5 −i5
φ_{4} 20 −4 0 0 0 −5 0 0 −1 0 0  1  0 −i5 i5
φ_{5} 21 5  1 1 5 −4 1 1 0 −1 −1 0  1  0  0
φ_{6} 21 5 −1 1 −5 −4 1 1 0 −1 1  0 −1 0  0
φ_{7} 28 4  4 0 4  3  3 −2 0 0 0 −1 −1 −1 −1
φ_{8} 28 4 −4 0 −4 3  3 −2 0 0 0 −1 1  1  1
φ_{9} 56 8  0 0 0  6 −4 1 0 0 0 −2 0  0  0
φ_{10} 84 −4 4 0 −4 9 −1 −1 0 0 0  1 −1 1  1
φ_{11} 84 −4 −4 0 4  9 −1 −1 0 0 0  1  1 −1 −1
φ_{12} 126 6  6 −2 −6 1  1 1 0 0 0  1  1 −1 −1
φ_{13} 126 6 −6 −2 6  1  1 1 0 0 0  1 −1 1  1
φ_{14} 252 −12 0 0 0  2  2 2 0 0 0 −2 0  0  0
φ_{15} 288 0  0 0 0 −12 −2 −2 1 0 0  0  0  0  0

Appendix C 174

Table C.3: Decomposition Matrix for U_{3}(5) : 2

Principal Block

1_{A} 1_{B} 20_{A} 20_{B} 28_{A} 28_{B} 56
1_{A} 1 0 0 0 0 0 0
1_{B} 0 1 0 0 0 0 0
20_{A} 0 0 1 0 0 0 0
20_{B} 0 0 0 1 0 0 0
28_{A} 0 0 0 0 1 0 0
28_{B} 0 0 0 0 0 1 0
56 0 0 0 0 0 0 1
125_{A} 1 0 1 1 1 0 1
125_{B} 0 1 1 1 0 1 1

Non-Principal Blocks

21_{A} 84_{A} 21_{B} 84_{B} 126_{A} 126_{B} 252 288
21_{A} 1 0
84_{A} 0 1
105_{A} 1 1
21_{B}  1 0
84_{B}  0 1
105_{B}  1 1
126_{A}  1
126_{B}  1
252  1
288  1

Appendix D

Character Tables and Decomposition Matrix for HS

Table D.1 gives the ordinary irreducible character table for HS taken from Conway et al. (1985), Table D.2 gives the 3-modular Brauer character table for HS taken from Parker and Table D.3 gives the decomposition matrix for HS. We make use of the following notation when using algebraic integers:

i5 = i\sqrt{5}

and

b11 = \frac{−1 + i\sqrt{11}}{2}
∗∗ = \frac{−1 − i\sqrt{11}}{2}.

Appendix D 176

Table D.1: Ordinary Character Table for HS

1A 2A 2B 3A 4A 4B 4C 5A 5B 5C 6A 6B
χ_{1} 1 1 1 1 1 1 1 1 1 1 1 1
χ_{2} 22 6 −2 4 −6 2 2 −3 2 2 −2 0
χ_{3} 77 13 1 5 5 5 1 2 −3 2 1 1
χ_{4} 154 10 10 1 −2 6 −2 4 4 −1 1 1
χ_{5} 154 10 −10 1 −10 −2 2 4 4 −1 −1 1
χ_{6} 154 10 −10 1 −10 −2 2 4 4 −1 −1 1
χ_{7} 175 15 11 4 15 −1 3 0 5 0 2 0
χ_{8} 231 7 −9 6 15 −1 −1 6 1 1 0 −2
χ_{9} 693 21 9 0 21 5 1 −7 3 −2 0 0
χ_{10} 770 34 −10 5 −14 2 −2 −5 0 0 −1 1
χ_{11} 770 −14 10 5 −10 −2 −2 −5 0 0 1 1
χ_{12} 770 −14 10 5 −10 −2 −2 −5 0 0 1 1
χ_{13} 825 25 9 6 −15 1 1 0 −5 0 0 −2
χ_{14} 896 0 16 −4 0 0 0 −4 1 1 −2 0
χ_{15} 896 0 16 −4 0 0 0 −4 1 1 −2 0
χ_{16} 1056 32 0 −6 0 0 0 6 −4 1 0 2
χ_{17} 1386 −6 18 0 6 −2 −2 11 6 1 0 0
χ_{18} 1408 0 16 4 0 0 0 8 −7 −2 −2 0
χ_{19} 1750 −10 10 −5 −10 6 2 0 0 0 1 −1
χ_{20} 1925 5 −19 −1 5 5 −3 0 5 0 −1 −1
χ_{21} 1925 5  1 −1 −35 −3 1 0 5 0 1 −1
χ_{22} 2520 24 0 0 24 −8 0 −5 0 0 0 0
χ_{23} 2750 −50 −10 5 10 2 2 0 0 0 −1 1
χ_{24} 3200 0 −16 −4 0 0 0 0 −5 0 2 0

(continued)

Appendix D 177

Table D.1 cont.

7A 8A 8B 8C 10A 10B 11A B^{∗∗} 12A 15A 20A B^{∗∗} HS.2
χ_{1} 1 1 1 1 1 1 1 1 1 1 1 1 :
χ_{2} 1 0 0 0 1 −2 0 0 0 −1 −1 −1 :
χ_{3} 0 1 −1 −1 −2 1  0 0 −1 0  0  0  :
χ_{4} 0 0 0 0 0  0  0 0 1  1 −2 −2 :
χ_{5} 0 0 2 −2 0  0  0 0 −1 1  0  0  ·|
χ_{6} 0 0 −2 2 0  0  0 0 −1 1  0  0  ·|
χ_{7} 0 −1 1 1 0  1 −1 −1 0 −1 0  0  :
χ_{8} 0 −1 −1 −1 2  1  0 0 0  1  0  0  :
χ_{9} 0 1 −1 −1 1 −1 0 0 0  0  1  1  :
χ_{10} 0 −2 0 0 −1 0  0 0 1  0  1  1  :
χ_{11} 0 0 0 0 1  0  0 0 −1 0 i5 −i5 ·|
χ_{12} 0 0 0 0 1  0  0 0 −1 0 −i5 i5  ·|
χ_{13} −1 1 1 1 0 −1 0 0 0  1  0  0  :
χ_{14} 0 0 0 0 0  1 b11 ∗∗ 0  1  0  0  ·|
χ_{15} 0 0 0 0 0  1 ∗∗ b11 0  1  0  0  ·|
χ_{16} −1 0 0 0 2  0  0 0 0 −1 0  0  :
χ_{17} 0 0 0 0 −1 −2 0 0 0  0  1  1  :
χ_{18} 1 0 0 0 0  1  0 0 0 −1 0  0  :
χ_{19} 0 −2 0 0 0  0  1 1 −1 0  0  0  :
χ_{20} 0 1 1 1 0  1  0 0 −1 −1 0  0  :
χ_{21} 0 1 −1 −1 0  1  0 0 1 −1 0  0  :
χ_{22} 0 0 0 0 −1 0  1 1 0  0 −1 −1 :
χ_{23} −1 0 0 0 0  0  0 0 1  0  0  0  :
χ_{24} 1 0 0 0 0 −1 −1 −1 0  1  0  0  :

Appendix D 178

Table D.2: Modular Character Table for HS mod 3

1A 2A 2B 4A 4B 4C 5A 5B 5C 7A
φ_{1} 1 1 1 1 1 1 1 1 1 1
φ_{2} 22 6 −2 −6 2 2 −3 2 2 1
φ_{3} 49 1  5  5 −3 1 −1 4 −1 0
φ_{4} 49 1  5  5 −3 1 −1 4 −1 0
φ_{5} 77 13 1  5 5 1 2 −3 2 0
φ_{6} 154 10 10 −2 6 −2 4 4 −1 0
φ_{7} 154 10 −10 −10 −2 2 4 4 −1 0
φ_{8} 154 10 −10 −10 −2 2 4 4 −1 0
φ_{9} 231 7 −9 15 −1 −1 6 1 1 0
φ_{10} 321 −15 −3 −3 5 1 −4 6 1 −1
φ_{11} 693 21 9 21 5 1 −7 3 −2 0
φ_{12} 748 28 −8 −8 0 −4 −2 −2 −2 −1
φ_{13} 770 −14 10 −10 −2 −2 −5 0 0 0
φ_{14} 770 −14 10 −10 −2 −2 −5 0 0 0
φ_{15} 825 25 9 −15 1 1 0 −5 0 −1
φ_{16} 1176 −24 −12 12 4 0 1 6 1 0
φ_{17} 1253 −11 5  1 −7 1 3 −12 −2 0
φ_{18} 1386 −6 18 6 −2 −2 11 6 1 0
φ_{19} 2520 24 0 24 −8 0 −5 0 0 0

(continued)

Appendix D 179

Table D.2: cont.

8A 8B 8C 10A 10B 11A B^{∗∗} 20A B^{∗∗} HS.2
φ_{1} 1 1 1 1 1 1 1 1 1 :
φ_{2} 0 0 0 1 −2 0 0 −1 −1 :
φ_{3} −1 1 1 1  0 b11 ∗∗ i5 −i5 ·|
φ_{4} −1 1 1 1  0 ∗∗ b11 −i5 i5  ·|
φ_{5} 1 −1 −1 −2 1  0 0 0  0  :
φ_{6} 0 0 0 0  0  0 0 −2 −2 :
φ_{7} 0 2 −2 0  0  0 0 0  0  ·|
φ_{8} 0 −2 2 0  0  0 0 0  0  ·|
φ_{9} −1 −1 −1 2  1  0 0 0  0  :
φ_{10} −1 1 1 0  2  2 2 2  2  :
φ_{11} 1 −1 −1 1 −1 0 0 1  1  :
φ_{12} −2 0 0 −2 2  0 0 2  2  :
φ_{13} 0 0 0 1  0  0 0 i5 −i5 ·|
φ_{14} 0 0 0 1  0  0 0 −i5 i5  ·|
φ_{15} 1 1 1 0 −1 0 0 0  0  :
φ_{16} 2 0 0 1 −2 −1 −1 −3 −3 :
φ_{17} −1 −1 −1 −1 0 −1 −1 1  1  :
φ_{18} 0 0 0 −1 −2 0 0 1  1  :
φ_{19} 0 0 0 −1 0  1 1 −1 −1 :

Appendix D 180

Table D.3: Decomposition Matrix for HS Principal Block

1 22 154_{a} 321 748 1176 1253
1 1 0 0  0 0  0  0
22 0 1 0  0 0  0  0
154_{a} 0 0 1  0 0  0  0
770_{a} 0 1 0  0 1  0  0
1408 1 0 1  0 0  0  1
1750 0 1 1  1 0  0  1
1925_{a} 1 0 0  0 1  1  0
2750 0 0 0  1 0  1  1
3200 1 1 0  0 1  1  1

Non-principal blocks

49_{a} 49_{b} 77 154_{b} 154_{c} 770_{a} 770_{b} 231 825 693 1386 2520
77 0 0 1 0  0  0  0
154_{b} 0 0 0 1  0  0  0
154_{c} 0 0 0 0  1  0  0
175 1 1 1 0  0  0  0
770_{b} 0 0 0 0  0  1  0
770_{c} 0 0 0 0  0  0  1
896_{a} 1 0 1 0  0  0  1
896_{b} 0 1 1 0  0  1  0
1925_{b} 0 0 1 1  1  1  1
231  1 0
825  0 1
1056  1 1
693  1
1386  1
2520  1

Appendix E

Results on Induction and Restriction Between U_{3}(5) : 2 and

HS

In this appendix we give a list of some of the results on 3-modular structures determined on the way to proving the main results on the structure of the PIMs of the Higman–Sims group. U_{3}(5) : 2 is always taken to be the extension of the 3-dimensional projective special unitary group over a field of 25 elements by C_{2} in which the character 28_{a} of U_{3}(5) splits, and the characters 28_{b} and 28_{c} fuse. We also give a brief description of how to calculate the composition factors of a character using the package PARI.

E.1 The PIMs of U_{3}(5) : 2

We give the structure of the PIMs for the principal block of U_{3}(5) : 2.

\begin{matrix}1_{A} \\ 20_{A} \\ 56 & 28_{A} \\ 20_{B} \\ 1_{A} \end{matrix}  \begin{matrix}1_{B} \\ 20_{B} \\ 56 & 28_{B} \\ 20_{A} \\ 1_{B} \end{matrix}  \begin{matrix}28_{A} \\ 20_{B} \\ 56 & 1_{A} \\ 20_{A} \\ 28_{A} \end{matrix}

\begin{matrix}28_{B} \\ 20_{A} \\ 56 & 1_{B} \\ 20_{B} \\ 28_{B} \end{matrix}  \begin{aligned}& 20_{A} \\ 1_{B} & 56 & 28_{A} \\ 20_{B} & 20_{A} & 20_{B} \\ 1_{A} & 56 & 28_{B} \\ & 20_{A} \end{aligned}  \begin{aligned}& 20_{B} \\ 1_{A} & 56 & 28_{B} \\ 20_{A} & 20_{B} & 20_{A} \\ 1_{B} & 56 & 28_{A} \\ & 20_{B} \end{aligned} ; \begin{aligned}& 56 \\ 20_{A} & 20_{B} \\ 1_{A} & 1_{B} & 28_{A} & 28_{B} & 56 \\ 20_{A} & 20_{B} \\ & 56 \end{aligned}

E.2 Calculating the Composition Factors of a

Character Using the Package PARI

The following discussion applies to both ordinary characters and Brauer characters. Let χ be a character of G and let X be the irreducible character table of G (we assume the rows of the character table are indexed by the irreducible characters of G and the columns are indexed by the conjugacy classes of G). The composition factors of χ are determined by finding the unique expression of χ as a linear combination of the rows of X. If χ has large dimension, as is often the case with induced and restricted characters, then determining the composition factors of χ can be a very difficult and time-consuming task. We have therefore made use of the linear system solving function of the arithmetic package PARI, using it to solve the system Ax = b where A = X^{T} and B = χ^{T}.

The first step is to store the matrix X^{T} in a form that PARI can read. We do this by typing the matrix into a file, which will be called filename, as follows: the matrix is begun with an open square bracket ([) followed by the first entry of the first row of X. The entries of a row are separated by a comma (,) and the rows themselves must be separated by a semicolon (;) (no return characters should be introduced—tempting though it is to start a new row on a new line). The last entry of the last row is followed by a closed square bracket (]) to indicate the end of the matrix X. Immediately following this we type a tilde (∼), this is the transposing operation in PARI.

The file is read into PARI by using the command \r filename. PARI then responds with “%1=” followed by the transposed matrix. This allows for the matrix to be checked and indicates that this piece of information has been stored as %1 (or %r if it is the rth piece of data PARI has dealt with on this run).

Next the character χ is typed in, again starting with an open square bracket, separating entries by a comma, ending with a closed square bracket, and then typing a tilde so that PARI considers χ as a column vector. PARI will then respond with “%2=” followed by the row vector and its tilde. To solve the system we then type Gauss(%1, %2). PARI will then respond with “%3=” followed by a row vector and its tilde, which is the solution of the system. The row vector then gives the composition factors of χ.

With this technique we were able to calculate the composition factors of all the induced and restricted characters.

For more details on running PARI, see the PARI manual, Cohen et al. (1992).

E.3 Structures of the Simple HS-Modules Re-

stricted to U_{3}(5) : 2

1↓ = 1_{A}.

22↓ = 21_{B} ⊕ 1_{B}.

49_{a}↓ = \begin{matrix}1_{A} \\ 20_{A} \\ 28_{A} \end{matrix}.

49_{b}↓ = \begin{matrix}28_{A} \\ 20_{B} \\ 1_{A} \end{matrix} .

77↓ = 56 ⊕ 21_{A}.

154_{a}↓ = 126_{A} ⊕ 28_{A}.

Appendix E 184

154_{b}↓ = 1_{B} ⊕ \begin{matrix}28_{B} \\ 20_{A} \\ 1_{B} & 56 \\ 20_{B} \\ 28_{B} \end{matrix} or 154_{b}↓ = 28_{B} ⊕ \begin{matrix}1_{B} \\ 20_{B} \\ 28_{B} & 56 \\ 20_{A} \\ 1_{B} \end{matrix} .

154_{c}↓ = 28_{B} ⊕ \begin{matrix}21_{B} \\ 84_{B} \\ 21_{B} \end{matrix} .

321↓ = 252 ⊕ \begin{matrix}20_{B} \\ 1_{A} & 28_{B} \\ 20_{A} \end{matrix} .

748↓ = 288 ⊕ 126_{A} ⊕ 126_{B} ⊕ \begin{matrix}56 & 28_{B} \\ 20_{B} & 20_{A} \\ 28_{B} & 56 \end{matrix} .

770_{a}↓ = 288 ⊕ 252 ⊕ 126_{A} ⊕ 84_{A} ⊕ 20_{A}. 770_{b}↓ = 288 ⊕ 252 ⊕ 126_{A} ⊕ 84_{A} ⊕ 20_{B}.

1176↓ = 288 ⊕ 252 ⊕ 252 ⊕ 126_{B} ⊕ \begin{matrix}84_{B} \\ 21_{B} \\ 84_{B} \end{matrix} ⊕ \begin{matrix}20_{A} \\ 28_{A} & 1_{B} \\ 20_{B} \end{matrix} .

1253↓ = 288 ⊕ 288 ⊕ 252 ⊕ 84_{B} ⊕ \begin{matrix}84_{A} \\ 21_{A} \\ 84_{A} \end{matrix} ⊕ \begin{matrix}56 \\ 20_{A} & 20_{B} \\ 56 \end{matrix} .

E.4 Structures of Some Simple U_{3}(5) : 2 Mod-

ules Induced to HS

1_{A}↑ = 1 ⊕ \begin{matrix}49_{b} \\ 77 \\ 49_{a} \end{matrix} .

1_{B}↑ = 154_{b} ⊕ 22.

Appendix E 185 20_{A}↑ = 770_{a} ⊕ 321 1253 1176 . 20_{B}↑ = 770_{b} ⊕ 1176 1253 321 . The principal block component of 21_{B}↑ has structure 22 748 22 . The second block component of 28_{A}↑ has structure \begin{matrix}49_{a} \\ 77 \\ 49_{b} \end{matrix} . The principal block component of 28_{B}↑ has structure \begin{aligned}& 748 \\ 1 & 1176 & 22 \\ & 748. \end{aligned} .

Appendix F

Relations for the Basic Algebra of the Principal Block of HS

In this appendix we present a list of the generating relations found when determining the basic algebra of the principal block of HS by comparing the structures of KQ with P_{S} for each simple module S in the principal block.

\begin{aligned}c̄a & = & 0 \\ d̄b & = & 0 \\ āc & = & 0 \\ b̄d & = & 0 \\ yd̄a & = & 0 \\ zēa & = & 0 \\ xc̄b & = & 0 \\ zēb & = & 0 \\ ādȳ & = & 0 \\ b̄dȳ & = & 0 \\ b̄bb̄a & = & 0 \\ c̄bb̄a & = & 0 \end{aligned}  \begin{aligned}d̄bb̄a & = & 0 \\ ēbb̄a & = & 0 \\ āaāb & = & 0 \\ c̄aāb & = & 0 \\ d̄aāb & = & 0 \\ ēaāb & = & 0 \\ ābb̄c & = & 0 \\ b̄bb̄c & = & 0 \\ d̄bb̄c & = & 0 \\ ēbb̄c & = & 0 \\ ȳxx̄x & = & 0 \\ z̄xx̄x & = & 0 \\ āaād & = & 0 \\ b̄aād & = & 0 \end{aligned}  \begin{aligned}c̄aād & = & 0 \\ ēaād & = & 0 \\ x̄xx̄y & = & 0 \\ z̄xx̄y & = & 0 \\ āaāe & = & 0 \\ b̄aāe & = & 0 \\ c̄aāe & = & 0 \\ d̄aāe & = & 0 \\ x̄xx̄z & = & 0 \\ ȳxx̄z & = & 0 \\ cx̄xx̄ & = & 0 \\ dȳxx̄ & = & 0 \\ ez̄xx̄ & = & 0 \end{aligned}

\begin{aligned}aābb̄a & = & 0 \\ bb̄aāb & = & 0 \\ cx̄xx̄x & = & 0 \\ xx̄xx̄x & = & 0 \\ dȳxx̄y & = & 0 \\ yȳxx̄y & = & 0 \end{aligned}  \begin{aligned}ez̄xx̄z & = & 0 \\ zz̄xx̄z & = & 0 \\ x̄xx̄xx̄ & = & 0 \\ ȳxx̄xx̄ & = & 0 \\ z̄xx̄xx̄ & = & 0 \end{aligned}  \begin{aligned}āaābb̄ & = & 0 \\ b̄aābb̄ & = & 0 \\ c̄aābb̄ & = & 0 \\ d̄aābb̄ & = & 0 \\ ēaābb̄ & = & 0 \end{aligned}^{\begin{aligned}āa & = & k_{1}ābb̄a \\ ēc & = & r_{1}z̄x \\ ēd & = & s_{1}z̄y \\ d̄d & = & s_{2}ȳy + k_{2}ȳxx̄y \\ c̄e & = & t_{1}x̄z \\ d̄e & = & t_{2}ȳz \\ yȳ & = & α_{1}xx̄ + k_{3}xx̄xx̄ \\ zz̄ & = & α_{2}xx̄ + k_{4}xx̄xx̄ \\ cx̄ & = & α_{3}dȳ \\ ez̄ & = & α_{4}dȳ \\ xc̄ & = & β_{1}yd̄ \\ zē & = & β_{2}yd̄ \\ cc̄ & = & γ_{1}aā + k_{5}aābb̄ \\ dd̄ & = & γ_{2}aā + γ_{3}bb̄ + k_{6}aābb̄ \\ eē & = & γ_{4}aā + γ_{5}bb̄ + k_{7}aābb̄ \\ dd̄a & = & λ_{1}bb̄a \\ eēa & = & λ_{2}bb̄a \\ bb̄b & = & λ_{3}aāb \\ cc̄b & = & λ_{4}aāb \\ eēb & = & λ_{5}aāb \\ xc̄c & = & r_{2}xx̄x \\ yd̄c & = & r_{3}xx̄x \\ zz̄x & = & r_{4}xx̄x \\ yȳx & = & r_{5}xx̄x \\ cc̄c & = & r_{6}bb̄c \end{aligned}  \begin{aligned}dd̄c & = & r_{7}bb̄c \\ ez̄x & = & r_{8}bb̄c \\ cx̄x & = & r_{9}bb̄c \\ dȳx & = & r_{10}bb̄c \\ xc̄d & = & s_{3}xx̄y \\ zz̄y & = & s_{4}xx̄y \\ yȳy & = & s_{5}xx̄y \\ cc̄d & = & s_{6}aād \\ ez̄y & = & s_{7}aād \\ dȳy & = & s_{8}aād \\ cx̄y & = & s_{9}aād \\ zēe & = & t_{3}xx̄z \\ yȳz & = & t_{4}xx̄z \\ zz̄z & = & t_{5}xx̄z \\ bb̄e & = & t_{6}aāe \\ eēe & = & t_{7}aāe \\ cx̄z & = & t_{8}aāe \\ dȳz & = & t_{9}aāe \\ ez̄z & = & t_{10}aāe \\ c̄dȳ & = & α_{5}x̄xx̄ \\ d̄dȳ & = & α_{6}ȳxx̄ \\ ēdȳ & = & α_{7}z̄xx̄ \\ ādd̄ & = & β_{3}ābb̄ \\ āeē & = & β_{4}ābb̄ \\ b̄bb̄ & = & β_{5}b̄aā \end{aligned}  \begin{aligned}b̄cc̄ & = & β_{6}b̄aā \\ b̄eē & = & β_{7}b̄aā \\ c̄cc̄ & = & β_{8}c̄bb̄ \\ c̄dd̄ & = & β_{9}c̄bb̄ \\ c̄eē & = & β_{10}c̄bb̄ \\ x̄yd̄ & = & β_{11}c̄bb̄ \\ d̄cc̄ & = & β_{12}d̄aā \\ d̄dd̄ & = & β_{13}d̄aā \\ d̄eē & = & β_{14}d̄aā \\ ȳyd̄ & = & β_{15}d̄aā \\ ēbb̄ & = & β_{16}ēaā \\ ēcc̄ & = & β_{17}ēaā \\ ēdd̄ & = & β_{18}ēaā \\ ēeē & = & β_{19}ēaā \\ z̄yd̄ & = & β_{20}ēaā \\ c̄bb̄c & = & r_{11}x̄xx̄x \\ d̄aād & = & s_{10}ȳxx̄y \\ ēaāe & = & t_{11}z̄xx̄z \\ yȳxx̄ & = & α_{8}xx̄xx̄ \\ zz̄xx̄ & = & α_{9}xx̄xx̄ \\ bb̄aā & = & β_{21}aābb̄ \\ cc̄bb̄ & = & β_{22}aābb̄ \\ dd̄aā & = & β_{23}aābb̄ \\ eēaā & = & β_{24}aābb̄ \end{aligned}}

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