Buildings and Kneser graphs

(1)

Buildings and Kneser graphs

Citation for published version (APA):

Güven, Ç. (2012). Buildings and Kneser graphs. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR721532

DOI:

10.6100/IR721532

Document status and date: Published: 01/01/2012

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

(2)

(3)

A catalogue record is available from the Eindhoven University of Technology Library. ISBN: 978-90-386-3067-0

The research which leaded to this thesis was financed by Lex Schrijver, who was rewarded with Spinoza Prize of the Netherlands Organization for Scientific Research (NWO) in 2005.

The cover of this thesis is designed by Atike Dicle Pekel Duhbacı. Printed by Print Service at Eindhoven University of Technology.

(4)

(5)

(6)

Buildings and Kneser Graphs

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op woensdag 25 januari 2012 om 16.00 uur

door

Çiçek Güven Özçelebi

(7)

(8)

P

REFACE

The following questions with their possible generalizations initiated years of mathemati-cal research, including this thesis:

If C is a collection of mutually intersecting k-subsets of a fixed n-set, how big can C be? And in the extreme case, what is the structure of C?

According to Erd˝os [53], this question was answered by Erd˝os, Ko, and Rado, already

in 1938, but they first published this result in 1961 [54]: For 2k≤ n, one must have

|C| ≤ n−1

k−1

, and if equality holds then C is the collection of all k-subsets containing some

fixed element of the given n-set for 2k< n. For 2k = n, this bound can be obtained by

picking one k-set from each complementary pair in any way.

The Kneser graph K(n, k) is the graph with as vertices the k-subsets of a fixed n-set,

where two k-subsets are adjacent when they are disjoint. In this terminology, Erd˝os, Ko,

and Rado found the largest cocliques (independent sets of vertices) in K(n, k).

Many people have studied generalizations and variations of this problem, and that is also what we shall do in this thesis.

The question that initiated this research is the following:

“ Try to describe Erd˝os-Ko-Rado sets with maximal size in the various geometries arising from a spherical diagram by circling any set of nodes.”

Here ‘circling a certain set of nodes’ in a diagram means defining the type of flags of the geometry which will be the vertices of the graph. This problem generalizes the

classical Erd˝os, Ko, and Rado problem, which is about sets, to a problem about buildings

of spherical type.

The natural generalization of being disjoint as sets is being ‘far apart’ as geometrical objects. We have objects and some kind of a distance function, and define a Kneser graph with our objects as vertices, two objects being adjacent when they have maximum distance. The goal is always to find the maximal size of a coclique in such a graph, and to characterize the cases that reach this maximum.

Our objects will usually be flags of some fixed type in a finite building. This thesis consists of three parts and seven chapters.

Part one is the introduction and consists of two chapters.

In Chapter 1, relevant definitions for the content of this thesis are given.

In Chapter 2, we present our first generalization of the Kneser graph, which we call

generalized Kneser graphs, GK(n : k1, . . . , kl). These graphs are defined over inclusive

(9)

the conditions under which these graphs are connected are determined. Descriptions for maximal coclique sizes for generalized Kneser graphs in many cases are given, the existence of relations between the maximal coclique sizes for different examples of those graphs are shown.

Part two consists of Chapter 3, Chapter 4, and Chapter 5. In this part, further gen-eralizations of Kneser graphs are introduced, from finite sets to buildings of spherical type, and from being disjoint as sets to being far apart as geometrical objects. Chapter 3 describes this generalization. We introduce Kneser graphs on buildings in this chapter.

While studying generalizations of the Erd˝os-Ko-Rado theorem and the chromatic

number of Kneser-type graphs, one needs information about maximal cocliques of near-maximal size. In Chapter 4 we describe a simple construction that in the most interesting cases produces all such near-maximal cocliques. This chapter treats the results of the paper [22].

In Chapter 5, we work on maximal sizes of cocliques for the point-hyperplane graphs. This chapter is about the results of the paper [11]. Our result gives an upper bound for the coclique size for Kneser graphs on point hyperplane flags, and characterizes the case when equality holds.

Part three consists of Chapter 6 and Chapter 7. In this part, other parameters and eigenvalues of Kneser type graphs are calculated.

In Chapter 6, the Smith Normal Forms (SNF) of some Kneser type graphs are con-jectured and proved. A symmetry relation is generalized which holds in strongly regular graphs with prime power eigenvalues. So once the p-rank is known, the SNF is known in some cases because of this symmetry. We conjecture and come up with partial results for some graphs related to some generalized quadrangles.

In Chapter 7, four geometric objects are taken, and their collinearity graphs are exam-ined. These geometries are parapolar spaces. With the help of this fact, the parameters of the related distribution diagrams are calculated. Knowing those parameters enables us to calculate the eigenvalues of some Kneser type graphs to be powers of q. We make use of Delsarte’s Linear Programming bound and Hoffman bound to come up with bounds on the sizes of some subconfigurations.

(10)

C

ONTENTS

Preface vii

List of Figures xiii

List of Tables xv

I

Introduction

1

1 Preliminaries 3

1.1 Graphs . . . 3

1.1.1 Regular partitions . . . 6

1.2 Field with one element and q-analogue . . . . 8

1.3 Groups . . . 8

1.3.1 Coxeter groups . . . 9

1.3.2 Groups with a Tits system . . . 11

1.3.3 Finite simple groups of Lie type . . . 13

1.4 Geometric objects . . . 14

1.5 Buekenhout-Tits geometries . . . 14

1.5.1 Point-line geometries . . . 15

1.5.2 Chamber complexes and foldings . . . 16

1.5.3 Buildings . . . 19

1.5.4 Projective spaces . . . 21

1.5.5 Polar spaces . . . 22

1.6 Association schemes . . . 26

1.6.1 Delsarte’s linear programming bound . . . 29

2 A Generalization of Kneser Graphs on Increasing Sequences of Sets 31 2.1 Introduction . . . 31

2.1.1 Codes . . . 31

2.1.2 (Classical) Kneser graphs . . . 32

2.1.3 Johnson graphs . . . 32

2.1.4 The relation between J (n, k) and K(n, k) . . . . 33

2.1.5 History of the Kneser graphs . . . 33

2.2 Generalized Kneser graphs . . . 34

2.2.1 Connectedness of the generalized Kneser graphs . . . 35

(11)

2.3 Maximal cocliques in GK(n : k1, . . . , kl) . . . 42

2.3.1 Foldings . . . 45

II

A Unifying Approach: From Sets to Groups

49

3 Generalizations to Buildings of Spherical Type 51 3.1 Understanding the graphs . . . 51

3.1.1 q-Kneser graphs . . . 51

3.1.2 Kneser graphs for Coxeter groups . . . 52

3.1.3 Generalization of Kneser graphs over buildings of spherical type . 53 3.1.4 Taking sums over the Weyl group . . . 55

3.2 Cocliques on flags of P G(4, q) . . . . 57

3.2.1 Graphs on point-hyperplane pairs . . . 57

3.2.2 Graphs on point-line flags . . . 57

3.2.3 Graphs on point-plane flags . . . 58

3.2.4 Graphs on line-plane flags . . . 60

4 Unique Coclique Extension Property 61 4.1 Introduction . . . 61

4.1.1 Method . . . 61

4.1.2 Matroids . . . 62

4.1.3 Statement of the main theorem . . . 63

4.2 Subspaces of a projective space . . . 64

4.3 Points in a polar space . . . 65

4.4 Totally singular lines in an orthogonal space . . . 65

4.5 Minuscule weights . . . 66

4.6 Adjoint representation . . . 67

4.7 Nonexamples . . . 68

5 Maximal Cocliques in Point-Hyperplane Graphs 71 5.1 Introduction . . . 71

5.1.1 Rank 1 matrices . . . 72

5.1.2 The thin case . . . 72

5.2 Maximum-size cocliques . . . 73

5.3 Maximum number of points . . . 74

5.4 Classification of cocliques for n_{≤ 4 . . . .} 77

5.4.1 P G(2, q) . . . . 78

5.4.2 P G(3, q) . . . . 78

III

Calculating Some Other Parameters and Eigenvalues

81

6 Smith Normal Forms of Some Kneser Graphs for Buildings 83 6.1 Introduction . . . 83

6.1.1 Preliminaries . . . 83

6.1.2 Review of the related problems . . . 87

(12)

CONTENTS xi

6.3 Non-collinearity graphs of generalized quadrangles . . . 89

6.3.1 Non-collinearity graph of GQ(q, q2) . . . 90

6.3.2 Non-collinearity graph of GQ(q, q) . . . . 90

6.3.3 Non-collinearity graph of GQ(q2, q) . . . . 92

6.3.4 Non-collinearity graph of GQ(q2, q3) . . . 92

6.3.5 Non-collinearity graph of GQ(q3, q2) . . . 92

6.4 Graphs with prime power eigenvalues and SNF . . . 93

6.4.1 Oppositeness graphs . . . 93

6.4.2 Bipartite graph of disjoint point and lines in P G(2, q) . . . . 94

7 From graphs of Lie type to Kneser graphs on Buildings 95 7.1 Introduction . . . 95

7.2 Preliminaries . . . 96

7.2.1 Four diagrams . . . 96

7.2.2 Graphs of Coxeter type . . . 97

7.2.3 Graphs of Lie type . . . 99

7.2.4 Relations between Γ(G, G_S, r) and Γ(W, W_S, r) . . . 102

7.2.5 Parapolar spaces . . . 103

7.2.6 Reading the diagrams and chain calculus . . . 104

7.3 Four point-line geometries . . . 107

7.3.1 The graph of Coxeter and Lie types E_7,1 . . . 107

7.3.2 The graph of Coxeter and Lie types E6,2 . . . 115

7.3.3 The graph of Coxeter and Lie types E8,8 . . . 119

7.3.4 The graph of Coxeter and Lie types F4,1 . . . 124

7.4 Using the Hoffman bound and the DLPB bound . . . 129

7.5 Eigenvalue results for the Kneser graphs on buildings . . . 129

7.5.1 Eigenvalues of K(A_2d−1(q),_{{d}) . . . 130} Abstract 131 Acknowledgements 133 Curriculum Vitae 135 Bibliography 137 Index 144

(13)

(14)

L

IST OF

F

IGURES

1.1 the Petersen graph with vertices labeled as 2-subsets of the set{1, 2, 3, 4, 5} 4

1.2 the distance distribution diagram of the Petersen graph . . . 7

1.3 the diagrams of the irreducible finite Coxeter systems . . . 10

1.4 for A2, positive roots r, s, and r + s . . . . 11

1.5 a simplicial 3-complex . . . 16

1.6 the diagram of the cube . . . 21

6.1 distribution diagram of the incidence graph of points and lines of P G(2, q) 94 7.1 the diagrams E_6,2, E_7,1, E_8,8, and F_4,1 . . . 96

7.2 the diagram of D₆where node number 3 is circled . . . 99

7.3 distribution diagram of the E_7,1(1) graph . . . 108

7.4 distribution diagram of the graph of Lie Type E_7,1(q) . . . 111

7.5 distribution diagram of the E6,2(1) graph . . . 115

7.6 distribution diagram of the E8,8(1) graph . . . 120

(15)

(16)

L

IST OF

T

ABLES

2.1 diameters of generalized Kneser graphs . . . 37

2.2 maximum coclique sizes . . . 42

5.1 maximal coclique classification for P G(3, q) . . . . 78

7.1 the degrees of the finite irreducible Coxeter systems . . . 99

7.2 parameters for Chevalley groups . . . 102

7.3 the bound given by DLPB and actual Y_Afor the graph of Coxeter type E_7,1 110

(17)

(18)

P

ART

I

NTRODUCTION

(19)

(20)

1

P

RELIMINARIES

In this Chapter, we introduce the general notions that are relevant to the content of this thesis. We give the definitions that are most relevant to a chapter within the chapter.

This book is self contained up to some background in elementary linear algebra, graph theory and group theory.

For graph theory, see [8], [59], [21], [60], and [24].

For standard theory of linear algebraic groups see [72], for the standard theory on groups of Lie type and finite groups of Lie type, see [36] and [37], for classical groups see [34]. For geometry of classical groups, see [92].

We define Coxeter groups, groups with (B, N )-pairs, buildings with related geomet-rical objects, and association schemes here, but for a complete understanding of these objects, see [97], [98], [84], [103], [27] and [1]. For the theory of Coxeter groups and Tits systems, [18] is a well known source.

For coset graphs for parabolic subgroups of groups of Lie type, see [21], Chapter 10. We refer to this chapter quite often in this thesis.

1.1 Graphs

The problems that are in the focus of this thesis can be considered as problems of alge-braic graph theory, which is the study of graphs using algealge-braic methods.

DEFINITION 1.1.0.1. In a graph Γ, a path of length i between two vertices u and v is

a sequence of distinct vertices u = u0, u1, . . . , ui = v such that for any k, 0≤ k ≤ i − 1,

the vertex u_k is adjacent to u_k+1. A walk is a path in which vertices or edges may be

repeated.

DEFINITION1.1.0.2. A graph is connected when it has one connected component, that is,

(it is non-empty and) for any pair of vertices, there is a path joining them.

All graphs of our interest are finite, without loops, without multiple edges, and undi-rected.

DEFINITION1.1.0.3. A (proper) vertex coloring of a graph Γ = (V, E) is a mapϕ : V → C

from the set V of vertices to a set C of colors, such that for u, v∈ V , if u is adjacent to v,

(21)

A proper vertex coloring of a graph Γ gives a partition of the set of vertices. For any

c∈ C, we call ϕ−1(c) a color class.

DEFINITION1.1.0.4. The chromatic number of a graph Γ is the smallest number of colors

needed for a proper vertex coloring of Γ, and is denoted byχ(Γ).

DEFINITION 1.1.0.5. A coclique in a graph is a set of vertices, such that no two vertices

in the set are adjacent. The maximum size of a coclique in a graph Γ is denoted byα(Γ).

Each color class is a coclique.

REMARK1.1.1. For Γ = (V, E),

|V | ≤ α(Γ)χ(Γ). (1.1)

DEFINITION1.1.0.6. A graph of order v is called strongly regular with parameters v, k,λ, µ

whenever it is not complete or edgeless and i . each vertex is adjacent to k vertices,

ii . for each pair of adjacent vertices there areλ vertices adjacent to both,

iii . for each pair of non-adjacent vertices there areµ vertices adjacent to both.

EXAMPLE1.1.0.7. The Petersen graph is strongly regular with parameters (10, 3, 0, 1).

{1,2} {3,4} {3,5} {4,5} {1,5} {2,3} {1,4} _{2,4} {2,5} {1,3}

FIGURE1.1: the Petersen graph with vertices labeled as 2-subsets of the set{1, 2, 3, 4, 5}

DEFINITION1.1.0.8. For any pair of vertices u, v in a connected graph Γ, the length of the

shortest path from u to v in Γ is called the distance between them, which we denote by

d(u, v). The diameter of a connected graph Γ is the maximal distance occurring in Γ.

DEFINITION 1.1.0.9. For any pair of vertices u, v in a connected graph Γ, any path of

length d(u, v) is called a geodesic between them. A subset C of Γ is called convex or

geodetically closed if for any pair of vertices in C, all geodesics between them are in C.

(22)

1.1 GRAPHS 5

DEFINITION1.1.0.10. [21] A connected graph Γ is called distance-regular if it is regular

of valency k, and if for any two points u, v at distance i = d(u, v), there are precisely c_i

neighbors of v in Γi−1(u) and bi neighbors of v in Γi+1(u). The sequence

ι(Γ) :={b0, b1, . . . , bd−1; c1, . . . , cd}

where d is the diameter of Γ, is called the intersection array of Γ; the numbers ci, bi,

and aiwhere

ai:= k− bi− ci (i = 0, . . . , d)

is the number of neighbors of v in Γ_i(u) for d(u, v) = i, are called the intersection

numbers of Γ. Clearly

b₀= k, b_d = c₀= 0, c₁= 1.

The size of Γ_i(u) is denoted by k_i. The valency of Γ is k = k₁.

For a distance-regular graph with adjacency matrix A, the matrices A_i, where (A_i)_u_,v=

1 if d(u, v) = i and (A_i)_u_,v= 0 otherwise, satisfy the relations [21, p. 127] :

A₀= I , A₁= A, (1.2)

AA_i= b_i₋₁A_i₋₁+ a_iA_i+ c_i+1Ai+1, (1.3)

A0+ A1+· · · + Ad= J . (1.4)

This means that we can write each Ai as a polynomial in A, and that the minimal

polynomial of A has degree d + 1.

Let us define group actions here since we will define graphs based on them.

DEFINITION1.1.0.11. A group action of a group G on a set X is a map from G× X to X ,

written as g x for g_{∈ G, x ∈ X , satisfying the following properties:}

(i) (g₁g2)x = g1(g2x)for all g1, g2in G, x∈ X ,

(ii) 1x = x for all x_{∈ X , where 1 is the unit element of G.}

If a group acts on a set X it automatically also acts on the subsets of X . A group action is transitive if for any a, b in A, there exists a g in G such that ga = b. A group action is

faithful if for any g, h_{∈ G, g 6= h, there exists an a in A such that ga 6= ha.}

DEFINITION 1.1.0.12. A permutation group G is a group of permutations of a set X . It

acts on the elements of the set X naturally. An orbit of G is a set G x =_{{g x | g ∈ G}. The}

stabilizer of x, which is denoted by G_x is the subset of G fixing x, i.e._{{g | g x = x}. The}

orbits of G on X_{× X are called orbitals.}

DEFINITION1.1.0.13. An automorphism of a (finite) graph Γ = (V, E) is a permutation

σ of the vertex set V , such that for any edge e = (u, v), σ(e) = (σ(u), σ(v)) is also an edge.

The set of automorphisms of a given graph, under the composition operation, forms a group, which is known as the automorphism group of the graph, denoted by Aut(Γ).

(23)

DEFINITION 1.1.0.14. A graph Γ is vertex-transitive if for any two vertices v₁and v₂of

Γ, there existsσ∈ Aut(Γ) such that σ(v1) = v2.

DEFINITION 1.1.0.15. A graph Γ is edge-transitive if for any two edges e1 and e2of Γ,

there existsσ∈ Aut(Γ) such that σ(e1) = e2.

DEFINITION 1.1.0.16. A graph Γ is flag-transitive if for any two edges e₁= (v₁, u₁) and

e₂= (v₂, u₂) of Γ, there existsσ∈ Aut(Γ) such that σ(v1) = u1andσ(v2) = u2.

DEFINITION 1.1.0.17. A graph Γ is distance-transitive if for any two pairs of vertices

u, v and u′, v′ where d(u, v) = d(u′, v′) = i, there existsσ∈ Aut(Γ) such that σ(u) =

u′,σ(v) = v′.

DEFINITION1.1.0.18. Let G be a group, H be a subgroup of it, and S be any subset of G.

One can define a graph Γ(G, H, S) (when HSH = HS−1H) called coset graph of G on H

(with respect to S) on the left cosets of H in G. Two cosets g1Hand g2H will be adjacent

in this graph if and only if H g₂−1g1H⊆ HSH.

The group G will act as a group of automorphisms of Γ(G, H, S) by left

multipli-cation: for any edge (g₁H, g₂H), g(g₁H, g₂H) = (g g1H, g g2H) is also an edge, since

H(g₂−1g−1)(g g₁)H = H g₂−1g₁H _{⊆ HSH. This action is transitive: for any g}₁H, g₂H,

(g₂g−1₁ )g₁H = g₂H. The stabilizer of the vertex H, is the subgroup H.

Later on, we will define such graphs for finite groups of Lie type, where S will be a 1-set, and H will be a parabolic subgroup.

1.1.1 Regular partitions

DEFINITION 1.1.1.1. For a graph Γ = (V, E), we say a partition P of V is regular (or

equitable) if for any Vi, Vjin P, and any vertexν∈ Vi, ei, j, the number of neighbors ofν

in Vjis independent of the choice ofν.

Suppose A is a symmetric real matrix, whose rows and columns are indexed by a

set V = _{{1, 2, . . . , n}. Let P be a partition of V with parts V}_i for 1_{≤ i ≤ m. Let A be}

partitioned according to P into block matrices, that is:

A =     A1,1 · · · A1,m .. . . .. ... Am,1 · · · Am,m     ,

where Ai, jis the submatrix of A formed by rows in Viand columns in Vj.

The characteristic matrix S is the n_{× m matrix, whose j’th column is the}

character-istic vector of V_j for j = 1, . . . , m. Define n_i=_|V_i_{|, K =diag(n}₁, . . . , n_m). Let b_i_{, j} denote

the average row sum of A_i_{, j}. Then the matrix B = (b_i_{, j}) is called the quotient matrix and

we have

K B = S⊤AS, S⊤S = K.

Let A be the adjacency matrix of a graph Γ with the set V of vertices where|V | = n.

(24)

1.1 GRAPHS 7

DEFINITION1.1.1.2. [21] The distribution diagram of Γ with respect to a regular

parti-tion P consists of a number of balloons b_V_i, one for each V_i ∈ P, and a number of lines

lViVj joining the two balloons, bVi and bVj one for each pair{Vi, Vj} for which ei, j 6= 0.

Lines l_V_i_V_i are not drawn. In the balloon b_V_i,_|V_i_{| is written, and at the V}_i end of l_V_i_V_j, e_i_{, j}

is written. The number e_V_i_V_i is written next to b_V_i.

EXAMPLE1.1.1.3. When a graph is distance regular, the distribution diagram where the

partition P is based on the distance to a fixed vertex, is called a distance distribution

diagram.

0 2 1 2 1 1 3 6

FIGURE1.2: the distance distribution diagram of the Petersen graph

When the partition is regular, the row sum of the matrix A_i_{, j}is constant, and is equal

to e_i_{, j}for all i, j_{∈ {1, . . . , m}. The matrix (e}_i_{, j})_V_i_,V_j_∈Pis the quotient matrix of Γ.

Then, A_i_{, j}1 = e_i_{, j}1 for i, j = 0, . . . , d, where 1 is the all 1’s vector. Thus,

AS = SB which leads to the following well known result:

LEMMA1.1.2. If for a regular partition P, the vector v is an eigenvector of B, for an

eigen-valueλ, then S v is an eigenvector of A for the same eigenvalue λ.

Proof. Bv =λv implies AS v = SBv = λS v. For a distribution diagram, one can consider the balloons as vertices and lines as edges. The graph obtained is called a distribution graph.

For more details, see [61], [24].

DEFINITION1.1.1.4. Given a coset graph Γ = (G, H, r), the associated double coset graph

∆ = DC(G, H, r) is the distribution graph of Γ with respect to the partition_{{H gH|g ∈ G}}

of the vertex set of Γ.

The graph ∆ has a regular partition. The vertex set of ∆ is H\G/H = {H gH | g ∈ G}

and two vertices H g1H and H g2Hare adjacent if and only if H g1H⊆ H g2H r H.

The vertex H r H is the unique neighbor of H.

PROPOSITION 1.1.3. [21] Let Γ = (G, H, r) be a coset graph of G on H, and let ∆ =

DC(G, H, r) be the associated double coset graph of G on H. Then, for arbitrary g1, g2

in G:

1. The cosets g₁H and g₂H are at distance i in Γ if and only if H and H g−1₂ g₁H have distance i in ∆.

2. The graph Γ is connected if and only if ∆ is connected; in this case, Γ and ∆ have the same diameter.

3. G in its action by left multiplication acts distance-transitively on Γ if and only if ∆ is a walk, possibly with loops added.

(25)

1.2 Field with one element and q-analogue

Let q be a prime power pt for some prime p. We denote the finite field of order q by

F_q. A field with one element does not exist, since in a field, the identity of multiplication

and addition should be distinct. However, the suggestive name field with one element is used for a mathematical object, which behaves like a field, and has characteristic 1. It

is denoted byF₁. In 1956, Jacques Tits suggested studying the mathematics ofF₁[95].

DEFINITION 1.2.0.5. A q-analogue of a mathematical expression E is an expression Eq

which is parameterized by a variable q, and whose limit as q approaches to 1 is_E.

EXAMPLE1.2.0.6. Let [n]_qbe1−q n 1−q. Since lim q→1 1− qn 1− q = n,

[n]qcan be regarded as a q-analogue of the number n. For an n-dimensional vector space

overFq, the number of projective points (one-dimensional subspaces) in it is 1−q

n

1−q. We

consider it as a q-analogue of the set of size n.

EXAMPLE1.2.0.7. There is a q-analogue of n! (n-factorial) which we will call [n]|_q. Since

lim_q_→1[1]_q[2]_q[3]_q. . . [n]_q= n! we can define this q-analogue of n! as follows:

[n]|_q:= [1]q[2]q[3]q. . . [n]q.

Conversely, one can consider the combinatorial case of q = 1 as a limit of q-analogs

as q_{→ 1.}

1.3 Groups

A group G is called a simple group if its only normal subgroups are the trivial group

and the group itself, where a subgroup H of G is a normal subgroup if for any g_{∈ G,}

its left coset gN and right coset N g are equal as sets. Finite simple groups are the building blocks of all finite groups. By Jordan-Hölder Theorem, any finite group G can be broken down into uniquely determined simple components which are the factors of a composition series for G [70], [73], [74].

Since, the groups of our interest for this thesis are examples of those, we will give the following theorem, which gives the complete classification of finite simple groups.

THEOREM1.3.1. [4] Every finite simple group is one of 26 sporadic simple groups or is an

example (up to isomorphism) of at least one of the following four categories: • cyclic groups of prime order,

• alternating groups of degree at least 5,

• simple groups of Lie type, including both the classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field, and the exceptional and twisted groups of Lie type (including the Tits group).

In this thesis we will mostly be concerned with the simple groups of Lie type - see also Section 1.3.3.

(26)

1.3 GROUPS 9

1.3.1 Coxeter groups

DEFINITION1.3.1.1. A Coxeter group is a group W together with a set R of generators

of W, such that a presentation of W is given by:

〈r ∈ R | r2_{= 1, (rs)}mrs_{= 1 for all r, s}∈ R 〉

where mrs is the order of rs in W . The matrix M :={mi j}1≤i, j≤nis called the Coxeter

matrix. The pair (W, R) is also called a Coxeter system.

DEFINITION 1.3.1.2. The Coxeter-Dynkin diagram (Coxeter diagram) of a Coxeter

sys-tem (W, R) is a labeled graph with vertex set R, where two vertices r, s are joined by an

edge labeled mrs. Edges labeled 2 are omitted. Labels 3 are omitted, and when mi j= 4,

there is a double edge.

This diagram determines W uniquely.

A Coxeter group is of spherical type if the group is finite. The possible connected components of the Coxeter diagrams of Coxeter groups of spherical type are given in Fig-ure 1.3. A Coxeter system is irreducible if its diagram is connected. The Coxeter group of a disconnected diagram is the direct product of the Coxeter groups of its components.

EXAMPLE 1.3.1.3. The Dihedral group D_2m is an example of the Coxeter groups. Its

diagram is called I₂(m). Its Coxeter group presentation is the following:

D2m:=〈s, t|s2= t2= (st)m= 1〉.

DEFINITION1.3.1.4. For a Coxeter system (W, R) for any element w∈ W, the length of w,

denoted by l(w) is the length of the shortest expression of w as a product of factors in R. In case W is finite, which is the case of our interest, it has a unique longest element

w0. This element satisfies l(w0w) = l(ww0) = l(w0)− l(w) for w ∈ W ([18], Chapter IV,

Exercise 22). It follows that w₀has order (at most) 2 and that conjugation by w₀induces

a graph automorphism on the Coxeter diagram.

The following theorem classifies all finite Coxeter groups.

THEOREM 1.3.2. [48] Finite Coxeter groups are those whose diagrams are given by the

disjoint unions of the diagrams of An, Cn, Dn, E6, E7, E8, F4, H3, H4, I (m)

2 .

Root systems

Let the Euclidean inner product be denoted by (_{·, ·).}

DEFINITION 1.3.1.5. A reduced root system is a finite collection Φ of non-zero vectors

spanningRl_{for some l}_{≥ 1 such that:}

i. ifα∈ Φ then Rα ∩ Φ = {α, −α}, ii. ifα, β∈ Φ then 2(β,α) (α,α) ∈ Z, and iii. ifα, β∈ Φ then wα(β ) := β− 2(β,α) (α,α)α∈ Φ.

(27)

FIGURE1.3: the diagrams of the irreducible finite Coxeter systems

Here, wα’s are the reflections in the plane perpendicular toα.

DEFINITION1.3.1.6. A subcollection ∆ of a root system Φ is called a fundamental system

of roots when it has the properties:

i . the collection ∆ is a basis ofRl_,

ii . each root when written as a linear combination of vectors in ∆, has either only nonnegative or nonpositive coefficients.

Each root system has a fundamental system of roots I . Given a root system Φ one can

choose a set of positive roots. This is a subset Φ+of Φ such that:

i . for each rootα∈ Φ exactly one of the roots α, −α is contained in Φ+_,

ii . for any two distinct rootsα, β∈ Φ+_{such that}_{α + β is a root, α + β is a root in Φ}+_.

(28)

1.3 GROUPS 11

a fundamental system I , one has Φ+, the set of roots with positive coefficients, given Φ+,

one has I , the set whose elements cannot be written as the sum of two elements of Φ+.

Let R ={wα|α ∈ Φ+}. The group W := 〈wα|α ∈ Φ〉 is called the Weyl group of the root

system and (W, R) is a Coxeter system [21] (p. 310). For a subset J of a fundamental

system of roots, letR_J be the space spanned by J , ΦJ= Φ∩ RJ, and WJ be the subgroup

of W generated by the fundamental reflections wr for r∈ J.

PROPOSITION 1.3.3. [36] The set Φ_J is a system of roots inR_J, J is a fundamental system

in ΦJ. The Weyl group of Φ is WJ.

DEFINITION1.3.1.7. The subgroups W_J and their conjugates are called the parabolic

sub-groups of W .

Most of the finite Coxeter systems can be described in terms of their root systems.

The exceptions are the groups called H₃and H₄, and the family I₂(m) for m_{6= 2, 3, 4, 6}

[21], (p. 310).

EXAMPLE1.3.1.8. For the diagram A₂, we can name the positive roots as r, s, r + s and

the negative roots as−r, −s, −rs.

The corresponding Weyl group is: {1, wr, ws, wrws, wswr, wrwswr = wswrws} =

Sym(3), 1 = 1, wr = (12), ws = (23), wrws = (132), wswr, = (123), wrwswr =

(12)(23)(12) = (13).

In Figure 1.4, the generators are the reflections that are perpendicular to r and s.

wr

ws

FIGURE1.4: for A₂, positive roots r, s, and r + s

In Chapter 7, we will define root system graphs and work on some examples of those.

1.3.2 Groups with a Tits system

DEFINITION1.3.2.1. A group G is called a group with a Tits system (B, N , W, R) , if

i . the Group G has fixed subgroups B and N that generate it,

ii . the group H := B∩ N is normal in N, W := N/H,

(29)

iv . if n_i ∈ N maps to w_i under natural homomorphism of N into W , and if n is any element of N , then

BniB.BnB⊆ BninB∪ BnB,

v . if niis as above, then niBni6= B.

The group W is called the Coxeter group of the Tits system.

The Coxeter diagram of G is the Coxeter diagram of the Coxeter group (W, R). If for

a Tits system, the related Coxeter system has diagram Xn, the Tits system is said to be of

type Xn. The rank of G is the size of R.

All Chevalley groups are groups with Tits systems (see [36] Proposition 8.2.1, Theo-rem 13.5.4.), and every finite group with a Tits system of rank at least 3 is a Chevalley group [97].

The subgroup B is sometimes called the Borel subgroup, H is sometimes called the

Cartan subgroup, and W is called the Weyl group.

The following proposition describes the subgroups of G.

PROPOSITION 1.3.4. [36] Let G be a group with a (B, N )-pair. Then for each subset J of I ,

let N_J be the subgroup of N that is mapped to W_J under the natural homomorphism. Then,

BN_JB is a subgroup of G.

The following proposition describes the relation between the subgroups of G and the elements of W .

PROPOSITION 1.3.5. [36] Let G be a group with a (B, N )-pair. Let n, n′ be elements of N .

Then BnB = Bn′B if and only if n, n′ are mapped to the same element of W under natural homomorphism from N into W . Thus, there is a one to one correspondence between the double cosets of B in G and the elements of W .

From now on, we use the notation BwB for any double coset BnB where n∈ N, when

wis the image of n under the homomorphism described above.

THEOREM1.3.6. [21] [36] Let G be a group with Tits system (B, N , W, R) and let I , J⊆ R.

Then:

i . the pair (W, R) is a Coxeter system, ii . G = BN B,

iii . the set GJ = BWJB is a subgroup of G. Conversely, any subgroup of G containing B is of this form,

iv . each subgroup G_J is equal to its normalizer, hence, distinct subgroups G_I, G_J can not be conjugate in G,

v . the subgroup GJ for distinct subsets J of I are all distinct, furthermore, GI∩GJ= GI∩J. Thus, the subgroups GJ form a lattice isomorphic to the lattice of all subsets of I , vi . the map W_IwW_J 7→ G_IwG_J is a bijection from W_I\W/W_Jonto G_I\G/G_J, in particular,

if for w, w′∈ W, we have BwB = Bw′B, then w = w′,

(30)

1.3 GROUPS 13

viii . if g B g−1⊆ G_I, then g∈ G_I (g∈ G), ix . GIwGJ = BWIwWJB (w∈ W).

REMARK1.3.7. [21] W = N/(B∩ N), and R is uniquely determined by R = {r ∈ W\{1} |

B∪ BrB is a group} so that, instead of giving the Tits system (B, N, W, R), it suffices to

give the (B, N )-pair (B, N ). The groups with Tits systems are also called groups with (B, N )-pairs.

Let G be a group with a (B, N )-pair and Weyl group W . The Bruhat decomposition of G:

G =[· BwB

is a decomposition of G as a disjoint union double cosets of the form BwB, where w is in

W (see [18], Chapter IV).

A subgroup that contains B is called a standard parabolic subgroup, that is, it is of

the form G_I for some I_{⊂ R. There is a bijection between the set of subsets of R, and the}

set of standard parabolic subgroups of G. Hence, for a Chevalley group with diagram X_n,

there are 2n_{such standard parabolic subgroups. Any conjugate of a standard parabolic}

subgroup is called a parabolic subgroup.

1.3.3 Finite simple groups of Lie type

Finite simple groups of Lie type are subgroups of the group of invertible matrices over a

finite field namely G L(n, q), hence G_{≤ GL(n, q) for some n ≥ 2 and prime power q.}

They are classified as the classical groups (projective special linear groups,

or-thogonal groups, symplectic groups, unitary groups), and exceptional groups and

twisted groups. The non-twisted groups are also called Chevalley groups.

Projective special linear groups have diagram A_n, orthogonal groups O_2n+(q) have

di-agram D_n, O_2n+1(q) have diagram B_n, symplectic groups ) S p_n(q) have diagram C_n, and

the exceptional groups E6, E7, E8, F4, G2are named after their diagrams.

The groups2A_n,2D_n,2E6,3D4, are called Steinberg groups. The groups2B2(22n+1),

2_F

4(22n+1), and2G2(32n+1) are known as Suzuki-Ree groups. Together, these are called

twisted Chevalley groups. The group2F4(2) is not simple. Its derived group2F4(2)′is

known as the Tits group.

History

For more details about the history of the classification of finite simple groups, see[90]. Our aim here, is to have a short look at the story behind the construction of the mathe-matical objects of our interest, namely, buildings. Groups with (B, N )-pairs are groups of automorphisms of these geometries.

Since finite simple groups are the buildings blocks of finite groups, understanding and classifying them is a good start for understanding all finite groups.

Let us have a brief look at the time line. In 1832 Galois introduced the concept of

a normal subgroup and found the simple groups An(n = 5) and PS L2(Fp), (p≥ 5). In

1861 Mathieu found the first two Mathieu groups M11, M12, the first sporadic simple

groups. In 1870, Jordan began to build a database for finite simple groups. He listed the alternating and projective special groups as simple groups. In 1873 Mathieu found three

(31)

more Mathieu groups M22, M23, M24. In 1892, Hölder underlined the problem of giving

an overview of all finite simple groups. In the same year, Cole determined all finite simple groups up to order 500 except a few cases. In 1905, Dickson introduced the simple groups

of type G2 over finite fields. Chevalley introduced Chevalley groups in 1955. Steinberg

in 1959, Suzuki in 1960, Ree in 1961 introduced, respectively, Steinberg, Suzuki and Ree groups. Tits introduced (B, N )-pairs for groups of Lie type and found the Tits group in

1964. Janko constructed the Janko group J1in 1965, which is a sporadic group. In 1968,

Higman and Sims constructed the Higman-Sims group. Same year, Conway discovered

the three Conway groups. In 1969, the Suzuki sporadic group, the Janko group J₂, the

Janko group J₃, the McLaughlin group, and the Held group were found. Fischer found the

three Fischer groups in 1971. In 1972, Lyons found the Lyons group and in 1973 Rudvalis found the Rudvalis group. In 1973, Fisher discovered the baby monster group and used this with Griess to discover the Monster group. As a result, Thompson discovered the Thompson sporadic group and Norton discovered the Harada-Norton group. In 1974, Tits showed that groups with (B, N )-pairs of rank at least 3 are groups of Lie type. In 1976, O’Nan introduced the O’Nan group, and Janko introduced the last sporadic group,

the Janko group J4. Frobenius, Dedekind, Burnside, Moore, Disckson, Killing, Cartan,

Hall, Schur, Zassenhaus, Miller, Fitting, Grün, Wielandt, Suzuki, Chevalley, Steinberg, Ree, Gorenstein, Lyons, Aschbacher, Smith, Solomon are some of the many people who contributed to the process of classification and construction of finite simple groups.

The classification process mostly took place in between 1955 and 1983. The proof is believed to be finished as of 2004 [3], and from then on, people focused more on understanding and improving the proof.

1.4 Geometric objects

In this thesis, we deal with some graphs related to finite simple groups of Lie type. To understand these groups better, Tits [97] came up with some geometrical objects called buildings. Buildings will be defined in this section. We will define some other geomet-rical objects in this section to be able to talk about buildings. Groups with (B, N )-pairs are groups of automorphisms of buildings. Similarly, Weyl groups are groups of auto-morphisms of subcomplexes called apartments. See [97], or [36](Chapter 15), for more detail. The content of the following three sections is based on this chapter.

For finite geometries in general, we refer to [33].

1.5 Buekenhout-Tits geometries

Most of the definitions you will see in this section will be used in Chapter 7. Much more about incidence geometries can be found at [30].

DEFINITION1.5.0.1. A Buekenhout-Tits geometry (or an incidence geometry) Γ(X ,∗, t)

with type set I is a set X of objects together with a map t : X _{→ I that assigns a type}

to each object, and a symmetric relation∗ called incidence such that two objects of the

same type are never incident. A geometry can be considered as a|I|-partite graph Γ over

X, where the incident pairs are adjacent.

(32)

1.5 BUEKENHOUT-TITS GEOMETRIES 15

DEFINITION1.5.0.2. A flag is a subset of X where any pair of elements are incident. For a

flag F , the image J of F under t is the type of F . A chamber is a maximal flag of type I . Γ is thick if every non-maximal flag is contained in at least 3 chambers of Γ. The

rank of a geometry is the size of the type set I .

For any i_{∈ I, when the number of chambers in Γ containing a given flag of type I\{i}}

is 2, the geometry is thin.

A morphism from Γ to Γ′ is a map α from X to X′, for which incidence of a pair

of elements implies the incidence of their images, moreover when a pair of elements are of the same type, so are their images. An isomorphism of two Buekenhout-Tits geometries is a bijection on their sets of objects, which preserves incidence and types. An

automorphism is an isomorphism on the geometry.

The rank (corank) of a flag F is|F| (|I\t(F)|).

DEFINITION1.5.0.3. If F is a flag of Γ, then the residue ΓF of F in Γ is the geometry over

I\t(F), whose elements are all elements which are incident to all elements of the flag F.

In ΓF, the incidence relation and the type map are inherited from the geometry Γ(X ,∗, t).

The incidence graph of Γ(X ,∗, t) is an I-partite graph on X , where two vertices are

adjacent when they are incident.

A geometry is connected when its incidence graph is connected. A geometry is

residually connected when the residue of every flag of co-rank at least 2 is connected,

and residue of every flag of corank 1 is non-empty.

For an element x of X , the set consisting of x and all the elements incident with x is

denoted by x∗. For a subset Y of X , Y∗=∩_x_∈Yx∗. Using this notation, the set of objects

of ΓF is F∗\F.

DEFINITION1.5.0.4. If F is a flag of Γ(X ,∗, t), and J is a subset of I, the J-shadow Sh_J(F )

(on P) is t−1(J )∩ F∗, i.e. the set of all J -flags that are incident with F . The J -space is

the set of all J -flags equipped with all J -shadows of all possible flags in Γ.

Point-line geometries and chamber complexes are examples of Buekenhout-tits ge-ometries.

1.5.1 Point-line geometries

DEFINITION1.5.1.1. A point-line geometry (P, L,∗) is an incidence system (or geometry)

of rank two, consisting of a set of points denoted by P, a set of lines denoted by L and an

incidence relation∗ ⊆ (P × L)∪(L×P) which is symmetric, (so if a point is on a line, then

the line is passing through this point and vice versa) and moreover, any pair of points in

Plie together on at most one line of the geometry. An incidence system is linear if any

pair of distinct points in the system are on exactly one line.

If two points are on the same line, they are collinear, if two lines have a point in common, they intersect, if not, they are skew.

By definition, the length of shortest cycle (girth) of the incidence graph of a linear point-line geometry can not be 4. So it is at least 6.

(33)

DEFINITION 1.5.1.2. A geometry of rank two with types called points and lines is called

a generalized m-gon for m ≥ 2 if the incidence graph has diameter m and girth 2m.

A generalized polygon is called regular of order (s, t) for s, t cardinal numbers finite or infinite if for any point, there are t + 1 lines passing through it and for any line, there are

s +1 points on it.

THEOREM1.5.1. [55] For a regular generalized m-gon, for s, t > 1, both finite, m∈ {2, 3,

4, 6, 8}.

Usually, lines will have at least two points, and in such a situation the lines can be identified with the set of points that are incident with them, and considered as subsets of

P. A line is called thick if there are at least three points on it, a line is called thin if there

are exactly two points on it.

DEFINITION1.5.1.3. For a point-line geometry, the collinearity graph is the graph whose

vertices are the points of the geometry. In this graph any two vertices are adjacent if and only if the corresponding pair of points is collinear.

1.5.2 Chamber complexes and foldings

We take a set Θ with a partial order⊆ on it.

DEFINITION1.5.2.1. The set Θ is a simplex if it is isomorphic to the set of all subsets of

some set, partially ordered by inclusion.

For n (0 _{≤ n ≤ ∞), take a (n + 1)-set S. Call the collection of all its subsets S .}

Elements of S are the vertices, ordered pairs are edges, and subsets of S are the faces

of the n-dimensional simplex (or n-simplex )_{S . Any set that is isomorphic to a power}

set of an (n + 1)-set with a partial order_{⊆ defined on them is called an n-dimensional}

simplex.

FIGURE1.5: a simplicial 3-complex

DEFINITION1.5.2.2. The set Θ is called a (simplicial) complex if for any A∈ Θ, the set of

elements B such that B_{⊆ A forms a simplex, and for any A, B in Θ, they have a greatest}

lower bound A_{∩ B.}

(34)

DEFINITION 1.5.2.3. For any A∈ Θ, its rank is the number of elements B minimal with

respect to properties B⊆ A, B 6= 0.

Thus, the set of elements B ⊆ A is isomorphic to the set of subsets of a set with

cardinality rank A. The rank of A is denoted by rk A.

For a simplex A in Θ, the set of simplices containing A is called star of A, S t(A). Star of A is a complex itself with the order induced from the original complex, but not a subcomplex of it, unless A is 0. For any element B in S t(A), its rank there is the

codimension of A in B, which is denoted by cod im_BA.

DEFINITION1.5.2.4. A complex is called a chamber complex if every element is contained

in an element of maximal dimension called a chamber, and if given two chambers C, C′,

there exists a sequence of chambers C = C₀, C₁, . . . , C_m= C such that

cod imCi−1(Ci−1∩ Ci) = cod imCi(Ci−1∩ Ci)≤ 1 for all i = 1, 2, . . . , m.

Such a sequence is called a gallery. Two chambers C, C′ are adjacent if cod im(C_∩

C′) = 1. So, a gallery is a sequence of chambers where any pair of consecutive chambers

are either adjacent or identical.

DEFINITION1.5.2.5. In a chamber complex, a set L of chambers is called convex , if every

minimal gallery whose extremities belong to L has all its terms in L.

DEFINITION1.5.2.6. A chamber complex is called thin when each simplex of codimension

1 in any chamber is contained in exactly two chambers. A chamber complex is called

thick when each simplex of codimension 1 in any chamber is contained in at least three

chambers.

Let Θ′be a chamber complex.

DEFINITION1.5.2.7. A mapα : Θ→ Θ′ is a morphism of chamber complexes ifα(C) is

a chamber of Θ′for any chamber of C of Θ, and for each chamber C of Θ,α induces an

isomorphism on the simplices defined by C,α(C).

LEMMA1.5.2. [36] Let Θ, Θ′be two chamber complexes in which each element of

codimen-sion 1 is contained in at most 2 chambers, and letα, β be two morphisms of Θ into Θ′ in-jective on the set of chambers. Suppose there exists a chamber C_{∈ Θ such that α(A) = β(A)} for all A_{⊆ C. Then α = β.}

DEFINITION1.5.2.8. For a thin chamber complex Θ, an endomorphismα on it is called a

folding if it satisfies:

i . α2₌_α,

ii . Each chamber inα(Θ) is the image of exactly two chambers of Θ.

DEFINITION 1.5.2.9. A set of chambers in a chamber complex is called convex if every

(35)

PROPOSITION 1.5.3. [36] Letα be a folding of Θ and C, C′be a pair of adjacent chambers of Θ such thatα(C′) = C. Suppose there is a foldingβ such that β (C) = C′. Then,β has the same property for any other adjacent chambers of this type, if D, D′ are adjacent and

α(D′) = D, thenβ (D) = D′.

The folding β of 1.5.3, which, if exists, is determined uniquely by α, is called the

opposite folding ofα.

PROPOSITION 1.5.4. [36] Let α, β be opposite foldings of Θ. Then there exists an

auto-morphism ρ of Θ which coincides with β on α(Θ) and with α on β (Θ). Also ρ2 _{is the}

identity.

Abstract Coxeter complexes

DEFINITION 1.5.2.10. An abstract Coxeter complex is a thin chamber complex Σ such

that given any pair C, C′of adjacent chambers, there exists a foldingα of Σ with α(C′) =

C.

A retraction will be called a retraction.

PROPOSITION 1.5.5. [36] Let Σ be an abstract Coxeter complex, C a chamber in Σ, and

S(C) the simplex of all faces of C. Then there exists a unique idempotent morphism of Σ onto S(C).

Let us call this mapρC, the retraction of Σ onto the simplex of faces of C.

The following relations is an equivalence relation: given A, B_{∈ Σ, we write A s B if}

ρC(A) =ρC(B).

LEMMA1.5.6. [36] The equivalence relation defined on Σ is independent of the chamber.

The equivalent elements of Σ have the same type. Each chamber has one face of each type.

LEMMA1.5.7. [36] Let Σ be an abstract Coxeter complex and let C be a chamber of Σ. Let

γ be an endomorphism of Σ leaving invariant the type of each face of C. Then, γ preserves the type of each element of Σ.

Endomorphisms and automorphisms of the kind discussed in 1.5.7 are called

type-preserving.

LEMMA1.5.8. [36] The automorphismsρ defined in 1.5.4 are type preserving.

Let W (Σ) be the group generated by all the automorphisms of Σ of the kind described in 1.5.4.

PROPOSITION1.5.9. [36] Let Σ be an abstract Coxeter complex. Then, W (Σ) is the group of

all type-preserving automorphisms of Σ.

THEOREM1.5.10. [36] Let Σ be an abstract Coxeter complex and let C be a chamber of Σ.

Then the reflections in the faces of codimension 1 in C generate the group W (Σ). Moreover, W (Σ) is a Coxeter group with respect to these generators.

(36)

1.5.3 Buildings

Jacques Tits introduced the geometries called buildings as simplicial complexes with a family of subcomplexes.

DEFINITION1.5.3.1. A building is a pair Ω,A where Ω is a chamber complex, and A is a

set of subcomplexes, called apartments, satisfying the following axioms: i . The chamber complex Ω is thick.

ii . The apartments of Ω are thin chamber complexes.

iii . Given any two chambers C, C′ in Ω, there exists an apartment Σ_{∈ A such that}

C_{∈ Σ and C}′_{∈ Σ.}

iv . If A, A′ are elements of Ω which are contained in each of the apartments Σ, Σ′ _∈

A , there exists an isomorphism between Σ, Σ′_{leaving invariant A, A}′_{and all their}

faces.

It follows that any two apartments of a building Ω are isomorphic. Let G be a group with a (B, N )-pair as described in Section 1.3.2.

EXAMPLE1.5.3.2. [36] Let Ω be the set of left cosets gG_J for all g∈ G, and all subsets J

of I . Let Σ₀be the subset of Ω consisting of all elements nG_J for all n_{∈ N and all J ⊂ I.}

Then gΣ₀ is the set of all cosets gnG_J for all n_{∈ N and all J. Let A be the family of}

subsets gΣ₀of Ω for all g_{∈ G. Then, Ω is a building and, A is a set of apartments in Ω.}

The building constructed in this way will be called Ω(G; B, N ).

We pick a building, an apartment of it, and a chamber of that apartment. We call

them respectively Ω, Σ, and C. For each element A∈ Ω there exists an apartment Σ′

containing A and C, by axiom (iii) in Definition 1.5.3.1. By (i v), there is an

isomor-phism Σ′→ Σ which leaves invariant all faces of C. By Lemma 1.5.2, there is only one

such isomorphism. The image of A under this isomorphism is an element of Σ, which is

independent of the choice of Σ′, by (i v). This image will be called retΣ,C(A).

LEMMA1.5.11. [36] The map

Ω_{→ Σ,}

A→ retrσ,C(A)

is a retraction from Ω onto Σ.

THEOREM1.5.12. [36] The apartments of a building are abstract simplicial complexes.

DEFINITION1.5.3.3. Let A, A′be two elements of a building Ω. Then A, A′are said to have

the same type in Ω if they have the same type in any apartment containing A, A′.

The following theorems are converses of each other, relating Ω(G; B, N ) and G.

THEOREM1.5.13. [36] In the building Ω(G; B, N ) the group G operates by left

multiplica-tion as a group of type preserving automorphisms, which is transitive on the pairs (C, Σ) where C is a chamber and Σ is an apartment containing C.

(37)

THEOREM1.5.14. [36] Let (Ω,A ) be a building and G a group of type preserving automor-phisms of Ω which is transitive on the pairs (C, Σ), with C∈ Σ, where C is a chamber and

Σ is an apartment of Ω. Let C0, Σ0be a fixed chamber and apartment with C0∈ Σ0, let B

be the stabilizer of C0 in G and N be the stabilizer of Σ0. Then the subgroups B, N form

a (B, N )-pair in G. Moreover, W = N/(B∩ N) is isomorphic to the group W(Σ) of type

preserving automorphisms of each apartment Σ of Ω.

A building is called spherical if its apartments are finite. Buildings of spherical type are the ones of our interest.

The type of the building is that of the Coxeter group. The types of spherical buildings

are the types of finite irreducible Coxeter groups. The thick buildings of type A_nare the

projective spaces. The buildings of type Bn, Cnand Dnare polar spaces.

Tits proved that a finite building whose associated Coxeter group is an indecompos-able Weyl group of rank at least 3 must be a building Ω(G; B, N ) where G is a finite Chevalley group of twisted group, and deduced that the only finite simple groups with a (B, N )-pair of rank at least 3 are the finite Chevalley groups and twisted groups which have this property [97].

In his famous lecture notes, Tits [97] phrased these under two main problems: “(A) Determination of the buildings of rank _{≥ 3 and irreducible, spherical type, other} than H₃and H₄. Roughly speaking, those buildings all turn out to be associated to simple algebraic or classical groups. An easy application provides the enumeration of all finite groups with (B, N )-pairs of irreducible type and rank≥ 3 up to normal subgroups contained in B.

(B) Determination of all isomorphisms between buildings of rank_{≥ 3 and spherical type,}

associated with algebraic or classical simple groups, and in particular, the determination of the full automorphism group of these buildings."

More information about buildings can be found at, [97], [33], [36], [27], [86], and [103].

Buekenhout-Tits diagrams

DEFINITION1.5.3.4. A diagram for a geometry Γ is a labeled complete graphD on the

type set, where each label _D_{i j} is a class of rank 2 geometries. The graph_{D is called a}

Buekenhout-Tits diagram X_nof the geometry Γ (with rank n) when for any flag of type

I_{\{i, j}, the residue Γ}_F belongs to the class of geometries_D_{i j}. Sometimes, Γ is called a

geometry of type X_n.

Our classes of geometries will be self-dual, so that we need not worry about directing

the edges of D. Indeed, each class Di j will be the class of all generalized m-gons, for

some fixed m = mi j. Now the edge i j will be labeled by the number m.

Conventionally, edges labeled 2 are omitted. In that case, every i object is incident with every j object. The label 3 is omitted. Here, the i and j objects form the points and lines of a projective plane. Instead of a label 4 one draws a double edge. Here, the i and

jobjects form the points and lines of a generalized quadrangle.

EXAMPLE1.5.3.5. The geometry of 8 corners, 12 edges and 6 faces of a cube satisfies the

(38)

1 2 3

FIGURE1.6: the diagram of the cube

Diagram Xn,i is diagram Xn with type i circled, i.e., selected as the point type, and

here n indicates the rank, i.e. the number of nodes.

For each choice of a type i₀in I , a geometry Γ(X ,_{∗, t) with type set I gives a point-line}

geometry where P = t−1(i₀) is the set of objects of type i₀, and L is the set of i₀-shadows

of flags which are of cotype i₀in Γ. When we say l is a line, we consider it as the set of

points that it is incident with it. For instance, consider the diagram of the cube (Figure

1.6 ). Let the set of points be t−1(1). The lines of the geometry are the 1-shadows of

{2, 3}-flags. Here a line is a set of two points.

DEFINITION 1.5.3.6. The point-line geometry obtained from a geometry of type X_n by

choosing type i as the point type described as above is called the incidence system of

type X_n,i, and this choice is indicated in the diagram by circling the node representing

the point type. The incidence system of type X_n,i, associated with a thick building of type

Xnis called a Lie incidence system of type Xn,i (or Xn,i(K) where K is the underlying

field).

1.5.4 Projective spaces

Here we introduce projective geometries. They are examples of incidence geometries. More detailed information about these objects can be found in [32] and in [68].

Let V be an (n + 1)-dimensional left vector space over a fieldF.

DEFINITION1.5.4.1. A projective space P G(n,F) is an incidence geometry (X ,_{∗, t) with}

type set I of rank n. The set X consist of subspaces of V except the trivial and full

subspace,_{∗ is symmetrized strict inclusion, I = {1, 2, . . . , n} and t maps every element of}

X to its vector space dimension.

We are interested in finite projective spaces, overF_q. For any finite prime power q,

for the unique fieldF_qwe denote the corresponding projective space by P G(n, q).

Rank 1 or vector space dimension 1 subspaces are called points, and vector space dimension 2 subspaces are called lines. Projective dimension is one less than vector space dimension.

The vectors of V (n + 1,F)_{\{0} are called equivalent if and only if x = k y for some}

k ∈ F\{0}. So the point set of PG(n, F) is the set of all equivalence classes under this

relation. Let P(x) be the equivalence class of a point x. Then, P(x1), . . . , P(xk) are

linearly independent in the projective space if the corresponding vectors are linearly

independent in the vector space. A subspace of P G(n,F) of dimension r is a set of all

points, whose corresponding vectors form a (r + 1)-dimensional subspace of V (n + 1, q) together with 0.

We will give an axiomatic definition of projective space and the projective planes, which are projective spaces of dimension 2, here.

A projective space is a point-line geometry S = (P, L,∗) that satisfies the following