Buildings and Kneser Graphs

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

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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

geboren te ˙Izmir, Turkije Dit proefschrift is goedgekeurd door de promotor:

prof.dr. A.E. Brouwer PREFACE

The following questions with their possible generalizations initiated years of mathematical 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 Erdos˝ [53], this question was answered by Erdos,˝ Ko, and Rado, already in 1938, but they first published this result in 1961 [54]: For 2k n, one must have n 1 ≤ C − , and if equality holds then C is the collection of all k-subsets containing some | |≤ k 1 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, Erdos,˝ 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 Erdos,˝ 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 series of subsets of a finite set of respective sizes k for 1 i l. In this chapter, i ≤ ≤ viii PREFACE 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 generalizations 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 Erdos-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 conjectured 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 examined. 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. CONTENTS

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 2.2.2 Results about the diameter ...... 37 x CONTENTS

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 PG(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 PG(2, q) ...... 78 5.4.2 PG(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 6.2 Main theorem ...... 88 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 PG(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, GS, r) and Γ(W, WS, 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 E7,1 ...... 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(A2d 1(q), d ) ...... 130 − { } Abstract 131

Acknowledgements 133

Curriculum Vitae 135

Bibliography 137

Index 144 xii CONTENTS LISTOF FIGURES

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 PG(2, q) 94

7.1 the diagrams E6,2, E7,1, E8,8, and F4,1 ...... 96 7.2 the diagram of D6 where node number 3 is circled ...... 99 7.3 distribution diagram of the E7,1(1) graph ...... 108 7.4 distribution diagram of the graph of Lie Type E7,1(q) ...... 111 7.5 distribution diagram of the E6,2(1) graph ...... 115 7.6 distribution diagram of the E8,8(1) graph ...... 120 7.7 distribution diagram of the F4,1(1) graph ...... 124 xiv LIST OF FIGURES LISTOF TABLES

2.1 diameters of generalized Kneser graphs ...... 37 2.2 maximum coclique sizes ...... 42

5.1 maximal coclique classiﬁcation for PG(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 YA for the graph of Coxeter type E7,1 110 7.4 the bound given by DLPB and actual YA for the graph of Coxeter type E6,2 116 7.5 the bound given by DLPB and actual YA for the graph of Coxeter type E8,8 121 7.6 the bound given by DLPB and actual YA for the graph of Coxeter type F4,1 125 xvi LIST OF TABLES PART I

INTRODUCTION

1 PRELIMINARIES

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 geometrical 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 algebraic graph theory, which is the study of graphs using algebraic 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 = u , u ,..., u = v such that for any k, 0 k i 1, 0 1 i ≤ ≤ − the vertex uk is adjacent to uk+1. A walk is a path in which vertices or edges may be repeated.

DEFINITION 1.1.0.2. 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 undirected.

DEFINITION 1.1.0.3. A (proper) vertex coloring of a graph Γ=(V, E) is a map ϕ : 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, ∈ then ϕ(u) = ϕ(v). 6 4 PRELIMINARIES

A proper vertex coloring of a graph Γ gives a partition of the set of vertices. For any 1 c C, we call ϕ− (c) a color class. ∈ DEFINITION 1.1.0.4. The chromatic number of a graph Γ is the smallest number of colors needed for a proper vertex coloring of Γ, and is denoted by χ(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 by α(Γ).

Each color class is a coclique.

REMARK 1.1.1. For Γ=(V, E), V α(Γ)χ(Γ). (1.1) | |≤ DEFINITION 1.1.0.6. 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 are λ vertices adjacent to both,

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

EXAMPLE 1.1.0.7. The Petersen graph is strongly regular with parameters (10, 3, 0, 1).

{1,2}

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

{1,4} {2,4}

{2,5} {1,3}

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

DEFINITION 1.1.0.8. 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.

For any ﬁxed vertex x Γ, let Γ (x) be the set of vertices that are at distance l to x. ∈ l 1.1 GRAPHS 5

DEFINITION 1.1.0.10. [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 ci 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 ai where a := k b c (i = 0, . . . , d) i − i − i is the number of neighbors of v in Γi(u) for d(u, v) = i, are called the intersection numbers of Γ. Clearly

b0 = k, bd = c0 = 0, c1 = 1.

The size of Γi (u) is denoted by ki . The valency of Γ is k = k1.

For a distance-regular graph with adjacency matrix A, the matrices Ai, where (Ai)u,v = 1 if d(u, v)= i and (Ai)u,v = 0 otherwise, satisfy the relations [21, p. 127] :

A0 = I, A1 = A, (1.2)

AAi = bi 1Ai 1 + aiAi + ci+1Ai+1, (1.3) − −

A + A + + A = J. (1.4) 0 1 · · · d

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.

DEFINITION 1.1.0.11. 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)(g g )x = g (g x) for all g , g in G, x X , 1 2 1 2 1 2 ∈ (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 = h, there exists an a in A such that ga = ha. ∈ 6 6 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 Gx = g x g G . The { | ∈ } stabilizer of x, which is denoted by G is the subset of G ﬁxing x, i.e. g g x = x . The x { | } orbits of G on X X are called orbitals. × DEFINITION 1.1.0.13. 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). 6 PRELIMINARIES

DEFINITION 1.1.0.14. A graph Γ is vertex-transitive if for any two vertices v1 and v2 of Γ, there exists σ Aut(Γ) such that σ(v )= v . ∈ 1 2

DEFINITION 1.1.0.15. A graph Γ is edge-transitive if for any two edges e1 and e2 of Γ, there exists σ Aut(Γ) such that σ(e )= e . ∈ 1 2

DEFINITION 1.1.0.16. A graph Γ is ﬂag-transitive if for any two edges e1 =(v1, u1) and e =(v , u ) of Γ, there exists σ Aut(Γ) such that σ(v )= u and σ(v )= u . 2 2 2 ∈ 1 1 2 2 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 exists σ Aut(Γ) such that σ(u) = ∈ u′, σ(v)= v′.

DEFINITION 1.1.0.18. Let G be a group, H be a subgroup of it, and S be any subset of G. 1 One can deﬁne a graph Γ(G, H, S) (when HSH = HS− H) called coset graph of G on H

(with respect to S) on the left cosets of H in G. Two cosets g1H and g2H will be adjacent 1 in this graph if and only if H g− g H HSH. 2 1 ⊆ The group G will act as a group of automorphisms of Γ(G, H, S) by left multiplication: for any edge (g1H, g2H), g(g1H, g2H)=(g g1H, g g2H) is also an edge, since 1 1 1 H(g2− g− )(g g1)H = H g2− g1H HSH. This action is transitive: for any g1H, g2H, 1 ⊆ (g2 g1− )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 V , V in P, and any vertex ν V , e , the number of neighbors of ν i j ∈ i i,j in Vj is independent of the choice of ν.

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 for 1 i m. Let A be { } i ≤ ≤ partitioned according to P into block matrices, that is:

A A 1,1 · · · 1,m  . . .  A = . .. . ,    Am,1 Am,m   · · ·  where Ai,j is the submatrix of A formed by rows in Vi and columns in Vj. The characteristic matrix S is the n m matrix, whose j’th column is the character- × istic vector of V for j = 1, . . . , m. Deﬁne n = V , K =diag(n ,..., n ). Let b denote j i | i | 1 m i,j the average row sum of Ai,j. Then the matrix B =(bi,j) is called the quotient matrix and we have KB = S⊤AS, S⊤S = K.

Let A be the adjacency matrix of a graph Γ with the set V of vertices where V = n. | | Let P, A and V be as deﬁned above for 1 i, j m. i,j i ≤ ≤ 1.1 GRAPHS 7

DEFINITION 1.1.1.2. [21] The distribution diagram of Γ with respect to a regular partition P consists of a number of balloons b , one for each V P, and a number of lines Vi i ∈ l joining the two balloons, b and b one for each pair V , V for which e = 0. Vi Vj Vi Vj { i j} i,j 6 Lines l are not drawn. In the balloon b , V is written, and at the V end of l , e Vi Vi Vi | i| i Vi Vj i,j is written. The number e is written next to b . Vi Vi Vi EXAMPLE 1.1.1.3. 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

FIGURE 1.2: the distance distribution diagram of the Petersen graph

When the partition is regular, the row sum of the matrix Ai,j is constant, and is equal to ei,j for all i, j 1, . . . , m . The matrix (ei,j)V ,V P is the quotient matrix of Γ. ∈{ } i j ∈ Then, Ai,j1 = ei,j1 for i, j = 0, . . . , d, where 1 is the all 1’s vector. Thus, AS = SB which leads to the following well known result:

LEMMA 1.1.2. If for a regular partition P, the vector v is an eigenvector of B, for an eigenvalue λ, then Sv is an eigenvector of A for the same eigenvalue λ. Proof. Bv = λv implies ASv = SBv = λSv. 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].

DEFINITION 1.1.1.4. 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 HgH g G { | ∈ } of the vertex set of Γ. The graph ∆ has a regular partition. The vertex set of ∆ is H G/H = HgH g G \ { | ∈ } and two vertices H g H and H g H are adjacent if and only if H g H H g HrH. 1 2 1 ⊆ 2 The vertex HrH 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 1. The cosets g1H and g2H are at distance i in Γ if and only if H and Hg2− 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. 8 PRELIMINARIES

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

Fq. 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 by F1. In 1956, Jacques Tits suggested studying the mathematics of F1 [95]. DEFINITION 1.2.0.5. A q-analogue of a mathematical expression is an expression E Eq which is parameterized by a variable q, and whose limit as q approaches to 1 is . E 1 qn EXAMPLE 1.2.0.6. Let [n] be − . Since q 1 q − 1 qn lim − = n, q 1 1 q → − [n]q can be regarded as a q-analogue of the number n. For an n-dimensional vector space 1 qn over F , the number of projective points (one-dimensional subspaces) in it is − . We q 1 q consider it as a q-analogue of the set of size n. −

EXAMPLE 1.2.0.7. There is a q-analogue of n! (n-factorial) which we will call [n]q| . Since limq 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.

THEOREM 1.3.1. [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. 1.3 GROUPS 9

1.3.1 Coxeter groups

DEFINITION 1.3.1.1. 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 n is 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 system (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 D2m is an example of the Coxeter groups. Its (m) diagram is called I2 . Its Coxeter group presentation is the following:

D := s, t s2 = t2 =(st)m = 1 . 2m 〈 | 〉 DEFINITION 1.3.1.4. 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 w . This element satisfies l(w w)= l(ww )= l(w ) l(w) for w W ([18], Chapter IV, 0 0 0 0 − ∈ Exercise 22). It follows that w0 has order (at most) 2 and that conjugation by w0 induces 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 (m) disjoint unions of the diagrams of An, Cn, Dn, E6, E7, E8, F4, H3, H4, I2 .

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 spanning Rl for some l 1 such that: ≥ i. if α Φ then Rα Φ= α, α , ∈ ∩ { − } ii. if α, β Φ then 2(β,α) Z, and ∈ (α,α) ∈ iii. if α, β Φ then w (β) := β 2(β,α) α Φ. ∈ α − (α,α) ∈ 10 PRELIMINARIES

FIGURE 1.3: the diagrams of the irreducible ﬁnite Coxeter systems

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

DEFINITION 1.3.1.6. 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 of Rl ,

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 root α Φ exactly one of the roots α, α is contained in Φ+, ∈ − ii . for any two distinct roots α, β Φ+ such that α + β is a root, α + β is a root in Φ+. ∈ If a set of positive roots Φ+ is chosen, elements of Φ+ are called the negative roots.Given − 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 = w α Φ+ . The group W := w α Φ is called the Weyl group of the root { α| ∈ } 〈 α| ∈ 〉 system and (W, R) is a Coxeter system [21] (p. 310). For a subset J of a fundamental system of roots, let R be the space spanned by J, Φ =Φ R , and W be the subgroup J J ∩ J J of W generated by the fundamental reﬂections w for r J. r ∈

PROPOSITION 1.3.3. [36] The set ΦJ is a system of roots in RJ , J is a fundamental system in ΦJ . The Weyl group of Φ is WJ .

DEFINITION 1.3.1.7. The subgroups WJ and their conjugates are called the parabolic subgroups of W.

Most of the ﬁnite Coxeter systems can be described in terms of their root systems. The exceptions are the groups called H and H , and the family I (m) for m = 2, 3, 4, 6 3 4 2 6 [21], (p. 310).

EXAMPLE 1.3.1.8. 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, w , w , w w , w w , w w w = w w w = { r s r s s r r s r s r s} Sym(3), 1 = 1, wr = (12), ws = (23), wr ws = (132), wswr , = (123), wr wswr = (12)(23)(12)=(13). In Figure 1.4, the generators are the reﬂections that are perpendicular to r and s.

FIGURE 1.4: 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

DEFINITION 1.3.2.1. 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, ∩ iii . R is a set of involutions w i I generating W ((W, R) is a Coxeter System here), { i| ∈ } 12 PRELIMINARIES

iv . if n N maps to w under natural homomorphism of N into W, and if n is any i ∈ i element of N, then Bn B.BnB Bn nB BnB, i ⊆ i ∪ v. if n is as above, then n Bn = B. i i i 6 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 NJ be the subgroup of N that is mapped to WJ under the natural homomorphism. Then, BNJ B 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 ∈ w is the image of n under the homomorphism described above.

THEOREM 1.3.6. [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 = BNB,

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

iv . each subgroup GJ is equal to its normalizer, hence, distinct subgroups GI , GJ 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 . themapW wW G wG is a bijection from W W/W onto G G/G , in particular, I J 7→ I J I \ J I \ J if for w, w′ W, we have BwB = Bw′B, then w = w′, ∈ vii . BrBwB = BrwB if and only if l(rw) > l(w); otherwise, BrBwB = BwB BrwB, ∪ 1.3 GROUPS 13

1 viii. if gBg− G , then g G (g G), ⊆ I ∈ I ∈ ix. G wG = BW wW B (w W). I J I J ∈ REMARK 1.3.7. [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 for some I R. There is a bijection between the set of subsets of R, and the I ⊂ set of standard parabolic subgroups of G. Hence, for a Chevalley group with diagram Xn, 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 GL(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, orthogonal 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 An, orthogonal groups O2n(q) have diagram Dn, O2n+1(q) have diagram Bn, symplectic groups ) Spn(q) have diagram Cn, and the exceptional groups E6, E7, E8, F4, G2 are named after their diagrams. 2 2 2 3 2 2n+1 The groups An, Dn, E6, D4, are called Steinberg groups. The groups B2(2 ), 2 2n+1 2 2n+1 F4(2 ), and G2(3 ) are known as Suzuki-Ree groups. Together, these are called 2 2 twisted Chevalley groups. The group F4(2) is not simple. Its derived group F4(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 mathematical 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 A (n = 5) and PSL (F ), (p 5). In n 2 p ≥ 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 14 PRELIMINARIES

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 J1 in 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 geometrical 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 automorphisms 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].

DEFINITION 1.5.0.1. 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. When we say geometry, we mean a Buekenhout-Tits geometry. 1.5 BUEKENHOUT-TITS GEOMETRIES 15

DEFINITION 1.5.0.2. 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) ). | | | \ |

DEFINITION 1.5.0.3. 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 Γ , the incidence relation and the type map are inherited from the geometry Γ(X , , t). F ∗ 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 Y x∗. Using this notation, the set of objects ∩ ∈ of Γ is F ∗ F. F \

DEFINITION 1.5.0.4. If F is a flag of Γ(X , , t), and J is a subset of I, the J-shadow ShJ (F) 1 ∗ (on P) is t− (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 geometries.

1.5.1 Point-line geometries

DEFINITION 1.5.1.1. 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 P lie 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. 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. 16 PRELIMINARIES

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.

THEOREM 1.5.1. [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.

DEFINITION 1.5.1.3. 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. ⊆ DEFINITION 1.5.2.1. 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 ) . Any set that is isomorphic to a power S set of an (n + 1)-set with a partial order deﬁned on them is called an n-dimensional ⊆ simplex.

FIGURE 1.5: a simplicial 3-complex

DEFINITION 1.5.2.2. 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. ∩ It follows that, the simplex Θ has a unique minimal element denoted by 0. 1.5 BUEKENHOUT-TITS GEOMETRIES 17

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 = 0. ⊆ 6 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, St(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 St(A), its rank there is the codimension of A in B, which is denoted by codimBA.

DEFINITION 1.5.2.4. 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,..., Cm = C such that

codim (C C )= codim (C C ) 1 Ci 1 i 1 i Ci i 1 i − − ∩ − ∩ ≤ for all i = 1, 2, . . . , m.

Such a sequence is called a gallery. Two chambers C, C′ are adjacent if codim(C ∩ C′)= 1. So, a gallery is a sequence of chambers where any pair of consecutive chambers are either adjacent or identical.

DEFINITION 1.5.2.5. 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.

DEFINITION 1.5.2.6. 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.

DEFINITION 1.5.2.7. A map α : Θ Θ′ is a morphism of chamber complexes if α(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).

LEMMA 1.5.2. [36] Let Θ, Θ′ be two chamber complexes in which each element of codimension 1 is contained in at most 2 chambers, and let α, 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. ⊆ DEFINITION 1.5.2.8. For a thin chamber complex Θ, an endomorphism α on it is called a folding if it satisﬁes:

i . α2 = α,

ii . Each chamber in α(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 gallery of minimal length between any pair of chambers in it has all its terms in the set. 18 PRELIMINARIES

PROPOSITION 1.5.3. [36] Let α be a folding of Θ and C, C′ be a pair of adjacent chambers of Θ such that α(C′)= C. Suppose there is a folding β 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, then β(D)= D′.

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

PROPOSITION 1.5.4. [36] Let α, β be opposite foldings of Θ. Then there exists an automorphism 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 folding α 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 map ρC , the retraction of Σ onto the simplex of faces of C. The following relation s is an equivalence relation: given A, B Σ, we write A s B if ∈ ρC (A)= ρC (B).

LEMMA 1.5.6. [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.

LEMMA 1.5.7. [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.

LEMMA 1.5.8. [36] The automorphisms ρ 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.

PROPOSITION 1.5.9. [36] Let Σ be an abstract Coxeter complex. Then, W(Σ) is the group of all type-preserving automorphisms of Σ.

THEOREM 1.5.10. [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. 1.5 BUEKENHOUT-TITS GEOMETRIES 19

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

DEFINITION 1.5.3.1. A building is a pair Ω, where Ω is a chamber complex, and is a A 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 such that ∈ A C Σ and C′ Σ. ∈ ∈

iv. If A, A′ are elements of Ω which are contained in each of the apartments Σ, Σ′ ∈ , there exists an isomorphism between Σ, Σ′ leaving invariant A, A′ and all their A 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.

EXAMPLE 1.5.3.2. [36] Let Ω be the set of left cosets gG for all g G, and all subsets J J ∈ of I. Let Σ be the subset of Ω consisting of all elements nG for all n N and all J I. 0 J ∈ ⊂ Then gΣ is the set of all cosets gnG for all n N and all J. Let be the family of 0 J ∈ A subsets gΣ of Ω for all g G. Then, Ω is a building and, is a set of apartments in Ω. 0 ∈ A 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 (iv), there is an isomorphism 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 (iv). This image will be called retΣ,C (A).

LEMMA 1.5.11. [36] The map Ω Σ, → A retr (A) → σ,C is a retraction from Ω onto Σ.

THEOREM 1.5.12. [36] The apartments of a building are abstract simplicial complexes.

DEFINITION 1.5.3.3. 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.

THEOREM 1.5.13. [36] In the building Ω(G; B, N) the group G operates by left multiplication 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. 20 PRELIMINARIES

THEOREM 1.5.14. [36] Let (Ω, ) be a building and G a group of type preserving automor- A 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 C , Σ be a ﬁxed chamber and apartment with C Σ , let B 0 0 0 ∈ 0 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 An are the projective spaces. The buildings of type Bn, Cn and Dn are 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 H3 and 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

DEFINITION 1.5.3.4. A diagram for a geometry Γ is a labeled complete graph on the D type set, where each label is a class of rank 2 geometries. The graph is called a Di j D Buekenhout-Tits diagram Xn of the geometry Γ (with rank n) when for any ﬂag of type I i, j , the residue Γ belongs to the class of geometries . Sometimes, Γ is called a \{ } F Di j geometry of type Xn. Our classes of geometries will be self-dual, so that we need not worry about directing the edges of . Indeed, each class will be the class of all generalized m-gons, for D Di j 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 j objects form the points and lines of a generalized quadrangle.

EXAMPLE 1.5.3.5. The geometry of 8 corners, 12 edges and 6 faces of a cube satisﬁes the axioms given by diagram in Figure 1.6 : 1.5 BUEKENHOUT-TITS GEOMETRIES 21

1 2 3

FIGURE 1.6: 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 i0 in I, a geometry Γ(X , , t) with type set I gives a point-line 1 ∗ geometry where P = t− (i0) is the set of objects of type i0, and L is the set of i0-shadows of ﬂags which are of cotype i0 in 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 1.6 ). Let the set of points be t− (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 Xn by choosing type i as the point type described as above is called the incidence system of type Xn,i , and this choice is indicated in the diagram by circling the node representing the point type. The incidence system of type Xn,i , associated with a thick building of type Xn is 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 ﬁeld F.

DEFINITION 1.5.4.1. A projective space PG(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, over Fq. For any finite prime power q, for the unique field Fq we denote the corresponding projective space by PG(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 = ky 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 PG(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 ∗ three conditions: 22 PRELIMINARIES

i . any two points are connected by exactly one line,

ii . for any four distinct points a, b, c, d if ab intersects dc then ac intersects d b,

iii . any line is incident with at least three points.

Projective planes are examples of projective spaces that satisfy the following:

i . any two points are connected by exactly one line,

ii . two lines intersect at least at one point,

iii . any line is incident with at least three points,

iv . there are at least two lines.

Since ii holds also for lines, we added this condition (iv) to make sure that it is a plane. For a subset A of P if every line meeting A in two points is contained in A then we call S(A)=(A, L′, ′) where L′ is the set of lines on points of A, and ′ is the incidence relation ∗ ∗ in between A and L′, a subspace of the projective space. The subspace S(A) is a space in its own sense. Empty set, the set of all points, the set of all points on a line or a singleton are all subspaces. For a subset B of the point set, B is the smallest subspace containing B. 〈 〉 A set A P is called linearly independent if for any subset A′ A and point p A A′, ⊂ ⊂ ∈ \ we have p not in the span A′ of A′. In other words, A A′ = A′. 〈 〉 ∩ 〈 〉 A basis of a projective space is a set of linearly independent points whose span is the whole space. The number of such points is always constant and this number is called the rank of the space which is denoted by rk(S). A hyperplane is a subspace of codimension 1. If H is a hyperplane and L a line not contained in H, then H L is a point. ∩ A ﬁnite projective space of rank at least 3 is PG(n, q)[101].

The number of rank k subspaces in PG(n, q) (where 1 n): ≤ For any k such that 0 k n these numbers are called Gaussian coefﬁcients: ≤ ≤ n n n 2 n k 1 n (q 1)(q q)(q q ) (q q − ) := − − − · · · − . k (qk 1)(qk q)(qk q2) (qk qk 1) q − − − · · · − −

1.5.5 Polar spaces Veldkamp [102] described and studied polar spaces in his thesis. Later on, Tits classified and redefined these objects for rank at least 3. A polar space of rank 2 is a generalized quadrangle, which is introduced by Tits [96]. So polar spaces are defined separately for rank smaller than 3 and rank at least 3 cases. In this thesis we shall only consider finite generalized quadrangles.

DEFINITION 1.5.5.1. A partial linear space is a geometry with points and lines where two distinct points are joined by at most one line and lines have at least two points. 1.5 BUEKENHOUT-TITS GEOMETRIES 23

DEFINITION 1.5.5.2. A generalized quadrangle (generalized 4-gon) is a partial linear space such that given a point p outside a line l there is a unique point on l collinear with p, and such that each point is on at least two lines. Two kinds of generalized quadrangles (which are dual to each other) are the complete bipartite graphs, and the grids. In a grid, the set of lines can be partitioned into two parts,

L1 and L2, where for each point there are two lines passing through, one from each line set. In grids, if two line sets L1 and L2 are of different sizes, then the number of points on lines is not constant. In complete bipartite graphs, the set of points is partitioned into two, namely P1 and P2. If P1 and P2 have different number of points, then number of lines passing through each point is not constant. Apart from complete bipartite graphs and their duals grids, all generalized quadrangles have orders [32] :

THEOREM 1.5.15. Let G be a generalized quadrangle in which there is a line with at least three points and a point on at least three lines. Then the number of points on a line, and the number of lines through a point, are constants. So apart from the two cases above, one can consider a generalized quadrangle, as a generalized 4-gon, such that all lines have (s + 1) points, and all points are on (t + 1) lines where s and t are ﬁnite or inﬁnite cardinal numbers. ‘Generalized quadrangle of order (s, t)’ is abbreviated by GQ(s, t).

DEFINITION 1.5.5.3. A polar space of rank n 3 is an incidence geometry (X , , t) with ≥ ∗ type set I = 1, 2, . . . , n , where is symmetrized strict inclusion, that consists of a set 1 { } ∗ P = t− (1), the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms: i . The incidence system obtained by considering all subspaces strictly in one subspace of size at least 2 is isomorphic to a projective space of (projective) dimension k with 1 k (n 1). ≤ ≤ − ii . The intersection of two subspaces is always a subspace or empty.

iii . For each point p not in a subspace A of dimension of (n 1), there is a unique − subspace B containing p of dimension (n 1) such that A B is (n 2)-dimensional. − ∩ − B is the subspace that is the union of the lines joining p to the points of A.

iv . There are at least two disjoint subspaces of dimension (n 1). − The varieties of (projective) dimension n 1 in a polar space of rank n will be referred − to as the generators or maximals. The subspaces of (projective) dimension 2 as the planes of the polar spaces. In a projective space, any pair of points is joined by a line, which might not be the case in a polar space. For the construction of the classical polar spaces, we will deﬁne sesquilinear, quadratic and bilinear forms here.

DEFINITION 1.5.5.4. A map V V F is a sesquilinear form relative to the ﬁeld auto- × → morphism θ if f satisﬁes: f (x + y, z)= f (x, z)+ f (y, z), 24 PRELIMINARIES

f (x, y + z)= f (x, y)+ f (x, z),

f (ax, y)= a f (x, y),

f (x, ay)= aθ f (x, y). for all x, y, z in V and a in F.

The set Rad(V )= v V f (x, v)= 0 x V is called the radical of V with respect to { ∈ | ∀ ∈ } the sesquilinear form f . The radical is always a subspace of V. Some types of sesquilinear forms are the following:

if Rad(V)= 0, then f is called non-degenerate,

if θ is an identity automorphism, then, f is called a bilinear form,

if f is bilinear and f (x, y)= f (y, x) for all x, y in V , then, f is called symmetric,

if f is bilinear and f (v, w)= 0 implies f (w, v)= 0, then it is called reﬂexive.

if f is bilinear and f (v, v)= 0 for all v V , then it is called alternating (or symplectic), ∈ if f is bilinear and f (x, y)= f (y, x) for all x, y in V, then f is called skew-symmetric, − if θ 2 is the identity automorphism of F but θ is not, and if also f (x, y)=(f (y, x))θ x, y, then f is called Hermitian-symmetric form, ∀ if char(F) = 2, and f is non-degenerate and symmetric, or if char(F) = 2, and f is 6 non-degenerate, symmetric, and f (x, x)= 0 for all x V, it is called orthogonal ∈ form,

if f is non-degenerate and skew-symmetric, then f is called symplectic form,

if f is non-degenerate and Hermitian-symmetric, then it is called unitary form.

Reflexivity allows us to define orthogonality: two vectors v and w are orthogonal with respect to the reflexive bilinear form if and only if: f (v, w)= 0 or f (w, v)= 0.

2 DEFINITION 1.5.5.5. A quadratic form is a map Q : V F such that Q(αu)= α Q(u) (for → u V, α F) with the property that B, which is deﬁned by B(u, v)= Q(u+v) Q(u) Q(v) ∈ ∈ − − is a bilinear form.

We have B(u, u)= 2Q(u), so if char (F) = 2 by knowing one of B or Q, we will have 6 the other, otherwise, many quadratic forms can lead to the same B. A quadric is the set of zeros of a quadratic form. A point v is a singular point (or isotropic point) with respect to Q if Q(v)= 0. With respect to a sesquilinear form f , a point x is singular if f (v, x)= 0 for all v in V, and is isotropic if f (x, x)= 0. A subspace W is isotropic with respect to a form if it has a isotropic vector. A subspace W of V is called totally singular if Q(W) = 0 and totally isotropic if B(W, W) = 0 where B is the bilinear form related to Q. (In odd or 0 characteristic, totally singular subspaces are precisely totally isotropic subspaces.) The (vector space) dimension of the maximal totally singular subspaces is called the Witt index. 1.5 BUEKENHOUT-TITS GEOMETRIES 25

If U is a subspace of V , then U⊥ denotes the subspace of V consisting of the vectors orthogonal to each element of U. The quadratic form Q is called non-degenerate if Q does not vanish on any point in V ⊥. A bijective linear map in between two vector spaces both of which are equipped with a form is an isometry if it transforms the first form to the second. For any non-degenerate Hermitian, quadratic, alternating or symmetric form f , an isometry between two subspaces U1 and U2 of V extends to an isometry of V (Witt’s theorem). This implies, every totally isotropic subspace is a subspace of a totally isotropic subspace of maximal dimension. There are different types of quadrics in even and odd dimensions. In even (v.s.) dimension n > 0, there are two types of non-degenerate quadrics: hyperbolic quadrics and elliptic quadrics. in odd (v.s.) dimensions n > 1, there is one type of non-degenerate quadric. Hyperbolic quadrics are an orthogonal direct sum of hyperbolic lines, and elliptic quadrics are the rest. The maximal totally singular subspaces have dimension m for a hyperbolic quadric in n = 2m dimensions, and m 1 for an elliptic quadric in n = 2m dimensions, and m for a − non-degenerate quadric in n = 2m + 1 dimensions. We have the classification of all polar subspaces of dimension at least 3. In the (thick) finite case, all polar spaces of rank at least 3 are called classical polar spaces. Here is a list of all the classical polar spaces with their ranks. Consider (S, , t) with I = 1, 2, . . . , n to ∗ { } be the incidence system, where V is a vector space with a non-degenerate form with Witt index at least 2, S is the set of all non-trivial totally isotropic subspaces, is symmetrized ∗ inclusion, t maps elements of S to their dimension. This is the classical polar space of rank n induced by the form. The polar spaces induced by non-degenerate quadratic forms are the quadrics.

Q+(2n + 1, q): non-singular hyperbolic quadric in PG(2n + 1, q) for some n 1, giving ≥ a hyperbolic polar space of rank n + 1.

Q(2n, q): non-singular parabolic quadric in PG(2n, q) for some n 2, giving a parabolic ≥ space of rank n.

Q−(2n + 1, q): non-singular elliptic quadric in PG(2n + 1, 1), for some n 2, giving a ≥ hyperbolic polar space of rank n.

W(2n + 1, q): Polar space consisting of the points of PG(2n + 1, q) together with the totally isotropic subspaces of a non-singular symplectic polarity of PG(2n + 1, q), giving a symplectic polar space of rank n.

H(2n, q2): non-singular Hermitian variety in PG(2n, q2) for some n 2 giving a Hermi- ≥ tian polar space of rank n.

H(2n + 1, q2): non-singular Hermitian variety in PG(2n + 1, q2) for some n 1, giving ≥ a Hermitian polar space of rank n + 1.

Here is the classiﬁcation of classical generalized quadrangles as in [30] Chapter 9: The classical generalized quadrangles are the ones that are associated with classical groups. There are three families of them: 26 PRELIMINARIES

(i) The points and lines of non-singular quadrics of Witt index 2 in some projective spaces give generalized quadrangles. These are: (1) Q+(3, q) in the projective space PG(3, q) with parameters : s = q, t = 1, v = (q + 1)2, b = 2(q + 1) where b is the number of lines.

(2) Q(4, q) in PG(4, q), the parameters are : s = t = q, v = b =(q + 1)(q2 + 1)

2 3 2 (3) Q−(5, q) in PG(5, q) with parameters s = q, t = q , v = (q + 1)(q + 1), b = (q + 1)(q3 + 1). These quadrics have the following canonical equations:

+ Q : X0X1 + X2X3 = 0, 2 Q : X0 + X1X2 + X3X4 = 0,

Q− : F(X0, X1)+ X2X3 + X4X5 = 0, where F is an irreducible homogeneous polynomial in X0, X1 over Fq.

(ii) A Hermitian variety H(n, q2) of the projective space PG(n, q2) has projective index 1 if and only if n is 3 or 4. Then the points and lines of H form the following generalized quadrangles: (1) s = q2, t = q, v =(q2 + 1)(q3 + 1), b =(q + 1)(q3 + 1) when n = 3,

(2) s = q2, t = q3, v =(q2 + 1)(q5 + 1), b =(q3 + 1)(q5 + 1) when n = 4. H has the canonical equation: X q+1 + X q+1 + + X q+1 = 0. 0 1 · · · n (iii) The points of PG(3, q) together with totally isotropic lines with respect to a symplectic polarity, form a generalized quadrangle W3(q) with parameters s = t = q, v = b = (q + 1)(q2 + 1). The symplectic polarity stated above has the following canonical bilinear form: X Y X Y + X Y X Y = 0. 0 1 − 1 0 2 3 − 3 2 The generalized quadrangle derived from Q(4, q) is always isomorphic with the dual of

W3(q), and they are both self-dual and thus isomorphic to each other if and only if q is even.

1.6 Association schemes

DEFINITION 1.6.0.6. A d-class (symmetric) association scheme (X , R) is a set of points X , along with d + 1 binary (symmetric) relations R = R = I, R ,..., R which gives a { 0 1 d } partition of X X such that the following holds: × There exists (d + 1)3 nonnegative integers pl for which 0 i, j, l d and for any i j ≤ ≤ (x, y) R , there are exactly pl elements z X such that (x, z) is in R , (z, y) in R . ∈ l i j ∈ i j An association scheme (X , R) is called metric association scheme, if the distance relation d which is deﬁned by: d(x, y)= i (x, y) R is a metric. ⇔ ∈ i 1.6 ASSOCIATION SCHEMES 27

For each relation Ri for a symmetric scheme, there is an undirected graph Ri on the set of points of the scheme, for a pair of points x, y, (x, y) Ri if and only if x and y are ∈ 0 adjacent in Ri. For that graph Ri , the valency vi is equal to pii .

DEFINITION 1.6.0.7. A connected simple graph Γ=(V, E) of diameter d forms a d-class metric association scheme together with graph distance, when the vertices of the graph are considered as the points of the scheme and a pair of them are i’th associates if and only if their mutual distance is i. Then the graph is a distance-regular graph. If the diameter of a distance regular graph is two, then it is called strongly regular. By the triangle inequality, for distance regular graphs, pk = 0 if i + j < k or j i > k i j | − | i+j and pi j > 0. When a group G acts transitively on a ﬁnite set X , one can take the partition of X X × into G orbits as the set of relations, and obtain an association scheme. One can ﬁx a relation, and make a graph whose adjacency condition is having this relation. This graph admits G as a group of automorphisms.

EXAMPLE 1.6.0.8. Consider the set X of 2-sets of the set [5] = 1, 2, 3, 4, 5 . The sym- { } metric group of order ﬁve Sym(5) acts on it transitively. The group Sym(5) has three orbits on X X . One consists of identical pairs of 2-sets, one consists of intersecting pairs × of 2-sets, one consists of disjoint pairs of 2-sets. Once we take these orbits as relations

R0, R1, R2, we obtain an association scheme. Each of these relations, when taken as the adjacency condition, gives us a graph on the elements of X . If we choose the ﬁrst relation as the adjacency condition, we obtain a graph without edges (our graphs are loopless), if we choose the second relation as the adjacency condition, we obtain the complement of the Petersen graph, if we choose the third relation as the adjacency condition, we obtain the Petersen graph itself.

In Chapter 7, we will work on association schemes that are related to certain graphs.

EXAMPLE 1.6.0.9. The Hamming scheme H(n, 2) has vertex set the binary vectors of length n, say Qn. For a pair x, y of elements in Qn, (x, y) R if they differ in i coordinate ∈ i positions. Then, in the graph Ri a pair of vertices are adjacent if and only if they are at distance i in R1. This deﬁnes a metric called the Hamming metric: if a pair of vertices x, y are at distance i in R1, we say dH (x, y)= i. The graph R1 is called the Hamming graph.

EXAMPLE 1.6.0.10. The Johnson scheme J(n, k) has point set X equal to the set of all k-subsets of an n-set N. The points of J(n, k) can also be thought as the vectors in V(n, 2) of weight k, where weight is the nonzero entries of the vector. For a pair of k-sets x and y, (x, y) is in R if they share k i elements, or as vectors, if they differ in 2i coordinate i − positions. This deﬁnes the Johnson metric d (x, y) for which d (x, y)= 1 d (x, y). J J 2 H The graph R1 is a distance regular graph which is known as the Johnson graph, and denoted by J(n, k).

DEFINITION 1.6.0.11. Let A =(X , R) be a d-class symmetric association scheme. For each R , there is a symmetric zero-one matrix A in Mat (C). For all i, 0 i d, A (u, v)= 1 i i X ≤ ≤ i if (u, v) R , 0 otherwise. These matrices called the associate matrices and they have ∈ i the following properties: 28 PRELIMINARIES

i . A0 = I,

d ii . i=0 Ai = J, P iii . A A = d ph A i, j where 0 i, j d. i j h=0 i j h ∀ ≤ ≤ P The associate matrices are (0, 1)-matrices, so they are linearly independent by (ii) and we see that the product of any two of them lies in the linear span of A ,..., A by { 0 d } (iii).

The Ai form the basis of an algebra, which is known as the Bose-Mesner algebra. That algebra is commutative since its basis elements are symmetric.

There is another basis E0 = (1/n)J, E1,..., Ed (J is the all 1’s matrix) for the Bose- Mesner algebra. That is the basis of minimal idempotents (see [23]) which exists because the Ai commute and are simultaneously diagonalizable. These matrices satisfy the following properties:

i . Ei Ej = δi j Ei,

d ii . i=0 Ei = I, P iii . Ak Ei = Pik Ei,

iv . A = d P E i, j 0 i, k d, k i=0 ik i ∀ ≤ ≤ P or alternatively, if entrywise (Schur) matrix multiplication is denoted by ‘ ’ then, the ◦ relations between the two bases can be summarized as follows as well:

i . E E = qk E (qk ’s are known as Krein parameters), i ◦ j i j k i j P ii . A A = δ A , i ◦ j i j i d iii . Ek =(1/n) i=0 QikAi , P iv . E A =(1/n)Q A . i ◦ k ki k k The numbers pi j are called the intersection numbers. We can relate those matrices shortly to the basis of the Bose-Mesner algebra as follows:

DEFINITION 1.6.0.12. For an association scheme with associate matrices A , A ,..., A { 0 1 d } and minimal idempotent matrices E , E ,..., E , the matrix P which satisﬁes the con- { 0 1 d } dition:

(A0, A1,..., Ad )=(E0, E1,..., Ed )P is called the ﬁrst eigenmatrix, and the matrix which satisﬁes the condition:

1 (E , E ,..., E )= X − (A , A ,..., A )Q 0 1 d | | 0 1 d is called the second eigenmatrix of the association scheme. 1.6 ASSOCIATION SCHEMES 29

The i’th column of the matrix P consists of the eigenvalues of Ai . There is a matrix Q which is related to P by the relation PQ = QP = nI, and Q0i is the multiplicity of the eigenvalue Pi j of Aj. The relation between P and Q matrices can be summarized by the following: PQ = QP = nI. Normalized P, which is denoted by Pˆ is the matrix obtained by dividing the rows of P by its ﬁrst row. Similarly normalized Q which is denoted by Qˆ is the matrix obtained by dividing the rows of Q by its ﬁrst row. The matrices Pˆ and Qˆ are related by the equation Pˆ⊤ = Qˆ. k Let Li be the matrix whose (k, j)-entry Li(k, j) = pi j . The eigenvalues of Li are the eigenvalues of Ai [21] p. 45.

1.6.1 Delsarte’s linear programming bound

DEFINITION 1.6.1.1. The distribution vector of Y , where Y X for a d-class association ⊆ scheme (X , R) is a vector, whose entries are: (Y Y ) R a = | × ∩ i | . i Y | | For an n-class symmetric association scheme (X , R), there is a relation called Del- sarte’s linear programming bound (DLPB) [49], which states the following:

THEOREM 1.6.1. If a is the distribution vector of a non-empty subset Y of an association scheme (X , R), then aQ 0. ≥ To be precise, if we want to maximize the size of the set A, where for any pair of elements x, y in A, (x, y) R for some R R = R ,..., R , the bound is given by ∈ ik ik ∈ { i1 il } max a under the conditions a = 1, a 0, aQ 0, a = 0 for R R R. i 0 i ≥ e ≥ j j ∈ \ Proof.P Let y be the characteristic vector of Y . Then e a = (Y Y ) R / Y = yA y⊤/ Y i | × ∩ i| | | i | | For all E , yE 2 0. i k ik ≥ 2 yE =(yE )(yE )⊤ k ik i i = yEi Ei⊤ y⊤ 2 = yEi y⊤

= yE y⊤ 0 i ≥ since E is symmetric and idempotent. E =( Q A )/ X . Then, i i j j,i j | | P yE y⊤ =(1/ X )y( Q A )y⊤ i | | j,i j Xj

In Chapter 7, we will make use of DLPB to obtain bounds for the sizes of sub- conﬁgurations of certain graphs. Distribution diagrams with respect to regular partitions are used to demonstrate the structure of the graphs in terms of given relations. In the distribution diagram, the equivalence classes are named K for i in 0, 1, 2, . . . , n . The Quotient matrix is the i { } matrix D for which Di j is the number of neighbors in Kj of a point p in Ki. Eigenvalues of the matrix D are also eigenvalues of the graph itself. A way of proving this in the context of association schemes is the following: For an association scheme, let D, and A be deﬁned as above. The A have a span- i { i } ning set of common eigenvectors. Let v be one of them, and θ0 = 1, θ1 = θ,..., θn be the corresponding eigenvalues for v. Let θ~ = 1, θ,..., θ . { n} For all i, n

AAi = Di j Aj Xi=0 since AA = # z (x, z) R , (y, z) R . Hence, i x,y { | ∈ 1 ∈ i } n

AAi (v)= Di j Aj(v), Xj=0

θθi = Di j θj, Xj=0

θθ~ = D⊤θ~, eigenvalues of D and transpose of D are the same, so, the eigenvalue θ of A is an eigenvalue of D as well, moreover, once the eigenvectors of D⊤ is known, the eigenvalues of all Ai are known, hence, the matrix P. Graphs with regular partitions gives symmetric association schemes. See Lemma 1.1.2. 2 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCESOF SETS

2.1 Introduction

In this chapter, starting from the ﬁfties, you will see the story of Johnson and Kneser graphs. Moreover, we will deﬁne a generalized version of the Kneser graph. Charac- terization of connectedness conditions for these graphs and sizes of largest cocliques for some of these graphs are given here.

2.1.1 Codes Coding theory deals with problems that occur during transfer of data through noisy chan- nels. Through a noisy channel, information that arrives to the receiver might have errors. The question for the received information is, if it is correct, and if not, where the error is. The data should be transferred in such a way that the above two problems can be solved and the size of the transmitted data should not be inefficiently large when compared to the original data. n DEFINITION 2.1.1.1. A code C is a subset of a finite set X = Q for some fixed n and some alphabet Q of size q. Elements of X are called words, and elements of C are called codewords.

DEFINITION 2.1.1.2. If v =(v1, v2,..., vn), and u =(u1, u2,..., un) are elements of X , then we deﬁne the Hamming distance between them as d(u, v)= ♯ i u = v . The function { | i 6 i } d : X X N is called the Hamming metric. × → If a received word is not one of the codewords, then it is decoded as the codeword whose distance to this word is minimum in Hamming metric. This is nearest neighbor decoding. The minimum distance in a code is related to its error correcting capacity. More precisely, if the minimum distance d satisﬁes d 2e + 1, then nearest neighbor decoding ≥ will be correcting s errors where s e. ≤ For any word v in Qn, a ball with radius e and center v includes all words that are at most distance e from v. 32 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCES OF SETS

n Let Be,v be a ball with radius e and center v in Q . The size of B in Qn is e n (q 1)i because a word should differ in i coordinate e,v i=0 i − positions to be at a certainP distance i from v where i satisﬁes 0 i e. There are n ≤ ≤ i ways of determining these i positions, and in each position, there might be q 1 different − letters. With a given minimum distance, how large can a code can be? Hamming gave an answer to that question.

PROPOSITION 2.1.1. [79] Let X be as above. For any code C where d 2e + 1, ≤ V (n, q) C | |. | |≤ B | e,v| This is known as the Hamming bound.

Proof. The e balls are disjoint because d 2e+1, and they are deﬁned for all codewords ≤ in C, so the following equality holds: v C Be,v = C . Be,v . But v C Be,v V (n, q)= |∪ ∈ | | | | | |∪ ∈ |≤ qn. Combining these two, C V (n, q) / B . | | ≤ | | | e,v |

2.1.2 (Classical) Kneser graphs Kneser graphs on sets are also called classical Kneser graphs.

DEFINITION 2.1.2.1. Let, for integers m, n with 1 m n/2, the graph K(n, m) have as ≤ ≤ vertices the m-subsets of a ﬁxed n-set, adjacent when disjoint (this is equivalent to having maximal distance in J(n, m)). This graph is known as the Kneser graph.

For example, K(5, 2) is the Petersen graph. n k The Kneser graph is regular of valency − . It is strongly regular if k = 2, and n 5. k ≥ 2.1.3 Johnson graphs In Example 1.6.0.10 we deﬁned the Johnson graph in relation to the Johnson scheme.

DEFINITION 2.1.3.1. Let X be a set of size n and let 0 k n. The Johnson graph J(n, k) ≤ ≤ is the graph whose vertices are the k-subsets of X , where a pair of vertices are adjacent when they have k 1 elements in common. − For example, J(n, 1) is the complete graph. The Johnson graph is connected. Suppose we have two k-sets A, B, sharing k m − elements. There is a path from A to B, such that along the path, the vertices have respectively k, k 1, k 2, . . . , k m elements common with A. For each k-set, the number of − − − k n k k-sets that are at distance i to it is − : the k-sets that are at distance i to a ﬁxed k i i k-set A, have k i elements in common − with A, and these k i elements can be chosen k − − in A in ways. The other i elements are chosen from the complement of A, which is k i of size n − k. − Consider a k-set B in Γi (A). One can count the number of neighbors of it in Γi 1(A) − as follows. For any element C in Γi 1(A) that is adjacent to B, A B C, indeed, − ∩ ⊂ (A C) (A B) = 1. This one element can be chosen among the i elements in A B. | ∩ \ ∩ | \ 2.1 INTRODUCTION 33

Moreover, (B A) (C A) = 1, and this element can be chosen among i elements in B A. | 2\ \ \ | \ So, there are i neighbors of B in Γi 1(A). One can also count number of neighbors of B in Γ (A) by similar counting arguments.− For any such neighbor C of B, (A B) (A C) = i+1 | ∩ \ ∩ | 1, and we choose this element among k i elements in A B. Now, we have to choose − ∩ the element in C B, and we have to choose it in (A B)C , which is of size n k i. \ ∪ − − So there are (k i)(n k i) neighbors of B in Γ (A). Hence, the Johnson graph is − − − i+1 distance-regular, with parameters c = i2, b =(k i)(n k i). i i − − − For a pair of sets at distance i, A B n, 2k (k i) n, so i n k, and i k | ∪ |≤ − − ≤ ≤ − ≤ hence, diameter d =min(k, n k). − The Johnson graph J(n, 2) is strongly regular for any n.

2.1.4 The relation between J(n, k) and K(n, k) If two k-sets meet in k i points where 0 i k, then, the distance between them is − ≤ ≤ i. For a pair of k-sets, being at mutual distance k in J(n, k) means being disjoint, that is, being adjacent in K(n, k). That is how J(n, k) and K(n, k) are related. Later, we will generalize this relation further.

2.1.5 History of the Kneser graphs Hamming introduced Hamming distance in his work “Error detecting and error correcting codes" [67] in 1950. By then, he was working in Bell Telephone Laboratories. The error detection programs of the computers that Hamming was using was not efﬁcient. After each error detection, the program was pausing. So, he developed a method to encode the input and correct the isolated errors, and continue to run.

That is how the C2(7, 4, 3) Hamming code was born. Association schemes were introduced by Bose and Shimamoto in 1952 [17]. In 1959, Bose and Mesner introduced the Bose-Mesner algebra of the association schemes [16]. Delsarte’s famous linear programming bound was introduced in 1973 in his thesis [49]. This was improving the bound given by Hamming on the size of a code. Delsarte made use of the theory of association schemes to introduce his bound. In 1989, Brouwer, Cohen, and Neumaier in their book Distance regular graphs [21], discussed Hamming and Johnson graphs as distance regular graphs with classical parameters. Kneser graphs are named after Martin Kneser, who ﬁrst investigated them in 1955 [77]. n 1 Erdos,˝ Ko & Rado [54] showed that the maximal cocliques in K(n, m) have size − m 1 and, if n > 2m, they are precisely the collections of all m-sets containing a ﬁxed element. − Kneser [77] conjectured, and Lovász [78] proved that K(n, m) has chromatic number n 2m + 2 by making use of topological methods; Lovász’s proof of Kneser conjecture − led to the discipline of topological combinatorics, which deals with problems in discrete mathematics by making use of topological concepts or theorems. Other proofs of the Kneser conjecture were given by Bárány [6], Greene [64] and Matoušek [80]. Lately, Pabon and Vera worked on the diameter of the Kneser graphs [99] and for K(n, k) where n > 2k they showed, the diameter of the graph is (k 1)/(n 2k) + 1. ⌈ − − ⌉ We discuss this in Section 2.2.2. 34 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCES OF SETS

So far, we had seen, the story of the Kneser graphs are somehow related to coding theory, the theory of association schemes, and topology.

2.2 Generalized Kneser graphs

In the following chapters, we will introduce a generalized version of the Kneser graphs, whose vertices are flags of a geometry. For a pair of flags, the condition of being adjacent will then be being mutually ‘far apart’ in a certain precise technical sense. Here we define graphs for the incidence geometry on the proper non-empty subsets of an n-set. The orders of the sets define their types. A pair of objects are incident if one is a proper subset of the other, hence a pair of objects of the same type are never incident.

This geometry can be regarded as the “thin n-vector space over F1". Let X be a n-set. We denote the complement of a set A in X as AC .

DEFINITION 2.2.0.1. A generalized Kneser graph is a graph that we denote here by GK(n : k ,..., k ). Each vertex correspond to a series of sets, (S ,..., S ) where 1 l 1 l ; ⊂ S S X and 0 < k < < k < n, X = n, S = k for all i, (1 i l), k = 0 1 ⊂···⊂ l ⊂ 1 · · · l | | | i | i ≤ ≤ 0 and k = X . l+1 | | Two vertices (S ,..., S ) and (T ,..., T ) are adjacent when for all i 1, 2, . . . , l 1 l 1 l ∈ { } either the condition S T = or S T = X holds. i ∩ j ; i ∪ j We always take the set [n] := 1, 2, . . . , n as our n-set. We denote our vertices by { } ordered l-tuples of sets for the graph GK(n : k1,..., kl ).

PROPOSITION 2.2.1. The following properties hold for the generalized Kneser graphs:

i. GK(n : k) ∼= K(n, k) if n > 2k,

ii. GK(2k : k) = 1 2k K , the graph consisting of 1 2k components consisting of a ∼ 2 k 2 2 k single edge,

iii. GK(n : k ,..., k ) = GK(n : n k ,..., n k ). 1 l ∼ − l − 1

Proof. (i) Vertices of GK(n : k) are k-subsets of an n-set. A pair of vertices of GK(n : k) are adjacent either if the corresponding k-sets are disjoint or their union is the n-set. That is, if n > 2k, the condition of being adjacent in GK(n : k) and K(n, k) are the same. (ii) Vertices of GK(2k : k) are the k-subsets of an 2k-set and each vertex in the graph is of degree one since it is only adjacent to its complement. There are 2k k-sets in a 2k-set k and they constitute 1 2k K graphs with their complements. (iii) There is a one to one 2 k 2 correspondence between the vertices GK(n : k1,..., kl ) and GK(n : n k1,..., n kl ). c c − − For any vertex (A1,..., Al ) in GK(n : k1,..., kl ), (Al ,..., A1) is the corresponding vertex in GK(n : n kl ,..., n k1). For a pair of vertices (A1,..., Al ), (B1,..., Bl ) of an n- − − c c c c set that are adjacent in GK(n : k1,..., kl ), (Al ,..., A1) and (Bl ,..., B1) are adjacent in GK(n : n k ,..., n k ). − l − 1 2.2 GENERALIZED KNESER GRAPHS 35

PROPOSITION 2.2.2. The number of vertices v in the generalized Kneser graph l ki+1 GK(n : k1,..., kl ) is = . i 1 ki Q Proof. In the graph GK(n : k1,..., kl ), the vertices are series of sets, (S1,..., Sl ) for n kl+1 which S1 Sl X and Si = ki . For Sl , the largest set, there are = ;⊂ ⊂···⊂ ⊂ | | kl kl ki ways of choosing the kl set. For any ki set Si, there are ways of choosing a ki 1 set ki 1 − − inside Si, so it adds a multiple of that size to the vertex number.

REMARK 2.2.3. The vertex set of GK(n : 1, 2, . . . , n 1) can be identiﬁed with the set of − all permutations, and that the number of vertices in this case is n!.

PROPOSITION 2.2.4. The generalized Kneser graph GK(n : k1,..., kl ) is a regular graph. Deﬁne i′ = j ifki + kj < n, ki + kj+1 n. Let mi be max(ki 1, n ki +1). ≥ − − ′ Then the valency of the graph is:

l n k m l n k m − i′ − i − i′ − i i=1 = i=1 . ki mi n ki ki Q − Q − − ′ Proof. Suppose we have vertex u =(S1, S2,..., Sl ) in GK(n : k1,..., kl ). We will count number of its neighbors v = (T1, T2,..., Tl ). Because of the inclusive structure of the l sets, for each i supposing we know the ﬁrst i 1 sets, we will ﬁnd out in how many ways − can we have a ki set. Once we do this for all ki, we will be done. For any ki-set, we know ki 1 elements of it. The number ki +1 corresponds to the size − ′ of the smallest set in u that intersects the ki set. This set’s complement will be a subset of the ki-set. For mi , the number of elements in the ki set we already know, we take the maximum of n ki +1 and ki 1. − ′ − The largest set that do not intersect the k -set is S . We choose the points in Sc , but, i i′ i′ mi of them are already chosen.

2.2.1 Connectedness of the generalized Kneser graphs

Let f be an element of Sym(n). For a vertex s =(S1, S2,..., Sl ), the image of s under f is deﬁned as follows: f (s)=(f (S1), f (S2),..., f (Sl )).

THEOREM 2.2.5. The group Sym(n) acts transitively on ordered pairs of adjacent vertices.

Proof. Take two pairs of adjacent vertices (x1, x2) and (y1, y2). We have to show there is f in Sym(n) so that, f ((x1, x2))=(f (x1), f (x1))=(y1, y2). First, for a pair of partitions of the n-set X , listed as (q1,..., qs) and (p1,..., ps) where p = q for all i, for 1 i s, we can come up with a permutation f on X for which f | i | | i | ≤ ≤ on qi is a bijection to qi, f (qi)= pi . 1 1 1 In GK(n : k1, k2,..., kl ) suppose our pairs of adjacent vertices are ((S1, S2 , ... Sl ), 1 1 1 2 2 2 2 2 2 (T1 , T2 ,... Tl )) and ((S1 , S2 ,... Sl ), (T1 , T2 ,... Tl )). For each ordered pair, we will come up with a partition of the above form. One partition will be mapped to the other, this will imply, one ordered pair is mapped to the other by a permutation, that is Sym(n) acts transitively on ordered pairs of adjacent vertices. Here we see how to make a partition related to a pair of adjacent vertices. We can make an ordering on the set of elements of X for each a pair. Let x, y be two elements. 36 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCES OF SETS

i . x in Si, y not in Si, then x < y,

ii . x, y in Si, but there is Sj such that x in Sj, y not in Sj, then x < y,

iii . x, y in Si, but there is Tj such that x in Tj, y not in Tj, then x > y,

iv . x in Ti, y not in Ti , then x > y,

v . x, y in Ti, but there is Tj such that x in Tj, y not in Tj, then x > y,

vi . x, y in Ti, but there is Sj such that x in Sj, y not in Sj, then x < y,

vii . otherwise, x, y they are equivalent (that happens when all sets Si , or Tj including x also includes y and vice versa).

Write the elements of X consecutively, according to the above ordering, starting from the least element(s), by making use of parenthesis indicating the start and end of the Si′s and Tj’s. Now we can think two vertices as series of disjoint sets, each set in the series consists of equivalent elements. by above argument there is an f mapping one series to 1 the other. Moreover, our partition of X indeed partites each Si and Tj, moreover maps Si 2 1 2 to Si , and Ti to Ti .

EXAMPLE 2.2.1.1. We will give an example of a partition deﬁned above. Let X = a, b, c, d, { e, f , g . Let our graph be GK(7 : 1, 4, 6). Consider the vertices ( d , d, a, f , b , d, a, f ,b, } { } { } { c, g ) and ( e , e, g, c, b , e, g, c, b, f , a ). So here, S = d , S = d, a, f , b , S = } { } { } { } 1 { } 2 { } 3 d, a, f , b, c, g , T = e , T = e, g, c, b , T = e, g, c, b, f , a . The above ordering rules { } 1 { } 2 { } 3 { } give us the following partition, where things in the same set are equivalent:

d < a, f < b < c, g < e { } { } { } { } { }

REMARK 2.2.6. For each pair of adjacent vertices, there is a unique partition of the set [n] as described above. So, the number of edges of the graph is equal to the number of such partitions for each generalized Kneser graph. For instance, for GK(7 : 1, 4, 6), the number of edges is 7 6 4 3 /2. 1212

THEOREM 2.2.7. The generalized Kneser graph Γ = GK(n : k1,..., kl ) is connected unless there exists i, j such that ki + kj = n and n > 2. Proof. Let X =[n] be the underlying set, let the sequence S S S determines 1 ⊂ 2 ⊂···⊂ l a vertex x = (S , S , , S ) of Γ. An ordering e , e ,..., e of X is compatible with x 1 2 · · · l 1 2 n if every S is an initial segment, so S = e , e ,..., e . It is clear that the ordering i i { 1 2 ki } determines the flag x. The reverse ordering en, en 1,..., e1 determines a flag y and x − and y are adjacent in Γ. If we interchange two adjacent elements ei and ei+1, then either x (or better, the flag determined by the new sequence) stays the same (this is the case when i k , , k or x changes to another vertex say x′, but now y doesn’t change, 6∈ { 1 · · · l } since now n i k , , k . This implies x and x′ have the common neighbor y, and − 6∈ { 1 · · · l } since every two sequences can be connected by a sequence of transpositions of adjacent elements the graph is connected. 2.2 GENERALIZED KNESER GRAPHS 37

For GK(2k : k), we have 2k /2 disconnected edges, thus, so many components. k More generally, when we have GK(n : k1,..., kl ) and when for each ki there exists some ki′ such that ki + ki′ = n, for each vertex, there is a unique vertex that it is adjacent to, so number of connected components is equal to the number of edges of the graph.

Let k1′ , k2′ ,... km′ be the maximal subset of k1, k2,..., kl , such that, for all ki′ in it, { ˆ } {ˆ } there is some k in k′ , k′ ,..., k′ such that k′ + k = n. i′ { 1 2 l } i i′

COROLLARY 2.2.8. The number of connected components of the graph GK(n : k1,..., kl ) is equal to the number of vertices of the graph GK(n : k′ , k′ ,..., k′ ) when k′ , k′ ,... k′ is 1 2 m { 1 2 m} as above.

Proof. Take a vertex v in the graph GK(n : k ,..., k ). Let k′ be in k′ , k′ ,..., k′ . In 1 l i { 1 2 m} the connected component including v, there is a path between any vertex u and v. Let ˆ A be the k′-set of v. Let B = X A be the k′-set of any vertex adjacent to v. Then in i \ i the path joining v and u, the vertices will be alternating vertices containing A and B, and any vertex not having one of them can not be in this component. Moreover, this is true for all ki′. So, for this component we will have an alternating pair A′1, A′2 ..., A′m and B′ , B′ ..., B′ (of respective sizes k′ , k′ ,..., k′ , n k′ , n k′ , . . .¸, n k′ ) of list of 1 2 m { 1 2 m} { − 1 − 2 − m} sets. For each component such a pair exist, and the number of such pairs is the number of vertices in GK(n : k1′ , k2′ ,..., km′ ).

2.2.2 Results about the diameter

We want to ﬁnd the diameter of the graph GK(n : k1,..., kl ). In Table 2.1, diameter of some graphs are given. The diameter for the graph GK(n : k) is already known. For k n/2, the diameter of GK(n : k ,..., k ) is the same as the l ≤ 1 l diameter of GK(n : kl ). The rest is veriﬁed by computer. When a graph is disconnected, we write ‘DC’ instead. For GK(n : k1,..., kl ), instead of k1, k2,..., kl we wrote shortly k1 . . . kl .

TABLE 2.1: diameters of generalized Kneser graphs

n: 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1111111111111 2 -1DC2222222222 3 - -12DC32222222 4 - - -123DC432222 5 - - - -1224DC5332 6 - - - - -12235DC64 7 ------122236DC 12 -DCDC 2 2 2 2 2 2 2 2 2 2 13 - -DC 4DC 3 2 2 2 2 2 2 2 14 - - -DC 4 5DC 4 3 2 2 2 2 15 - - - -DC 4 4 5DC 5 3 3 2 16 - - - - -DC 4 4 4 5DC 6 4 17 ------DC 4 4 4 5 6DC Continued on next page 38 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCES OF SETS

Table 2.1 – continued from previous page n: 2 3 4 5 6 7 8 9 10 11 12 13 14 23 - -DCDCDC 3 2 2 2 2 2 2 2 24 - - - 4DC 5DC 4 3 2 2 2 2 25 - - - - 4DC 6 6DC 5 3 3 2 26 - - - - - 4DC 6 4 7DC 6 4 27 ------4DC 6 4 5 7DC 34 - - - 2DCDCDC 4 3 2 2 2 2 35 - - - -DC 5DC 6DC 5 3 3 2 36 - - - - - 5 6DC 7 7DC 6 4 37 ------4 6DC 8 5 8DC 45 - - - - 2 3DCDCDC 5 3 3 2 46 - - - - - 3DC 6DC 7DC 6 4 47 ------DC 6 7DC 8 8DC 56 - - - - - 2 2 4DCDCDC 6 4 57 ------2 4DC 7DC 8DC 67 ------2 3 3 5DCDCDC 123 - -DCDCDC 3 2 2 2 2 2 2 2 124 - - -DCDC 5DC 4 3 2 2 2 2 125 - - - -DCDC 6 6DC 5 3 3 2 126 - - - - -DCDC 6 4 7DC 6 3 127 ------DCDC 6 4 5 7DC 134 - - - DC DC DC DC 4 3 2 2 2 2 135 - - - -DC 6DC 6DC 5 3 3 2 136 - - - - -DC 6DC 7 7DC 6 4 137 ------DC 6DC 8 5 8DC 145 - - - - DC 5 DC DC DC 5 3 3 2 146 - - - - - DC DC 7 DC 7 DC 6 4 147 ------DC 6 7DC 8 8DC 156 - - - - - DC 4 5 DC DC DC 6 4 157 ------DC 5 DC 7 DC 8 DC 167 ------DC 4 4 5 DC DC DC 234 - - - DC DC DC DC 4 3 2 2 2 2 235 - - - - DC DC DC 6 DC 5 3 3 2 236 - - - - - 5DCDC 7 7DC 6 4 237 ------4DCDC 8 5 8DC 245 - - - - DC DC DC DC DC 5 3 3 2 246 - - - - - 6DC 7DC 7DC 6 4 247 ------DC DC 7 DC 8 8 DC 256 - - - - - DC DC 6 DC DC DC 6 4 257 ------6 DC DC 8 DC 8 DC 267 ------DC DC 6 7 DC DC DC 345 - - - - DC DC DC DC DC 5 3 3 2 346 - - - - - DC DC DC DC 7 DC 6 4 347 ------DC 6 DC DC 8 8 DC 356 - - - - - 5 DC DC DC DC DC 6 4 Continued on next page 2.2 GENERALIZED KNESER GRAPHS 39

Table 2.1 – continued from previous page n: 2 3 4 5 6 7 8 9 10 11 12 13 14 357 ------DC 7 DC 8 DC 8 DC 367 ------6 DC DC 8 DC DC DC 456 - - - - - 3 DC DC DC DC DC 6 4 457 ------DC DC DC DC DC 8 DC 467 ------DC 6 DC DC DC DC DC 567 ------2 4 DC DC DC DC DC 1234 - - - DC DC DC DC 4 2 2 2 2 2 1235 - - - - DC DC DC 6 DC 5 3 3 2 1236 - - - - - DC DC DC 7 7 DC 6 4 1237 ------DC DC DC 8 5 8 DC 1245 - - - - DC DC DC DC DC 5 3 3 2 1246 - - - - - DC DC 7 DC 7 DC 6 4 1247 ------DC DC 7 DC 8 8 DC 1256 - - - - - DC DC 6 DC DC DC 6 4 1257 ------DC DC DC 8 DC 8 DC 1267 ------DC DC 6 7 DC DC DC 1345 ------DC DC DC 5 3 3 2 1346 ------DC DC DC 7 DC 6 4 1347 ------DC 6 DC DC 8 8 DC 1356 - - - - - DC DC DC DC DC DC 6 4 1357 ------DC DC DC 8 DC 8 DC 1367 ------DC DC DC 8 DC DC DC 1456 - - - - - DC DC DC DC DC DC 6 4 1457 ------DC DC DC DC DC 8 DC 1467 ------DC 7 DC DC DC DC DC 1567 ------DC 5 DC DC DC DC DC 2345 - - - - DC DC DC DC DC 5 3 3 2 2346 - - - - - DC DC DC DC 7 DC 6 4 2347 ------DC DC DC DC 8 8 DC 2356 - - - - - DC DC DC DC DC DC 6 4 2357 ------DC DC DC 8 DC 8 DC 2367 ------DC DC DC 8 DC DC DC 2456 - - - - - DC DC DC DC DC DC 6 4 2457 ------DC DC DC DC DC 8 DC 2467 ------DC DC DC DC DC DC DC 2567 ------DC DC DC DC DC DC DC 3456 - - - - - DC DC DC DC DC DC 6 4 3457 ------DC DC DC DC DC 8 DC 3467 ------DC DC DC DC DC DC DC 3567 ------DC DC DC DC DC DC DC 4567 ------DC DC DC DC DC DC DC 12345 - - - - DC DC DC DC DC 5 5 3 2 12346 - - - - - DC DC DC DC 7 DC 6 4 12347 ------DC DC DC DC 8 8 DC Continued on next page 40 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCES OF SETS

Table 2.1 – continued from previous page n: 2 3 4 5 6 7 8 9 10 11 12 13 14 12356 - - - - - DC DC DC DC DC DC 6 4 12357 ------DC DC DC 8 DC 8 DC 12367 ------DC DC DC 8 DC DC DC 12456 - - - - - DC DC DC DC DC DC 6 4 12457 ------DC DC DC DC DC 8 DC 12467 ------DC DC DC DC DC DC DC 12567 ------DC DC DC DC DC DC DC 13456 - - - - - DC DC DC DC DC DC 6 4 13457 ------DC DC DC DC DC 8 DC 13467 ------DC DC DC DC DC DC DC 13567 ------DC DC DC DC DC DC DC 14567 ------DC DC DC DC DC DC DC 23456 - - - - - DC DC DC DC DC DC 6 4 23457 ------DC DC DC DC DC 8 DC 23467 ------DC DC DC DC DC DC DC 23567 ------DC DC DC DC DC DC DC 24567 ------DC DC DC DC DC DC DC 34567 ------DC DC DC DC DC DC DC 123456 - - - - - DC DC DC DC DC DC 6 4 123457 ------DC DC DC DC DC DC DC 123467 ------DC DC DC DC DC DC DC 123567 ------DC DC DC DC DC DC DC 124567 ------DC DC DC DC DC DC DC 134567 ------DC DC DC DC DC DC DC 234567 ------DC DC DC DC DC DC DC

Let us have an overview of the diameter result of the classical Kneser graph from [99]. We will reformulate the names according to our notation.

LEMMA 2.2.9. [91] For two vertices A, BinK(n, k), if they are joined by a path of length 2p, then A B k (n 2k)p. | ∩ |≥ − − COROLLARY 2.2.10. [99] Let A, B be two vertices of K(n, k) joined by a path of length 2p+1. Then, A B (n 2k)p. | ∩ |≤ − PROPOSITION 2.2.11. [99] Let n, k be positive integers where n > 2k. Let k 2, and ≥ n 3k 1. Then the diameter of the Kneser graph is equal to 2. − ≥− LEMMA 2.2.12. [99] Let A, B be two distinct vertices of the graph K(n, k) where 1 n 2k < ≤ − k 1. If A B = s then − | ∩ | k s s d(A, B)= min 2 − , 2 + 1 { ⌈ n 2k ⌉ ⌈ n 2k ⌉ } − − Let us give a sketch of the proof here: 2.2 GENERALIZED KNESER GRAPHS 41

Proof. Take a pair of vertices A and B, call C = A B, C = s, X (A B) = D, D = k s∩ | | \ ∪ | | n 2k + s. Make partitions of A B and B A to − pieces of size n 2k (or less for − \ \ ⌈ n 2k ⌉ − the last one). We will come up with a path from− A to B, consisting of vertices called k s X = A, X ,..., X = B where t = 2 − , alternating between two types of vertices. 0 1 t n 2k ⌈ − ⌉ Odd types namely X2i 1 for all i k s/(n 2k) has ﬁrst i 1 parts of A B, last − ≤ ⌈ − − ⌉ − \ t i parts of B A and D, even types, namely X for each i has ﬁrst i parts of B A, last − \ 2i \ t i parts of A B and C. Take a subset D′ of D of size s. Let A′ be (B A) D′. Then − \ \ ∪ A and A′ are adjacent, now we will come up with a path from B to A′, and then to A. Call s′ = A′ B = k s Previous argument between A and B, gives us a path of length k s′ | ∩ s | − s k s 2 − = 2 between A′ and B, so we have d(A, B) min (2 + 1, 2 − ). ⌈ n 2k ⌉ ⌈ n 2k ⌉ ≤ ⌈ n 2k ⌉ ⌈ n 2k ⌉ −We deduce− it can not be smaller by making use of Lemma 2.2.9. − − This construction gives us a path for GK(n : n k) simply by taking complements of − all sets in the path.

THEOREM 2.2.13. [99] For a Kneser graph K(n, k), let n > 2k.Then, the diameter of the k 1 Kneser graph is − + 1 n 2k ⌈ − ⌉ This is proved by showing the maximal possible value of d(A, B) min (2 s + ≤ ⌈ n 2k ⌉ k s k 1 − 1, 2 − ) for all s is − + 1. n 2k n 2k ⌈ − ⌉ ⌈ − ⌉ To generalize this proof to generalized Kneser graphs, we have to come up with a nice path between any two nonadjacent vertices. Then, to generalize Lemma 2.2.9, we have to draw a connection between the existence of a path of certain path between two vertices and how far they are, that is the intersection sizes of the corresponding subsets.

EXAMPLE 2.2.2.1. For GK(5 : 1, 3), for any vertex x, there are 1, 4, 10, 14, 1 vertices at respective distance 0, 1, 2, 3, 4 to x. So the diameter is 4. For GK(7 : 1, 4), diameter is 5, and distribution is 1, 9, 30, 69, 30, 1. In both cases, there is a unique vertex at maximal distance.

There are case distinctions for generalized Kneser graphs for diameter. Keep in mind diameter is deﬁned for connected graphs. The diameter of GK(n : k) is known. The next simplest case is GK(n : 1, k). Consider the graph GK(n : 1, k) where n/2 < k < n 1 (for k < n/2, the diameter of − GK(n : 1, k) and the diameter of GK(n : k) are the same, for n = 2k, or k = n 1 graph − is not connected). We want to ﬁnd the diameter of the graph.

CONJECTURE 2.2.14. The diameter of GK(n : 1, k) where n/2 < k < n 1 and n = 2k a − − is 4 unless a 2/7k. ≤ Make a partition of the n-set 0, 1, . . . , n 1 into three sets A , A , A of respective { − } 1 2 3 sizes 1, k 1, n k(= k a). − − − For any vertex v =(C, D) in the graph we will describe it in terms of the intersection sizes of C and D with A for i 1, 2, 3 . For (C, D) we have (c , c , c ; d , d , d ) where i ∈{ } 1 2 3 1 2 3 c = Ai C , d = Ai D for i 1, 2, 3 . i | ∩ | i | ∩ | ∈{ } Seing the diameter is at least 4 is easy: Fix a vertex x =(A , A A ). It is of the form (1, 0, 0; 1, k 1, 0). 1 1 ∪ 2 − Take any vertex y of the form (1, 0, 0; 1, a 1, k a) x and y are not adjacent, since − − they have the same 1-set. 42 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCES OF SETS

The vertices at distance 1 to x are of the form (0, 0, 1; 0, a, k a). For any vertex x − 1 here, x1 is nonadjacent to y, since 1-set of x1 is in the k set of y. The vertices at distance 2 to x are of the form (1, 0, 0; 1, k 1 a+i, a i) or of the form (0, 1, 0; 1, k 1 a+i, a i), − − − − − − where 0 i < a. The vertex y can not be adjacent to a vertex of these forms, 1-set of y ≤ is in the k-sets of these vertices. That shows, diameter is at least 4. There are examples of GK(n : 1, k) with diameter more than 4, such as: GK(7 : 1, 4), GK(9 : 1, 5), GK(11 : 1, 6), GK(12 : 1, 7), GK(13 : 1, 7), GK(14 : 1, 8), GK(15 : 1, 8), GK(16 : 1, 9), GK(17 : 1, 9) (calculated by computer).

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

In Table 2.2, you see maximal coclique sizes of some of generalized Kneser graphs. These graphs can be grouped a little bit further. The unﬁlled cells does not fall into any case of our classiﬁcation, or are not covered by the propositions at the end of this chapter, or simply not calculated.

TABLE 2.2: maximum coclique sizes

n: 345 6 7 8 9 10 11 12 111111 1 1 1 1 1 21345 6 7 8 9 10 11 123 6 8 10 12 14 16 18 20 22 3- 1 4 10 15 21 28 36 45 55 13 - 6 12 30 45 63 84 108 135 165 23 - 6 15 30 45 63 84 108 135 165 123 - 12 30 60 90 126 168 216 270 330 4- - 1 5 15 35 56 84 120 165 14 - - 10 22 60 140 224 336 480 660 24 - - 12 45 90 210 336 504 720 990 124 - - 30 90 180 420 672 1008 1440 1980 34 - - 8 30 70 140 224 336 480 660 134 - - 30 90 210 420 672 1008 1440 1980 234 - - 30 90 210 420 672 1008 1440 1980 1234 - - 60 180 420 840 1344 2016 2880 3960 5- - - 1 6 21 56 126 210 330 15 - - - 15 37 105 280 630 1050 1650 25 - - - 22 105 ? 560 1260 2100 3300 125 - - - 60 210 420 1120 2520 4200 6600 35 - - - 30 90 280 560 1260 2100 3300 135 - - - 90 ? 840 1680 3780 6300 9900 235 - - - 90 315 840 1680 3780 6300 9900 1235 - - - 180 630 1680 3360 7560 12600 19800 45 - - - 10 45 140 315 630 1050 1650 145 - - - 60 ? 560 1260 2520 4200 6600 245 - - - 90 315 840 1890 3780 6300 9900 1245 - - - 180 630 1680 3780 7560 12600 19800 345 - - - 60 210 560 1260 2520 4200 6600 Continued on next page 2.3 MAXIMAL COCLIQUES IN GK(n : k1,..., kl ) 43

Table 2.2 – continued from previous page n: 345 6 7 8 9 10 11 12 1345 - - - 180 630 1680 3780 7560 12600 19800 2345 - - - 180 630 1680 3780 7560 12600 19800 12345 - - - 360 1260 3360 7560 15120 25200 39600 6- - - - 1 7 28 84 210 462 16 - - - - 21 58 ? 504 1260 2772 26 - - - - 37 210 ? 1260 3150 6930 126 - - - - 105 420 ? 2520 6300 13860 36 - - - - 60 210 840 ? 4200 9240 136 - - - - 210 ? 2520 5040 12600 27720 236 - - - - 180 840 2520 5040 12600 27720 1236 - - - - 420 1680 5040 10080 25200 55440 46 - - - - 45 210 560 1575 3150 6930 146 - - - - 210 840 ? 6300 12600 27720 246 - - - - 270 1260 ? 9450 18900 41580 1246 - - - - - 630 2520 18900 37800 83160 346 - - - - 210 840 2520 6300 12600 27720 1346 - - - - 630 2520 7560 18900 37800 83160 2346 - - - - 630 2520 7560 18900 37800 83160 12346 - - - - 1260 5040 15120 37800 75600 166320 56 - - - - 12 63 224 630 1386 2772 156 - - - - 105 ? ? 3150 6930 13860 256 - - - - 210 840 ? 6300 13860 27720 1256 - - - - 420 1680 ? 12600 27720 55440 356 - - - - 180 840 2520 6300 13860 27720 1356 - - - - 630 2520 7560 18900 41580 83160 2356 - - - - 630 2520 7560 18900 41580 83160 12356 - - - - 1260 5040 15120 37800 83160 166320 456 - - - - 90 420 1260 3150 6930 13860 1456 - - - - 420 1680 5040 12600 27720 55440 2456 - - - - 630 2520 7560 18900 41580 83160 12456 - - - - 1260 5040 15120 37800 83160 166320 3456 - - - - 420 1680 5040 12600 27720 55440 13456 - - - - 1260 5040 15120 37800 83160 166320 23456 - - - - 1260 5040 15120 37800 83160 166320 123456 - - - - 2520 10080 30240 75600 166320 332640

the ﬁrst column stands for the types for vertices, instead of writing k , k ..., k for { 1 2 m} GK(n : k1, k2 ..., km), we shortly wrote k1k2 . . . km the ﬁrst column stands for the value of n.

i . The graph GK(n : k , k ..., k ) is isomorphic to GK(n : n k ,..., n k ). 1 2 r − r − 1 n 1 ii . For the graph GK(n : k) with n 2k, α(GK(n : k)) = − (the classical Erdos-Ko-˝ ≥ k 1 Rado result). −

iii . For the graph GK(n : 1, 2, . . . , n 1), α(GK(n : 1, 2, . . . , n 1)) = n!/2. More − − generally, for GK(n : k1, k2 ..., kr ) if ki + kr+1 i = n for all i then the Kneser graph − 44 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCES OF SETS

has valency 1 and a coclique of largest possible size can be obtained by taking one

vertex of each edge. More generally, if there is an i with ki +kr+1 i = n (the middle node or a pair of symmetric nodes), then the Kneser graph is bi−partite, and again the maximum coclique size is half the total number of vertices. (Any acyclic graph is bipartite, so suppose there is a cycle in the graph. A vertex F in the cycle will

have a ki -set A and any vertex adjacent to it will have a kr+1 i-set B =[n] A. Any − \ path starting from F will consist of vertices including A as the ki set and B as the kr+1 i set in an alternating way. So any cycle including F will be an even cycle. Since− there is no odd cycle in the graph, it is bipartite. The graph is bipartite and regular, so the sizes of the two parts are the same. The maximal coclique will have size half of the number of the vertices.) iv. If n k +k then in pairs of adjacent vertices, the k -sets of one are disjoint from all ≥ i r i sets in the other, and therefore also the kj-sets with j < i. Now ki 1 just contributes − a factor ki , in this case, ki 1 − s=i ks α(GK(n : k1,..., ki 1, ki ,..., kr )) = α(GK(n : ki,..., kr )) , − ks 1 Ys=1 − v . Lower and upper bound: Vertices that are nonadjacent remain nonadjacent when

they are extended. That is, in GK(n : k1, k2 ..., kl 1) if u =(S1,..., Sl 1) and v = − − (T1,..., Tl 1) are nonadjacent, then u′ =(S1,..., Sl 1, Sl ) and v′ =(T1,..., Tl 1, Tl ) − − − are nonadjacent in GK(n : k1, k2 ..., kl 1, kl ). This gives both lower and upper bounds. Also, α(GK(k : k , k ..., k )) − α(GK(n : k , k ..., k )) for k n, since 1 2 l ≤ 1 2 l ≤ any pair of nonadjacent vertices in the ﬁrst graph will remain nonadjacent in the second graph: we take [k] and [n] as the universal sets, there are increasing sequence of subsets of k, for which there exists corresponding vertices in both graphs.

For a pair of those in α(GK(k : k1, k2 ..., kl )), if they are nonadjacent, then the corresponding pair of vertices in α(GK(n : k1, k2 ..., kl )) are also nonadjacent. Here are a few constructions for generating cocliques.

EXAMPLE 2.3.0.2. An explicit example of a coclique of size 12 for GK(5 : 1, 3) is the following: ( 1 , 1, 2, 3 ), ( 2 , 1, 2, 3 ), ( 3 , 1, 2, 3 ), ( 1 , 1, 2, 4 ), ( 2 , 1, 2, 4 ), { } { } { } { } { } { } { } { } { } { } ( 4 , 1, 2, 4 ), ( 1 , 1, 3, 4 ), ( 3 , 1, 3, 4 ), ( 4 , 1, 3, 4 ), ( 2 , 2, 3, 4 ), ( 3 , 2, 3, 4 ), { } { } { } { } { } { } { } { } { } { } { } { } ( 4 , 2, 3, 4 ). { } { } Another example ( 1 , 1, a, b ), ( 2 , 1, 2, c ), ( 3 , 1, 2, 3 ), ( 4 , 1, 2, 4 ), { } { } { } { } { } { } { } { } ( 5 , 1, 2, 5 ). This coclique reaches the upper bound as we will see by the Proposition { } { } 2.3.2. This example can be generalized as follows: for a set of size 2n+1, the way to obtain a coclique in GK(2n + 1 : 1, n + 1) coclique is to take all (n + 1)-sets missing an element, with all points as singletons for each of those, which gives a coclique of size (n+1) 2n . n+1 This construction gives cocliques of size 2 for GK(3 : 1, 2), so it is not maximal since ( 1 , 1, 2 ), ( 1 , 1, 3 ), ( 2 , 1, 2 ) is of size 3. But for GK(5 : 1, 3), the example is of { } { } { } { } { } { } size 12, GK(7 : 1, 4) the coclique obtained is of size 60, for GK(9 : 1, 5), the coclique is of size 280.

EXAMPLE 2.3.0.3. An explicit example for a coclique in GK(7 : 4) of size 15 is 1, 2, , , { ∗ ∗} 1, 3, 4, 5 , 1, 3, 4, 6 , 1, 3, 4, 7 , 1, 3, 5, 6 , 1, 4, 5, 6 . { } { } { } { } { } 2.3 MAXIMAL COCLIQUES IN GK(n : k1,..., kl ) 45

According to (v) in the above description, being adjacent in GK(7 : 1, 4) implies being adjacent in GK(7 : 4) and conversely, being nonadjacent in GK(7 : 4) implies being nonadjacent in GK(7 : 1, 4). So, the GK(7 : 4) example of size 15 gives us a lower bound for GK(7 : 1, 4) of 60. Similarly, being adjacent in GK(8 : 1, 5) implies being adjacent in GK(8 : 5), so being nonadjacent in GK(8 : 5) implies being nonadjacent in GK(8 : 1, 5), α(GK(8 : 5)) = 21, we have 5 points to choose, so 105 is a lower bound for α(GK(8 : 1, 5)).

EXAMPLE 2.3.0.4. α(GK(6 : 1, 4)) α(GK(5 : 1, 3)) + α(GK(6 : 3)), (equality holds by ≥ Proposition 2.3.2). To obtain a coclique example of size 22 in GK(6 : 1, 4), take all 4-sets including ta ﬁxed element, say ‘6’, and for any such 4-set A, take the vertex ( 6 , A) in { } GK(6 : 1, 4). Then, take a coclique of size α(GK(5 : 1, 3)) in GK(5 : 1, 3), for any vertex ( x , B) there, take the vertex ( x , B 6 ) GK(6 : 1, 4). This makes a coclique of size { } { } ∪{ } 22 in (GK(6 : 1, 4). For GK(7 : 1, 5) one can mimic the same argument as above, which implies α(GK(7 : 1, 5)) 37. ≥ The coclique given by the above construction does not reach the upper bound for GK(5 : 1, 3). The coclique obtained by this method is of size 10 = α(GK(5 : 2)) + α(GK(4 : 1, 2)), but there is a coclique of size 12 as we had seen above.

2.3.1 Foldings Here, we use maps that are called foldings which helps us to count the size of maximal cocliques for generalized Kneser graphs. The collection of cocliques in those graphs are invariant under foldings. Foldings are morphisms defined on chamber complexes. We define here foldings as maps which are idempotent and have preimages of size at most 2. We can make use of them to have an easier counting method for generalized Kneser graphs. First we will define a folding on the set of subsets of some n-set. Then we will see our map induces a folding on the set vertices of generalized Kneser graphs, and in the end it induces a folding on the set of cocliques of the graph. The simple folding ϕ where ϕ(j)= i and ϕ(x)= x for x = j (denoted by j i) on 6 → [n] is the idempotent map defined on [n] and then on subsets of [n] as follows:

ϕ(A) := ϕc c A, ϕc / A c c A, ϕc A , ϕA = A { | ∈ ∈ } { | ∈ ∈ } | | | | [ For instance, j i maps i, j, a, b to itself, j, a, b, c to i, a, b, c , a, b, c, d to → { } { } { } { } itself, where i, j a, b, c, d = . { }∩{ } ; The map ϕ map preserves cardinalities and inclusion. Deﬁnition on subsets of [n], naturally extends to increasing sequences of subsets of [n], (that is vertices of generalized Kneser graphs). Let C be a coclique of largest possible size in a generalized Kneser graph.

PROPOSITION 2.3.1. The map ϕ induces a cardinality-preserving map on the collection of all cocliques of a generalized Kneser graph, if we deﬁne the image of a coclique as:

ϕ(C) := ϕ(F) F C, ϕ(F) / C F F C, ϕ(F) C , { | ∈ ∈ } { | ∈ ∈ } [ This map is mapping cocliques of the Kneser graph to cocliques of the same size. 46 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCES OF SETS

Proof. By definition, ϕ(C) = C . | | | | First, we will see, for a pair of sets, ϕ either increases the intersection size or keeps it constant ( ϕ(A) ϕ(B) A B ). Suppose we have two distinct k-sets A, B. For the | ∩ | ≥ | ∩ | map j i, if j / A B, nothing changes in our sets, intersection size remains the same. → ∈ ∪ Suppose first j A B, if also i A B, nothing changes, intersection size remains the ∈ ∩ ∈ ∩ same, if i / A B, intersection size remains the same, if i A B, the intersection size ∈ ∪ ∈ \ increases. If j A B, and if i / A B, intersection size remains the same, if i A B, ∈ \ ∈ ∪ ∈ \ intersection size remains the same, i B A, intersection size increases, if i A B, ∈ \ ∈ ∩ intersection size remains the same. Finally, if j / (A B), nothing changes. So, similarly ∈ ∪ for vertices, a folding can only increase the number of points that the sets of equal size have in common in the sequences of sets. That is, for a pair of nonadjacent vertices F, F ′, ϕ(u) and ϕ(v) will be nonadjacent as well. The images of cocliques have to be cocliques. For a coclique C, suppose there are two vertices in ϕ(C) that are far apart. Since ϕ either keeps constant or increases intersection sizes of sets, those can not be of the form ϕ(F), ϕ(F ′). Then they are of the form F, ϕ(F ′). The vertex F is in ϕ(C), then ϕ(F) C. The vertices F and F ′ are in the coclique, they ∈ are not adjacent. Hence there are A F, B F ′, which are the specified k-sets of F and F ′ ∈ ∈ which are intersecting. But since ϕ(F ′) and F are adjacent, then ϕ(B) and A are disjoint, that is, ϕ(B) A < A B . This can happen only, j A B, and i / A and i / B. But then, | ∩ | | ∩ | ∈ ∩ ∈ ∈ B and ϕ(A) are disjoint. Since similar arguments holds for all not far specified sets of the flag pair F and F ′ we can say, F ′ and ϕ(F) are adjacent. These are both in the coclique C, so this is a contradiction.

Now we can apply foldings and shift a coclique to a lexicographically minimal form. We choose an order on the elements on the set, then use only foldings that makes things smaller in our order. if there exists a coclique C of a certain size, then there also exists a coclique of this same size that is invariant for all foldings j i where i < j. The image → of a set i, j,... under a folding map i j is itself. If two elements meet before, they { } → will meet in the image as well. For a coclique, the size of C and φ(C) are the same. moreover, distances inside the image are not larger then distances in C. Given a coclique in a Kneser graph for a Coxeter group, the image under folding is again a coclique, not smaller than the original one. when we apply folding, we will recognize, since things get lexicographically smaller, after ﬁnitely many steps, the result is stable, and we will obtain a coclique which is invariant under these foldings. This means, if there is a coclique C in a Kneser graph, then there is a lexicographically minimal coclique of the same size which can be obtained from C, by a series of foldings. If the transitive closure of a vertex under all foldings is not a coclique, then this vertex is not in C. Without loss of generality, C is lexicographically minimal, we can not make it smaller. W.l.o.g. ϕC = C.

EXAMPLE 2.3.1.1. Consider GK(6 : 2). Suppose we have minimal C we can not shift it any more to the left. Our coclique should be closed under folding. So, if some object is foldable to another thing that is far apart from it, the ﬁrst object is not in C. For instance, by 3 1 and 4 2, 3, 4 is mapped to 1, 2 which is adjacent to it. So C does not → → { } { } contain 3, 4 , or any other 2-set for which j i 3 by similar arguments. Hence, for { } ≥ ≥ now, all pairs in C should be of the form 2, i or 1, i . { } { } 2.3 MAXIMAL COCLIQUES IN GK(n : k1,..., kl ) 47

Similarly, if we apply 2 1 and 4 3 to 2, 4 , we obtain 1, 3 , which is adjacent → → { } { } to it. Since 2, 4 can be mapped to some other flag that is far apart from it, it can not be { } in C. But similarly, 2, 5 and 2, 6 can not be in C. { } { } Maximal cocliques can be 1, 2 , 1, 3 , 1, 4 , 1, 5 , 1, 6 or 1, 2 , 2, 3 , 1, 3 . { } { } { } { } { } { } { } { } EXAMPLE 2.3.1.2. Take GK(6 : 1, 4). Suppose ( 3 , 3, 4, 5, 6 ) is in C. It can be mapped { } { } to ( 1 , 1, 2, 5, 6 ) by foldings 3 1, 4 2. Image of ( 3 , 3, 4, 5, 6 ) should also be { } { } → → { } { } in C. But ( 3 , 3, 4, 5, 6 ) and ( 1 , 1, 2, 5, 6 ) are adjacent, so can not be in C simulta- { } { } { } { } neously. This is a contradiction, so ( 3 , 3, 4, 5, 6 ) is not in C. Similarly, we can show, { } { } ( 2 , 2, 3, 4, 6 ) is not in C, since it conflicts with ( 1 , 1, 3, 4, 5 ), and ( 3 , 1, 3, 5, 6 ) { } { } { } { } { } { } contradicts with ( 1 , 1, 3, 4, 5 ). { } { } The elements of C are of the form ( , 1, , , ) (but not ( 3 , 1, 3, 5, 6 )) or {∗} { ∗ ∗ ∗} { } { } ( , 2, 3, 4, 5 ). {∗} { } If ( 1 , 1, a, b, c ) is not in C, then we add it. We can not add it only if ( , 2, 3, 4, 5 ) { } { } {∗} { } is in C. In particular, ( 2 , 2, 3, 4, 5 ) conflicts with ( 1 , 1, 3, 4, 6 ), ( 3 , 2, 3, 4, 5 ) { } { } { } { } { } { } conflicts with ( 1 , 1, 4, 5, 6 ), ( 4 , 2, 3, 4, 5 ) conflicts with ( 1 , 1, 2, 3, 6 ) and fi- { } { } { } { } { } { } nally ( 5 , 2, 3, 4, 5 ) conflicts with ( 1 , 1, 2, 3, 6 ). This implies, we can replace ( , { } { } { } { } {∗} 2, 3, 4, 5 ) by ( 1 , 1, , , ) without decreasing the size of C. { } { } { ∗ ∗ ∗} We already accepted all ten vertices of the form ( 1 , 1, , , ). { } { ∗ ∗ ∗} Now, let us look at ( 2 , 1, 2, , ). Let us see if ( 2 , 1, 2, , ) conflict with any- { } { ∗ ∗} { } { ∗ ∗} thing of the form ( , 1, , , ). {∗} { ∗ ∗ ∗} Now, we replace ( 3 , 1, 3, 4, 5 ), ( 4 , 1, 3, 4, 5 ), ( 5 , 1, 3, 4, 5 ), ( 3 , 1, 3, 4, 6 ), { } { } { } { } { } { } { } { } ( 4 , 1, 3, 4, 6 ), ( 6 , 1, 3, 4, 6 ) by ( 2 , 1, 2, 4, 6 ), ( 2 , 1, 2, 3, 6 ), ( 2 , 1, 2, 3, 4 ), { } { } { } { } { } { } { } { } { } { } ( 2 , 1, 2, 5, 6 ), ( 2 , 1, 2, 3, 5 ), ( 2 , 1, 2, 4, 5 ). { } { } { } { } { } { } We have everything of the form ( 2 , 1, 2, , ). { } { ∗ ∗} Now finally we can add ( 3 , 1, 2, 3, 4 ) if it does not conflict with ( 5 , 1, 2, 5, 6 ). { } { } { } { } So we get

C = ( 1 , 1, , , ), ( 2 , 1, 2, , ), ( 3 , 1, 2, 3, , ), ( , 1, 2, 3, ) { { } { ∗ ∗ ∗} { } { ∗ ∗} { } { ∗ } {∗} { ∗} which is of size 22. n PROPOSITION 2.3.2. α(GK(n : 1, n 2)) = 2 + for n 4. − 3 ≥ Proof. For n = 4 the graph is bipartite, number of half of the vertices is 6, that is 2+ 4 3 n 1 obeying the statement. For n > 4 we show α(GK(n : 1, n 2)) = − + α(GK(n : − 2 1, n 3)) and then the result follows by induction. − n 1 First of all, α(GK(n : 1, n 2)) ( − )+ α(GK(n : 1, n 3)). To see that, take a − ≥ 2 − coclique of maximal size in GK(n : 1, n 3)). Rename any point i in any vertex of this − coclique with i +1, and add ‘1’ to the (n 3)-set of the vertices. This will still be coclique − in GK(n : 1, n 2). Then add all vertices having the 1-set 1 to this coclique. There are n 1 − { } − of these. This is a coclique of the required size. 2 Conversely, let C be a coclique of maximal size closed for all foldings j i where → i < j. n 1 If C contains all − vertices containing 1 , then every other vertex in C must have 2 { } a ‘1’ in its (n 2)-set (otherwise we can find two adjacent vertices in C), and C arises as − sketched and has the claimed size. If C does not contain all vertices containing 1 , then w.l.o.g. say ( 1 , 1, 4, 5, . . . , n ) { } { } { } is missing. We will simply write for each vertex (A, B) in our graph, (A, [n] B). So instead \ 48 A GENERALIZATION OF KNESER GRAPHS ON INCREASING SEQUENCES OF SETS of ( 1 , 1, 4, 5, . . . , n ) we will write ( 1 , 2, 3 ). For this notation, keep in mind for { } { } { } { } ( a , b, c ) a, b and c has to be distinct. Pick C of maximum size, and given that size { } { } with maximum number of vertices containing 1 . { } If ( h , 1, i ) C then also ( 1 , j, n ) C, since we can shift ( h , 1, i ) down to { } { } ∈ { } { } ∈ { } { } ( 1 , j, n ) (for instance by h 1, n i, j h when j > h), and C is closed for all { } { } → → → foldings. Suppose ( 1 , a, b ) / C. Since C is maximal, we cannot add ( 1 , a, b ) and it must { } { } ∈ { } { } conflict with at least two vertices ( a , 1, i ) or ( b , 1, i ) in C. But then ( 1 , j, n ) { } { } { } { } { } { } ∈ C for all j (1 < j < n) and in particular a, b < n. Since no two vertices in C are adjacent, and we can take j = a or j = b, we must have i = n (otherwise ( a , 1, i ) is adjacent to { } { } ( 1 , a, n )), and we see that 1 , a, b / C implies that ( a , 1, n ), ( b , 1, n ) C. { } { } { } { } ∈ { } { } { } { } ∈ On the other hand, ( a , 1, n ) C implies ( 1 , a, c ) / C for all c = 1, a, n. It { } { } ∈ { } { } ∈ 6 follows that if ( 1 , a, b ) / C for some a, b, then ( 1 , a, b ) / C for all a, b with { } { n 2} ∈ { } { } ∈ 1 < a, b < n. That is − missing elements. The possibly conflicting elements that are 2 present are of the form ( a , 1, n ), and there are at most n 2 of those. Since n > 4 we { } { } − can remove all vertices ( a , 1, n ) and add all vertices ( 1 , a, b ) without decreasing { } { } { } { } the size of C. Contradiction.

n CONJECTURE 2.3.3. α(GK(n : 2, n 3)) = 6 + 4 for n 6. − 5 ≥ n CONJECTURE 2.3.4. α(GK(n : 3, n 4)) = 20 + 15 for n 8. − 7 ≥ 2i 2i n CONJECTURE 2.3.5. α(GK(n : i, n 1 i)) = + for n 2(i + 1). − − i i 1 2i+1 ≥ − CONJECTURE 2.3.6. α(GK(2m + 1 : i, m + 1)) = mv/(2m + 1) where v is the total number of vertices, and 1 i m 1. ≤ ≤ − CONJECTURE 2.3.7. α(GK(2m : 1, m + 1))=(m 1)v/2m. − PART II

AUNIFYING APPROACH: FROM SETS TO GROUPS

3 GENERALIZATIONS TO BUILDINGSOF SPHERICAL TYPE

Most of the research in this thesis is initiated by the following question:

" Try to describe potentially maximum Erd˝os-Ko-Rado sets in the various geometries arising from a spherical diagram by circling any set of nodes."

To answer this question, we will look at spherical diagrams (see Figure 1.3) and try to find the size of maximum set of particular flags which are mutually not far apart. The meaning of “being far apart" changes up to the diagram and up to the flag types that we choose by circling certain nodes of the diagrams.

3.1 Understanding the graphs

Classical Kneser graphs, generalized Kneser graphs and q-Kneser graphs that are defined below are specific examples of graphs that are defined in this chapter in relation to buildings. After defining Kneser graphs on buildings, we will briefly review the generalized Kneser graphs by examples, taking the new definition into account.

3.1.1 q-Kneser graphs

One can generalize Kneser graphs by working on vector spaces over a ﬁnite ﬁeld Fq instead of sets.

DEFINITION 3.1.1.1. Let for integers m, n with 1 m n/2, K(n, m, q) be the graph on ≤ ≤ the m-spaces in an n-space, adjacent when intersecting trivially. This graph is known as the q-Kneser graph.

In Section 2.1.4, we draw a relationship between the Johnson graphs and the Kneser graphs. The q-analogue of Johnson graph is called Grassmann graph, and a similar relationship exists in between the q-Kneser graphs and the Grassmann graphs. 52 GENERALIZATIONS TO BUILDINGS OF SPHERICAL TYPE

Let V be a vector space of dimension n over Fq. The graph with as vertices the subspaces of dimension m of V , where two subspaces are adjacent when they meet in a hyperplane of each is called the Grassmann graph, Grassmann (n, m)q. This graph is distance-regular, with diameter d = min(e, n e). − Two vertices are adjacent in K(n, m, q) when they are at maximal distance in Grass- mann (n, m)q. n 1 Cocliques in K(n, m, q) have size at most − with equality only for the col- m 1 − q lections of all m-spaces containing a fixed point and (in case n = 2m) for the collections of all m-spaces contained in a fixed hyperplane. The upper bound was proved for n > 2m by Hsieh [71], and for n = 2m by Greene & Kleitman [63]. Equality holds only for all m-spaces on a fixed point (or its dual, in case n = 2m). This was proved for n > 2m+1 and for n = 2m+1, q > 2 by Hsieh [71], for n = 2m+1, q = 2 by Godsil & Newman [62], for n = 2m by Frankl & Wilson [56]. Blokhuis et al. [10] showed for q 3 and n 2m + 1, or q = 2 and n 2m + 2, that ≥ ≥ ≥ n m + 1 K(n, m, q) has chromatic number − . 1 q Eisfeld et al. [52] showed for m = 2, and Blokhuis et al. [12] showed for m = 3 and m m 1 for m < q log q q that K(2m, m, q) has chromatic number q + q − . − Hence, the chromatic number of the q-Kneser graph are settled for n > 2m except when n = 2m + 1 and q = 2.

3.1.2 Kneser graphs for Coxeter groups

Let W be a ﬁnite irreducible Coxeter group, let its longest elements be called w0, and let R be the generating set of involutions as usual.

DEFINITION 3.1.2.1. For any J R let X be the subgroup of W generated by R J. The ⊆ \ graph whose vertices are the cosets wX for w W, and for which a pair of vertices uX , vX 1 ∈ are adjacent when X v− uX = Xw0X is called a W-Kneser graph.

If J w0 = J, then the W-Kneser graph for J has degree 1, and questions about maximal cocliques and chromatic number are trivial. This means that, here only Coxeter groups with diagram An, Dm (m odd) or E6 are of interest. For a Coxeter group of type Xn, with flag type J defining the type of (vertices) flags, we denote the graph by K(Xn(1), J).

EXAMPLE 3.1.2.2. The classical Kneser graphs are a special case of the Kneser graphs for Coxeter groups. Let W be Sym(n), the symmetric group on an n-set, say [n]. A suitable set of generators is R := r1,..., rn 1 , where ri is the transposition (i(i + 1)). The { − } Coxeter diagram is An 1. Let J = ri , where n 2i. Then the W-Kneser graph for J is − { } ≥ the classical Kneser graph K(n, i).

EXAMPLE 3.1.2.3. Let W be a group of type A5(1), that is Sym(6). Let R be as above. The graph K(A5(1), 4) has the 4-subsets of [6] as vertices. 3.1 UNDERSTANDING THE GRAPHS 53

For W, when J = (45) , X = R J = (12), (23), (34), (56) , X stabilizes the 4-set { } 〈 \ 〉 〈 〉 F = 1, 2, 3, 4 . { } Vertices are of the form wX , cosets of X in W, and can be represented by the shortest element in the coset. Number of cosets is W / X = 6!/(4!2)= 15. | | | | The cosets:

X , (45)X , (34)(45)X , (23)(34)(45)X , (12)(23)(34)(45)X , correspond to the vertices of a coclique, and the corresponding 4-sets are: 1, 2, 3, 4 , { } 1, 2, 3, 5 , 1, 2, 4, 5 , 1, 3, 4, 5 , 2, 3, 4, 5 . { } { } { } { }

EXAMPLE 3.1.2.4. We will describe the graph K(A3(1), 2). Let W be a group of type A3(1), that is Sym(4)= (12), (23), (34) . For W, when J = (23) , X = R J = (12), (34) , X 〈 〉 { } 〈 \ 〉 〈 〉 stabilizes the 2-set 1, 2 . { } Vertices are of the form wX , cosets of X in W, and can be represented by the shortest element in the coset. The number of cosets is W / X = 4!/4 = 6. | | | | The cosets of an example coclique are:

X , (23)X , (34)(23)X , and the corresponding 2-sets are: 1, 2 , 1, 3 , 1, 4 . { } { } { }

In each of the above two examples, J is a 1-set. There are partial results for J > 1. | | In Chapter 2, we introduced generalized Kneser graphs which are the W-Kneser graphs for the Coxeter group An. The graphs that we called GK(n : k1,..., kl ) will be called K(An 1(1), rk ,..., rk ) from now on. In Chapter 2 we gave sizes of largest cocliques in − { 1 l } many cases (including cases where J > 1). | | EXAMPLE 3.1.2.5. Let W have Coxeter diagram D where n is odd, n 3, and let r be n ≥ one of the two vertices not ﬁxed under conjugation by w (n’th or (n 1)’th node). 0 − The W-Kneser graph for r has as vertices the binary vectors of length n and of even { } weight, where two such vectors are adjacent when they have Hamming distance (n 1). n−2 According to Kleitman [76], maximal cocliques in this graph have size at most 2 − s − 1 2 when n = 2s + 1. To see how this bound is attained, ﬁrst, one can throw away 2 s a coordinate so that, the even weight restriction is not required. Now, we have length (n 1) in the codewords, and the distance allowed is (n 3). To obtain this, we take all − − codewords of weight i for all i (n 3)/2. ≤ −

EXAMPLE 3.1.2.6. Let W have Coxeter diagram E6 and let r be an end node not ﬁxed under conjugation by w . The W-Kneser graph for r is the collinearity graph of the 0 { } unique generalized quadrangle with parameters GQ(2, 4), the complement of the Schläﬂi graph. Maximal cocliques have size 6, and the chromatic number is 6 [19].

3.1.3 Generalization of Kneser graphs over buildings of spherical type In the most general case, our graphs are related to Coxeter groups and groups with Tits systems, in particular, ﬁnite simple groups of Lie type. 54 GENERALIZATIONS TO BUILDINGS OF SPHERICAL TYPE

The vertices of the graph and the distance function

Let G be a ﬁnite simple group of Lie type, W be its Weyl group, with R as above. Let J R and put X = R J , a subgroup of W, and P = BXB, a parabolic subgroup ⊆ 〈 \ 〉 of G. The cosets gP with g G are called objects of type J or J-objects . ∈ Earlier we said, in the generalization of Kneser graphs, we substitute sets with cosets, and we substitute being disjoint for sets with being "far apart" for cosets. A geometric way of expressing this is to talk about buildings with a W-valued distance function ∂ . Distances between two J-objects are double cosets X wX with w W. For instance, if ∈ J = R i.e., the objects of our interest are the chambers of the building, the cosets gB 1 with g G, the distance ∂ between them is ∂ (gB, hB) = w when Bh− gB = BwB. ∈ More generally, two conjugate parabolic subgroups x P and y P, where P = BSB for some 1 S < W, are said to be in relation SwS when Py− x P = BSwSB. Instead of indexing relations with double cosets, one can also index them using the unique shortest element of the double coset.

w Two types I and J are called opposite when J = I 0 . An I-object gP and an I ′-object 1 g′ P′ are called opposite when I and I ′ are opposite and moreover P g− g′ P′ = Pw0 P′. 1 Two I-objects gP and g′ P are called far apart when P g− g′ P = Pw0 P. The oppositeness graph for opposite types I, J is the graph with as vertices the I-objects and J-objects, where vertices are adjacent when opposite.

The graph

Given a group G with a (B, N)-pair and Weyl group (W, R) and J R one has a building ⊆ with set of chambers G/B and standard apartment WB/B. The following graph Γ is a generalization of Kneser graphs over buildings of spherical type, due to the distance relation deﬁned above between the vertices.

DEFINITION 3.1.3.1. The Kneser graph for type I Γ is the graph with as vertices the I-objects, adjacent when far apart. That is, the vertices of Γ are the cosets gP with g G, 1 ∈ where gP is adjacent to g′ P when P g′ g− P = Pw0 P.

Here, as described above, being far apart is having distance w0, the longest word of W, and adjacency as defined above, is a natural generalization of the concept of Kneser graphs to this setting: two objects adjacent when they are opposite or as far apart as w0 possible for such objects. When I = I, and in particular when w0 induces the trivial graph automorphism, oppositeness graph and the Kneser graph coincide. Suppose we made a particular circling of the nodes. We take an arbitrary flag F of that type, find out the stabilizer P of that flag, and create the graph defined above for this P. For each flag F0, there is a g for which gF = F0. This g is not unique, indeed, all elements in gP wil map F to F0. Thus, the coset of P corresponding to F is gP. That is how the flags and cosets are related.

For Lie group of type Xn(q), with flag type J defining the type of (vertices) flags, we denote the graph by K(Xn(q), J).

EXAMPLE 3.1.3.2. The q-Kneser graphs are a special case of Kneser graphs for groups of Lie type. 3.1 UNDERSTANDING THE GRAPHS 55

Take the Coxeter diagram is An 1. Let G = GL(n, q). Then B is the subgroup of non- singular upper triangular matrices,− N is the subgroup of monomial matrices, W is the symmetric group Sym(n) (the permutation matrices having 1 non-zero entry (1) at each row and column), and R = r1,..., rn 1 , where ri is the transposition (i, i + 1). Let { − } J = r , where n 2i. Then the Kneser graph for J is the q-Kneser graph K(n, i, q). { i} ≥

A useful graph defined on the geometries: We related classical Kneser graphs to Johnson graphs; for a pair of vertices, being at mutual distance k in the J(n, k) is equivalent to being adjacent in the K(n, k). Similarly, we related q-Kneser graphs to Grassmann graphs, being at mutual distance k in the Grassmann graph is equivalent to being adjacent in the K(n, m, q). Now, we will draw such a relation for Kneser graphs of flags. Again, the related graphs have the same set of vertices, and for a pair of vertices, being at maximum possible mutual distance in this graph is equivalent to being adjacent in the Kneser graph. Make a graph Γ, whose vertices are the maximal flags (or chambers) of the geometry, and they are adjacent when the maximal flags meet in I 1 elements. Name the | |− diameter of this graph as d. The Kneser graph on maximal flags is the graph which can be obtained by taking vertices of Γ, and putting an edge between a pair of vertices if and only if their mutual distance is d in Γ (that is what we called far apart before). The above graph Γ is for the maximal flags. We will make use of Γ again for any Kneser graph on other flags. Suppose we choose our flag type for vertices to be J I. Then we have some flags ⊂ of the corresponding type as vertices of our graph. There are for sure some maximal flags which cover our flags, since J I. A flag F consists of a subset of X , which include ⊂ at most one element of each type, and all elements in the flag are incident. There is a maximal flag F ′ which can be obtained by adding elements of the missing types of F to it, which are incident to all element of F. Two vertices F1 and F2 are adjacent in ΓJ , when there are two corresponding maximal flags F1′ and F2′ in Γ that are adjacent. Name the diameter of ΓJ as dJ . The Kneser graph for this flag type is the graph obtained by joining only the vertices whose mutual distance is dJ in ΓJ .

3.1.4 Taking sums over the Weyl group

For any Kneser type graph Γ = K(Xn(q), J) on a building related to the group G of Lie type, there is a corresponding W-Kneser graph Σ= K(Xn(1), J). Our aim is to discover the relationships between the cocliques of Γ and Σ. For instance, if C is a coclique in Σ, C = wX w A where X is the stabilizer of one ﬂag and w is the shortest element of { | ∈ } wX , then F = BwP w A is also a coclique in Γ. { | ∈ } Here are some examples, where a maximal coclique in a Kneser graph for a Coxeter group is related to a maximal coclique in a Kneser graph for a group of Lie type. The conditions of extension is the topic of the next chapter. The number of maximal ﬂags is (see [21], Proposition 10.7.3. (i) or [36] Section 8.6): l(w) G/B = BwB/B = Σw W q . (3.1) | | | · | ∈ w[W ∈ 56 GENERALIZATIONS TO BUILDINGS OF SPHERICAL TYPE

CONJECTURE 3.1.1. There is a maximal coclique of largest size in a Kneser graph for a group of Lie type, with a set A, (A W), such that, if we take the sum ⊂ l(w) Σw Aq ∈ then we will have the size of the coclique.

EXAMPLE 3.1.4.1. For A2, we saw that the corresponding Weyl group is Sym(3). There, the elements have lengths 0, 1, 1, 2, 2, 3, the size of S3 = 6. According to Equation 3.1

q0 + q1 + q1 + q2 + q2 + q3 = q3 + 2q2 + 2q + 1 =(1 + q)(1 + q + q2) is the number of maximal flags in A2(q). Number of maximal flags is 3! = 6 for the Coxeter group case. The list of maximal flags is the following for A (1): ( 1 , 1, 2 ), 2 { } { } ( 1 , 1, 3 ), ( 2 , 1, 2 ), ( 2 , 2, 3 ), ( 3 , 1, 3 ), ( 3 , 2, 3 ). { } { } { } { } { } { } { } { } { } { }

For A2(q), the number of maximal flags that are mutually not far apart is three in the Coxeter group case (e.g. ( 1 , 1, 2 ), ( 2 , 1, 2 ), ( 1 , 1, 3 )), and 2q + 1 in the group { } { } { } { } { } { } of Lie type case. To count this, take a fixed flag (m, L), then take all flags (r, M) where r = m (there are (q2 1)/(q 1) 1 lines including m except L), or take the flags where − − − M = L (there are (q2 1)/(q 1) 1 points in L except m), or take the flag r = m, M = L. − − − l(w) 0 1 1 Obeying the conjecture, Σw Aq = w A BwB/B = q + q + q = 1 + 2q. The ∈ | · | corresponding set of words in the Weyl groupS is∈ A = 1, r, s . { }

EXAMPLE 3.1.4.2. Look at A3 with three generating involutions r = (12), s = (23), t = (34) of W. Take a starting flag F =(1, 1, 2 , 1, 2, 3 ). Here, r represents changing the 0 { } { } point in F0, s represents changing the line, t represents changing the plane. The stabilizer of (point, line, plane) flag F0 is the Borel group B. The stabilizer of (point,plane) flag ( 1 , 1, 2, 3 ) is B BsB. { } { } ∪ The total number of (point, plane) flags is the number of cosets of the group 1, s in { } S4, that is, it is 4!/2 = 12. Each such coset can be represented by the shortest element inside, that is for example, represent w 1, s = w, ws by w when l(w) < l(ws). { } { } We have the following coclique

( 1 , 1, , ), ( 2 , 1, 2, ), ( 3 , 1, 2, 3 ) { } { ∗ ∗} { } { ∗} { } { } of size α(K(A (1), 1, 3 )). The right-reduced coset representatives, the shortest elements 4 { } in 1, s , t, ts , st, sts , r, rs , rt, r ts , sr, srs {{ } { } { } { } { } { }} are respectively: 1, t, st, r, rt, sr.

The corresponding coclique in the Kneser graph for the group of Lie type has size 3q2 + 2q + 1 (See Table 5.1). 3.2 COCLIQUES ON FLAGS OF PG(4, q) 57

EXAMPLE 3.1.4.3. Consider the diagram A3. In the Coxeter group case, our group G is Sym4. Consider the graph on point,line-flags. Our basis flag is ( 1 , 1, 2 ). The involution (12) represents changing the point, { } { } and (23) represents changing the line, and (34) generates the stabilizer (S = (34) , 〈 〉 P = BSB, for the W-Kneser graph, P = (34) ). The number of point-line flags is the 〈 〉 number of the cosets of (34) inside S , which is 12. An example of maximal coclique is 〈 〉 4 ( 1 , 1, 2 ), ( 2 , 1, 2 ), ( 1 , 1, 3 ), ( 3 , 1, 3 ), ( 1 , 1, 4 ), ( 4 , 1, 4 ). { } { } { } { } { } { } { } { } { } { } { } { } The list A of shortest elements of the Weyl group mapping ( 1 , 1, 2 ) to the flags of { } { } l(w) type 1, 2 have lengths 0, 1, 1, 2, 2, 3, 2, 3, 3, 4, 4, 5. Then, the sum Σw A q = 1 + 2q + { } ∈ 0 4 2 3q2 + 3q3 + 2q4 + q5 = . This is the number of flags of type 1, 2 . 2 q 1 q { } A coclique of flags that are mutually non-far is: ( 1 , 1, 2 ), ( 2 , 1, 2 ), ( 1 , 1, 3 ), { } { } { } { } { } { } ( 3 , 1, 3 ), ( 1 , 1, 4 ), ( 4 , 1, 4 ). The list B of shortest elements of the Weyl group { } { } { } { } { } { } mapping ( 1 , 1, 2 ) to those flags have lengths respectively 0, 1, 1, 2, 2, 3. When we take { } {l(w) } 2 3 the sum Σw Aq , the result is 1 + 2q + 2q + q . ∈ As n and q increases, it is not as easy as the above examples to see what happens in the group of Lie type case.

3.2 Cocliques on ﬂags of PG(4, q)

In this section, we give maximal coclique sizes for Kneser type graphs on certain ﬂag types in PG(4, q).

3.2.1 Graphs on point-hyperplane pairs These graphs in the more general case is the focus of Chapter 5. The adjacency condition for a pair of vertices (ﬂags) (P, H), (P′, H′) is having P outside H′ and having P′ outside H. The upper bound for the sizes of maximal cocliques is proved to be 1+2q+3q2 +4q3, and the cocliques for which this upper bound is attained have been characterized in [11]. (See Theorem 5.1.1.)

3.2.2 Graphs on point-line flags The adjacency condition for the Kneser graph on point-line flags of PG(4, q) is the lines’ being disjoint. The points of the flags do not play a role for adjacency. Consider the q-Kneser graph on lines for PG(4, q). There, pair of lines are nonadjacent as long as they intersect. Consider a pair of intersecting lines L1 and L2. If any line intersecting both, does not pass through their intersection, then it is in the plane spanned by these two lines. Pick such a line and call it L3. We can only extend the coclique on L1, L2, and L3 by taking all the lines on this plane L , L and L . Any line outside this plane will be 〈 1 2〉 3 adjacent to every line on this plane. Any line hitting this plane at a point will be adjacent to a line of the plane. For this graph, a maximal coclique consists of lines passing through a fixed point. This coclique can be extended to a maximal coclique C′ for a Kneser graph on point-line flags, simply by taking for any line L in C, all (p, L) pairs where p is any point on L. The 2 3 coclique C′ will be of size (1 + q)(1 + q + q + q ). 58 GENERALIZATIONS TO BUILDINGS OF SPHERICAL TYPE

3.2.3 Graphs on point-plane ﬂags

For the Kneser graph Γ on point-plane pairs of PG(4, q), the adjacency condition for a pair of vertices (p1, Π1), (p2, Π2) is, the planes’ intersection to be a point, say p, different 2 3 2 from p1 and p2. Here the maximal cocliques have size (q + q + 1)(q + q + q + 1) [13]. We had seen by Proposition 2.3.2 that, in the thin case, maximal cocliques have size 12, and Example 2.3.0.2 is an explicit coclique example of that size in the thin case. In the thin case, there are several nonequivalent examples reaching the upper bound, but in the thick case, there is only a single type.

Lower bound

First, we will see α(K(A (q), 1, 3 )) (q2 + q + 1)(q3 + q2 + q + 1). By taking all planes 4 { } ≥ in a hyperplane in PG(4, q) with all points on them, one obtains a coclique, since in an hyperplane here, any pair of planes has to meet at least at a line. This coclique has the desired size.

Upper bound

To see why (q2 + q + 1)(q3 + q2 + q + 1) is the largest possible coclique, we will classify planes involved in a maximal coclique C. Let for a plane Π, for distinct points a and b, (a, Π) and (b, Π) are both in C. Let c be an arbitrary point in Π. For any other vertex (p∗, Π∗) in C, either Π Π∗ is at least a line, or Π Π∗ is a point and this point is p∗. In ∩ ∩ both cases, (p∗, Π∗) is also nonadjacent to (c, Π). So if for a plane Π and a pair of distinct points a, b on it, (a, Π) and (b, Π) are both in C, then for any other c Π, (c, Π) is also ∈ in C. such a plane we call a heavy plane. Any heavy plane is seen q2 + q + 1 times in C, once for each point on it. Otherwise, a plane which is involved in C is seen only once. In that case, we call it a light plane. Two heavy planes have at least one line in common. Otherwise, since all the points of heavy lines are involved in the coclique, there would be an adjacent pair of vertices on those planes, which is a contradiction. If all planes are heavy, then in PG(4, q) we have a set of planes pairwise intersecting in a line. In the dual, this means, having a set of lines, pairwise lying on a plane. that is, either all of the lines are passing through a ﬁxed point or they all lie on a ﬁxed plane. Consecutively in these cases there can be at most q3 + q2 + q + 1 and q2 + q + 1 such planes. So our example is optimal and unique if all the planes are heavy. A light plane Π (with the involved point a) intersect all heavy planes on a line or on the point a. Otherwise, for the points on the heavy plane outside the intersection, the corresponding vertices would be adjacent to (a, Π). Suppose Π meets a heavy planes on a line, and another on a point. Then, this line has to be on a, otherwise, we would have a pair of intersecting planes in a point a, and a third plane, intersecting both in lines which do not intersect, which is impossible. That is, the intersections of a light plane Π with the heavy planes all contain a, or it intersect all heavy planes on lines. Case (i) Suppose all heavy planes contain a. In the residue of a, all heavy planes are a set of pairwise intersecting lines. They are either on a point or on a plane. In both cases, there are q2 + q + 1 of them. One has to add at least q3(q2 + q + 1) light planes to reach (q2 + q + 1)(q3 + q2 + q + 1). 3.2 COCLIQUES ON FLAGS OF PG(4, q) 59

Case (ii) Suppose all heavy planes meet Π in a line. Consider two heavy planes Π1, Π2 intersecting Π in different lines. The planes Π1, Π2, and P are three planes mutually intersecting in a line. Then they are in a hyperplane H = Π, Π = Π, Π . So, either 〈 1〉 〈 2〉 all heavy planes intersect Π in the same line, and there are at most q2 + q + 1 of them (number of planes on a line in PG(4, q)), or they do not intersect Π in the same line. For the above Π1 and Π2 suppose there is another heavy plane call it Π3. This plane intersect all Π1, Π2, and Π in a line. Since Π1, Π2, Π span H together and they all mutually intersect in a line, Π3 is in H as well. Hence in that case (heavy planes do not intersect Π in the same line), all heavy planes are in the same hyperplane H on Π. Suppose all planes are in the same hyperplane H on Π. For the light plane Π, (a, Π) in C implies for any other b Π, (b, Π) can not be in C; (b, Π) should be conflicting ∈ with a vertex in C. So, there is a flag (c, Π0) in C which is adjacent to (b, Π): b is not in the plane Π , c is not in the plane Π and Π Π is a single point (but all heavy planes 0 ∩ 0 meet Π in a line, so Π0 is light). That is, Π0 is not in H (any pair of distinct planes in a solid intersect in a line). Then H intersect Π0 in a line L. But as we had seen earlier, for a light plane, there are two options: either all heavy planes contain its flag point, or it intersect it in a line, necessarily L (since Π0 intersect all heavy planes in a line, and all heavy planes are inside H, and Π0 intersect H in L, so Π0 intersects all heavy planes in L). In H, there are q + 1 planes on L. So in the latter case, we have at most q + 1 heavy planes. For the former case, all heavy planes contain c, and on c, there are q2 + q + 1 planes in the residue of c. So in all the above cases, at least q3(q2 + q + 1) light planes are needed to reach the desired coclique size. We had seen earlier, if a light plane exists, either a heavy plane meets it at a point, or if all the heavy planes meet it at a line, then there is another light plane intersecting it in a point. So if every two flag planes in C meet in a line all planes are heavy, but now we are considering otherwise. So take a pair of flags in C, whose planes intersect in a point, call them (a, Π1) and (b, Π2). At least one of a, b should be at the intersection, say a is in the intersection. If (c, Π) is in C where a, b are not in Π, then (c is in Π or Π Π is a line) 1 1 ∩ and (c is in Π2 or Π2 Π is a line). If c is in Π2, then Π Π1 is a line L, and Π= c, L . 2 ∩ 2 ∩ 〈 〉 There are q lines in Π2 not on a, and there are q +q points on Π1 except b. Hence there 2 2 2 2 are q (q + q) such flags. Similarly, there are q (q + q) flags whose plane intersect Π2 in a line and point is in Π different from a. Since a is not in Π, Π Π and Π Π can not 1 ∩ 1 ∩ 2 be lines simultaneously. The number of planes on a, or b is at most (2q2 + 1)(q2 + q + 1): in PG(4, q) the number of planes on a fixed point is (q2 + 1)(q2 + q + 1), for a and b, it is (2q2 + 1)(q2 + q + 1), since we take out the ones counted twice, the ones on bot a and b. But for q 4, the sum (2q2 + 1)(q2 + q + 1)+ 2q2(q2 + q) < (q3 + 1)(q2 + q + 1). ≥ If this situation never occurs for a = b, then any two light flags with the same point a have planes meeting in a line, so that there are at most q2 +q +1 light flags on a, and the estimate becomes 2q2(q2 +q)+2(q2 +q +1). But this is smaller than (q3 +1)(q2 +q +1) for all q. So, the situation with a = b does occur, and our estimate becomes 2q2(q2 +q)+(q2 + 1)(q2 + q + 1), and this is smaller than (q3 + 1)(q2 + q + 1) also for q = 3. It follows that q = 2. The term 2q2(q2 + q) came from counting planes not on a with a point on one side, and a line on the other side. But such flags may conflict. Two flags with distinct points and distinct lines with points on the same side will always conflict, so that this bound can 60 GENERALIZATIONS TO BUILDINGS OF SPHERICAL TYPE be replaced by 2(q2 + q), and the contradiction is also there for q = 2.

3.2.4 Graphs on line-plane flags Let Γ be the graph with as vertices the line-plane flags (incident line-plane pairs) in PG(4, q), where two flags (L, A), (L′, A′) are adjacent when they are in general position, 2 i.e., when L A′ = L′ A = 0. The maximum-size cocliques in Γ have size (q + q + ∩ ∩ 1)(q3 + 2q2 + q + 1) [9]. Brouwer and Blokhuis proved cocliques of size (q2 + q + 1)(q3 + 2q2 + q + 1) in Γ can be constructed in the following four ways. (i) Fix a solid S , and take all (L, A) with A S , together with (ia) all (L, A) with 0 ⊆ 0 L = A S and P L, for some fixed point P S , or (ib) all (L, A) with L = A S and ∩ 0 0 ⊆ 0 ⊆ 0 ∩ 0 L A , for some fixed plane A S . ⊆ 0 0 ⊆ 0 (ii) Or, fix a point P , and take all (L, A) with P L, together with (iia) all (L, A) with 0 0 ⊆ A = L + P and A S for some fixed solid S P , or (iib) all (L, A) with A = L + P and 0 ⊆ 0 0 ⊇ 0 0 L A for some fixed line L P . 0 ⊆ 0 ⊇ 0 This can be generalized further:

CONJECTURE 3.2.1. [19] For K(A (q), n, n + 1 ) the size of a maximal coclique can be at 2n { } 2n 1 q2n 1 most − − + qn. n 1 q 1 − q − 4 UNIQUE COCLIQUE EXTENSION PROPERTY

4.1 Introduction

The famous Erdos-Ko-Rado˝ result bounds the size of a collection of pairwise intersecting k-subsets of an n-set, and determines the cases with equality. This can be viewed as a result about maximal cocliques in the Kneser graph K(n, k). Kneser graphs can be generalized from sets to vector spaces over Fq, moreover, they can be generalized in the theory of buildings. In the spirit of Erdos-Ko-Rado˝ we want to find information about maximal cocliques in this general buildings setting, about their size (when finite) and structure. While studying generalizations of the Erdos-Ko-Rado˝ theorem and the chromatic number of Kneser-type graphs, one needs information about maximal cocliques of near- maximum size, cf. [11, 12]. In this chapter, we describe a simple construction that in the most interesting cases produces all such near-maximum cocliques. We say that a pair (Γ, Σ) consisting of a graph and an induced subgraph has the unique coclique extension property when every maximal coclique C of Σ is contained in a unique maximal coclique D of Γ. Necessarily D consists of all vertices x of Γ such that C x is a coclique, and this property claims that D thus defined is a coclique. ∪{ } Kneser graphs on buildings induce Kneser graphs on apartments. In general the maximal cocliques of an apartment are much easier to classify than the maximal cocliques of the building. The present chapter considers the question when a maximal coclique of an apartment extends uniquely to a maximal coclique of the building. That is what we call unique coclique extension property above. This will turn out to be the case when the corresponding representation has minuscule weight and also when the diagram is simply laced and the representation is adjoint.

4.1.1 Method We will embed the vertices of the graph to a vector space and define a bilinear map on the vector space. This map will define adjacency for a pair of vertices. Suppose V, V ′ are vector spaces and we have a bilinear map µ : V V V ′ that is × → reflexive. For A V, write A⊥ = v V µ(v, a)= 0 for all a A . If the vertex set V Γ of ⊆ { ∈ | ∈ } Γ can be embedded into the projective space PV corresponding to V in such a way that 62 UNIQUE COCLIQUE EXTENSION PROPERTY

vertices x, y are non-adjacent in Γ if and only if µ(x, y)= 0, then D = C⊥ V Γ(D consist ∩ of vertices that are nonadjacent to all elements of C). If also D C (the linear span of ⊆ 〈 〉 C in V -that is µ(v , v ) = 0 for v , v D, any pair of elements in D are nonadjacent) 1 2 1 2 ∈ then it follows that D is a coclique. Below we show that this setup works for the Kneser graph Γ on the objects of some type J in a building of spherical type, with as subgraph Σ that induced on such objects inside a fixed apartment, for suitable choices of J. Given a group G with (B, N)-pair and Weyl group (W, R) and J R one has a building with set of chambers G/B and standard ⊆ apartment WB/B. Put X = R J , a subgroup of W. Put P = BXB, a parabolic subgroup 〈 \ 〉 of G. Let W have longest element w0. The vertices of Γ are the cosets gP with g G, 1 ∈ where gP is adjacent to g′ P when P g′ g− P = Pw0 P. The vertices of Σ are the cosets wP with w W. ∈ We can embed VΓ in a vector space as follows: Let F be an algebraically closed field. Let λ be the smallest dominant weight which has zeroes and ones on the basis of the fundamental weights, such that X = Wλ. Let V = V (λ), the irreducible FG-module of highest weight λ. Now P is the stabilizer in G, which is an algebraic group over F, in its action on PV of the projective point v , where v V (λ) is the highest weight vector. 〈 λ〉 λ ∈ And thus the vertex gP can be identified with the projective point gv . 〈 λ〉

4.1.2 Matroids Here we give a very short introduction to matroids. For more information on matroid theory, see [35] or [87], whose deﬁnitions we followed in this section.

DEFINITION 4.1.2.1. Let S be a set. A matroid M on S is a pair (S, ), where is a I I non-empty family of subsets of S (called independent sets) satisfying

(i) if I and J I, then J , ∈I ⊆ ∈I (ii) (the exchange property) if I , I and I < I , then there exists e I I such 1 2 ∈I | 1| | 2| ∈ 2\ 1 that I e . 1 ∪{ }∈I Elements of are ﬁnite sets. I Given a matroid, an subset of S not in is called a dependent set. For U S, a I ⊆ subset B of U is called a base of U if B is inclusionwise maximal independent subset of U. That is, B and there is no Z such that B Z U. ∈I ∈I ⊂ ⊂ Condition (ii) guarantees under condition (i) that for any subset U of S, any two bases of U have the same size, which is the rank of U, denoted by rM (U). Maximal independent sets are called bases of matroid M. Consider a family of subsets of S, B whose elements are bases of a matroid on S. Then

(a) no element of contains any other, B (b) of B , B , x B B , then there exists y B B such that (B y ) x . 1 2 ∈B ∈ 1\ 2 ∈ 2\ 1 2\{ } ∪{ }∈B One can come up with new matroids given a set of matroids by making use of union operation on sets. For more information about matroid union, see [87] Chapter 42. The following result is due to Edmonds [50]: Let M = (S , ),..., M = (S , ) be 1 1 I1 k k Ik 4.1 INTRODUCTION 63 matroids. Deﬁne the union of these matroids as M M =(S . . . S , . . . ), 1 ∨···∨ k 1 ∪ ∪ k I1 ∨ ∨Ik (Si need not to be disjoint) where := I I I ,..., I . I1 ∨···∨Ik { 1 ∪···∪ k | 1 ∈I1 k ∈Ik} THEOREM 4.1.1. (matroid union theorem) Let M =(S , ),..., M =(S , ) be matroids 1 1 I1 k k Ik with rank functions r ,..., r respectively. Then M M is a matroid again with rank 1 k 1 ∨···∨ k function given by:

r(U)= minT U ( U T + r1(T S1)+ + rk(T Sk)) ⊆ | \ | ∩ · · · ∩ for U S S . ⊆ 1 ∪···∪ k in particular, Let M and M be two matroids on the set I, with rank functions r , r . 1 2 M1 M2 Let M be the matroid whose independent sets are the unions of an independent set in

M1 and an independent set in M2 (this is different from taking the union of the set of independent sets of M1 and M2 to be the set of independent sets of M). Then M has rank function r(X )= minB X X B + rM (B)+ rM (B). ⊆ | \ | 1 2 EXAMPLE 4.1.2.2. If S is a family of vectors of size n, in a vector space V, and is the set I of linearly independent subsets of S, then (S, ) is matroid. Such a matroid is called a I vector matroid. Given a matrix M over a field F or division ring D, one finds a matroid on the set of columns. Elements of the matroid are some sets of columns of the matrix, which are linearly independent. A vector matroid (S, ) is isomorphic to the matrix I matroid on the matrix M whose columns are the elements of the family S. Since we will be working on matroids related to vector spaces defined over division rings, we make the following remark.

REMARK 4.1.2. Let D be a division ring, and A an m n matrix with entries in D. Consider × the left vector space spanned by the rows of A and the right vector space spanned by the columns of A. These have the same dimension. 1 i (Note that a matrix like (over the quaternions) has left row rank and right j k column rank 2, while its transpose has row- and column rank 1.) Indeed, row manipulation is done using invertible matrices P in front, column manipulation is done using invertible matrices Q behind, and PAQ has the same row- and column rank as A, and may be taken in diagonal form, showing that both values are equal.

4.1.3 Statement of the main theorem We show that a suitable bilinear map µ can be constructed if λ is a minuscule weight, and for a simply laced diagram also in case of the adjoint representation. In the former case C⊥ C , in the latter case C⊥ V Γ C . This results in ⊆ 〈 〉 ∩ ⊆ 〈 〉 THEOREM 4.1.3. Let X be one of A (1 i n), B , B , C , D , D , D ,E , n,i n,i ≤ ≤ n,1 n,n n,1 n,1 n,2 n,n 6,1 E6,2,E7,1,E7,7,E8,8, G2,1. Then the pair (Γ, Σ) has the unique coclique extension property, when Γ is the Kneser graph on the objects of type i in a building with diagram X and Σ { } n is the subgraph induced on a apartment. The same holds when Γ is the Kneser graph on the objects of type 1, n in a building with diagram A . { } n 64 UNIQUE COCLIQUE EXTENSION PROPERTY

4.2 Subspaces of a projective space

Fix integers m, n, where 1 m 1 n. Let N = 1, . . . , n . Let F be a division ring, let ≤ ≤ 2 { } V be an n-dimensional left vector space over F with basis e ,..., e , and let PV be the { 1 n} corresponding projective space. Each m-set K N determines an m-space φK PV by ⊆ ≤ φK = e i K . 〈 i | ∈ 〉 PROPOSITION 4.2.1. Let C be a maximal collection of pairwise intersecting m-subsets of N. Let D be the collection of all m-spaces in PV that meet φK for all K C. Then D is a ∈ maximal collection of pairwise intersecting m-subspaces of PV. (Here and elsewhere, dimensions are vector space dimensions.) First proof (for the case of a ﬁeld F). Let V be the exterior algebra of V. Map m- subspaces of PV to projective points in P( V ) via V ψ : U = u ,..., u u . . . u . 〈 1 m〉 7→ 〈 1 ∧ ∧ m〉

Now U U′ = 0 if and only if ψU ψU′ = 0. For M = i ,..., i N with i < . . . < i ∩ 6 ∧ { 1 m}⊆ 1 m let e = e . . . e . The e form a basis for the degree m part of V . The m-space M i1 ∧ ∧ im M U with ψU = α e intersects φK for all K C if and only if α V= 0 whenever M M M ∈ M is disjoint fromP some K C. (Indeed, if K M = , then the coefﬁcient of eK M in ∈ ∩ ; ∪ ψU ψφK is α .) Since C is maximal, this condition is equivalent to ψU ψφC . If ∧ ± M ∈ 〈 〉 U, U′ D, then ψU ψφC implies ψU ψU′ = 0 so that U U′ = 0. ∈ ∈ 〈 〉 ∧ ∩ 6 Second proof (for the general case). Given a matrix A with entries in F, let its rank be the dimension of the left vector space spanned by its rows, or, equivalently, the dimension of the right vector space spanned by its columns.

Given a matrix A with entries in F and columns indexed by N, we find a matroid MA on N by letting the rank of M N be the rank of the submatrix of A on the columns ⊆ indexed by M. Since V has a fixed basis (e ) , we can regard each v V as a row vector. Given a i i ∈ subspace U of V , let A(U) be a matrix of which the rows form a basis of U. We find a matroid MU on N by taking MU = MA(U). It is independent of the choice of A(U). An m-space U meets φK if and only if no basis of M is disjoint from K. Hence U D U ∈ if and only if each basis of MU meets all elements of C, i.e., if and only if each basis of MU belongs to C. Now the claim follows from:

EMMA L 4.2.2. Let U, U′ be disjoint subspaces of V . Then MU , MU′ have disjoint bases.

Proof. Let M be the union matroid of MU , MU′ , that is the matroid of which the independent sets are the unions of an independent set of MU and one of MU′ . As is well known (see, e.g., [87], Ch. 42), the rank function of M is given by

r(K)= min K L + rU (L)+ rU (L), L K | \ | ′ ⊆ where rU , rU′ are the rank functions of MU , MU′ . Let dim U = m, dim U′ = m′. We have to show that r(N)= m + m′, that is, that

r (L)+ r (L) L + m + m′ n U U′ ≥ | | − 4.3 POINTS IN A POLAR SPACE 65

for all L N. But r (L)+ r (L) r (L) and m + m′ r (L) n L since the ⊆ U U′ ≥ U+U′ − U+U′ ≤ − | | subspace of U + U′ consisting of the vectors vanishing in L is contained in the subspace of V of such vectors.

4.3 Points in a polar space

Consider the noncollinearity graph Γ on the points of a polar space. The maximal cocliques are the maximal totally isotropic subspaces. 2 2 2 In particular, consider a polar space of type Bn, Cn, Dn, A2n 1, A2n, or Dn+1. All maximal totally isotropic subspaces have dimension n. − In these cases, an apartment has size 2n and is linearly independent. The graph Σ is isomorphic to nK2, the disjoint union of n copies of K2. Every maximal coclique C of Σ has size n, and the linear span C is the unique maximal totally isotropic subspace 〈 〉 containing C. It follows that the unique coclique extension property holds in these cases.

For the classical generalized hexagon of type G2, the points are the points of the polar space of type B3, and the Kneser graphs of both coincide. The apartments differ (that of G2 being a hexagon) but it easily follows that the unique coclique extension property also holds for type G2,1.

4.4 Totally singular lines in an orthogonal space

Fix an integer n 2. Let N = 1, 1′, 2, 2′,..., n, n′ . It is provided with an involution ′ ≥ { } that interchanges i and i′. Let V bea2n-dimensional vector space with basis e s N { s | ∈ } and let Q be the quadratic form on V deﬁned by Q(x)= x1 x1′ + . . . + xn xn′ . Now (V,Q) is a nondegenerate orthogonal space with maximal Witt index. Each pair s, t N with { }⊆ t = s′ determines a line in PV by φ s, t = e , e . This line will be totally singular when 6 { } 〈 s t 〉 t = s′. 6

PROPOSITION 4.4.1. Let C be a maximal collection of pairs s, t in N such that t = s′ and { } 6 if s, t C then s′, t′ C. Let D be the collection of all totally singular lines in PV that { }∈ { } 6∈ meet (φK)⊥ for all K C. Then L⊥ M = 0 for any two lines L, M D. ∈ ∩ 6 ∈

Note that L⊥ M = 0 if and only if L M ⊥ = 0. ∩ 6 ∩ 6 Proof. As before, let ψ map subspaces of PV to points in P( V ). For two lines L, M we have L M ⊥ = 0 whenever ψL ψ(M ⊥)= 0. For K C weV have ψ((φK)⊥)= eN K . ∩ 6 ∧ ∈ \ ′ It follows that the line L with ψL = αP eP (with P running over the 2-subsets of N) intersects (φK)⊥ for K C if and onlyP if α = 0. Let L D, and suppose ψL involves ∈ K′ ∈ ei,i′ for some i. Since L is totally singular, it follows that ψL also involves ej,j′ for some j = i. The collection C contains precisely one of s, t and s′, t′ whenever t = s′, so we 6 { } { } 6 may assume that C contains i, j and i, j′ , so that α i ,j = α i ,j = 0. Let L = u, v { } { } { ′ } { ′ ′} 〈 〉 with u = u e and v = v e where u = 0. Then v = 0 and u = u = 0 so that s s s s i′ i′ 6 j j′ ψL does notP involve e , aP contradiction. So, α = 0 when P = i, i′ . We proved that j,j′ P { } ψL ψφC . Now the proof ﬁnishes as before. ∈ 〈 〉 66 UNIQUE COCLIQUE EXTENSION PROPERTY

PROPOSITION 4.4.2. Let C be a maximal collection of pairwise intersecting 2-subsets s, t of { } N with t = s′. Let D be the collection of all totally singular lines in PV that meet φK for all 6 K C. Then D is a maximal collection of pairwise intersecting totally singular lines in PV. ∈ Proof. This is trivial: one ﬁnds either all lines in a totally singular plane or all totally singular lines on a ﬁxed point.

4.5 Minuscule weights

Let the setup be as in the introduction: G a group of Lie type, W the Weyl group with longest element w0, B a Borel subgroup, F an algebraically closed ﬁeld, λ a dominant weight, Wλ its stabilizer in W, V = V (λ) the irreducible FG-module of highest weight λ, P = BWλB a parabolic in G, Γ the graph with vertices gP for g G, where gP is adjacent 1 ∈ to g′ P when P g− g′ P = Pw P, Σ the induced subgraph with vertices wP for w W. Let 0 ∈ us call a reﬂexive bilinear map µ : V V V ′ (for some vector space V ′) good when × → two vertices x, y of Γ are adjacent in Γ if and only if µ(x, y) = 0. 6 PROPOSITION 4.5.1. Let C be a maximal coclique in Σ. If λ is a minuscule weight, then there is a good bilinear map µ such that C⊥ C . ⊆ 〈 〉

Proof. First an ad hoc proof: try all minuscule weights one by one. The cases are: An,j (1 j n), B , C , D , D , E , E . ≤ ≤ n,n n,1 n,1 n,n 6,1 7,7 We already did A (using the bilinear map : V V V , where V = j V and V n,j ∧ × → 0 ∧ 0 0 is the natural module for SLn+1 of dimension n + 1), and CnV,1 and Dn,1 (using the natural symplectic or symmetric bilinear form µ : V V F). × → n In the case of Bn,n we have the spin module of dimension 2 , the vertices of Γ are the maximal totally singular subspaces, and being adjacent is having zero intersection. Let µ be the exterior product. Again V Σ is a basis for V , and maximality of C implies C⊥ C ⊆ 〈 〉 as in the case of An,j. n 1 In the case of Dn,n we have the half-spin module of dimension 2 − . If n is even, we can again use the exterior product, and all is as for Bn,n. So let n be odd. Now any two maxes (maximal totally singular subspaces) of the same type meet, and we want to describe meeting in precisely one point. Take the 2n coordinate hyperplanes Hi. Two maxes S, T meet in a single point when there is an i for which S T H = 0. So for µ we can take the vector ((S H ) (T H )) . Since VΣ is ∩ ∩ i ∩ i ∧ ∩ i i a basis for V , this suffices. (We need a linear map S S Hi. Now if S T Hi = 0 then n n 1 7→ ∩ ∩ ∩ S Hi . Define ρ : V0 − V0 on the basis vectors eM by ρ(eM )= eM i if i M and 6⊆ ∧ →∧ \{ } ∈ ρ(e )= 0 if i M. If S H then ρ( S)= (S H ), otherwise ρ( S)= 0. Therefore, M 6∈ 6⊆ i ∧ ∧ ∩ i ∧ ρ( S) ρ( T) = 0 if and only if S T H = 0.) ∧ ∧ ∧ 6 ∩ ∩ i Remain the cases E6,1, E7,7. The idea of the proof for An generalizes to the present setting. Assume we have a good bilinear map µ. The module V is a direct sum of 1-dimensional weight spaces permuted by W and spanned by the vertices of Σ, so that dim V = V Σ . Pick a basis vector e for | | s each s Σ. For x VΓ write x = x e . If x C⊥, then 0 = µ(x, e )= x µ(e , e ) ∈ ∈ s s ∈ c s s c for each c C. We would like to concludeP x = 0 whenever µ(e , e ) = 0,P but perhaps ∈ s s c 6 the vectors µ(es, ec) are linearly dependent. So add to the conditions on µ that for fixed c the nonzero vectors µ(e , e )(s VΣ) are linearly independent. Then indeed it follows s c ∈ 4.6 ADJOINT REPRESENTATION 67 that if x = 0 then µ(e , e )= 0 for all c C. Since C is maximal, s C, and we showed s 6 s c ∈ ∈ that x C . ∈ 〈 〉 It remains to construct a suitable µ. The dual of V (λ) has highest weight w ( λ), so that V is certainly self-dual when 0 − w = 1, that is, in the cases B , C , D (n even), E . In these cases we can take V ′ = F. 0 − n n n 7 The condition on linear independence is automatically satisfied since the graph Σ has valency 1. That settles E7,7.

For E6,1 the dual module is that of E6,6. Now the map µ can be defined by µ(x, y)= z, where z is the unique symplecton containing x, y when x, y are noncollinear, and z = 0 otherwise. The apartment is the Schläfli graph (27 vertices, valency 16). For fixed c there are 10 nonadjacent vertices, and the 10 symplecta µ(c, s) are distinct and hence linearly independent.

4.6 Adjoint representation

Assume G has simply laced diagram.

PROPOSITION 4.6.1. Let C be a maximal coclique in Σ. If the weight λ corresponds to the adjoint representation, then there is a good bilinear map µ such that C⊥ V Γ C . ∩ ⊆ 〈 〉

This covers Dn,2 (n 4), E6,2, E7,1, E8,8, and An, 1,n (n 2). ≥ { } ≥ Proof. The module V = V (λ) carries the structure of a Lie algebra g. Let Φ be the root system. Then dim V = Φ + ℓ where ℓ is the Lie rank of G, the dimension of a Cartan | | subalgebra h. Pick a Chevalley basis consisting of e for s Φ and e for 1 i ℓ, where s ∈ i ≤ ≤ V Σ is the collection of rootspaces e for s Φ, and h is spanned by the e . 〈 s〉 ∈ i The module V is self-dual. Define µ : V V F by µ(e , e )= 0 unless a, b Φ and × → a b ∈ a + b = 0, and µ(ea, e a) = 1. Now µ is good (since it is G-invariant), and we have to − show that C⊥ V Γ C . The graph Σ has valency 1, so that C contains exactly one ∩ ⊆ 〈 〉 from each pair of opposite root vectors. If x C⊥, where x = x + x e with x h, ∈ 0 s s 0 ∈ then 0 = µ(x, ec)= x c. So if xs = 0 then s C. Suppose x0 = 0. P − 6 ∈ 6 Let r Φ, and consider [e , [e , v]] for arbitrary v V. If v = e for some s Φ, ∈ r r ∈ s ∈ then r, s span a 2-dimensional root system, and we see that this is zero, unless r + s = 0, in which case it is 2e . If v = h h, then [e , [e , v]] = 0. More generally, for v = − r ∈ r r h + vses, we find [er , [er , v]] = 2v r er . Let us call an element x of a Lie algebra − − extremalP when [x, [x, y]] is a multiple of x for all y. (The usual definition adds a requirement for the case of characteristic 2 that we don’t need.) We just observed that er is extremal.

Assume now that x V Γ. Then x is extremal. Let t Φ be a root with t(x ) = 0 and ∈ ∈ 0 6 et C. (If only t(x0) = 0, then replace t by t.) Now [x, [x, et ]] is a multiple of x. 〈 〉 6∈ 6 − 2 On the other hand, the coefﬁcient of et in this expression is t(x0) = 0 (since xs x s = 0 6 − for all s Φ). This is a contradiction. ∈ 68 UNIQUE COCLIQUE EXTENSION PROPERTY

4.7 Nonexamples

The unique extension property does not hold for objects of arbitrary type. We give coun- terexamples for types An 1, i,n i (1 < i < n/2), B3,2, D4, 3,4 , and D5,3. − { − } { } PROPOSITION 4.7.1. The unique coclique extension property does not hold for the Kneser graph on the flags of type i, n i in a building with diagram An 1, where 1 < i < n/2. { − } − Proof. Let V be a vector space of dimension n with basis e1,..., en. Put u = e1 + e2, v = e1 + en. Let A = u, e3,..., ei+1 and A′ = v, en 1,..., en i+1 so that A and A′ are 〈 〉 〈 − − 〉 i-spaces. Let B = u, e3,..., en i , en and B′ = v, en 1,..., ei+2, e2 so that B and B′ are 〈 − 〉 〈 − 〉 (n i)-spaces. Now the flags F = (A, B) and F ′ = (A′, B′) are adjacent since A B′ = − ∩ A′ B = 0. Note that F is mapped to F ′ by the coordinate permutation (2, n)(3, n 1) .... ∩ − The graph Σ has valency 1 and each choice of one vertex from each edge of Σ yields a maximal coclique C. We can find a maximal coclique C compatible with F, F ′ when in no edge of Σ both endpoints are adjacent to either F or F ′. Let N = 1, . . . , n . The vertices { } of Σ are pairs (S , S ) where S = e i I and I = i, J = n i, I J. The unique I J I 〈 i | ∈ 〉 | | | | − ⊆ neighbour of (SI , SJ ) is (SN J , SN I ). If (SI , SJ ) is adjacent to F, then SI B = SJ A = 0, \ \ ∩ ∩ so that I = j, n i + 1, . . . , n 1 and J = j, i + 2, . . . , n , where j 1, 2 . If (S , S ) { − − } { } ∈{ } I J is adjacent to F ′, then I = k, i + 1,...,3 and J = k, n i,...,2 where k 1, n . { } { − } ∈ { } Altogether 4 vertices in Σ are adjacent to either F or F ′, and for i > 1 this set of 4 does not contain any edge. B nonexample: Let char F = 2 and let V = F 7, with basis vectors e , 1 i 7. Pro- 3,2 6 0 i ≤ ≤ vide V with the nondegenerate quadratic form Q(x) = x x + x x + x x x2. The 0 1 2 3 4 5 6 − 7 geometry of totally singular subspaces of (V0,Q) has type B3. For type B3, the adjoint representation is 2V , corresponding to B , the geometry of totally singular lines. Con- ∧ 0 3,2 sider the apartment with points e , 1 i 6, and let C consist of the six totally singular 〈 i〉 ≤ ≤ lines 13, 14, 25, 26, 35, 36, where i j abbreviates e , e . Pick u = e , v = e + e + e and 〈 i j〉 1 3 4 7 u′ = e , v′ = e + e + e . Then u, v and u′, v′ are totally singular and adjacent in the 2 5 6 7 〈 〉 〈 〉 Kneser graph and both nonadjacent to all vertices of C.

Planes in D4 nonexample: Let N = 1, 2, 3, 4, 1′, 2′, 3′, 4′ and let ′ be the involution on { 8 } N that maps i to i′. Let char F = 2 and let V = F , with basis vectors e (i N) and 0 i ∈ provided with nondegenerate quadratic form Q(x)= x1 x1′ + x2 x2′ + x3 x3′ + x4 x4′ . The geometry of totally singular subspaces of (V0,Q) has type D4. Consider the apartment with points e , i N. Let π = e + e , e + e , e and π′ = e + e , e + e , e . 〈 i 〉 ∈ 〈 1 2 1′ 2′ 4〉 〈 1′ 3 1 3′ 4′ 〉 Then π, π′ are totally singular planes, adjacent in the Kneser graph on the planes in D4 (since π⊥ π′ = 0). One checks that there is no edge in the subgraph Σ induced on the ∩ apartment such that both ends are adjacent to π or π′, so there is a maximal coclique C compatible with both π and π′, and the unique coclique extension property fails. This was the case D4, 3,4 . Of course this also means that it fails for Dn,3 for all n 5. { } ≥ If the unique coclique extension property fails for objects of some type J in a building of w spherical type with Coxeter system (W, R), where J 0 = J, and J J ′ R, then it also ⊆ ⊆ fails for objects of type J ′.

PROPOSITION 4.7.2. Let Γ, Γ′ be the Kneser graphs on the objects of type J, J ′, respectively, in a building of spherical type. Let Σ, Σ′ be the subgraphs of Γ, Γ′, respectively, induced w on an apartment. If J 0 = J, and J J ′, and the pair (Γ′, Σ′) has the unique coclique ⊆ extension property, then also (Γ, Σ) has this property. 4.7 NONEXAMPLES 69

Proof. Let (W, R) be the Coxeter system of the building, and put X = R J , X ′ = R J ′ , 〈 \ 〉 〈 \ 〉 and P = BXB, P′ = BX ′B, so that the vertices of Γ, Γ′ can be viewed as left cosets of P, P′, respectively. The subgraphs Σ, Σ′ are now induced on W P/P and W P′/P′. Since J J ′, so that X ′ X and P′ P, we have a canonical map φ : Γ′ Γ ⊆ ≤ ≤ 1 → sending gP′ to gP′ P = gP. This φ is a homomorphism: if Bw0B P′ g− hP′, then also 1 ⊆ Bw0B P g− hP. ⊆ w Since J 0 = J, if a′ is a vertex of Γ′ and a, b are adjacent vertices of Γ, where φ(a′)= a, then there is a vertex b′ of Γ′, adjacent to a′, with φ(b′)= b. That is, edges can be lifted. (Indeed, let a′ = y P′, a = y P, b = zP. Since y P, zP are adjacent, we have Bw0B 1 1 1 1 ⊆ Pz− y P, so that z− y Pw P = BXw XB = BXw B Pw B, and w B = p− z− yB for ∈ 0 0 0 ⊆ 0 0 some p P. It follows that the vertex a′ = y P′ is adjacent to b′ = zpP′.) ∈ 1 Let C be a maximal coclique in Σ. Then φ− C is a coclique in Σ′. If wP C, there is 6∈ 1 a vP C adjacent to wP, and we can lift this edge and ﬁnd a neighbour of wP′ in φ− C. ∈ 1 Hence φ− C is a maximal coclique in Σ′. If y, z are two vertices of Γ such that both C y and C z are cocliques, then 1 1 1 1 ∪{ } ∪{ } both φ− C φ− (y) and φ− C φ− (z) are cocliques in Γ′, and by the unique coclique ∪ ∪ 1 1 extension property of (Γ′, Σ′), no vertex of φ− (y) is adjacent to a vertex of φ− (z). But then also y and z are nonadjacent. 70 UNIQUE COCLIQUE EXTENSION PROPERTY 5 MAXIMAL COCLIQUES IN POINT-HYPERPLANE GRAPHS

5.1 Introduction

In this chapter, we prove an Erdos-Ko-Rado-type˝ theorem for the Kneser graph on the point-hyperplane ﬂags in a ﬁnite projective space.

Let V be a vector space of finite dimension n > 0 over Fq, and consider in the associated projective space PV the flags (P, H) consisting of an incident point-hyperplane pair (that is, with P H), and call two such flags (P, H) and (P′, H′) adjacent when P H′ ⊆ 6⊆ and P′ H. We study maximal cocliques in this graph Γ(V ). 6⊆ Let F = (X1, ..., Xn 1) be a chamber (maximal flag) in PV , that is, Xi is a subspace of V of vector space− dimension i for all i, and X X for i j. From F we construct i ⊆ j ≤ the coclique CF = (P, H) i : P Xi H . This coclique is maximal, and CF = 2 { |n ∃ 2 ⊆ ⊆ } | | 1 + 2q + 3q + ... +(n 1)q − . − For a coclique C in Γ(V), define Z(C) = P (P, H) C for some H and Z′(C) = H { | ∈ } { | (P, H) C for some P . ∈ } 2 THEOREM 5.1.1. Let C be a coclique in Γ(V ) and let Z = Z(C). Let f (n) := 1 + 2q + 3q + n 2 2 n 2 +(n 1)q − and g(n) := 1 + q + q + + q − . Then · · · − · · · (i) C f (n), and equality holds iff C = C for some chamber F, and | |≤ F (ii) Z g(n), and for q > 2 equality holds iff Z is a hyperplane. | |≤ Note that g(n) is the number of points in a hyperplane, and f (n) = q f (n 1)+ − g(n). Part (i) was conjectured, and partial results were obtained, in Mussche [?]. Both inequalities are sharp, and the theorem characterizes equality in (i). What about equality in (ii)? In examples of type CF the set Z is a hyperplane, and equality holds. When q = 2 there are further examples of equality, described below.

EXAMPLE 5.1.0.1. (n = 3) In the plane, take three points Pi (i = 1, 2, 3) in general position. Let C consist of the ﬂags (Pi, Pi + Pi+1) (indices mod 3). Then C is a coclique, and n 1 Z = 3 = 2 − 1. | | − (This example also arises from the trivial one point coclique for n = 2 by the construction in Example 5.1.0.3.) 72 MAXIMAL COCLIQUES IN POINT-HYPERPLANE GRAPHS

EXAMPLE 5.1.0.2. (n = 4) Consider a plane π and a point outside π. For each point P ∞ in π, let P′ be the third point of the line P + . Label the points of π with the integers ∞ mod 7, so that the lines become 1, 2, 4 (mod 7). The seven ﬂags (i′, i′, i+1, i+2, i+4 ) { n 1 } 〈 〉 form a coclique C, and Z = 7 = 2 − 1. | | − (This is an honest sporadic example.)

EXAMPLE 5.1.0.3. Let H be a hyperplane in V , and let D be a coclique in Γ(H) with n 2 Z(D) = 2 − 1. Let G be a hyperplane of H, such that G is disjoint from Z(D). Let Q | | − be a point outside H. Let C consist of the flags (P, H) with P G, and (P, N + Q) with ⊆ (P, N) D, and (Q, G + Q). Then C is a coclique, and Z(C) = Z(D) G Q , so that ∈n 1 ∪ ∪{ } Z = 2 − 1. | | − (We’ll see later that this only gives something for n 5.) ≤ EXAMPLE 5.1.0.4. Let T be a t-space in V, 0 < t < n. Let D be a coclique in Γ(T). Let n 1 t E be a coclique in Γ(V /T) such that Z(E) has maximal size 2 − − 1. Construct a − coclique C in Γ(V ) by taking (i) all flags (P, H) with P T H, (ii) the flags (P, K) ⊆ ⊆ where (P, K T) D, and (iii) the flags (Q, H) where (Q + T, H) E. Then Z(C) = t t∩ n 1∈t n 1 ∈ | | (2 1)+ 2 (2 − − 1)= 2 − 1. − − − Note that the cocliques CF are of this form (for every T in F). The above describes all cases of equality:

PROPOSITION 5.1.2. Let V be an n-dimensional vector space over F2, and let C be a coclique n 1 in the graph Γ(V ). Put Z := P (P, H) C for some H . Then Z 2 − 1 with equality { | ∈ } | |≤ − if and only if Z is a hyperplane, or C arises by the construction of Example 5.1.0.2, 5.1.0.3 or 5.1.0.4.

5.1.1 Rank 1 matrices The graph Γ(V ) can be described in terms of rank 1 matrices. Represent points by column vectors p and hyperplanes by row vectors h, then a point-hyperplane pair can be represented by the rank 1 matrix ph, and an incident point-hyperplane pair by the rank 1 matrix ph with hp = 0, that is, with tr ph = 0. Rank 1 matrices that differ by a constant only represent the same point-hyperplane pair. Given two incident point-hyperplane pairs x = ph and x′ = p′h′, they are nonadjacent when (hp′)(h′p)= 0, i.e., when tr x x′ = 0. Thus, ﬁnding the maximal cocliques in Γ(V) is equivalent to ﬁnding the intersections of the maximal totally isotropic subspaces for the symmetric bilinear form (x, y)= tr xy on the space Mn(Fq) (of matrices of order n over Fq) with the space of trace 0 rank-1 matrices.

The extremal example CF is conjugate to the example of all rank-1 strictly upper- triangular matrices.

5.1.2 The thin case There is a thin analog (q = 1 version) of our problem. Consider for an n-set V the pairs (P, H), where P = 1, H = n 1, and P H V . Call (P, H) and (P′, H′) adjacent when | | | | − ⊆ ⊂ P H′ and P′ H, that is, when (P′, H′)=(V H, V P). Here the graph is the union of n6⊆ 6⊆ \ \ components K . A maximal coclique is obtained by taking a single vertex from each 2 2 5.2 MAXIMUM-SIZE COCLIQUES 73

K , so that C n = f (n) and Z n. Here g(n)= n 1, and Z g(n) holds only 2 | |≤ 2 | |≤ − | |≤ for n = 1, 2.

5.2 Maximum-size cocliques

Proof of Theorem 5.1.1. The problem is self-dual, so all that is proved for points and hyperplanes, also holds for hyperplanes and points. In particular, g(n) will also be an upper bound for the number Z′(C) of hyperplanes involved in C. | | We may assume that C is maximal. It follows that C has a certain linear structure:

LEMMA 5.2.1. Let C be a maximal coclique in Γ(V ), and let H Z′(C). Then H has a ∈ subspace S(H) such that (P, H) C if and only if P S(H). IfH, K Z′(C) then S(H) K ∈ ⊆ ∈ ⊆ or S(K) H (or both). ⊆ Proof. If (P, H), (Q, H) C and R is a point on the line P +Q, then also (R, H) C since ∈ ∈ C is maximal. If P is in S(H) but not in K, and Q is in S(K) but not in H, then (P, H) and (Q, K) are adjacent in Γ(V ), impossible. We use induction on n. If W is a subspace of V , then (P, H W) (P, H) C, P { ∩ | ∈ ⊆ W, W H is a coclique in the graph Γ(W). And if S is a subspace of V , then (P +S, H) 6⊆ } { | (P, H) C, P S, S H is a coclique in the graph Γ(V /S). ∈ 6⊆ ⊆ } Let the maximal dimension of S(H) (for varying H) be s, and let H be a hyperplane with dim S(H)= s. We have 1 s n 1. ≤ ≤ − LEMMA 5.2.2. If s = n 1, then C q f (n 1)+ g(n)= f (n) and Z = g(n), and Z = H. − | |≤ − | | Proof. If s = n 1 then S(H)= H, so that S(H) K for K = H and therefore S(K) H − 6⊆ 6 ⊆ for all K, i.e., Z = H and Z = g(n). The coclique C consists of the g(n) ﬂags (P, H) | | together with at most q f (n 1) ﬂags (P, K) with P H, K = H, so that C q f (n 1)+ − ⊆ 6 | |≤ − g(n)= f (n), as desired.

LEMMA 5.2.3. If s < n 1, then C < q f (n 1)+ g(n)= f (n). − | | − Proof. Count the three types of ﬂags: a: those involving H, b: those involving M with S(M) contained in H, c: those involving K with S(K) not contained in H (and hence containing S(H)). qs 1 qs 1 a: − ; b: atmost q f (n 1); c: atmost g(n s) − . q 1 − − q 1 − n 2 − q − 1 It follows that C q f (n 1)+ 2 − < f (n), as desired. | |≤ − · q 1 Since there is strict inequality here, equality− C = f (n) only occurs for s = n 1, | | − where there is a hyperplane H such that (P, H) C for all points P H, and all other ∈ ⊆ elements of C restrict to a coclique with equality in Γ(H). By induction it follows that equality implies that C = CF for some ﬂag F.

LEMMA 5.2.4. Let q > 2 and s < n 1. Then Z < g(n). − | | Proof. We estimate Z . For the flags (P, M) C with P H and M = H, the system | | ∈ ⊆ 6 (P, M H) forms a coclique in Γ(H), so the number of points P involved is at most ∩ g(n 1). For the flags (Q, K) C with Q H, the hyperplanes K contain S(H), and − ∈ 6⊆ 74 MAXIMAL COCLIQUES IN POINT-HYPERPLANE GRAPHS the flags (Q + S(H), K) form a coclique in Γ(V /S(H)), so there are at most g(n s) such s 1 − hyperplanes K. For each K there are at most q − points Q outside H with (Q, K) C. ∈ Finally, Z contains the points in S(H). Altogether, s s 1 q 1 Z g(n 1)+ g(n s) q − + − < g(n). | |≤ − − · q 1 − For q = 2 and s > 1 more work is required, because now the above estimate of Z is s 1 | | 2 − 1 larger than the desired upper bound g(n). − As observed, there are at most g(n s) hyperplanes K involved in flags (P, K) C − ∈ with P S(H) and S(H) K. If the number of such K is strictly smaller than this, or 6⊆ ⊆ s 1 S(K) H for one of them, then the upper bound on Z improves by 2 − , and Z < g(n). ⊆ | | | | Finally, if for at least two of them dim S(K) < s, then again the same holds. So, we may assume that there are precisely g(n s) such hyperplanes K, for none of them S(K) H, − ⊆ and for all of them with at most one exception dim S(K)= s. Let E(H) be the set of points P in S(H) that occur in a single flag only, namely in (P, H). Let e(H)= E(H) . The (last) term 2s 1 in the bound was an upper bound for | | − s 1 e(H)—the points of S(H) E(H) are already covered by the term g(n 1). If e(H) 2 − \ s 1 − ≤ then Z g(n) as desired. So, we may assume that e(H) 2 − + 1. | |≤ ≥ First consider the case s = n 2. We have g(n s) = g(2) = 1, and S(H) is a − − hyperplane in the unique K. Now if S(K) has dimension t( s), then S(H) and S(K) t 1 t≤1 have 2 − 1 points in common, S(K) contributes at most 2 − points outside H to Z − s t 1 | | and e(H) 2 2 − , proving the bound. So, we may assume 2 s n 3. ≤ − ≤ ≤ − Let be the collection of all hyperplanes H such that dim S(H) = s. All we have H said about H above holds for all H . Make a directed graph with vertex set and ∈ H H arrows H K when S(H) K. (As we have seen, now K H.) → ⊆ 6→ The outdegree of the graph on is g(n s) or g(n s) 1 at each vertex. It follows H − − − that the indegree at some vertex is at least g(n s) 1. − − First suppose that the indegree of H is at least g(n s). The number of points P − involved in flags (P, M) where M H is bounded above by g(n 1). On the other → s 1 s − hand it is bounded from below by (g(n s) 1)(2 − + 1)+(2 1) > g(n 1) since s 1 − − − − e(M) 2 − + 1 for each such M. This is a contradiction. It follows that all outdegrees ≥ and all indegrees of are precisely g(n s) 1. H − − s 1 s 2 Our estimate now becomes Z g(n 1)+(g(n s) 1)2 − + 2 − + e(H) and | |s ≤2 − − − s 2 Z g(n) will follow if e(H) 3 2 − . So, we may assume that e(H) 3 2 − +1 for all | |≤ ≤ · ≥ · H . Again we bound the number of points P from below. We find the contradiction ∈ H s 2 s (g(n s) 2)(3 2 − + 1)+(2 1) > g(n 1) if s n 4. So, we may assume that − − · − − ≤ − s = n 3. − Now S(H) has codimension 2 in each K, so codimension at most 2 in each S(K), so s 2 that dim(S(H) S(K)) s 2. It follows that e(H) 3 2 − , as desired. ∩ ≥ − ≤ · This finally proves part (ii) of the theorem.

5.3 Maximum number of points

Proof of Proposition 5.1.2. The inequality was shown already. So assume we have n 1 equality, i.e., Z = 2 − 1. Recall the above proof. Given a hyperplane H, consider the | | − 5.3 MAXIMUM NUMBER OF POINTS 75 three types of flags: (i) flags (Q, K) with Q H, (ii) flags (P, M) with P H and M = H, 6⊆ ⊆ 6 (iii) flags (P, H). Correspondingly we get three contributions to Z : the points Q from | | flags of type (i), the points P from flags of type (ii), and the points P from flags (P, H) that were not counted yet, i.e., that occur in flags (P, H) only. (The set of such P is called E(H), and has size e(H).) Let H be one of the hyperplanes with dim S(H) = s maximal. We find Z g(n s 1 n 1 s 1 s 1 | | ≤ − s)2 − +g(n 1)+e(H) 2 − +2 − 2. This estimate is precisely 2 − 1 too large, so it − ≤s 1 − − cannot be improved by 2 − . It follows that there are precisely g(n s) hyperplanes K on − S(H), and all except at most one have dim S(K)= s. None of these K satisfies S(K) H. s 1 ⊆ It also follows that e(H) 2 − . By Lemma 5.2.2 we may suppose that 1 s n 2. ≥ ≤ ≤ − LEMMA 5.3.1. If s = 1, then n 4. ≤ s 1 s Proof. For s = 1 the counting Z g(n s)2 − + g(n 1)+ 2 1 holds with equality. | |≤ − − − This means that if (P, H) C then P is the only point in H with this property, because ∈ s = 1, but also H is the only hyperplane on P with this property, because the counting is exact. Hence C involves equally many points as hyperplanes. Above we saw that if S(H) K, then S(K) H, which in this case simply means that for two flags (P, H) ⊆ 6⊆ and (Q, K) exactly one of P K and Q H holds. Now consider the point/hyperplane ⊆ ⊆ nonincidence matrix of PG(n 1, 2). The 2-rank of this matrix is n, but the submatrix M − corresponding to the points and hyperplanes in C has the property that M + M T = J I n 1 n 1 − of 2-rank 2 − 2. It follows that 2n 2 − 2, so n 4. − ≥ − ≤ Make a directed graph ∆ on the hyperplanes H occurring in flags (P, H) C with ∈ dim S(H)= s, writing an arrow H K if S(H) K. → ⊆ LEMMA 5.3.2. The graph ∆ is a tournament, and we have one of three cases, where k = n s 1 g(n s)= 2 − − 1 and v is the number of vertices: − − a) All indegrees and all outdegrees equal k and v = 2k + 1, b) All indegrees and all outdegrees equal k 1 and v = 2k 1, − − c) All indegrees are k 1 or k (and both occur) and all outdegrees are k 1 or k (and − − both occur) and v = 2k.

Proof. We saw that at each vertex H the outdegree equals either k or k 1, where − k = g(n s). If ∆ has v vertices, then at each vertex the indegree equals v 1 k or − − − v k. Since the average indegree equals the average outdegree, we have one of the − stated cases.

LEMMA 5.3.3. If H has outdegree k 1, so that one hyperplane K on S(H) has dim S(K)= s t 1 s−2 t < s, then e(H) 2 2 − 3.2 − . ≥ − ≥ Proof. The contribution to Z from hyperplanes K is now at most a := (g(n s) s 1 t 1 | | s t 1 − − 1)2 − + 2 − , so e(H) Z g(n 1) a = 2 2 − . ≥ | |− − − − s 1 LEMMA 5.3.4. If k = 1, and H is a vertex of ∆ with indegree 1, then e(H)= 2 − , and H also has outdegree 1.

Proof. If H has indegree 1, say M H, then S(M) is a hyperplane in H (since s = n 2) s 1 → s 1 − and covers at least 2 − 1 of the points of S(H). It follows that e(H)= 2 − , and H also − has outdegree 1 by Lemma 5.3.3. 76 MAXIMAL COCLIQUES IN POINT-HYPERPLANE GRAPHS

LEMMA 5.3.5. If H is a vertex of ∆ with indegree k, then each of its inneighbours has s 1 outdegree k. If k > 1 then e(M)= 2 − for each inneighbour M, and all inneighbours have s 1 the same 2 − 1 nonunique points. − Proof. Look at the coclique D in Γ(H) consisting of the ﬂags (P, M H) for M H. We s 1 ∩ → s have Z(D) g(n 1). Each M has e(M) 2 − unique points, and contributes 2 1 | |≤ − ≥ s 1 s 1 n 2 − points altogether to Z(D), so we ﬁnd at least k.2 − +(2 − 1)= 2 − 1 = g(n 1) − − s 1− points in Z(D). Equality must hold, so if k > 1 all inneighbours M have e(M)= 2 − . By Lemma 5.3.3 they all have outdegree k. Obviously, if k = 1, an inneighbour cannot have outdegree k 1. − LEMMA 5.3.6. If all indegrees and all outdegrees are k, then there is a subspace T of V of dimension s 1 such that for each H with dim S(H)= s we have S(H) E(H)= T. − \ Proof. The counting of the previous lemma shows for k > 1 that each inneighbour M s 1 of H has the same set S(M) E(M) of size 2 − 1. Since this covers S(H) E(H) which \ − \ has the same size, we conclude that S(M) E(M)= S(H) E(H) when there is an arrow \ \ M H. If k = 1, the same conclusion follows from Lemma 5.3.4. → s 1 This set of size 2 − 1 is the intersection of all S(H), so is a subspace. − LEMMA 5.3.7. If all indegrees and all outdegrees are k, and s > 1, then C arises by the construction of Example 5.1.0.4.

n s s 1 Proof. We have v = 2k+1 = 2 − 1 vertices H. Each vertex H contributes e(H)= 2 − − s 1 unique points to Z, and all vertices contribute the same set of T = 2 − 1 common n 1 | | − points, 2 − 1 = Z points altogether. Now consider another ﬂag (P, H) C (with − | | ∈ dim S(H) < s). The point P cannot be one of the unique points, so P T. This shows ⊆ that C arises as in Example 5.1.0.4.

LEMMA 5.3.8. If s = n 2 then C arises as in Example 5.1.0.1, 5.1.0.3 , or 5.1.0.4. − Proof. If s = n 2, then k = 1. Now ∆ has at most 3 vertices. If v = 3, then it is a − directed 3-cycle. By Lemma 5.3.7, either C arises by the construction of Example 5.1.0.4, or s = 1, n = 3, in which case we have Example 5.1.0.1. By Lemma 5.3.4, if v = 3, then v = 1. Let K be the unique hyperplane in Z′(C) with 6 S(H) K, and suppose dim S(K)= t. Then 1 t < s. If t = 1, then we have Example ⊆ ≤ 5.1.0.3 (with G = S(H) and Q = S(K)). Let t > 1. t 1 The subspace S(H) is a hyperplane in both H and K, so S(H)= H K, and the 2 − 1 ∩ n 2 t−1 points in H S(K) are not unique in H. By Lemma 5.3.3 we have e(H)= 2 − 2 − . ∩ n 2 − The situation is tight again, so the ﬂags (P, M) contribute precisely 2 − 1 points. Put − T := S(K) S(H), so that dim T = t 1. Since t > 1, T = 0. Compare (P, M) and (Q, K). ∩ − 6 If S(K) M, then S(M) K H = S(H). Since E(H)= S(H) T, this means that T M 6⊆ ⊆ ∩ \ ⊆ or S(M) T. This is Example 5.1.0.4 (applied to a coclique from Example 5.1.0.3). ⊆ Now assume 1 s n 3, so that k 3. ≤ ≤ − ≥ LEMMA 5.3.9. Case c) of Lemma 5.3.2 does not occur.

Proof. In case c), suppose that we have v = 2k vertices, b of which have outdegree k 1 (and indegree k) and v b of which have outdegree k (and indegree k 1). The − − − total number of arrows is vk b = vk (v b), so that b = v/2 = k. Let A be the set − − − 5.4 CLASSIFICATION OF COCLIQUES FOR n 4 77 ≤ of vertices with outdegree k, B that with outdegree k 1. By Lemma 5.3.5 we have all − arrows from A to B, and this ﬁlls up all outarrows in A. Then there are no arrows inside A, contradiction. So case c) does not occur.

LEMMA 5.3.10. Case b) of Lemma 5.3.2 does not occur. s 2 Proof. In case b) we have e(H) 3.2 − for each vertex H of ∆, by Lemma 5.3.3. ≥ s 2 s 2 The k 1 inneighbours of any vertex H contribute at least 3.2 − (k 1)+ 2 − 1 = n 3− s 2 n 3 s 2 n 2 − n 3 − s 2 3.2 − 5.2 − 1 points in H, so 3.2 − 5.2 − 1 2 − 1, that is, 2 − 5.2 − − − − − ≤ − ≤ and hence s n 3. Since we are assuming s < n 2 this means s = n 3. Now k = 3 ≥ − − − and all vertices have indegree and outdegree 2. The graph ∆ has 5 vertices. They can be labeled Hi, i = 0, 1, 2, 3, 4, such that Hi H j iff j = i + 1 or j = i + 2 (mod 5). Now → s 1 H contains S and S , both of dimension s = n 3. Since H contributes at least 2 − 2 H1 H0 − 1 points Q, we have S = S . So S + S is an (n 2)-space contained in the (n 2)- H1 6 H0 H0 H1 − − space H H . Hence S +S = H H . Since S H , also S = H H H . Now 1 2 H0 H1 1 2 H1 0 H0 0 1 2 ∩ ∩ n 4 6⊆ ∩ ∩ S and S meet in an (n 4)-space, so at least 2 − 1 points are not unique in H . The H0 H1 0 − n 2 s 1 −s 2 s s 1 n 1 n 5 standard counting gives Z (2 − 1)+(2.2 − +2 − )+(2 2 − )= 2 − 1 2 − , | |≤ − − − − contradiction. So case b) does not occur. So far we saw: either we have Example 5.1.0.1, 5.1.0.3 or 5.1.0.4, or s = 1, n = 4.

LEMMA 5.3.11. If s = 1, n = 4, then we have Example 5.1.0.2.

Proof. Let Z′ = Z′(C). Here Z = Z′ = 7, and ∆ is a tournament on seven vertices | | | | with all in and outdegrees equal to 3. We ﬁrst show that Z is a cap, that is, no three points in Z are collinear. Indeed, if (P, H), (Q, K), (R, L) C, and P,Q, R are collinear, and ∈ K H, then Q H but also P H, hence R H, that is L H. But now there can be no → ⊂ ⊂ ⊂ → consistent arrow between K and L. So the points form a cap of size 7. The only maximal caps of PG(3, 2) are the elliptic quadric (of size 5) and the complement of a plane (of size 8), cf. [68, Theorem 18.2.1], so there is a (hyper)plane π disjoint from Z and a point A not in Z and not in π. If (P, H), (Q, K) C and H K is a line in π, then (P, H), (Q, K) ∈ ∩ are adjacent, impossible. So the seven planes in Z′ meet π in seven different lines. The situation is self-dual, so there is a point not on any plane in Z′, necessarily the point A. So, the seven points of Z are the points different from A outside π, and the seven planes of Z′ are the planes different from π not on A. One now quickly establishes that we have Example 5.1.0.2. This ﬁnishes the proof of Proposition 5.1.2. Remarks. The construction of Example 5.1.0.4 does not change n s. The hyperplane − has n s = 1, in Example 5.1.0.3 we have n s = 2 and in Example 5.1.0.2 we have − − n s = 3. So, all examples of Z = g(n) have 1 n s 3. − | | ≤ − ≤ Example 5.1.0.3 needs as ingredient a hyperplane disjoint from Z(D). It follows that Z(D) cannot contain linear subspaces of dimension larger than 1. Hence the construction of Example 5.1.0.3 applies only for (i) the hyperplane example for n = 2, (ii) Example 5.1.0.1, (iii) Example 5.1.0.2. The resulting examples are Example 5.1.0.1, and two further examples with n = 4, s = 2 and n = 5, s = 3.

5.4 Classification of cocliques for n 4 ≤ In the preceding sections we found the maximum size of a coclique in the Kneser graph on the point-hyperplane flags of a projective space. In the special case of PG(3,q) (that 78 MAXIMAL COCLIQUES IN POINT-HYPERPLANE GRAPHS is, the case n=4), we can give a complete classification of all maximal cocliques in this graph. For n = 1, the graph has no vertices, and the unique maximal coclique is the empty set. For n = 2, the graph is the complete graph on q + 1 vertices, since here the flags are of the form (P, P), where P is any point in the space, which is a line. Hence, maximal cocliques are of size 1, and there are q + 1 of them.

5.4.1 PG(2, q) In PG(2, q) there are two cases of a maximal clique with Z of size 3:

i . q = 2 C = CF : take an incident point-line pair (P, L) and take the ﬁve ﬂags (P, M), M a line on P, (Q, L), Q a point on L,

ii . take a triangle P,Q, R , and take the three ﬂags (P, PQ ), (Q, QR ), (R, RP ). 〈 〉 〈 〉 〈 〉 〈 〉 5.4.2 PG(3, q) We will give the geometric description of the maximal cocliques here. The size of each type is given in Table 5.1

TABLE 5.1: maximal coclique classiﬁcation for PG(3, q)

type size number note 1 3q2 + 2q + 1 K 2,3 q2 + 4q + 1 1 qK 3 4 4q + 2 q5K 5 2q + 4 (q 1)q6K − 6,7 q + 5 1 (q 1)2q6K 3 − 8 6 1 (q 1)2q6K 4 − 9 6 (q 1)3q6K − 10 6 1 (q 1)3q6K for p = 2 3 − 6 11 7 1 (q 1)3q6K for p = 2 21 − the number K above is (q3 + q2 + q + 1)(q2 + q + 1)(q + 1)

i . C = CF . ii . Fix a hyperplane H, and let C consist of all ﬂags (P, H), and the hyperplanes ob-

tained by taking a triangle, take three corners of it P1, P2, P3, for the lines joining any two of those, take all hyperplanes on these lines, for such an hyperplane it includes one of the lines P , P or P , P , or P , P , let the point of the ﬂag in 〈 1 2〉 〈 2 3〉 〈 3 1〉 each case P1, P2, P3 respectively. iii . (this is the dual of the previous one) Take a point and all hyperplanes on it.

Then take three lines L1, L2, L3 in its residue, each two of these span a hyperplane 5.4 CLASSIFICATION OF COCLIQUES FOR n 4 79 ≤

H12, H23, H31, the points are on L1, L2 and L3 respectively with those hyperplanes for each ﬂag.

iv . Take three hyperplanes H , J , K , intersecting at a point S, take a point P in J H 0 0 0 1 0∩ 0 different from S, and a point P in H K different from S. The coclique consists 2 0 ∩ 0 of ﬂags (P, H ), P H K , (Q, K ), Q K J , (P , J ), (P , H) with H is any 0 ⊆ 0 ∩ 0 0 ⊆ 0 ∩ 0 1 0 2 hyperplane on the line P1 + P2, and (S, H) with H is any hyperplane on S + P2.

v . Take a plane H0, and a line L0 in H0, and a triangle P1, P2, P3 in H0 of which only the point P1 lies in L0, and a point Q outside H0. Put Li = Pi + Pi+1 mod 3, the coclique consists of the flags (P, H ) with P L and the flags (P , H) with L H 0 ⊂ 0 1 1 ⊂ and the three flags (Q, L0 + Q), (P2, L2 + Q), (P3, L3 + Q).

vi . Take a plane H0, and 3 linearly independent points on it P1, P2, P3, take a line L0 which does not include any of these 3 points, but lies in H0, take a point Q outside, call the lines L = P , P mod 3. Take all ﬂags (P, H ) where P is in L , take i 〈 i i+1〉 0 0 4 ﬂags: (Q, L ,Q ), (P , L ,Q ) for i = 1, 2, 3. There are all points except one on 〈 0 〉 i 〈 i 〉 some hyperplane and a point outside this hyperplane.

vii . (this is the dual of the previous one) Take a point P0 and 3 hyperplanes on it, H , H , H , where H H = L mod 3, and take a line L intersecting each H 1 3 3 i ∩ i+1 i 0 i at a point. Take a hyperplane H∗ not on P0, take all ﬂags (P0, H) where H is a hyperplane on P , take the hyperplanes (P, H∗) where P is any point in (H∗ L ), 0 ∩ 0 take the hyperplanes (L H∗, H ). i ∩ i viii . Take two skew lines, L, M and three distinct points P,Q, R on L and three distinct points A, B, C on M. The coclique consist of the six ﬂags (P, B + P + Q), (Q, C + P + Q), (R, B + C + R), (A, A + P + Q), (B, A + B + Q), (C, A + C + P).

ix . Take a line L and three distinct points P,Q, R on L, and take three distinct points A, B, C not on L such that A, B, C are not collinear but A, B, C, P are coplanar. The coclique again consist of six ﬂags (P, B + P +Q), (Q, C + P +Q), (R, B +C +R), (A, A+ P + Q), (B, A + B + Q), (C, A + C + P).

x . Take points Xi, Yi , S, T(i = 1, 2, 3) represented b y vectors xi, yi, s, t, where the four vectors xi, s are independent, and t = x1 + x2 + x3 + s and yi = xi + xi+1 + s. the coclique consist of the six ﬂags (Xi, Xi +Xi+1+S), (Yi+2, Xi +Xi+1 +Yi+2)(i = 1, 2, 3).

xi . Here we take a plane H0, a point Q not on it, and another point P outside H0, take a plane H on P, not on Q: take a permutation π on the set of points on the space fixing H0 (if we denote set of points of H0 by 0, 1, 2, 3, 4, 5, 6, π(i) = i + 1 mod 7) and Q. It is a permutation of order 7. The intersection of H and H0 is the line 1, 2, 4 . This line is carried to all other lines of H by powers of π. Let { } 0 (P + Q) H = 0 . The flags are the orbit of (P, H) under π. The image of P is ∩ 0 { } defined by the line joining Q and π(0).

For q = 1, only 1, 2, 3 and 4 holds. Final case exists only if q = 2. The case 10 is maximal only if q = 2. If q = 2, it extends to the final case. For q = 2, the cases 1, 2, 3, 6, and 11 6 attain g(3)= 7, and only for the first two, all flag points are on a hyperplane. 80 MAXIMAL COCLIQUES IN POINT-HYPERPLANE GRAPHS

There is a correspondence between the last cases in the classiﬁcations of cocliques in PG(2, 2) and PG(3, 2). We can rephrase the example of PG(2, 2) according to this correspondence. Take a projective plane over F , give the following naming to the points: P , P , P 2 { 1 2 3} be a line let this be H0. Take a point outside H0, call it Q, and another point say P outside H different from Q. P,Q will intersect H , say at P . 0 〈 〉 0 1 Deﬁne a permutation π on the set of points of the plane, sending P1 to P2, P2 to P3, P3 to P1 and Q to itself. Let a hyperplane joining P and P having the point P , Other lines are P , P, P , 3 4 { 5 2} P , P , P , P ,Q, P . { 1 4 5} { 4 2} Under this map π (P, H) has the orbit (P, P, P , P ), (P , P , P , P ), (P , P , P, P ), { 3 4} 4 { 1 4 5} 5 { 2 5} But then Z(C)= P, P , P and these are not on a line. { 4 5} PART III

CALCULATING SOME OTHER PARAMETERS AND EIGENVALUES

6 SMITH NORMAL FORMSOF SOME KNESER GRAPHS FOR BUILDINGS

6.1 Introduction

In this chapter, we will be working on the Smith Normal Forms of some strongly regular graphs, including the non-collinearity graphs of generalized quadrangles with classical parameters. Within the text, eigenvalues of a strongly regular graph are denoted by k, r, s with respective multiplicities 1, f , g being consistent with the general terminology. Let q be a prime power pt for some prime p, and t a positive integer. We denote the all-one vector by 1. We want to see the relation between the SNF of some strongly regular graphs of our interest (with prime power eigenvalues) and the eigenvalues of the graphs. The current focus is on the non-collinearity graphs of the generalized quadrangles with parameters: GQ(q, q), GQ(q, q2), GQ(q2, q), GQ(q2, q3), GQ(q3, q2), where q is as above. These graphs are all strongly regular with prime power eigenvalues. ± When the diameter of the collinearity graph is two, the Kneser graph on points and the non-collinearity graph coincide, since being far-apart will mean being at distance 2, which means being non-collinear.

6.1.1 Preliminaries

Here we state the propositions and theorems that we will make use of, and give the required deﬁnitions. The proofs of the theorems and propositions in this section can be found in [24].

DEFINITION 6.1.1.1. For an integral matrix A, there exist integral matrices P and Q with determinants 1, for which the matrix PAQ is a diagonal n n matrix, and the diago- ± × nal elements s , s ,..., s of PAQ satisfy the condition ‘s s s . . . s ’. This matrix PAQ, 1 2 n 1| 2| 3 | n which is unique up to signs, is called the Smith Normal Form (SNF) of A. The diagonal elements of PAQ are called the invariant factors (or elementary divisors) of the SNF. 84 SMITH NORMAL FORMS OF SOME KNESER GRAPHS FOR BUILDINGS

Through the text, to give the SNF of a matrix, we will write down the invariant factors to the power their multiplicities, instead of giving the matrix itself.

DEFINITION 6.1.1.2. The p-rank of a square integral matrix M, which is denoted by rkp M is the rank of the matrix M over the ﬁnite ﬁeld Fp.

PROPOSITION 6.1.1. [24]The p-rank of a square integral matrix M is the number of si not divisible by p which is denoted by rkp M.

It follows that if M is square and pe det M, then rk M n e. || p ≥ − The SNF and p-rank of a graph refers respectively to the SNF and p-rank of its adjacency matrix. The collinearity and non-collinearity graphs of generalized quadrangles GQ(q, q), GQ(q, q2), GQ(q2, q), GQ(q2, q3), GQ(q3, q2), are strongly regular graphs. Moreover, for the non-collinearity graphs, the eigenvalues are prime powers. The parameters (v, k, λ, µ) of the collinearity graphs of the above generalized quadrangles are known:

THEOREM 6.1.2. The collinearity graph of a ﬁnite generalized quadrangle GQ(s, t) is strongly regular with parameters:

(v, k, λ, µ)=((s + 1)(st + 1), s(t + 1), (s 1), (t + 1)). − The parameters of the complement graphs, that is the non-collinearity graphs are also known:

REMARK 6.1.3. The complement of a strongly regular graph with parameters (v, k, λ, µ), is again strongly regular with parameters:

(v, k, λ, µ)=(v, v k 1, v 2k + µ 2, v 2k + λ). − − − − − In particular, the noncollinearity graph of a ﬁnite generalized quadrangle GQ(s, t) is strongly regular with

(v, kλ, µ)=((s + 1)(st + 1), s2 t, s2 t st s + t, st(s 1)). − − − The eigenvalues of a strongly regular graph can be computed from its parameters:

THEOREM 6.1.4. Let the parameters of a strongly regular graph be (v, k, λ, µ). These parameters are related to its restricted eigenvalues r, s (r > s) and the respective multiplicities f , g in the following way:

(i) rs = µ k, r + s = λ µ, − − 1 (r+s)(v 1)+2k (ii) f , g = (v 1 − ). 2 r s − ∓ − REMARK 6.1.5. The parameter s in Theorem 6.1.2 and Theorem are different.

(v 1)s+k (v 1)r+k REMARK 6.1.6. There are alternative formulas for f and g: f = − and g = − s r r s since f + g + 1 = v and fr + gs + k = 0. − − 6.1 INTRODUCTION 85

REMARK 6.1.7. A generalized quadrangle with parameters (s, t) need not be determined by these parameters. For instance, GQ(q, q) can be obtained from Sp(4, q) and from O(5, q) which would be isomorphic only if q = 2e for some integer e [93]. So, although the eigenvalues will be the same, the graphs will be different.

EXAMPLE 6.1.1.3. Consider GQ(3, 3). One can obtain examples by making use of Sp(4, 3) (the isotropic points and totally isotropic lines in PG(3, 3) with a non-degenerate symplectic form, and from O(5, 3) -the isotropic points and totally isotropic lines in PG(4, 3) with a non-degenerate quadratic form. Their SNF are respectively [25] (110, 319, 910, 271), and (114, 311, 914, 271).

n For a submodule M of Z , we denote the Fp vector space obtained from M by coordi- natewise reduction mod p by M.

PROPOSITION 6.1.8. Let A be an integral matrix of order n, p a prime number and i a non- n i n n negative integer. Put M := M (A) := x Z p− Ax Z . Then, M F is an F vector i i { ∈ | ∈ } i ⊆ p p i space and the number of invariant factors of A divisible by p equals dimp Mi.

PROPOSITION 6.1.9. Let Abe an integral matrix of order n, p a prime number, and i a non- i negative integer. Put Ni := Ni (A) := p− Ax x Mi . Then the number of invariant factors i+1 { | ∈ } of A not divisible by p equals dimp Ni .

PROPOSITION 6.1.10. For a square integral matrix A with integral eigenvalue α with geometric multiplicity m, the number of invariant factors of A divisible by α is at least m.

n REMARK 6.1.11. [24] For A as in Proposition 6.1.10, one can show for W = x Q n { ∈ | Ax = αx , the α eigenspace over Q with dim (W)= m, for W ′ = W Z , that dim (W )= } Q ∩ p ′ m.

Let us see an example for which the above propositions help to determine the SNF.

EXAMPLE 6.1.1.4. Consider the collinearity graph of GQ(2, 4). The spectrum is 101, 120, 56 and the product of the invariant factors is − 101.120.( 5)6 = 3906250. By Proposition 6.1.10 number of invariant factors divisible − by for example 10 is at least 1, divisible by 1 at least 20 and divisible by 5 at least 6. − Hence, invariants are either 121, 55, 501 or 120, 56, 101. Claim: SNF of this graph is (120, 56, 101). Proof. Look at the eigenspaces. There is a 6-space W of eigenvectors for eigenvalue 5, let W be W reduced mod 5, which is still a 6-dimensional vector space (see Remark − 6.1.11). Similarly, the 1-space of eigenvalue 10 will still be a 1-space mod 5. Now the question is, can this 1-dimensional space be in W. Vector 1, the eigenvector of the eigenvalue 10, is orthogonal to W and W, so if it was in W, it should be orthogonal to itself which is not the case (since (1, 1) = 27 = 0 mod 5). For p = 5, i = 1, M is a 6 1 7-space, and upon reduction mod 5, it remains a 7-space, that is, by Proposition 6.1.8 the SNF is (120, 56, 101).

The following example by Brouwer is the motivation behind looking at strongly regular graphs with prime power eigenvalues: 86 SMITH NORMAL FORMS OF SOME KNESER GRAPHS FOR BUILDINGS

EXAMPLE 6.1.1.5. Consider the lines in PG(3, q). The graph qK(3 : 2) has the lines of this geometry as vertices. The adjacency condition for a pair of lines in this graph is to be disjoint. This graph is a strongly regular graph with prime power eigenvalues: q4, q, q2. − Hence, the determinant and the invariant factors are prime powers. i Let p occur as invariant factor with multiplicity ei. Brouwer proved 0 i t ei = f , 2t i 3t ei = g, e4t = 1, and ei = 0 for t < i < 2t, 3t < ≤ ≤ ≤ ≤ i < 4t, and i > 4Pt, e3t i = ei forP 0 i < t, so he showed how the multiplicities of the − ≤ invariant factors (si’s) and the multiplicities of the eigenvalues are related by making use of the structure of the graph.

We will apply Brouwer’s method to the generalized quadrangles stated above. For some of the graph examples in the following sections, the computer algebra system “Groups, Algorithms and Programming” (GAP) [58] is used to calculate the SNF of the related graph. If the automorphism group of a graph is known, by adding edge orbits, one can obtain the graph, and its SNF.To do that in GAP we used the package GRAPE [89].

THEOREM 6.1.12. (Lucas Theorem)[26] Let

i i m = i mi p and k = i ki p P P (0 m , k p) be the representations of m and k in base p, for some prime p. Then ≤ i ≤

m mi i mod p. k ≡ ki Q There is also a multinomial version of Lucas theorem. Let e be the digits of e in ki k base p. Then

m mi mod p. e ,...,e i e ,...,e 1 t ≡ 1i ti Q where mi = 0 if e + . . . + e = m . e ,...,e 1i ti i 1i ti 6 e n+1 THEOREM 6.1.13. [65] Let p be a prime and s, n, e be in N. Writeq = p and ν =(q − 1)/(q 1). The dimension r(q, n, s) of the F code of the design of points and projectively − p s-dimensional subspaces of PG(n, q) if e = 1 is

n s j n+1 n+i(p 1) jp − − − i=0 o j i i/p( 1) , ≤ ≤ − − j n P P n s n+1 which is equal to − when q = 2 and if s = n 1 is:, (i.e. for the square design of i=0 i − points and hyperplanes)P is

n+p 1 e − + 1 n .

COROLLARY 6.1.14. [26] The dimension of the Fq code spanned by the (characteristic vectors n+p 1 e e of the) complements of hyperplanes in PG(n, q) is − if q = p . n 6.1 INTRODUCTION 87

6.1.2 Review of the related problems

Smith Normal Forms of incidence matrices are of interest in many branches of mathematics like design theory, coding theory, representation theory, finite geometry, and algebraic graph theory. The incidence relation can be on vertices of a graph, on subsets of a finite set or the q-analogue of that, namely the subspaces of a finite vector space over Fq, subspaces of projective or affine spaces possibly of different dimensions, etc.. The Smith Normal Form and p-ranks were used to calculate bounds on the sizes of objects, to show existence conditions or to show some objects are not isomorphic. Here, we try to summarize what has been done so far about the p-ranks and Smith Normal Forms related to some objects. Hamada [66] was interested in p-ranks of matrices within the context of coding theory. Moorhouse described problems related to p-ranks of the incidence matrices of finite geometries, how they are useful and what methods can be used to obtain results for those ranks [81]. One can take a v set, have a look at all t-subsets and k-subsets of it where t < k, and let them be incident if the t-set is inside the k-set. The resulting incidence matrix is used by Wilson to work on t-(v, k, λ) designs [104]. Blokhuis and Calderbank [14] made use of the Smith Normal Form of the incidence matrix of a 2-(v, k, λ) design to obtain existence conditions for designs with given parameters. Xiang collected results on the p-ranks and Smith Normal Forms of some 2-(v, k, λ) designs [106], and geometries [105]. n The q-analogue of this problem deals with k-subspaces of Fq instead of k-subsets of an n-set. This is studied by Kantor [75], Yakir and Frumkin, [57]. The incidence matrices of graphs are the adjacency matrices. The p-ranks of strongly regular graphs were studied by Brouwer and van Eijl [25]. The p-ranks of distance regular graphs were studied by Peeters [83]. The Smith Normal Forms of the incidence matrices of points and projective (r − 1)-dimensional subspaces of PG(n, q) and of the incidence matrices of points and r- dimensional affine subspaces of AG(n, q) for all n, r, and arbitrary prime power q were calculated by Sin, Xiang and Chandler [38]. The p-ranks of the incidence matrix of intersecting linear subspaces of arbitrary dimensions are computed by Sin [88]. Blokhuis and Moorhouse [15] determined the p-rank of the incidence matrix of hyperplanes of PG(n, pe) and points of a non-degenerate quadric and made use of that to give bounds on some geometrical objects such as ovoids in the finite classical polar spaces. There exists a recent study of Arslan and Sin [2] on the p-rank of the point- hyperplane incidence of orthogonal geometries, based on a question introduced in [15]. In this paper, the p-rank of the point-hyperplane incidence matrix for the dual Hermitian generalized quadrangle DH(4, q2) is also given. Chandler and Xiang calculated the p-ranks of some relative difference sets they constructed, and by rank comparison they showed the inequality of those relative difference sets with the classical ones [40]. They made use of the Smith Normal Forms of some symmetric designs to show that the symmetric designs of interest are non-isomorphic in [41]. Concerning generalized quadrangles, Chandler, Sin and Xiang came up with a closed formula for the p-rank of the incidence matrix between the points and lines of the sym- 88 SMITH NORMAL FORMS OF SOME KNESER GRAPHS FOR BUILDINGS plectic generalized quadrangle over a field of odd order [39]. For the point-line incidence matrix of the classical generalized quadrangles, Sastry and Sin gave the 2-rank of every classical generalized quadrangle of order 2e [85], and for every classical generalized quadrangle of odd order, the 2-rank is determined by Bagchi, Brouwer and Wilbrink [5]. Caen and Moorhouse in an unpublished paper determined the p-rank of the point-line incidence matrix of the classical generalized quadrangles (of type Sp(4, p) or its dual O(5, p)) where p is any prime [31].

6.2 Main theorem

For the graph example related to PG(3, q), the key points of the proof are, graph’s being strongly regular, hence there is a quadratic equation concerning the adjacency matrix A, and all the eigenvalues’ being powers of q, which implies, the determinant’s being a prime power. When det A is a prime power, so are the invariant factors. In addition, v =(1,1) = 0 mod p, so vector 1 is not orthogonal to itself mod p. 6 Following the proof of Brouwer, one can come up with the following generalization:

THEOREM 6.2.1. Let p be a prime, A be the adjacency matrix of a strongly regular graph with parameters (v, k, λ, µ), and pa k, pb r, pc s, where a b + c and p ∤ v, and Ahas e || || || ≥ i invariant factors s with pi s , then e = 0 for min(b, c) < i < max(b, c) and b + c < i < a j || j i and i > a. Moreover, e(b+c) i = ei for 0 i < min(b, c). − ≤ Proof. Let the assumptions pa k, pb r, pc s , where a b + c and p ∤ v hold. Deﬁne || || || ≥ x := max(b, c), y := min(b, c), let X be the multiplicity of the eigenvalue related to px , y Y be the multiplicity of the eigenvalue related to p . Deﬁne mi := j i ej. ≥ Since the number of the invariant factors is equal to the size ofP the matrix, i ei = f + g + 1. Since the product of the invariant factors is the product of the eigenvalues,P

i iei =(f b+ gc+a). These can be written as i ei = X +Y +1, i iei =(xX + yY +a). PHence, (i y)e =(x y)X + a y. P P i − i − − HereP we will show m 1 and m X +1. Remember, pa k. The number of invariant a ≥ x ≥ || factors that are divisible by k is at least 1 by Proposition 6.1.10, hence ma 1. The i ≥ number mi is the number of invariant factors of A divisible by p and by Proposition 6.1.8 that is equal to dimp Mi. The eigenvectors corresponding to the eigenvalue e for which px e span a X -dimensional vector space V . The space V will still be a X -dimensional || vector space, and is a subspace of Mx . The question is, if the mod p reduction of the eigenvector 1 of eigenvalue k is in this vector space. The vector 1 is orthogonal to all eigenvectors with eigenvalues other than k, and (1, 1) = v = 0 mod p, (p ∤ v), that is 6 1 / V . Therefore, dim M = m X + 1. ∈ p x x ≥ Since the graph is strongly regular, A(A (r + s)I)= rsI + µJ. n (b+c i) − i − n i Now, we will show for ν Z , p− − A(p− (A (r+s)I))ν Z if p− (A (r+s)I)ν ∈ − ∈ − ∈ Zn, for 0 i < y which is the case here since p y r, and p y s. This means by deﬁnition, i ≤ | | p− (A (r + s)I)ν Mb+c i (A). − ∈ − (b+c i) i (b+c) p− − A(p− (A (r + s)I))ν = p− A(A (r + s)I)ν − − (b+c) = p− ( rsI + µJ)ν. − By Theorem 6.1.5 µ = rs+k, and since a b+c, pb+c divides µ and rs. So, the resulting n n i ≥ vector is in Z for ν Z . That is, p− (A (r + s))ν Mb+c i (A), for 0 i < y. ∈ − ∈ − ≤ 6.3 NON-COLLINEARITYGRAPHSOFGENERALIZEDQUADRANGLES 89

i Now, we will show Ni Mb+c i (A) for 0 i < y. If0 i < y, then p− (r + s)ν = 0 ⊆ − ≤ ≤ i mod p. Suppose ν Ni . So there is a ν1 Mi for which ν = p− Aν1 mod p. Then i i ∈ i ∈ i n ν = p− Aν1 + p− (r + s)ν1 = p− (A +(r + s))ν1 mod p. But, since p− Aν1 Z , then so i i ∈ does p− (A+(r + s)I)ν1, which implies, p− (A+(r + s)I)ν1 Mb+c i (A). The vector ν is i ∈ − in Mb+c i (A) since ν = p− (A +(r + s)I)ν1 mod p, that is Ni Mb+c i (A). − ⊆ − Here, we will come up with a lower bound on mb+c i by making use of the fact that − Ni Mb+c i (A) and Ma(A) Mb+c i (A) and Ma(A) contains vector 1 1. First, Ni does ⊆ − ⊆ − not contain 1. Vector 1 has coordinate sum v = 0 mod p, but the vectors in Ni have a i 6 coordinate sums divisible by p − , so they are 0 mod p, so 1 can not be in Ni. Hence we can deduce, dim(Ni)+ 1 dim(Mb+c i (A)). ≤ − All invariants are powers. The set S of invariants which are not divisible by pi+1 are i the ones less than or equal to p . There are e0 + . . . + ei of them. Size of S is equal to dimp(Ni) by Proposition 6.1.9. So, dimension of Ni is e0 + . . . + ei. The number of b+c i invariant factors divisible by p − (that is mb+c i ) is dimp(Mb+c i (A)). So,for0 i < y, i − − ≤ mb+c i k=0 ek + 1. − ≥ So forP all a for which 0 a < y, the inequality, e + e 1 is true. ≤ − 0 i a i i b+c a i ≥ These imply: P ≤ ≤ P ≥ −

(i y)e + (i xt)e + ym y (6.1) − i − i b+c+1 ≥ 0Xi y xt Xi (b+c) ≤ ≤ ≤ ≤

(i y)e + (i xt)e + ym +(x y)m y +(x y)(X + 1) (6.2) − i − i b+c+1 − xt ≥ − 0Xi y xt Xi (b+c) ≤ ≤ ≤ ≤

(i y)e + (i y)e + xtm (x y)X + x) (6.3) − i − i b+c+1 ≥ − 0Xi y xt Xi (b+c) ≤ ≤ ≤ ≤

(i y)e + (i y)e + xtm +(a x y)m (x y)X + a y (6.4) − i − i b+c+1 − − a ≥ − − 0Xi y xt Xi (b+c) ≤ ≤ ≤ ≤ But (i y)e = ((x y)X + a y). So equality holds everywhere. Therefore the i − i − − theoremP holds.

This theorem gives a symmetry but not the precise SNF. However it helps to find the actual SNF in may cases for small q. You will see in some of the following examples, once the p-rank is known, the exact SNF will be known. In particular, because of the pattern given by the theorem, when q = p, the SNF x f x 2 g x 3 x 4 1 follows from the p-rank. The SNF will be something like 1 , p − , (p ) − , (p ) , (p ) where x is the p-rank. So for those examples, where p = q, finding p-rank is equivalent to finding the SNF.

6.3 Non-collinearity graphs of generalized quadrangles

The vertices of these graph are the points of the geometry, which are adjacent when there is no line joining them. 90 SMITH NORMAL FORMS OF SOME KNESER GRAPHS FOR BUILDINGS

6.3.1 Non-collinearity graph of GQ(q, q2) The non-collinearity of the geometry is strongly regular with parameters (v, k, λ, µ) = ((q + 1)(q3 + 1), q4, q4 q3 + q2 q, q4 q3). − − − Since the number of the invariant factors is equal to the size of the matrix, i ei = f + g + 1. The number of p’s in the determinant of A is i iei = t(f + 2g + 4)P. Then, ie t e = (i t)e = t(g + 3). P i i − i i i − i P Since rsP = qP3, r + s = q2 q, the eigenvalues are q4, q2, and q with respective − − − multiplicities: 1, f = q3 q2 + q, and g = q4 + q2. − By Theorem 6.2.1, we have e0 + e1 + . . . + et = f and e2t + . . . + e3t = g and e4t = 1 and ei = 0 for t < i < 2t and 3t < i < 4t and i > 4t. Moreover, e3t i = ei for 0 i < t. − ≤ We will use the following lemma [25] in the examples below.

LEMMA 6.3.1. let A be adjacency matrix of the collinearity graph of GQ(s, t) where p s. Let | B be the sub-matrix of order s2 t induced by the rows and columns corresponding to points non-adjacent to a ﬁxed vertex v of the graph. Then rk (J I A)(non-collinearity graph) p − − is equal to rk (J I B). p − − This lemma says, the p-rank of the non-collinearity graph Γ of a generalized quadrangle GQ(s, t) where p s is equal to the p-rank of the local graph, that is, the graph induced | on the neighbors of a ﬁxed point in Γ.

EXAMPLE 6.3.1.1. For q = 2, there is a unique generalized quadrangle GQ(2, 4), and its non-collinearity graph is called the Schläﬂi graph. The eigenvalues are 161, 46, 220. We − will show that the SNF of this graph is, (16, 214, 40, 86, 161). x 20 x 6 x x 1 By Theorem 6.2.1 the SNF is (1 , 2 − , 4 − , 8 , 16 ). Here, 2-rank is the multiplicity x of the invariant factor 1. We know 2-rank of a symmetric integral matrix with zero diagonal is even [25], so, x 0, 2, 4, 6 . Lemma 6.3.1 applies to GQ(2, 4), where p = 2. But, the local graph of ∈{ } the non-collinearity graph is a strongly regular graph with parameters (16, 10, 6, 6)(its name is Clebsch graph) and its 2-rank is 6 [25]. That is, x is 6 by Lemma 6.3.1. Hence, SNF of this graph is, (16, 214, 86, 161).

EXAMPLE 6.3.1.2. SNF of non-collinearity graph of GQ(3, 9) and GQ(5, 25) are the following respectively: (119, 371, 92, 2719, 811), (185, 5565, 2520, 12585, 6251)(GAP)

2 CONJECTURE 6.3.2. The SNF of non-collinearity graph of GQ(p, p ) for prime p is: p(p 1)(p 2) p(p 1)(p 2) p(p 1)(p 2) p3 p2+p − − p4+p2 (p3 p2+p − − ) − − ((p0) − − 3 , (p1) − − − 3 , (p2) 3 , p(p 1)(p 2) p3 p2+p − − 1 (p3) − − 3 , (p4) )

6.3.2 Non-collinearity graph of GQ(q, q) The parameters of the non-collinearity graph of GQ(q, q) are: (v, k, λ, µ)=(q3 +q2 +q+1, q3, q3 q2, q3 q2), so by Theorem 6.1.5, rs = q2, r +s = 0. − − − Hence, the eigenvalues are k = q3, q, and q, with respective multiplicities, 1, f = − 1 (q3 + q) and g = 1 (q3 + 2q2 + q). 2 2 Similar to above, i ei = f + g + 1, where f and g are multiplicities of r and s and ie = t(f + g + 3),P hence (i t)e = 2t. i i − i P P 6.3 NON-COLLINEARITYGRAPHSOFGENERALIZEDQUADRANGLES 91

By Theorem 6.2.1, m3t = 1, for i > 3t, 2t < i < 3t ei = 0, for 2t < i < 3t ei = 0, 0 i 2t ei = f + g, and for 0 i < t, ei = e2t i. ≤ − P ≤ ≤

EXAMPLE 6.3.2.1. There is a unique GQ(2, 2), the generalized quadrangle on 15 points and 15 lines described by the pairs of a 6-set and the partitions of a 6-set into 3 pairs. We will show its non-collinearity graph has the following SNF: (14, 26, 44, 81). The non-collinearity graph has spectrum (81, 25, 29). According to Theorem 6.2.1, x 14 2x x 1 − its SNF is (1 , 2 − , 4 , 8 ). If we know the 2-rank, then we know the SNF. But the 2-rank, that is x here, is equal to the 2-rank of K K by Lemma 6.3.1, which is 4. So 2 × 4 the SNF is (14, 26, 44, 81). We are looking at the GQ(q, q) obtained from Sp(4, q), isotropic points and totally isotropic lines in PG(3, q) with a non-degenerate symplectic form.

EXAMPLE 6.3.2.2. The SNF of non-collinearity graphs of symplectic GQ(3, 3), GQ(5, 5) and GQ(7, 7) are respectively: (110, 319, 910, 271), (135, 585, 2535, 1251) and (184, 7231, 4984, 3431). (GAP).

PROPOSITION 6.3.3. The p-rank of the non-collinearity graph of the Sp(n+1, q) generalized p+n 1 t t quadrangle equals − , where q = p . n Proof. Let f be the symplectic form, so that f (x, y) = 0 when x and y are collinear. q 1 The entries of the non-collinearity matrix are f (x, y) − . Since f is non-degenerate by assumption, all linear forms on V are of the form l(x)= f (x, y), so that the question is q 1 about the dimension of the function space spanned by the l(x) − . q 1 q 1 Expanding ( a X ) − in terms of multinomial coefﬁcients one gets − ... a i i i j j j0 j q 1 sum of monomialsP X . . . X n where monomials will not appear when − P= 0 in F . 0 n j p q 1 If q = p then − clearly does not involve p and we get all monomials of total degree j p+n 1 p 1. The number of monomials in n + 1 variables of total degree p 1 equals − . − − n So, this settles q = p. If q = pt then the number of monomials that occur with non-zero coefﬁcients is p+n 1 t − , as follows by Lucas’ theorem. n For this argument, see [26].

q 1 EXAMPLE 6.3.2.3. q = 9, p = 3, t = 2, n = 3. We have ( − , j1, j2, j3)=(q 1)!/j0!j1!j2!j3! j0 − and ( 8 , 3, 0, 0) = 8!/5!3! = 56 = 2 mod 3 while ( 8 , 2, 1, 0) = 8!/5!2! = 168 = 0 5 5 mod 3. Looking at the non-zero terms we ﬁnd that there are precisely 100. Why? The multinomial version of Lucas says that one writes top and bottom in p-ary and expands the product. 8 = 2.3 + 2, 5 = 1.3 + 2 etc, so ( 8 , 3, 0, 0)=( 2 , 1, 0, 0)( 2 , 0, 0, 0)= 2 5 1 2 mod 3, while ( 8 , 2, 1, 0)=( 2 , 0, 0, 0)( 2 , 2, 1, 0) = 0. You see that in each digit 5 1 2 p+n 1 p+n 1 t position there are − possibilities, − in all. n n COROLLARY 6.3.4. The SNF of non-collinearity graph of symplectic GQ(p, p) for prime p is: 3 3 2 (1( p3 + 3p2 + 2p)/6, p(2p +p)/3, (p2)(p +3p +2q)/6, (p3)1). 92 SMITH NORMAL FORMS OF SOME KNESER GRAPHS FOR BUILDINGS

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

The parameters of the non-collinearity graph of GQ(q2, q) are: (v, k, λ, µ)=(q5 + q3 + q2 + 1, q5, q5 q3 q2 + q, q5 q3), so by Theorem 6.1.5, rs = − − − q3, r + s = q2 + q. − − The eigenvalues are k = q5, q, and q2, with respective multiplicities 1, f = q5 q4 + q3 − − and g = q4 + q2.

Similar to above, i ei = f + g + 1, where f and g are multiplicities of r and s and ie = t(f + 2g + 5)P, hence (i t)e = gt + 4t. i i − i P Deﬁne m as above, m P1 and m g + 1. i 5t ≥ 2t ≥ By Theorem 6.2.1, for i > 5t, for t < i < 2t, and for 3t < i < 5t ei = 0, m5t = 1, 2t i 3t ei = g, e3t i = ei for 0 i < t, and ﬁnally, 0 i

EXAMPLE 6.3.3.2. The SNF of non-collinearity graphs of the unitary generalized quadrangles GQ(4, 2), GQ(9, 3) and GQ(25, 5) are respectively: (114, 210, 46, 814, 321) and (169, 3120, 921, 2769, 2431), and (1525, 52100, 2525, 125525, 31251)(GAP).

2 CONJECTURE 6.3.5. The SNF of non-collinearity graph of (unitary) GQ(q , q) for a prime p is: 3 2 3 2 (p3+p2)(p 1) (p3+p2)(p 1) 4 2 4 (p +p )(p 1) 4 (p +p )(p 1) p4 − p5 p4+p3 p4 − 2 p +p p − 3 p − 5 1 (1 − 6 , p − − − 6 , (p ) − − 6 , (p ) − 6 , (p ) ).

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

The parameters of the non-collinearity graph of GQ(q2, q3) are: (v, k, λ, µ)=(q7 + q5 + q2 + 1, q7, q7 q5 + q3 q2, q7 q5), so by Theorem 6.1.5, rs = − − − q5, rs = q3 q2. Hence, the eigenvalues are k = q7, q3, and s = q2, with respective − − − multiplicities 1, f = q6 q5 + q4 q3 + q2 and g = q7 q6 + 2q5 q4 + q3. − − − − Similar to above, i ei = f + g + 1, where f and g are multiplicities of r and s and ie = t(3f + 2g + 7P), hence (i 2t)e = ft + 5t. i i − i P Deﬁne m as above, m 1P and m f + 1. i 7t ≥ 3t ≥ By Theorem 6.2.1, m7t = 1, m3t = f + 1, ei = 0 for 2t < i < 3t, 5t < i < 7t, and i > 7t, e7t = 1, ei = e5t i for 0 i < 2t, 0 i 2t = g, − ≤ ≤ ≤ 3t i 5t = f , P P According≤ ≤ to Theorem 6.2.1, SNF of this graph is of the form (x, y, g (x + y), − f (x + y), y, x, 0, 1). − 24 20 76 20 24 1 EXAMPLE 6.3.4.1. SNF of (unitary) GQ(4, 8) is 1 , 2 , 4 , 16 , 32 , 128 (GAP) SNF of (unitary) GQ(9, 27) is 1199, 3330, 91361, 2720, 81330, 243199, 21871 (GAP)

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

The parameters of the non-collinearity graph of GQ(q3, q2) are: (v, k, λ, µ)=(q8 + q5 + q3 + 1, q8, q8 q5 q3 + q2, q8 q5), so by Theorem 6.1.5, rs = − − − q5, r + s = q3 + q2. − − 6.4 GRAPHS WITH PRIME POWER EIGENVALUES AND SNF 93

The eigenvalues are q8, q2, and q3, with respective multiplicities 1, f = q8 q7 + q6 − − − q5 + q4 and g = q7 q6 + 2q5 q4 + q3 . − − Similar to above, i ei = f + g + 1, where f and g are multiplicities of r and s and ie = t(2f + 3g + 8P), hence (i 2t)e = t(g + 6). i i − i P Deﬁne m as above, m 1P and m g + 1. i 8t ≥ 3t ≥ By Theorem 6.2.1, m8t = 1, m3t = g + 1, ei = 0 for i > 8t, 5t < i < 8t, 2t < i < 3t, so, 0 i 2t ei = f , 3t i 5t ei = f P ≤ ≤ P ≤ ≤

6.4 Graphs with prime power eigenvalues and SNF

It is easier to find the SNF when the eigenvalues are powers of q. In that case, the determinants of the adjacency matrices are prime powers, hence the invariant factors are prime powers. So far we looked at strongly regular graphs with prime power eigenvalues since for such graphs we have a symmetry relation. Now we give examples of graphs with prime pover eigenvalues. Brouwer showed [20] for finite buildings of spherical type defined over Fq, for the graph defined over flags with the adjacency condition of being opposite, that the squares of the eigenvalues are powers of q. Here we summarize his notation and state his theorem.

6.4.1 Oppositeness graphs

These graphs are on buildings of spherical type. For a building of spherical type, let (W, R) be the related Coxeter group as usual. Conjugation by w0 gives a diagram automorphism, w0 where w0 is the longest element of R. So, for a flag type J, when we say J , what we mean is we take the nodes of the diagram determining the type, which are labeled with generators of the Coxeter group (W, R), take a conjugate by w0 for each element. The resulting generators will determine some other flag type. Here for a pair of flags to be opposite means, if one flag is of type J and the other is of type K, J w0 = K. The graphs will be on cosets of subgroups P of W, where J R, J ⊂ WJ = I and PJ = BWJ B. An object of type S J or cotype J is a coset of PJ . For a pair 〈 〉 \ 1 of objects gPJ , hPK the condition of being opposite is satisfying PJ gh− PK = PJ w0 PK , and having opposite types K, J. The graph ΓK,J is a bipartite graph having objects of cotype J and K as vertices, with the adjacency condition of being opposite objects. When

J = K, which is only possible when the diagram is symmetric, we call the graph ΓJ . The following is the main theorem of [20].

w0 THEOREM 6.4.1. Let G be deﬁned over Fq. Let J be a proper subset of S, and let K = J . 2 e Let Θ be an eigenvalue of ΓJ,K or, if J = K, of ΓJ . Then Θ = q for some integer e.

The next graph is not strongly regular, however, it has prime power eigenvalues. It is regular, and valencies are easy to calculate, so we know one eigenvalue and, since it is bipartite, its spectrum is symmetric that is for each eigenvalue θ, θ is also an − eigenvalue, with the same multiplicity. 94 SMITH NORMAL FORMS OF SOME KNESER GRAPHS FOR BUILDINGS

6.4.2 Bipartite graph of disjoint point and lines in PG(2, q) Let us call the bipartite graph of disjoint point and lines in PG(2, q) Γ. This graph has q3 1 q3 1 q2 1 2 − vertices. Each point has degree − − = q2. In ﬁgure 6.1, you see the q 1 q 1 q 1 − − − − distance distribution diagram of the graph Γ′ on points and lines of the space where adjacency condition is being incident. So, being far away in Γ′ means being adjacent in Γ. By making use of the distance distribution diagram we will calculate the eigenvalues. The eigenvalues of the distribution matrix is in general a subset of the eigenvalues of the graph. We will calculate the P, and Q matrices of the related association scheme which will give us the eigenvalues of Γ.

q+1 q(q+1) q2 1 q 1 q q+1

FIGURE 6.1: distribution diagram of the incidence graph of points and lines of PG(2, q)

The P matrix of the association scheme on the set of vertices with relation Ri being 1 q + 1 q(1 + q) q2  1 pq 1 pq  having distance i in Γ′ is: − −  1 1pq 1 pq   − − 2   1 q 1 q(1 + q) q   − − −  That is, the eigenvalues of Γ are q2, pq, pq, q2, with multiplicities 1, q2 + q, q2 + − − q, 1. 7

FROMGRAPHSOF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

7.1 Introduction

In this chapter, the collinearity graphs of some parapolar spaces, which are exceptional root filtration spaces as studied in [46]. are examined. These collinearity graphs’ distribution diagrams allow us to calculate the eigenvalues of the related Kneser type graphs, which we use to obtain bounds on the coclique sizes. The spaces involved are Lie incidence systems of types E6,2, E7,1, E8,8, and F4,1 (see definition 1.5.3.6). For finite simple groups of Lie type, and for the corresponding Coxeter groups, graphs of Lie type and graphs of Coxeter type, respectively are graphs defined on certain cosets, with the corresponding groups. They are studied in detail in Chapter 10 of [21]. There are Kneser graphs defined in connection to spherical buildings (see Section 3.1.3). We will work on four of those by making use of the relation between those graphs and the corresponding graphs of Lie type. These Kneser graphs and graphs of Lie type are defined on the same cosets of the related group of Lie type. Being at maximal distance in the graph of Lie type corresponds to being adjacent in the related Kneser graph. In Section 7.2, a brief introduction to the graphs of Coxeter and Lie type, and parapolar spaces is given. In Section 7.2.6 chain calculus is discussed, which is a tool to understand the geometry by use of the diagram. In Section 7.3, methods for calculating the parameters of the distribution diagrams are discussed. Moreover P and Q matrices, spectra of the collinearity graphs and eigenvalues of the related Kneser graphs are calculated. In Section 7.4, the Hoffman bound is used to obtain bounds on the sizes of cocliques of the four Kneser type graphs. This chapter ends with a summary of eigenvalue results about the Kneser graphs on flags of buildings. Paths, distances, cliques of the point-line geometry will refer to the paths distances, cliques of its collinearity graph. Similarly points and vertices will be used interchange- ably. 96 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

7.2 Preliminaries

The content of this section is mostly from [21] Chapter 10. Earlier in Chapter 1, we defined Buekenhout-Tits geometries. For basic definitions about them, see Section 1.5. Groups with Tits systems give rise to examples of these [97]. Coxeter groups and groups with Tits systems are also defined in Chapter 1.

Here we will be working on four of the Chevalley groups of types E6, E7, E8, F4 over the ﬁnite ﬁeld Fq which are examples of groups with Tits systems. They are denoted by E6(q), E7(q), E8(q), and F4(q).

7.2.1 Four diagrams

For a group G with a Tits system (B, N, W, R), if (W, R) is a Coxeter system of type Xn, then the Coxeter diagram of (W, R) is also the diagram of G.

Diagram Xn,i is diagram Xn with type i circled. The subscript n indicates the rank, i. e. the number of nodes of the diagram.

In Figure 7.1 below the diagrams commonly called E6,2, E7,1, E8,8, and F4,1 with nodes labeled as in [18] are given.

FIGURE 7.1: the diagrams E6,2, E7,1, E8,8, and F4,1 7.2 PRELIMINARIES 97

7.2.2 Graphs of Coxeter type

Deﬁne DI,J as follows:

D = w W l(iw)= l(w j) > l(w) for all i I and j J I,J { ∈ | ∈ ∈ } where ‘l’ refers to the length function of the Coxeter group. See Section 1.3.1 for an introduction to Coxeter groups.

THEOREM 7.2.1. ([21] 10.1.2 (iv)) There is a bijection between elements of DI,J and W W/W sending w D to W wW . The element of D that corresponds to W wW is I \ J ∈ I,J I J I,J I J the unique shortest element of WI wWJ . From now on, S R is the set R r where (W, R) is a Coxeter system with non-empty ⊆ \{ } R, and r is any ﬁxed element of R.

DEFINITION 7.2.2.1. Let (W, R) be a Coxeter system with non-empty R. Fix r R and ∈ put X = W . The graph ∆ := Γ(W, X , r) which has as vertices the left cosets wX (w S ∈ W), with the adjacency condition for any pair w1X , w2X of vertices, w1X and w2X are 1 adjacent if and only if w− w X rX is called a Coxeter graph. If the Coxeter system 2 1 ∈ has a diagram Xn, and r is numbered by i in the diagram, the related graph is denoted by Xn,i(1) and the graph is called a graph of Coxeter type Xn,i.

If (W, R) is reducible, and C is the connected component of the diagram Xn,i including r, then the Coxeter graph is isomorphic to (W , W , r), where D = C r [21]. C D \{ } The vertex w1X is adjacent to X if and only if w1 X rX . That is, for some x1, x2 X , ∈ 1 ∈ w1 = x1 r x2. Similarly, w2X is adjacent to w1X if and only if w2− w1 = x3 r x4 for some 1 1 x , x X , that is w = x r x x− r x− . More generally, vertices at distance k from X 3 4 ∈ 2 1 2 4 3 are the uX , where u has an expression as a product of factors in R with at most k factors equal to r.

The root system graph Let Λ be a fundamental system of roots in a root system Φ. Given Λ, there is a unique vector α Φ called the largest root such that each of its coefficients is not less than the ∈ corresponding coefficient of any other root [21]. Put Λ=Λ α . Suppose r = w (see ∪{ } α the Definitione 1.3.1.5) has a unique neighbor r in the Coxeter diagram. The stabilizer− of α Ý in W is precisely X = S so that, the cosets wX (w We) are in onee to one correspondencee 〈 〉 ∈ e with the roots that have the same length as α. Let Φ0 be this set of largest roots. Then the Coxeter graph Γ(W, X , r) can be described as a graph on Φ0, when two roots are adjacent where they make the same angle θ as α ande rα. This graph is called the root system graph for Φ. e e The association scheme related to a Coxeter graph The group W acts on the graph ∆ by left multiplication and this action is edge transitive. The stabilizer of the vertex X under this action is the subgroup X . The orbits of the stabilizer of the vertex X are the (sets of vertices in the) double cosets X wX . There is an association scheme on the vertices of ∆, where the orbitals of W on ∆ define the relations of the association scheme. 98 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

Since in each double coset there is a unique element from DS,S by Theorem 7.2.1, the relations can be labeled with elements from DS,S: let dI,J be the unique shortest element in WI wWJ . For a pair of vertices w1X and w2X in ∆ := Γ(W, X , r), we can say they are 1 in relation X wX (or in relation d ) when w− w X wX , For ∆ as above, r is in D . S,S 2 1 ∈ S,S Moreover, it is the shortest element of X rX . That is, a pair of vertices w1X and w2X are 2 adjacent in the Coxeter type graph (W, WS, r), when they are in relation r. Since r = 1, the relation it induces is symmetric, and the graph is undirected. For ∆, we deal with the distribution diagram of this association scheme with respect to the relation r. All the distribution diagrams for Coxeter graphs in this chapter are of this kind.

The Buekenhout-Tits geometry related to the Coxeter graph Given a Coxeter group (W, R) we find a geometry Γ(Y, , t) where the set of objects Y ∗ is the set of cosets wW (with S = R r ), for all r R and w W. The type function S \{ } ∈ ∈ maps wW to r , and cosets are incident when they have a nonempty intersection. The S { } Coxeter diagram of the Coxeter system is the Buekenhout-Tits diagram of the geometry. An object of type R I (cotype I) is a coset wW ; these can be identified with the flags \ I of corank i = I of the geometry mentioned above. | | DEFINITION 7.2.2.2. The shadow of an object of type R I on the graph of Coxeter type \ is the set of vertices incident with it, that is, the set of vertices contained in wWI X (the shadow of any vertex in (W, WS, r) is trivially itself).

This shadow induces a subgraph isomorphic to the Coxeter graph Γ(WI , WI r , r). \{ }

Calculating the number of vertices of ∆ For a ﬁnite irreducible Coxeter group, the method to calculate the number of vertices for ∆= Γ(W, W , r) where for any r R (equivalently the number of objects of type r in S ∈ { } the related Buekenhout-Tits geometry) is described below. For each ﬁnite Coxeter system (W, R), there is a sequence d ,..., d (where n = R ) 1 n | | of integers called its degrees, for which, the polynomial

n tdi 1 W(t)= tl(w) = − t 1 wXW Yi=1 ∈ − gives the size of the related Coxeter group when t is 1: W = W(1) = n d . The | | i=1 i number of objects of type r is [21] Q { } W | | . W | S| The number of vertices in relation i for i in DS,S to a given vertex is [21] (p. 308): W iW W W = | S S| = | S| = | S| ki 1 . WS iWS i− WS WiSi 1 S | | | ∩ | | − ∩ | REMARK 7.2.2. The second equality holds by Proposition 10.1.3 of [21]. Here, for kr 1 (with S = R r ) the set rSr− S corresponds to elements of R commuting with r. \{ } 1 ∩ Consider s, t S such that rsr− = t. But then s = t by [21] Theorem 10.1.2.(i). In the ∈ Coxeter diagram, this corresponds to the nodes not adjacent to r. 7.2 PRELIMINARIES 99

TABLE 7.1: the degrees of the ﬁnite irreducible Coxeter systems

Name degrees d (1 i n) i ≤ ≤ An 2, 3, . . . , n + 1 Bn 2, 4, 6, . . . , 2n D 2, 4, 6, . . . , 2n 4, 2n 2, n n − − E6 2, 5, 6, 8, 9, 12 E7 2, 6, 8, 10, 12, 14, 18 E8 2, 8, 12, 14, 18, 20, 24, 30 F4 2, 6, 8, 12 H3 2, 6, 10 H4 2, 12, 20, 30 I2(m) 2, m

EXAMPLE 7.2.2.3. Let us calculate the number of objects of type 3 (here 3 refers to the element of R corresponding to node 3 of the diagram) in the geometry associated to the

Coxeter system of type D6.

FIGURE 7.2: the diagram of D6 where node number 3 is circled

For this example, the degrees of the Coxeter systems D6, D3, and A2 are needed. The degrees of those geometries are respectively 2, 4, 6, 8, 10, 6 , 2, 3, 4 , and 2, 3 . { } { } { } The number of type 3 objects of the geometry D6,3 is:

2.4.6.8.10.6 v = = 160. 2.3.4.2.3

7.2.3 Graphs of Lie type

For each diagram Xn = An, Bn, Cn, Dn, E6, E7, E8, F4, G2, and every ﬁeld K, up to isomorphism, there is a unique Chevalley group G = Xn(K), due to Chevalley [42], cf. Carter [36]. These Chevalley groups are examples of groups with (B, N)-pairs, where B is a Borel subgroup. We deﬁned groups with (B, N)-pairs in Section 1.3.2. The nodes of the diagram Xn correspond to the elements of the generating set R of the Weyl group of Xn(K). G is transitive on the set of vertices and on the set of edges of Γ. Associated with Γ, there is a Coxeter graph ∆ := Γ(W, X , r). When ∆ is a Coxeter graph of type Xn,i , Γ is called a graph of Lie type Xn,i . 100 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

The association scheme related to the graph of Lie type

LEMMA 7.2.3. [21] There is a one to one correspondence between D and G G/G given S,S S\ S by d G dG ,d D . 7→ S S ∈ S,S

We can come up with an association scheme on the left cosets of GS: the pair (x P, y P) 1 is in relation Py− x P. Since in each double coset there is a unique element from DS,S, the relations can be labeled with elements from DS,S. We say a pair of vertices (x P, y P) 1 are in relation Ri, when i is the unique element of DS,S in Py− x P. For Γ, we deal with the distribution diagram of this association scheme with respect to the relation Rr . All the distribution diagrams for graphs of Lie type in this chapter are of this kind.

The Buekenhout Tits geometry related to the Graph of Lie type

We obtain a geometry with Buekenhout-Tits diagram Xn by taking as objects (flags) (of cotype I or of type R I) cosets of the form gG where G = BW B is a standard parabolic \ I I I subgroup of G = Xn(K). For instance, gGR r has type r (when the flag type is a one-set such as r , we say, it is of type r). So vertices\{ } of Γ are objects of type r as described { } above. A pair of objects are incident when they have a nonempty intersection. A chamber is a maximal flag, an object of cotype R. Chambers are in one to one correspondence with left cosets gB of B, and the elements of the chamber are the cosets gP where P is a standard maximal parabolic subgroup. The following is called Bruhat decomposition (cf. [28]):

G = BwB, · w[W ∈ and it provides the set of chambers with a G-invariant W-valued distance function: If g in BwB, then d(B, gB) = w. This induces a G-invariant distance function between the objects of the geometry given by d(gP, hQ)= w (P,Q are distinct standard maximal 1 parabolic subgroups) when w is the shortest element of W such that BwB meets Qh− gP. In the particular case where K is the finite field Fq, the geometry is finite, and we can compute the parameters (see section 7.2.3). The group G acts flag-transitively and we get an association scheme as described above. The relations in this scheme correspond to the relations in W [21].

DEFINITION 7.2.3.1. The shadow of the object gGI on Γ is the set of vertices incident with it (the set of vertices in gGI P).

This shadow induces a subgraph isomorphic to the graph of Lie type Γ(GI , GI r , r) if r I and a single point otherwise [21]. \{ } ∈ DEFINITION 7.2.3.2. A line of Γ is a subset of the form gB r P/P for some g G, i.e. the 〈 〉 ∈ shadow of an object of cotype r.

REMARK 7.2.4. [21] By deﬁnition of Γ, two points are adjacent if and only if they are collinear.

PROPOSITION 7.2.5. [21] Two adjacent vertices are on a unique line, and each line contains at least three points. 7.2 PRELIMINARIES 101

THEOREM 7.2.6. [21] (Strong gamma space property) Let x be a point and L be a line. Then either there is a relation R (that is for i D see Lemma 7.2.3) such that (x, y) R for i ∈ S,S ∈ i all y L, or, there are two relations R , R (i, j D ) such that (x, y ) R for a unique ∈ i j ∈ S,S 0 ∈ i point y L, while (x, y) R for all points y L y . In the latter case, l(j) > l(i). In 0 ∈ ∈ j ∈ \{ 0} particular, the distance d(x, y) for y L takes at most two values, and when it takes two ∈ values, then there is a unique point y L such that d(x, L)= d(x, y )= d(x, y) 1 for 0 ∈ 0 − y L y . ∈ \{ 0} In this context, a set of points is a subspace when for any pair of collinear points being in this set implies all the points on the line joining them are in this set was well. For a point-line geometry, the convex closure of a set of points C is the smallest convex subspace containing C.

Calculating the number of vertices of Γ In section 7.2.2, we had seen how to calculate the number of vertices of ∆, and the number of vertices in relation i (Ri) for i in DS,S to a given vertex. The analogues of these for the graphs of Lie type for Chevalley groups of type Xl over the finite field Fq is described below. For these groups there is a Tits system (B, N, W, R) with additional properties called a split saturated Tits system [21]. i . split: there is a normal subgroup U of B with B = UH, and U H = 1 (where ∩ { } H = B N), ∩ w ii . saturated: w W B = H. ∩ ∈ w 1 w 1 Here, A stands for wAw− and A stands for w− Aw if A is a subset of G invariant under conjugation by H. Remember for a Coxeter system (W, R) where W is a Weyl group, for the corresponding root system Φ, a root α is called a positive root, if when it is written as a linear combination of elements of the fundamental roots, all coefficients are nonnegative, and is this is denoted by α > 0. When the coefficients are all non-positive, we say it is a negative root and write α< 0. Now, choose a Cartan subgroup H and for α Φ call X the root subgroup with ∈ α respect to α. H normalizes each. Let N be the normalizer of H in G. Then, W = N/H permutes the X (α Φ) according to w X = X (w W). Now, U = X is a α ∈ α wα ∈ α>0 α subgroup of G normalized by H, so that B = UH is a subgroup of G withQB N = H. ∩ Given w W, set ∈ Uw− = Xα α>0,Ywα<0 which is a subgroup of U.

LEMMA 7.2.7. [21] Let r R, P = GS = BWS B. Each coset xP has a unique representation 1 1 ∈ x P = uw− P where w− D ,S and u Uw−. ∈ ; ∈ The Table 7.2 [21] includes parameters that we will make use of, for our calculations for the distribution diagrams of our graphs. In the table, ‘N’ refers to the number of positive roots, which is half of the total number of roots, and ‘d ’ for 1 i l are the i ≤ ≤ degrees of the Weyl group that we had already seen in Table 7.1. 102 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

TABLE 7.2: parameters for Chevalley groups

Name N degrees d (1 i n) i ≤ ≤ A l(l+1) 2, 3, . . . , l + 1 l 2 2 Bl , Cl l 2, 4, 6, . . . , 2l D l(l 1) 2, 4, 6, . . . , 2l 4, 2l 2, l l − − − E6 36 2, 5, 6, 8, 9, 12 E7 63 2, 6, 8, 10, 12, 14, 18 E8 120 2, 8, 12, 14, 18, 20, 24, 30 F4 24 2, 6, 8, 12 G2 6 2, 6

For a nontwisted Chevalley group Xl (q), one has [21]:

N l l(w) N q d U− = q , U = q , G = (q i 1). | w | | | | | d − Yi=1

PROPOSITION 7.2.8. [21] Let Γ=Γ(G, P, r) where G is a ordinary Chevalley group with Tits system (B, N, W, R), P is a maximal parabolic subgroup of G containing B, and r R is not ∈ contained in P.

i . The number v of vertices of Γ is:

v = G/P = ql(w), | | wXD ,S ∈ ;

ii. For i D the number k of vertices of Γ in relation R to a ﬁxed vertex is ∈ S,S i i

l(w) ki = q .

w DX,S WS i ∈ ; ∩

REMARK 7.2.9. [21] Using product formulae, for the orders of the groups involved, one obtains the following formula for the number of vertices v of Γ for the nontwisted Cheval- ley groups: l (qdi 1) v = i=1 − Ql 1 − (qei 1) i=1 − Q where di are degrees of the Weyl group, and ei are the degrees of the group WS.

7.2.4 Relations between Γ(G, GS, r) and Γ(W, WS, r)

Let Γ=Γ(G, GS, r) and ∆=Γ(W, WS, r) be as above. Many properties of Γ can be read of from the graph ∆, since Γ is the collinearity graph of the building associated with the (B, N)-pair of G, while ∆ is the collinearity graph of an apartment of this building [21]. 7.2 PRELIMINARIES 103

PROPOSITION 7.2.10. [7] The map ψ : W W/W G G/G deﬁned by ψ(W wW ) = S\ S → S\ S S S G wG where r R, (w W) is an almost isomorphism between the double coset graphs S S ∈ ∈ ∆ := DC(W, WS, r) and Γ := DC(G, GS , r) (see the Deﬁnition 1.1.1.4): it is a bijection of vertex sets, preserving edges, such that all edges of Γ that are not images of edges of ∆ are loops.e In particular, Γ ande ∆ have the same diameter, and Γ is distance transitive if and only if ∆ is. e e

For a vertex v in any graph Γ the set consisting of all neighbors of v and itself is denoted by v⊥. For a set of vertices A, A⊥ = v Av⊥. ∩ ∈ PROPOSITION 7.2.11. [21]

i. The map Φ : ∆ Γ deﬁned by Φ(wW )= wG (w W) is a distance and relation → S S ∈ preserving embedding. Put Σ = Φ[∆].

ii . Given points γ , γ Γ and lines L , L with γ L , γ L , there is a g G such 1 2 ∈ 1 2 1 ∈ 1 2 ∈ 2 ∈ that gΣ contains γ1, γ2 and a second point of each of L1 and L2.

iii . A line contains at most two points of Σ.

iv. IfwWS and vWS are adjacent vertices of ∆ both in double coset WS wXS, then the line on vGS and wGS is entirely contained in GS wGS.

v. Let vWS be a vertex not in WS wWS. then It has a neighbor in WS wWS if and only if vGS has a neighbor in GS wGS.

vi. Let d(Ws, vWS)= 2 and assume WS and vWS has precisely one common neighbor in ∆. Then, GS and vGS has precisely one common neighbor in Γ.

vii . Given m 2, there are γ , γ Γ, with d(γ , γ )= 2 and v⊥ v⊥ a polar space of ≥ 1 2 ∈ 1 2 1 ∩ 2 rank m if and only if there are δ , δ in ∆ with d(δ , δ )= 2 and δ⊥ δ⊥ a complete 1 2 1 2 1 ∩ 2 bipartite graph on 2m vertices (Km 2). ×

viii . There are γ , γ Γ with d(γ , γ )= 2 and γ⊥ γ⊥ a coclique of at least 2 elements 1 2 ∈ 1 2 1 ∩ 2 if and only if there are δ , δ ∆ with d(δ , δ )= 2 and δ⊥ δ⊥ a coclique of size 1 2 ∈ 1 2 1 ∩ 2 2.

7.2.5 Parapolar spaces A parapolar space ([47], [45]) is a point line geometry (P, L) which is connected, all of its lines have size at least three, and for which the following conditions are satisﬁed for its collinearity graph:

i. if x P and l L, if x⊥ l > 1, then x is collinear with all points of l (consider l ∈ ∈ | ∩ | as the set of points on it),

ii . the graph induced on x, y ⊥ is not a clique whenever x P, and y is in x⊥, { } ∈

iii . for x, y P, with distance 2, and x, y ⊥ > 1, then, x, y ⊥ is a non-degenerate ∈ |{ } | { } polar space of rank at least 2. 104 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

The incidence system of types D5,5(K), E7,1(K), E6,1(K), E6,2(K), E8,8(K) for a skew ﬁeld K are characterized as parapolar spaces by Cohen and Cooperstein in [45] and

F4,1(K) is characterized as a parapolar space by Cohen [43]. A pair of points having more than one common neighbor in a parapolar space is called a symplectic pair. For a symplectic pair, there exists a unique geodetically closed subspace S(x, y) of the set of points P which is isomorphic to a polar space [45]. A symplecton is a subspace isomorphic to a non-degenerate polar space of rank at least two that is the convex closure of any two of its points at distance two. It is deﬁned in the Cooperstein theory of parapolar spaces [45], [47].

7.2.6 Reading the diagrams and chain calculus In this section, chain calculus is introduced, which is a tool to understand the incidence information for a Buekenhout-Tits diagram.

Chain calculus A chain x x x is a sequence of elements of some residually connected Buekenhout- 0∗ 1 ···∗ n Tits geometry satisfying a given diagram X where for each i in 1, 2, . . . , n , xi 1 is inci- { } − dent with x . A corresponding sequence of types t -t - -t is given in which t is the i 0 1 · · · n i type of x . For a chain x x x , l is the length of the chain. i 0 ∗ 1 ···∗ l PROPOSITION 7.2.12. [29] Let Γ be a residually connected Buekenhout-Tits geometry of ﬁnite rank.

1 1 i . For any twodistincttypes i, j I, the subgraph induced on t− (i) t− (j) is connected. ∈ ∪ ii. If the types i, j belong to different connected components of the Buekenhout-Tits diagram, then each i object is incident with each j object.

Proof. (i) This is proved by induction on the rank. For rank two, the statement is true by definition: every object of type i and every object of type j are adjacent. Suppose our geometry is of rank n and for any k < n, the statement holds. Since the geometry is connected, for any x, y of types i and j respectively, there is a chain x = x x x = y 0∗ 1 ···∗ l in the geometry. For any x in the chain whose type is neither i nor j, look at Γ whose h xh rank is less than n, and which is connected by residual connectedness. Then, there is a walk in Γ , joining x to x . For this walk when there is an object of type different xh h 1 h+1 − 1 1 than i, or j, apply the same procedure by induction hypothesis till a walk in t− (i) t− (j) 1 1 ∪ is obtained. In the end a walk in t− (i) t− (j) joining x to y is obtained. ∪ (ii) This is proved by induction on the rank. For rank two, the statement is true by definition. Suppose our geometry is of rank n and for any k < n, the statement holds. By (i), for any x, y of types i, j respectively, there is a chain x = x0 x1 xl = y in 1 1 ∗ ···∗ t− (i) t− (j) joining them. Let this be the chain of shortest length. By contradiction, ∪ suppose l > 1. Let k be a third type different from i and j. Without loss of generality, suppose j and k are in different components of the Buekenhout-Tits diagram. In the chain, the types of the objects alternate between i and j. For x x x the type sequence 0 ∗ 1 ∗ 2 is i-j-i. In Γ , x x x can be replaced by x = x′ x′ x′ = x which is x1 0 ∗ 1 ∗ 2 0 0 ∗ 1 ∗···∗ n 2 a sequence of types i and k. Now, types of x3 and the two predecessors in the chain are k-i-j respectively. Since k and j are in different components of the diagram and by 7.2 PRELIMINARIES 105 the induction hypothesis any pair of objects of those types should be adjacent, one can omit the middle object. Now, types of x3 and the two predecessors in the chain are i-k-j respectively. One can omit the middle object once more. After doing this n times, one sees x0 is incident with x3, but then l was not minimal, contradiction, so l = 1. For the following statements, when a sequence of types t -t - -t is given, it is 0 1 · · · n claimed that, for any pair of objects x0 of type t0, xn of type tn, for the corresponding geometry, one can come up with a chain of the given sequence of types joining x0 to xn. To prove our claim the chains are modified keeping the ends fixed. To make such modifications the previous proposition is used. The method applied here is called chain calculus and it is due to Tits [94].

PROPOSITION 7.2.13. For 2 i n, 1-i-(i 1) holds for A . In particular for n 2, 1-2-1 ≤ ≤ − n ≥ holds.

Proof. This is proved this by induction. for n = 2, there is A2 and 1-2-1 holds in A2 by deﬁnition. Suppose the statement holds for any k < n. For i < n suppose there is a chain 1-2-1-i-(i 1). The residue of 1 is An 1. by induction there 1-i-(i 1) holds, which − − − corresponds to 2-(i + 1)-i here. So replace 1 with (i + 1). One can delete 2, since 1 and (i + 1) are in different components in its residue, and similarly i can be deleted. Finally look at the residue of (i+1), which is an Ai . By induction hypothesis, the statement holds there. So 1-i-(i 1) holds. Hence, (i + 1) can be replaced with i and obtain 1-i-(i 1). − − So, 1-2-1-i-(i 1) is deduced, which implies 1-i-(i 1). But by residual connectedness, − − one can make a chain between any pair of 1’s of the form 1-2- -2-1. So one can come · · · up with a chain of the form 1-2- -2-1-i-(i 1) for any object of type 1 and any object · · · − of type (i 1) starting and ending with this pair. And by repetitive applications of the − above argument, one can reduce it to a chain of the form 1-i-(i 1). Existence of 1-2-1 − for any pair of points in An means any pair of points are on a line in a projective space. As a result, for any object of type 1 and any object of type n 1 there is the chain 1- − 2-1-(n 1). In this chain, look at the residue of 1. It is An 1. There, by induction, the − − statement (1-(n 1)-(n 2) for An 1) holds. This corresponds to 2-n-(n 1) for An. So − − − − replace 1 with n. After that, delete 2, since in the residue of 2, 1 and n are in different components.

PROPOSITION 7.2.14. For D ,if n 3, n ≥ i . 1-(n 1)-n, − ii . if nis even, (n 1)-1-n, and if n is odd, n-1-n. −

Proof. This is proved by induction. When n = 3, there is A3 and everything holds by Proposition 7.2.13. Suppose for all k < n, the proposition holds. (i) As in the previous proof, it sufﬁces to show that 1-2-1-(n 1)-n implies 1-(n 1)-n. − − First, look at the residue of 1. It is a Dn 1. there, 1-(n 1)-(n 2) holds by induction − − − hypothesis, which corresponds to the nodes 2-n-(n 1) here, so replace 1 by n and obtain − 1-2-n-(n 1)-n. Then look at the residue of 2. There 1 and n are in different components, − so delete 2. Now our chain is 1-n-(n 1)-n. Then look at the residue of (n 1), it is − − An 1. There 1-2-1 holds, which corresponds to n-(n 2)-n here. So replace (n 1) with − − − (n 2) and get 1-n-(n 2)-n. Then look at the residue of n, which is an An 1. There − − − 106 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

1-(n 1)-(n 2) holds, so replace n with (n 1). Now there is 1-(n 1)-(n 2)-n. Finally, − − − − − delete (n 2) since in its residue n and (n 1) are in different components. − − (ii) If n is even, it sufﬁces to show that (n 1)-1-n-1-n implies (n 1)-1-n. First, − − (n 1)-1-n-1-n implies (n 1)-1-2-1-n, since the residue of n is a An 1, where 1-2-1 − − − holds. Then look at the residue of 1 which is a Dn 1. There, (n 2)-(n 1)-1 holds − − − by Proposition 7.2.14 (i), which corresponds to (n 1)-n-2 in our diagram, so there − is (n 1)-n-2-1-n. Then look at the residue of 2, for which 2 and 1 are in different − components, so delete 2, and obtain (n 1)-n-1-n. Then look at the residue 1, which is − a Dn 1. But (n 1) is odd, and there (n 1)-1-(n 1) holds, which corresponds to n-2-n − − − − here. So there is (n 1)-n-2-n. Then look at the residue of n, which is an An 1. There is − − 1-(i 1)-(i 2), which corresponds to (n 1)-1-2 here. So there is (n 1)-1-2-n. then − − − − delete 2, since in its residue 1 and n are disconnected. If n is odd, it sufﬁces to show that n-1-n-1-n implies n-1-n. One can start by showing n-1-n-1-n implies n-1-2-1-n. Look at the residue of n. It is An 1. There, 1-2-1 holds by Proposition 7.2.13. So replace n with 2. Second, n-1-2-1-n implies− n-1-2-(n 1)-n, since − residue of 1 is a Dn 1, there 1-(n 2)-(n 1) holds by Proposition 7.2.14 (i), which − − − corresponds to 2-(n 1)-n. So here replace 1 with (n 1). Third, n-1-2-(n 1)-n implies − − − n-1-(n 1)-n since in the residue of 2, 1 and (n 1) are disconnected, so delete 1. Then, − − n-1-(n 1)-n implies n-2-(n 1)-n, since the residue of 1 is a Dn 1. For Dn 1, (n 1)- − − − − − 1-(n 2) holds by induction hypothesis and this corresponds to n-2-(n 1) here. The − − next step is, showing n-2-(n 1)-n implies n-2-1-n. The residue of (n 1) is an An 1, − − − there 2-1-(n 1) should hold, (any 2 space and (n 1) space intersect at some point) − − correspond to 2-1-n here. Finally, n-2-1-n implies n-1-n, since in the residue of 2, n and 1 are disconnected, therefore delete 2.

PROPOSITION 7.2.15. For E6 there is 1-6-1, that is, any pair of objects of type 1 is in the residue of a type 6 object in E6. Proof. It sufﬁces to show 1-6-1-3-1 implies 1-6-1 by residual connectedness and induction. This is done in a few steps. In each step, the residue of the bold type in the diagram are considered. Step 1: 1-6-1-3-1 implies 1-6-2-3-1.

Residue of 1 is a D5. In D5, 1-4-5 holds by Proposition 7.2.14 (ii). This corresponds to 6-2-3 here. That is, one can replace 1 by 2 here. Step 2: 1-6-2-3-1 implies 1-6-2-1. In the residue of 3, 2 and 1 are in different components so delete it. Step 3: 1-6-2-1 implies 1-5-2-1.

In D5, 1-4-5 holds by Proposition 7.2.14 (ii). This corresponds to 1-5-2 here. That is, one can replace 6 by 5 here. Step 4: 1-5-2-1 implies 1-5-6-1.

In A5 there is 1-5-4 by Proposition 7.2.13 and this corresponds to 1-6-5 here. Step 5: 1-5-6-1 implies 1-6-1 In the residue of 5 6 and 1 are in different components, so one can delete 5.

Reading the diagrams Let us review some of the facts concerning the diagram for a Buekenhout-Tits geometry associated to group with a Tits system. For each node y in the diagram, there are 7.3 FOUR POINT-LINE GEOMETRIES 107

objects of type y in the geometry. These are cosets of the subgroup GR y . Let us call R y = I. Shadow of an object of that type in the Graph of Lie type Γ(G\{, G} , r) induces \{ } S a subgraph Γ(GI , GI r , r). If the diagram of the group GI is disconnected, and C is the \{ } set of nodes that are in the component containing r, then Γ(GI , GI r , r) is isomorphic to \{ } Γ(GC , GC r , r). Let us call the element of R represented by node i as ri, and let us call \{ } the component of the diagram of GR r including r as Ci . \{ i } We make use of the example E7,1 to tell the story more explicitly. In this case, we see C3 is A1, C4 is A2, C2 is A6, C5 is A4, C6 is D5, C7 is E6. The shadows of objects of types 3, 4, 2, 5, 6, 7 induces subgraphs, which are respectively graphs of Lie types A1,1, A2,1, A6,1, A4,1, D5,1, and E6,1. In the Buekenhout-Tits geometry E7,1, since it is a parapolar space, two points at distance two are together in a polar space, if the number of common neighbors is at least

2. The polar space in this geometry is a D5,1. We want to know how the lines on a fixed point behave in this geometry. To see that, we have to look at the shadow of an object of type r1 on the graph Γ(G, GR r , r2) which \{ 2} is Γ(GI , GI r , r2) where I = R r1 . Here, this graph is D6,6. When we apply the same \{ 2 } \{ } argument to see the how do the planes on a fixed point line flag behave, the result is the geometry A5,2. In the diagram of the Buekenhout-Tits geometry with diagram E6, the shadow of the residues of 1 and 6 objects are both of type D5,5. In E6,2 if two points are in the residue of a 6 flag, then they are also in the residue of a 1, 6 -flag. Chain calculus is applied here. { } Look at the residue of 6, which is a D5. In D5, 5-1-5 holds, which corresponds to 2-1-2 here, so, 2-6-2 implies 2-1-2. In other words, 2- 1, 6 -2. { } By the above argument, symplecta here are represented by the flag of type 1, 6 . The { } residue of a 1, 6 -flag is of type D . { } 4,1 In the next section, in relation to the shape of the diagram of E7,1 the way to calculate the parameters in the distribution diagram is discussed.

7.3 Four point-line geometries

7.3.1 The graph of Coxeter and Lie types E7,1 For this geometry, it is described in detail how to calculate the parameters of the distribution diagram. For the following three geometries, since the shapes of the distribution diagrams are the same, the calculations can be done accordingly.

The Coxeter graph E7,1(1) The following distribution of the Coxeter graph helps us to understand the structure of the corresponding graph of Lie type. In the Coxeter graph, the number of vertices is 126, the number of roots of the root system E7. That is the root system graph of E7. The parameters of the distance distribution of this graph are known (see [21]). The relations R for i in 0, 1, 2, 3, 4 are i { } respectively having inner product 2, 1, 0, 1, 2 with p , the ﬁxed point. − − 0 A presentation of the root system of type E7 is the following [21]: Take the standard basis of R7, e i Z , and standard inner product (, ). Fix a plane structure on Z , { i | ∈ 7} 7 with the naming of the lines 124, 137, 156, 235, 267, 457, 346. Then 108 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

Φ := 1 2e e e e e jklm is the complement of a line in the Fano plane with p2 {± i ± i ± j ± k ± l | the above naming of lines is a root system of type E . } 7 15 15

k=32 k3 =32 k4 =1 32 1 1 1 1 32

15 15

8 8

k2 =60

FIGURE 7.3: distribution diagram of the E7,1(1) graph

Here, the quotient matrix is the following: 0 32 0 0 0  1 15 15 1 0  D =  0 8 16 8 0 .    0 1 15 15 1     0 0 0 32 0    The eigenvalues of D are 32, 16, 4, 2, 4. − − Once D is known, P and Q matrices can be calculated.

Remember that P is the matrix whose i’th column consist of the eigenvalues of Ai, P is related to the Q matrix by the relation PQ = QP = nI, and Q0i is the multiplicity of the eigenvalue Pi j of Aj. So P and Q are the following respectively:

1 32 60 32 1  1 16 0 16 1  − − P =  1 4 10 4 1   −   1 2 0 2 1   − −   1 4 6 4 1   − −  1 7 27 56 35 7 27 7 35 1 − −  2 8 2 8  9 7 Q = 1 0 − 0  2 2 .  7 27 7 35   1 − −   2 8 2 8   1 7 27 56 35   − −  Linear programming bound can be used to obtain bounds on certain subsets. Let Y2 be a set, where for any (x, y) Y Y , (x, y) R . Then, a = 1, 0, Y 1, 0, 0 . ∈ 2 × 2 ∈ 2 { | 2|− } Delsarte’s linear programming bound 1.6.1 says, aQ 0. ≥ 7.3 FOUR POINT-LINE GEOMETRIES 109

For this example, this is equal to Y 1 6, so Y can be at most 7. So, there can | 2|− ≤ | 2| be at most 7 roots that are pairwise orthogonal. Consider a set Y where for any pair (x, y) Y , either (x, y) R or (x, y) R . 12 ∈ ∈ 1 ∈ 2 Hence, Y Y R Y Y R | 12 × 12 ∩ 1| + | 12 × 12 ∩ 2| = Y 1, Y 35. Y Y | 12|− | 12|≤ | 12| | 12| Similarly, Y is defined for distinct a, b, c in 1, 2, 3, 4 . abc { } By using the above description of the root system it is easy to see the actual size of YA where A 1, 2, 3, 4 . ⊆{ } For instance, for Y1, without loss of generality, suppose a set of size Y1, where any pair of roots have mutual inner product 1 includes v = 1 (2, 0, 0, 0, 0, 0, 0). None of p(2) 1 the 2ei has inner product 1 with v. For the roots derived from the lines of the ±p(2) Fano plane, to have a nonzero inner product with v, we have to consider the lines not on point 1 of the Fano plane, such as ‘235’, since to have an inner product 1 with v, the first coefficient should be 1. For this particular line, the corresponding roots are of the form 1 ( 1, 0, 0, 1, 0, 1, 1). Now, the four roots derived from lines not on p(2) ± ± ± ± point 1 of the Fano plane with all positive coefficients will have inner product 1 among themselves, or with v. Moreover, this set is maximal: any pair of roots derived from a line not including point 1 have two nonzero coinciding coordinate positions if they are derived from different lines. Their inner product should be 1, and the coefficient of the first coordinate position is positive, the other coinciding coordinate should have positive coefficients as well to have this inner product. We can not extend this set by including a pair of distinct roots obtained from the same line not including 1, since this would imply having a root with one negative coefficient, say i’th coefficient and this would conflict with one of the four roots already included, the one derived from the unique line except ‘235’ which does not include 1 and i simultaneously. Because of this correspondence,

Y1 = 5. The Table 7.3 gives the actual possible maximum size of those subconﬁgurations and the bound given by DLPB.

The graph of Lie type E7,1(q) The names to the parameters in the graph of Lie type are given as in ﬁgure 7.4.

There are 5 relations between any pair of vertices R0, R1, R2, R3, R4 respectively: identity, being at distance 1, being at distance 2 and having more than one common neighbor (spanning a symplecton), being at distance 2 and having one common neighbor, and being at distance 3. For a ﬁxed point p0, the set of points having relation i to p0 is called Ki for i 0, 1, 2, 3, 4 . ∈{ } We had seen earlier, by making use of product formulae for the orders of the groups, one can come up with a product rule for the number of vertices in the graph Xl,k [21] (p. 337):

l (qdi 1) v = i=1 − Q l 1 (q 1) − (qei 1) − i=1 − Q where di are the degrees of the Weyl group, and ei are the degrees of the Weyl group Ws. 110 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

TABLE 7.3: the bound given by DLPB and actual YA for the graph of Coxeter type E7,1

DLPB size

Y1 9 7 Y2 7 7 Y3 3 3 Y4 2 2 Y12 35 27 Y13 9 7 Y14 9 7 Y23 11 9 Y24 14 14 Y34 3 3 Y123 63 63 Y124 35 27 Y134 18 14 Y234 14 14 Y1234 126 126

The number of vertices of the graph is:

(q18 1)(q14 1)(q6 + 1) v = − − . (q4 1)(q 1) − −

All lines passing through a ﬁxed point p0 has q other points. Once the number of lines on p0 is known, the valency of the graph will be known. But the lines passing through a point show the structure of objects of type 6 in D6, since the residue of node 1 is a D6. On each line passing through p0, there are q other points. So the valency of the graph is: k = q(q5 + 1)(q4 + 1)(q3 + 1)(q2 + 1)(q + 1).

In the Coxeter graph, the number of common neighbors of p0 and any point in K3 is 1. By Proposition 10.6.8 (vi) of [21], it is 1 in the graph of Lie type as well. Since µ> 1 in the Coxeter graph, µ> 1 also in the graph of Lie type. When this is the case -number of common neighbors of a pair of points at distance 2 is greater than 1- they span a unique symplecton by deﬁnition, since E7,1 is a parapolar space, and it is a D5 for this diagram. As explained in Section 7.2.6, among the nodes of the diagram E7, only for node 6 the residue is a polar space and it is a D5. The graph on the points of the Coxeter graph related to D5 is a strongly regular, so is the graph of Lie type. The Coxeter graph is K2 2 2 2 2 with parameters (v, k, λ, µ)=(10, 8, 6, 8). In the graph× × × of× Lie type, the parameters are:

(q5 1)(q4 + 1) v = − , D5,1 (q 1) − q(q4 1)(q3 + 1) k = − , D5,1 (q 1) − 7.3 FOUR POINT-LINE GEOMETRIES 111

8 a3 a4

k 8 k3 k4 k k-b1 - -1 4 1 1 b3 c

1 b c3

: b2

FIGURE 7.4: distribution diagram of the graph of Lie Type E7,1(q)

q2(q3 1)(q2 + 1) λ = − + q 1, D5,1 q 1 − − µ =(q3 + 1)(q2 + 1)(q + 1). D5,1

The number of points in K that are adjacent to a point in K2 is µ because of geodetic closure. The number of points at distance 2 to a given point is v k 1 = q8 in the graph D5,1 − D5,1 − D5,1. To calculate the number of symplecta in as explained in Section 7.2.2, the degrees of

E7 and the degrees of D5 and A1 are used, which are in the residue of node 6 representing the symplecta of the geometry. But to calculate the number of symplecta incident with a certain point, look at the residue of the node representing the points (node 1 here), which is a D6. Then look at the residue of node 6 of E7 in D6, and make the calculations accordingly. So, the degrees of D6, D4 and A1 is used. The number of symplecta S in this geometry incident with a point is:

(q10 1)(q8 1)(q6 1)2(q4 1)(q2 1) #S = − − − − − . (q6 1)(q4 1)2(q2 1)2(q 1) − − − −

The points in distance 2 in the D5 graph are at distance 2 in the original graph as well because the corresponding D5 is geodetically closed in the original graph by 10. 6. 1 (ii) of [21]. For each symplecton, there are q8 of them. Hence:

8 k2 = q (#S), q8(q10 1)(q6 1)(q4 + 1) = − − . (q2 1)(q 1) − −

Since b1k = µk2 by regularity of the partition,

q7(q6 1)(q5 1) b = − − . 1 (q2 1)(q 1) − − 112 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

The number of common neighbors of two adjacent points p0 and p is denoted by λ. Two points are adjacent when they are collinear, and a third point’s being collinear to both of them means these three points span a plane. Apart from the points on the line p0 p, there 2 are q points on this plane. All such planes intersect at p0 p. To calculate λ the number of such planes should be calculated. The number of common neighbors on p p is (q 1). 0 − Add this to q2 times the number of planes to obtain λ. To calculate the number of planes of a line, look at the residue of a point line ﬂag, that is the residue of 1, 2 in the diagram. Then, one sees planes on it act like lines of { } an A5. So the number of lines on A5 is calculated. for that, the degrees of A5, A1 and A3 is used. Hence:

q2(q6 1)(q5 1)(q4 1)(q3 1)(q2 1) λ = q 1 + − − − − − , − (q4 1)(q3 1)(q2 1)2(q 1) − − − − then c = k 1 b λ = q16, 1 − − 1 −

k3 = c1k, = q17(q5 + 1)(q4 + 1)(q3 + 1)(q2 + 1)(q + 1),

k = v 1 k k k = q33. 4 − − − 2 − 3 For a point p1 in K3, with a neighbor p2 in K4, consider another neighbor of p2, p3 in K4. In the line joining p2 and p3, by 10.6.3 [21], either all of the points are collinear with p1, or a single point, that is, here p2 is collinear with p1. In the Coxeter graph, there is a single point in K4, so, the second case does not hold. That is, p2, and all other points on that line are collinear with p . that makes q points per line. Hence, c is k . 2 4 q

5 4 3 2 c4 =(q + 1)(q + 1)(q + 1)(q + 1)(q + 1),

c4 + a4 = k = q(q5 + 1)(q4 + 1)(q3 + 1)(q2 + 1)(q + 1), so,

k a = k , 4 − q q 1 = k − =(q5 + 1)(q4 + 1)(q3 + 1)(q2 + 1)(q + 1)(q 1). q −

Since b3k3 = c4k4, 16 b3 = q .

Consider a point p3 in K3. Its neighbors in K2 are counted. It has 1 neighbor, say p in K. Because of that, in the Coxeter graph, there are no triangles consisting of p3, p and another point of K. Triangles correspond to planes in the graph of Lie type, no plane 7.3 FOUR POINT-LINE GEOMETRIES 113

including p3 and p have any other point from K in the graph of Lie type. The number of planes that the line pp3 is on can be counted, and count the number of points from K2 per plane to count c3. Since any two planes would intersect at a line, and this line is pp3 here, a point from K2 would only be in a single such plane. Consider such a plane. Let p be the single point from K. Let p2 be a point from K2. Consider the line joining p and p2 in the plane. It has a single point from K, and all q other points are from K2 because of 2 10.6.3 of [21]. All the other q points in the plane are from K3. Per plane of that form, there are q points from K2, and the number of planes adjacent to a line is the number of lines in A5 as explained above for calculating λ. The number of neighbors of a point in K3 in K2 are: q(q6 1)(q5 1)(q4 1)(q3 1)(q2 1) c = − − − − − , 3 (q4 1)(q3 1)(q2 1)2(q 1) − − − − (q6 + q5 + q 1)(q4 1)(q3 + 1) a = − − , 2 (q 1) − q10(q4 1)(q3 + 1) b = − , 2 (q 1) −

a = k 1 c b , 3 − − 3 − 3 q6(q6 1)2 = − q5 q3 1. (q2 1)(q 1) − − − − − The distribution matrix of the graph helps us to ﬁnd the eigenvalues. The eigenvalues of D is a subset of eigenvalues of the graph. Eigenvalues of D are:

θ = (q5 + 1)(q3 + 1), 0 − θ = (q6 q4 q3 + q2 1)(q2 + 1), 1 − − − θ = (q7 + q5 q4 + q3 q2 + q 1)(q3 + 1)(q + 1), 2 − − − θ = (q4 + 1)(q3 + q 1)(q3 + 1)(q2 + 1)(q + 1) 3 − 5 4 3 2 θ4 = q(q + 1)(q + 1)(q + 1)(q + 1)(q + 1). Just as in the Coxeter graph, by knowing D, one can calculate P, and Q matrices. The ﬁrst row of Q consist of the multiplicities of the eigenvalues. The eigenvectors are:

q+1 q 1 1 1 1, − , − , − , q(q8 1) q5(q5 1)(q4+1) q10(q4+1)(q2+1)(q+1) q18 { − − } q6 q4 q3+q2 1 q+1 q6 q4+q3+q2 1 1 1, − − − , − , − − , − , { q(q5+1)(q4+1)(q3+1)(q+1) q5(q5+1)(q4+1) q10(q5+1)(q4+1)(q3+1)(q+1) q18 } q7+q5 q4+q3 q2+q 1 q9 q8 q6 q3 q+1 q7+q6 q5+q4 q3+q2+1 1 1, − − − , − − − − , − − − , , { q(q5+1)(q4+1)(q2+1) q5(q5+1)(q4+q2+1)(q4+1) q9(q5+1)(q4+1)(q2+1) q14 } q3+q 1 q4 1 q3 q2 1 1 1, − , − , − − , − , { q(q5+1) q4(q5+1) q6(q5+1) q9 } 114 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

1, 1, 1, 1, 1 . { } Hence, Pˆ, the normalized P is:

11 1 11 q3+q 1 q4 1 q3 q2 1 1 1 − − − − −  q(q5 +1) q4(q5+1) q6(q5+1) q9  q7+q5 q4+q3 q2+q 1 q9 q8 q6 q3 q+1 q7+q6 q5+q4 q3+q2+1 1 1 − − − − − − − − − −  q(q5+1)(q4+1)(q2+1) q5(q5+1)(q4+q2+1)(q4+1) q9(q5+1)(q4+1)(q2+1) q14   q6 q4 q3+q2 1 q+1 q6 q4+q3+q2 1 1   1 − − − − − − −   q(q5+1)(q4+1)(q3+1)(q+1) q5(q5+1)(q4+1) q10(q5+1)(q4+1)(q3+1)(q+1) q18  q+1 q 1 1 1  1 − − −   q(q8 1) q5(q5 1)(q4+1) q10(q4+1)(q2+1)(q+1) q18   − − 

The eigenvalues θ for i, j in 0, 1, 2, 3, 4 are the following: i j { }

θ0,0 = 1, θ1,0 = 1, θ2,0 = 1, θ3,0 = 1, θ4,0 = 1, 5 4 3 2 θ0,1 = q(q + 1)(q + 1)(q + 1)(q + 1)(q + 1), 4 3 3 2 θ1,1 =(q + 1)(q + q 1)(q + 1)(q + 1)(q + 1), 7 5 4 −3 2 3 θ2,1 =(q + q q + q q + q 1)(q + 1)(q + 1), 6 4 − 3 2 − 2 − θ3,1 =(q q q + q 1)(q + 1), −5 − 3 − θ4,1 = (q + 1)(q + 1), −q8(q10 1)(q4+q2+1)(q4+1) θ = − , 0,2 q 1 − q4(q8 1)(q5 1)(q4+q2+1) θ = − − , 1,2 q 1 − q3(q8 2q7+2q6 3q5+3q4 3q3+2q2 2q+1)(q5 1)(q+1) θ = − − − − − , 2,2 q 1 3 5 4 2 − θ3,2 = q (q 1)(q + q + 1), −3 5 − 4 2 θ4,2 = q (q + 1)(q + q + 1), 17 5 4 3 2 θ0,3 = q (q + 1)(q + 1)(q + 1)(q + 1)(q + 1), 11 4 3 2 3 2 θ1,3 = q (q + 1)(q q 1)(q + 1)(q + 1)(q + 1), 8 7 6 −5 −4 3 2 3 θ2,3 = q (q q + q q + q q 1)(q + 1)(q + 1), −7 6 −4 3 −2 2− − θ3,3 = q (q q + q + q 1)(q + 1), 7 −5 3 − θ4,3 = q (q + 1)(q + 1), −33 θ0,4 = q , 24 θ1,4 = q , −19 θ2,4 = q , 15 θ3,4 = q , −15 θ4,4 = q .

Multiplicities of the eigenvalues θi,j, for i = 0, 1, 2, 3, 4 respectively: 1,

q(q14 1)(q6 1) − − , (q4 1) − q2(q18 1)(q12 1) − − , (q4 1)(q2 1) − − 7.3 FOUR POINT-LINE GEOMETRIES 115

q3(q14 1)(q9+1)(q5+1)(q3+1) − , 2(q 1) −

q3(q14 1)(q9+1)(q6+1)(q5 1)(q4+1) − − . 2(q3+1)(q2+1)(q2 1)(q 1) − − 1 Now writing down Q is trivial since there is P, and Q = nP− , or one can write down Q by making use of multiplicities and Pˆ.

7.3.2 The graph of Coxeter and Lie types E6,2

The Coxeter graph E6,2(1)

Here, the parameters of the distribution diagram of the graph of Lie type E6,2(q) are calculated. The arguments that are used are similar to the arguments that are used for the graph of Lie type E7,1(q). The distribution diagram for the Coxeter graph can be found in [21]. In Figure 7.5 you see the distribution diagram for the Coxeter graph. For the graph of Lie type, the parameters are named as in Figure 7.4.

9 9

k=20 k3 =20 k4 =1 20 1 1 1 1 20

9 9

6 6 30

FIGURE 7.5: distribution diagram of the E6,2(1) graph

The quotient matrix is:

0 20 0 0 0  19910  D =  06860     01991     0 0 0 20 0    The eigenvalues of D are: 20, 10, 4, 2, 2. − − The P and Q matrices are:

1 20 30 20 1  1 10 0 10 1  − − P =  1 2 6 2 1   −   1 2 0 2 1   − −   1 4 6 4 1   − −  116 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

1 6 20 30 15  1 3 2 3 3  − − Q =  1 0 4 0 3  .  −   1 3 2 3 3   − −   1 6 20 30 15   − −  Linear programming bound is used to obtain bounds on certain subsets. The Table 7.4 gives the actual size of those subconﬁgurations and the bound given by DLPB.

TABLE 7.4: the bound given by DLPB and actual YA for the graph of Coxeter type E6,2

DLPB size

Y1 6 5 Y2 6 4 Y3 3 3 Y4 2 2 Y12 21 16 Y13 6 5 Y14 6 5 Y23 9 9 Y24 12 8 Y34 3 3 Y123 36 36 Y124 21 16 Y134 12 10 Y234 12 9 Y1234 72 72

The graph of Lie type E6,2(q) Since the distance distribution of the diagram and the set of relations is the same as the graph of Lie type E7,1(q), we imitate the methods we used for E7,1(q) here to calculate the parameters of E6,2(q). We do not repeat the arguments here, but give the parameters and refer to Section 7.3.1 for the details. The number of points of the geometry is:

(q12 1)(q6 + q3 + 1)(q4 + 1) v = − , (q 1) − The valency of the graph is:

q(q2 + 1)(q3 + 1)(q5 1) k = − . (q 1) −

The number of common neighbors of a point in K3 with the ﬁxed point is 1, just like in the Coxeter graph. 7.3 FOUR POINT-LINE GEOMETRIES 117

Here, symplecta are residues of a ﬂag of type 1, 6 . { } The number of symplecta on a point is:

(q5 1)(q6 1) #S = − − , (q 1)2 −

The symplecta here are D4′ s. The parameters of the graph D4,1 are:

(q3 + 1)(q4 1) v = − , D4,1 (q 1) − q(q3 1)(q2 + 1) k = − , D4,1 (q 1) − λ = q2(q + 1)2 + q 1, D4,1 − (q2 + 1)(q3 1) µ = − , D4,1 (q 1) − v k 1 = q6. D4,1 − D4,1 − µ = µ , D4,1 hence q6(q5 1)(q6 1) k =(#Sq6)= − − , 2 (q 1)2 − k2µ = kb1, hence, q5(q3 1)2 b = − , 1 (q 1)2 − (q3 1)2 λ := q 1 + q2 − , − (q 1)2 − k b λ 1 = q10, − 1 − − (k b λ 1)k = k , − 1 − − 3 q11(q5 1)(q2 + 1)(q3 + 1) k = − , 3 (q 1) − 10 b3 = q , k = v k k k 1 = q21, 4 − − 2 − 3 − k (q5 1)(q3 + 1)(q2 + 1) c = = − , 4 q (q 1) − q 1 a = k − =(q5 1)(q3 + 1)(q2 + 1), 4 q −

Now, once c3 is calculated, we are done by regularity of the partition.

q(q3 1)2 c = − , 3 (q 1)2 − 118 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

a = k 1 b c = q9 + 2q8 + 3q7 + 3q6 + 2q5 + q4 q3 q2 1, 3 − − 3 − 3 − − − k3c3 6 2 2 b2 = = q (q + q + 1)(q + 1), k2 a = k b µ = 2q7 + 2q6 + 3q5 + 2q4 + q3 q2 1. 2 − 2 − − − Eigenvalues of D are:

θ = q5 q3 q2 1, 0 − − − − θ = q5 q3 q2 1, 1 − − − θ = q7 + q6 + q5 + q4 q2 1, 2 − − θ = q8 + 2q7 + 2q6 + 3q5 + 2q4 + q3 1, 3 − 10 9 8 7 6 5 4 3 2 θ4 = q + q + 2q + 3q + 3q + 3q + 3q + 2q + q + q . The eigenvectors are:

q11(q 1) q8(q 1) q5(q 1) q12, − − , − , − − , 1 q5 1 (q5 1) q5 1 { − − − } q11( q5+q3+q2+1)(q 1) q8(q3 1) q5( q5 q3 q2+1)(q 1) q12, − − , − , − − − − , 1 (q5 1)(q3+1)(q2+1) (q6+q4+q3+q2+1)(q+1) (q5 1)(q2+1)(q3+1) {− − − } q8(q5+q4 1)(q 1) q5(q6 q5 q4 q2 q+1)(q 1) q3( q5+q+1)(q 1) q9, − − , − − − − − , − − , 1 (q5 1)(q3+1) (q5 1)(q3+1) (q5 1)(q3+1) { − − − } q5( q2 q+1) q3( q3+1) q2( q2+q+1) q6, − − , − , − , 1 (q3+1) (q2 q+1)(1+q) q3+1 {− − } 1, 1, 1, 1, 1 . { } So the normalized P is:

11 1 11 q2+q 1 (q3 1) q2 q 1 1 1 − − − − −  q(q3 +1) q3(q3+1) q4(q3+1) q6  (q5+q4 1)(q 1) (q6 q5 q4 q2 q+1)(q 1) ( q5+q+1)(q 1) 1 1 − − − − − − − − −  q(q5 1)(q3+1) q4(q5 1)(q3+1) q6(q5 1)(q3+1) q9  − − −  (q5 q3 q2 1)(q 1) (q3 1)(q 1) (q5+q3+q2 1)(q 1) 1   1 − − − − − − − − −   q(q5 1)(q3+1)(q2+1) q4(q5 1)(q3+1) q7(q5 1)(q3+1)(q2+1) q12  − q+1 − q− 1 − q+1 1  1 − − −   q(q5 1) q4(q5 1) q7(q5 1) q12   − − −  The eigenvalues θ for i, j in 0, 1, 2, 3, 4 are the following: i j { }

θ0,0 = 1, θ1,0 = 1, θ2,0 = 1, θ3,0 = 1, θ4,0 = 1, q(q5 1)(q3+1)(q2+1) θ = − , 0,1 (q 1) − (q5 1)(q2+1)(q2+q 1) θ = − − , 1,1 (q 1) − θ =(q5 + q4 1)(q2 + 1), 2,1 − θ = q5 q3 q2 1, 3,1 − − − 7.3 FOUR POINT-LINE GEOMETRIES 119

3 2 θ4,1 = (q + 1)(q + 1), −q6(q6 1)(q5 1) θ = − − , 0,2 (q 1)2 − q3(q3 1)2(q5 1) θ = − − , 1,2 (q 1)2 − q2(q3 1)(q6 q5 q4 q2 q+1) θ = − − − − − , 2,2 (q 1) − q2(q3 1)2 θ = − − , 3,2 q 1 − q2(q6 1) θ = − , 4,2 (q 1) − q11(q5 1)(q3+1)(q2+1) θ = − , 0,3 q 1 − q11(q5 1)(q2 q 1)(q2+1) θ = − − − , 1,3 q 1 5 5 − 2 θ2,3 = q ( q + q + 1)(q + 1), 4 −5 3 2 θ3,3 = q (q + q + q 1), 4 3 2 − θ4,3 = q (q + 1)(q + 1), −21 θ0,4 = q , 15 θ1,4 = q , −12 θ2,4 = q , 9 θ3,4 = q , −9 θ4,4 = q .

Multiplicities of the eigenvalues θi,j, for i = 0, 1, 2, 3, 4 respectively:

(q3 q2 + q)(q12 1)(q6 + q3 + 1)(q4 + 1)/(q 1)(q17 2q16 + q15 2q14 + q13 + q12 + − − − − − 4q11 + 2q10 + 4q9 + q8 q7 q5 + q3 + q2 + 1), − −

((q12 1)(q4 + 1)(q14 2q13 + q10 + 2q9 + 2q8 q7 + q6 q5 q4 + q2)(q5 1))/((q − − − − − − − 1)2(q15 2q14 + q11 + q10 + 3q9 + q8 + q7 2q5 + q3 + 1)(q2 + q + 1)), − −

(q19 2q18 +q17 2q16 +q15 +6q13 +q12 +5q11 3q8 q6 +q5 +q4 +q3)(q12 1)(q6 + − − − − − q3 +1)(q4 +1)/(q 1)(2q19 4q18 +4q15 2q14 +6q13 +2q12 +4q11 +2q10 +2q9 + − − − 2q8 + 4q7 4q5 + 2q4 + 2q3 + 2), −

(q22 q21 q20 q19 +6q16 +5q15 +7q14 +6q13 +q12 q11 2q9 q8 +4q7 +2q6 +2q5 + − − − − − − 2q4+q3)(q12 1)(q6+q3+1)(q4+1)/(q 1)(2q22 2q21 4q19 2q18+6q16+14q15+ − − − − − 20q14 +26q13 +22q12 +20q11 +10q10 +6q9 +2q6 +4q5 +6q4 +6q3 +4q2 +2q+2).

7.3.3 The graph of Coxeter and Lie types E8,8

Here the parameters of the distribution diagram of the graph of Lie type E8,8 are calculated. In Figure 7.6 you see the distribution diagram for the Coxeter graph. For the graph of Lie type, the parameters are named as in Figure 7.4. 120 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

27 27

k=56 k3 =56 k4 =1 56 1 1 1 1 56

27 27

12 12 126

FIGURE 7.6: distribution diagram of the E8,8(1) graph

The Coxeter graph E8,8(1) The quotient matrix is: 0 56 0 0 0  1 27 27 1 0  D =  0 12 32 12 0     0 1 27 27 1     0 0 0 56 0    The eigenvalues of D are 56, 28, 8, 4, 2. − − The P and Q matrices are:

1 56 126 56 1  1 28 0 28 1  − − P =  1 8 18 8 1   −   1 2 0 2 1   − −   1 4 6 4 1   − −  1 8 35 112 84  1 4 5 4 6  − − Q =  1 0 5 0 4  .  −   1 4 5 4 6   − −   1 8 35 112 84   − −  Linear programming bound is used to obtain bounds on certain subsets. There are ﬁve relations. The notation is as in the previous cases. The Table 7.4 gives the actual size of those subconﬁgurations and the bound given by DLPB.

The graph of Lie type E8,8(q) Since the distance distribution of the diagram and the set of relations is the same as the graph of Lie type E7,1(q), we imitate the methods we used for E7,1(q) here to calculate 7.3 FOUR POINT-LINE GEOMETRIES 121

TABLE 7.5: the bound given by DLPB and actual YA for the graph of Coxeter type E8,8

DLPB size

Y1 15 8 Y2 8 8 Y3 3 3 Y4 2 2 Y12 64 36 Y13 15 8 Y14 15 8 Y23 12 12 Y24 16 16 Y34 3 3 Y123 120 120 Y124 64 36 Y134 30 16 Y234 16 16 Y1234 240 240

the parameters of E8,8(q).

(q30 1)(q18 + q12 + q6 + 1)(q10 + 1) v = − , (q 1) − q(q14 1)(q9 + 1)(q5 + 1) k = − , (q 1) − q2(q9 1)(q8 + q4 + 1) λ = − + q 1, (q 1) − − The number of common neighbors of a point in K3 with the ﬁxed point is 1, just like in the Coxeter graph. Here, symplecta are residues of a ﬂag of type 1. The number of symplecta on a point is:

(q14 1)(q12 + q6 + 1)(q8 + q4 + 1) #S = − . (q 1) −

The symplecta here are D7′ s. The parameters of the graph D7,1 are: (q7 1)(q6 + 1) v = − , D7,1 (q 1) − q(q6 1)(q5 + 1) k = − , D7,1 (q 1) − q2(q5 1)(q4 + 1) λ = − + q 1, D7,1 (q 1) − − 122 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

(q6 1)(q5 + 1) µ = − , D7,1 (q 1) − µ = µ , D7,1 v k 1 = q12, hence D4,1 − D4,1 − q12(q14 1)(q12 + q6 + 1)(q8 + q4 + 1) k =(#Sq12)= − , 2 (q 1) − k2µ = kb1, hence, q11(q9 1)(q8 + q4 + 1) b = − , 1 (q 1) − k b λ 1 = q28, − 1 − − (k b λ 1)k = k , − 1 − − 3 q29(q14 1)(q9 + 1)(q5 + 1) k = − , 3 (q 1) − k = v k k k 1 = q57, 4 − − 2 − 3 − k (q14 1)(q9 + 1)(q5 + 1) c = = − , 4 q (q 1) − q 1 a = k − =(q14 1)(q9 + 1)(q5 + 1), 4 q − q(q9 1)(q8 + q4 + 1) c = − , 3 (q 1) − k3 b3 = k4c4, so 28 b3 = q . a = k 1 b c , 3 − − 3 − 3 q10(q5 1)(q6 1) a = ((q7 1)(q 1)+ q4) − − q9 q5 1, 3 − − (q 1)2 − − − − k c q18(q6 1)(q5 + 1) b = 3 3 = − , 2 k q 1 2 − a2 = k b2 µ, −22 −21 20 19 18 17 16 15 14 13 12 11 a2 = q + q + q + 2q + 2q + 3q + 3q + 3q + 3q + 3q + 3q + 3q + 2q10 + q9 + q8 + q7 + q6 q5 1. − − The eigenvalues are:

θ = (q9 + 1)(q5 + 1), 1 − θ = q14 q9 q5 1, 2 − − − θ =(q11 + q10 + q9 q6 + q3 1)(q5 + 1)(q3 + 1), 3 − − 7.3 FOUR POINT-LINE GEOMETRIES 123

(q9+2q8+2q7+q6+q5+q4+q3 q 1)(q7+1)(q5+1)(q3+1) θ = − − , 4 q+1

13 12 11 7 3 2 7 5 3 θ5 =(q + q + q + q + q + q + q)(q + 1)(q + 1)(q + 1).

The eigenvectors are:

q29(q 1) (q23(q2 q+1)(q2 1)) q14(q2 1) q30, − − , − − , − − , 1 , (q14 1) (q6+q3+1)(q14 1) (1+q)(q14 1) { − − − } q29(q14 q9 q5 1)(q 1) q23(q3 1)(q 1) q14(q14+q9+q5 1)(q 1) q30, − − − − − , − − , − − − , 1 , (q14 1)(q9+1)(q5+1) (q6 q3+1)(q14 1) (q14 1)(q5+1)(q9+1) {− − − − − } q23(q11+q10+q9 q6+q3 1)(q 1) q17(q8 q7+q6 2q5+q4 2q3+q2 q+1)(q2 1) q9(q 1)( q11+q8 q5+q2+q+1) q24, − − − , − − − − − , − − − , 1 , (q14 1)(q6 q3+1) (q6 q3+1)(q14 1) (q14 1)(q6 q3+1) { − − − − − − } q14( q3 q2+1) q9(q3 1) q5(q3 q 1) q15, − − , − − , − − − , 1 (q6 q3+1)(q+1) (q6 q3+1) (1+q)(1 q3+q6) {− − − − } 1, 1, 1, 1, 1 . { } So the normalized P is: 11 1 11 q3+q2 1 q3 1 q3 q 1 1 1 − − − − −  q(q6 q3+1)(q+1) q6(q6 q3+1) q10(1 q3+q6)(1+q) q15  − − − (q11+q10+q9 q6+q3 1)(q 1) (q8 q7+q6 2q5+q4 2q3+q2 q+1)(q2 1) (q 1)( q11+q8 q5+q2+q+1) 1 1 − − − − − − − − − − −  q(q14 1)(q6 q3+1) q7(q6 q3+1)(q14 1) q15(q14 1)(q6 q3+1) q24  − − − − − −  (q14 q9 q5 1)(q 1) (q3 1)(q 1) (q14+q9+q5 1)(q 1) 1   1 − − − − − − − − − −  q(q14 1)(q9+1)(q5+1) q7(q14 1)(q6 q3+1) q16(q14 1)(q5+1)(q9+1) q30  − − − −  q+1 (q2 q+1)(q2 1) q+1 1  1 − − − −   q(q14 1) q7(q14 1)(q6+q3+1) q16(q14 1) q30   − − − 

The eigenvalues θ for i, j in 0, 1, 2, 3, 4 are the following: i j { }

θ0,0 = 1, θ1,0 = 1, θ2,0 = 1, θ3,0 = 1, θ4,0 = 1, q(q14 1)(q9+1)(q5+1) θ = − , 0,1 q 1 − (q3+q2 1)(q14 1)(q2 q+1)(q5+1) θ = − − − , 1,1 q 1 11 10 −9 6 3 3 5 θ2,1 =(q + q + q q + q 1)(q + 1)(q + 1), 14 9 5 − − θ3,1 =(q q q 1), 9− − 5 − θ4,1 = (q + 1)(q + 1), −q12(q14 1)(q12+q6+1)(q8+q4+1) θ = − , 0,2 q 1 − q6(q14 1)(q6+q3+1)(q8+q4+1)(q3 1) θ = − − , 1,2 q 1 − q5(q8 q7+q6 2q5+q4 2q3+q2 q+1)(q8+q4+1)(q6+q3+1)(q2 1) θ = − − − − − , 2,2 q 1 − θ = q5(q8 + q4 + 1)(q6 + q3 + 1)(q3 1), 3,2 − − q5(q8+q4+1)(q6 q3+1)(q2 q+1)(q2 1) θ = − − − , 4,2 q 1 − q29(q14 1)(q9+1)(q5+1) θ = − , 0,3 (q 1) − q19(q14 1)(q5+1)(q3 q 1)(q3+1) θ = − − − , 1,3 q2 1 − 124 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

14 11 8 5 2 3 5 θ2,3 = q ( q + q q + q + q + 1)(q + 1)(q + 1), 13 −14 9 −5 θ3,3 = q (q + q + q 1), 13 9 5 − θ4,3 = q (q + 1)(q + 1), −57 θ0,4 = q , 42 θ1,4 = q , −33 θ2,4 = q , 27 θ3,4 = q , −27 θ4,4 = q .

Multiplicities of the eigenvalues θi,j, for i = 0, 1, 2, 3, 4 respectively:

q(q12 + 1)(q10 + 1)(q6 + 1),

q2(q14 1)(q10 1)(q10+1)(q8+q7 q5 q4 q3+q+1)(q8 q7+q5 q4+q3 q+1) − − − − − − − − , (q4 1)(q2 1) − −

q3(q14 1)(q12+1)(q6 q3+1)(q3+1)2(q5+1)2(q8+q7 q5 q4 q3+q+1) − − − − − , 2(q2 1) −

q3(q14 1)(q10+1)(q8 q4+1)(q8+q7 q5 q4 q3+q+1)(q6+q3+1)(q6+1)(q4+1) − − − − − . 2(q2 1) −

7.3.4 The graph of Coxeter and Lie types F4,1

Here the parameters of the distribution diagram of the graph F4,1 are calculated. In Figure 7.7 you see the distribution diagram for the Coxeter graph. For the graph of Lie type, the parameters are named as in Figure 7.4.

The Coxeter graph F4,1(1)

3 3

k=8 k3 =8 k4 =1 8 1 1 1 1 8

3 3

4 4 6

FIGURE 7.7: distribution diagram of the F4,1(1) graph 7.3 FOUR POINT-LINE GEOMETRIES 125

The quotient matrix is:

08000  13310  D =  04040     01331     00080    The eigenvalues of D are 8, 4, 4, 2, 0. − − The P and Q matrices are :

18 6 8 1  1 4 0 4 1  − − P =  1 0 2 0 1   −   1 2 0 2 1   − −   1 4 6 4 1   − −  14 9 8 2  1 2 0 2 1  − − Q =  1 0 3 0 2  .  −   1 2 0 2 1   − −   1 4 9 8 2   − −  Linear programming bound is used to give bounds on certain subsets. There are ﬁve relations. The notation is as in the previous cases. The Table 7.6 gives the actual size of those subconﬁgurations and the bound given by DLPB.

TABLE 7.6: the bound given by DLPB and actual YA for the graph of Coxeter type F4,1

DLPB size

Y1 3 3 Y2 4 4 Y3 3 3 Y4 2 2 Y12 8 6 Y13 3 3 Y14 3 3 Y23 6 5 Y24 8 8 Y34 3 3 Y123 12 12 Y124 8 8 Y134 6 6 Y234 8 8 Y1234 24 24 126 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

The graph of Lie type F4,1(q)

For this F4,1(q) case, the parameters of the distribution diagram were calculated also by Cohen, [44]. Since the distance distribution of the diagram and the set of relations is the same as the graph of Lie type E7,1(q), we imitate the methods we used for E7,1(q) here to calculate the parameters of F4,1(q). (q12 1)(q4 + 1) v = − , (q 1) − q(q4 1)(q3 + 1) k = − , (q 1) − q2(q3 1) λ = − + q 1, (q 1) − − Here, symplecta are B3, residues of a ﬂag of type 4, (octahedra in the Coxeter graph). The number of common neighbors of a point in K3 with the ﬁxed point is 1, just like in the Coxeter graph. The number of symplecta on a point is:

(q6 1) #S = − , (q 1) −

The parameters of the graph B3,1 are:

(q6 1) v = − , B3,1 (q 1) − q(q4 1) k = − , B3,1 (q 1) − λ = q2(q + 1)+ q 1, B3,1 − (q4 1) µ = − , B3,1 (q 1) − v k 1 = q5. B3,1 − B3,1 − µ = µ , B3,1 q5(q6 1) k =(#Sq5)= − , 2 (q 1) − k2µ = kb1, hence, q4(q3 1) b = − , 1 (q 1) − k b λ 1 = q7, − 1 − − (k b λ 1)k = k , − 1 − − 3 q8(q4 1)(q3 + 1) k = − , 3 (q 1) − 7.3 FOUR POINT-LINE GEOMETRIES 127

k = v k k k 1 = q15, 4 − − 2 − 3 −

k (q4 1)(q3 + 1) c = = − , 4 q (q 1) − q 1 a = k − =(q4 1)(q3 + 1), 4 q −

q(q3 1) c = − , 3 (q 1) − k3 b3 = k4c4 so 7 b3 = q , a = k 1 b c , 3 − − 3 − 3 a = q6 + q5 + 2q4 1, 3 − k c q4(q4 1) b = 3 3 = − , 2 k (q 1) 2 − a = k b µ, 2 − 2 − a =(q4 1). 2 − Eigenvalues of D are:

θ = (q3 + 1)(q2 + 1), 0 − θ = (q3 + 1), 1 − θ = q4 1, 2 − θ = q5 + 2q4 + q3 + q2 1, 3 − q(q3+1)(q4 1) θ = − . 4 q 1 − The eigenvectors are:

q5 q2 q6, − , q3, − , 1 , { q+1 q+1 } q8(q 1) q4 q9, − , 0, − , 1 , q4 1 (q2+1)(q+1) {− − } q7(q 1) q6 q3(q 1) q8, − , − , − − , 1 , (q2 q+1)(q+1) (q2 q+1)(q2+q+1) (q2 q+1)(q+1) { − − − } q5(q2+q 1) q3(q3 1) q2((q2 q 1) q6, − − , − − , − − − , 1 , q3+1 q3+1 (q2 q+1)(q+1) {− − } 1, 1, 1, 1, 1 . { } So the normalized P is: 128 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

11 1 11 q2+q 1 q3 1 q2 q 1 1  1 − − − − −  q(q3+1) q3(q3+1) q4(q3+1) q6 q 1 1 q+1 1 1 − − −  q(q3+1) q2(q2+q+1)(q2 q+1) q5(q3+1) q8   q+1 − 1 1   1 − 0 −  q(q4 1) q5(q2+1)(q+1) q9  −1 1 1 1   1 − −   q(1+q) q3 q4(1+q) q6    The eigenvalues θ for i, j in 0, 1, 2, 3, 4 are the following: i j { }

θ0,0 = 1, θ1,0 = 1, θ2,0 = 1, θ3,0 = 1, θ4,0 = 1, q(q4 1)(q3+1) θ = − , 0,1 q 1 − (q4 1)(q2+q 1) θ = − − , 1,1 q 1 4 − θ2,1 = q 1, −3 θ3,1 = (q + 1), −(q4 1)(q2 q+1) θ = − − − , 4,1 q 1 − q5(q6 1) θ = − , 0,2 q 1 − q2(q3 1)2 θ = − , 1,2 q 1 − θ = q3(q + 1), 2,2 − θ3,2 = 0, q2(q6 1) θ = − , 4,2 q 1 − q8(q4 1)(q3+1) θ = − , 0,3 (q 1) − q4(q4 1)(q2 q 1) θ = − − − , 1,3 q 1 3 4− θ2,3 = q (q 1), −3 2 − θ3,3 = q (q q + 1)(q + 1), q4(q4 −1)(q2 q+1) θ = − − − , 4,3 q 1 15 − θ0,4 = q , 9 θ1,4 = q , −7 θ2,4 = q , 6 θ3,4 = q , −9 θ4,4 = q .

Multiplicities of the eigenvalues θi,j, for i = 0, 1, 2, 3, 4 respectively: 1,

1 q11 + q8 + 1 q7 + 1 q5 + q4 + 1 q, 2 2 2 2 q14 + q12 + 2q10 + q8 + 2q6 + q4 + q2,

q15 + q13 + q11 + 2q9 + q7 + q5 + q3, 7.4 USING THE HOFFMAN BOUND AND THE DLPB BOUND 129

1 q11 + 1 q7 + 1 q5 + 1 q. 2 2 2 2

7.4 Using the Hoffman bound and the DLPB bound

We have four Kneser graphs on buildings for which we calculated eigenvalues, valencies and number of vertices. Hoffman bound (ratio bound) is used to have an upper bound on the size of cocliques.

THEOREM 7.4.1. [69] If X is a k-regular graph on v vertices, with least eigenvalue τ, then v α(X ) . ≤ 1 k − τ As it is shown above, for K(E (q), 1 ), K(E (q), 2 ), K(E (q), 8 ), and, K(F (q), 1 ), 7 { } 6 { } 8 { } 4 { } the smallest eigenvalues are respectively:

q24, q15, q42, q9. − − − − When Hoffman bound is applied to K(E (1), 1 ), K(E (1), 2 ), K(E (1), 8 ), and, 7 { } 6 { } 8 { } K(F (1), 1 ), the results are respectively: 63, 36, 120 and 12 (For these graphs, Hoffman 4 { } bounds gives V /2). | | PROPOSITION 7.4.2. Due to Hoffman bound,

α(K(E (q), 1 )) (q12 +q10 +q8 +q6 +q4 +q2 +1)(q6 +q3 +1)(q4 q2 +1)(q2 +q +1), 7 { } ≤ − α(K(E (q), 2 )) (q6 + q3 + 1)(q4 + 1)(q2 + q + 1)(q2 q + 1)(q + 1), 6 { } ≤ − α(K(E (q), 8 )) q42+q41+q40+q39+q38+q37+2q36+2q35+2q34+2q33+3q32+3q31+ 8 { } ≤ 4q30+4q29+4q28+3q27+4q26+4q25+5q24+5q23+5q22+4q21+5q20+5q19+5q18+4q17+ 4q16+3q15+4q14+4q13+4q12+3q11+3q10+2q9+2q8+2q7+2q6+q5+q4+q3+q2+q+1, and, α(K(F (q), 1 )) (q + 1)(q2 q + 1)(q2 + q + 1)(q4 + 1). 4 { } ≤ − Earlier, we made use of DLPB to have bounds on subconﬁgurations in the Coxeter graphs. Consider a coclique in the related Kneser graph, this consist of a set of vertices which has relation 1, 2 or 3 pairwise. That is, the bound on Y123 is the bound that we are looking for. But, DLPB and Hoffman bound gives the same bound.

7.5 Eigenvalue results for the Kneser graphs on buildings

In this section the eigenvalue results for the Kneser graphs on buildings are summarized. Being at maximal distance in the above four collinearity graphs is being adjacent it the corresponding Kneser graphs, namely K(F (q), 1 ), K(E (q), 2 ), K(E (q), 8 ), and 4 { } 6 { } 8 { } K(E (q), 1 ). Hence θ for any i 0, 1, 2, 3, 4 are the eigenvalues of those graphs. 7 { } i,4 ∈ { } Multiplicities are the same for the Graph of Lie type and the corresponding Kneser type graph. 130 FROM GRAPHS OF LIE TYPE TO KNESER GRAPHS ON BUILDINGS

In Section 6.4.1 the oppositeness graphs are deﬁned and the theorem of Brouwer is stated, which says, for these graphs, the squares of the eigenvalues are prime powers. These oppositeness graphs are the same as Kneser graphs for buildings when J = K.

Eigenvalues are given for K(An(q), 1, n ), K(A2n 1(q), n ), K(C2(q), 1, 2 ), { } − { } { } K(C (q), 1 ), K(C (q), n ), K(E (q), 2 ), K(E (q), 7 ), K(F (q), 1 ) in [20]. Among n { } n { } 6 { } 7 { } 4 { } these, the eigenvalues of K(F (q), 1 ) and K(E (q), 2 ), are calculated also above. The 4 { } 6 { } eigenvalues of K(E (q), 8 ), and K(E (q), 1 ) are also given. 8 { } 7 { }

7.5.1 Eigenvalues of K(A2d 1(q), d ) − { } For K(A2n 1(q), n )= K(2n, n, q), the eigenvalues are of the form − { } j n2 jn+j(j 1)/2 θ˜ =( 1) .q − − j − where 0 j n [20], [51], [100]. ≤ ≤

REMARK 7.5.1. For the graph Grassmann (n, e)q, Ad (as in Deﬁnition 1.1.0.10) is the adjacency matrix of K(n, e, q). Let Li be as in Section 1.6. The eigenvalues and the parameters bi and ci of the Grassman graph are known for all i 1, 2, . . . , d ([21], 9. 3. 3), so the eigenvalues of A can be calculated by Equations ∈{ } d 1.2, 1.3, and 1.4.

LEMMA 7.5.2. The eigenvalues of K(2d, d, q) are powers of q.

Proof. Since adjacency in K(2d, d, q) is the same as being at distance d in Grassmann (2d, d)q, we have to show, that the eigenvalues of the matrix Ad are powers of q. The eigenvalues of the Ai have the same set of multiplicities for each i, namely the dimensions of the eigenspaces. The multiplicities of the eigenvalues of the Grassmann (2d, d) are m =[ 2d ] [ 2d ] (0 j d) [21, p.269], and these are all distinct since q j j q j 1 q − − ≤ ≤ q j m . It follows that the eigenvalues are all integers [21, p.45]. || j Since the eigenvalues of Ai are equal to the eigenvalues of Li, for all i, it sufﬁces to show that the eigenvalues of Ld are powers of q. The determinant is the product of the eigenvalues, and the eigenvalues are integers.

If q is a power of the prime p, and the determinant of Ld is a power of q, it follows that the eigenvalues of Ld are powers of p. This implies that they in fact are powers of q. a Since pd,b is the number of z for a pair of points x, y for which ∂ (x, y)= a, ∂ (x, z)= d, and ∂ (z, y)= b, this number is 0 by the triangle inequality when a + b < d. Hence for a non-zero pd,b a + b d, in other words, Ld is lower right triangular, so its determinant ≥ i is the product of the (second) diagonal entries, that is pd,d i . i d Q − d Since ki pd, d i = kd pi, d i [21], 2. 1. 1 (iv) and ki = b0 . . . bi 1/c1 . . . ci , and pi, d i = − − − − k pd i d i, d i kd cd cd 1...cd i+1 cd cd 1 . . . cd i+1/c1 . . . ci, for such a diagonal entry: p = − = − − . In d, d i ki b0...bi 1 − − − − our particular case, that of Grassmann (2d, d)q, the numbers kd and bi /cd i are powers i − of q (for all i) ([21], 9. 3. 2 (iii), [21], 9. 3. 3), so that pd, d i is a power of q, as desired. − ABSTRACT

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? This question was answered by Erdos,˝ Ko, and Rado, (already in 1938, but they first n 1 published this result in 1961): One must have C − , and if equality holds then C | |≤ k 1 is the collection of all k-subsets containing some fixed element− of the given n-set. 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, Erdos,˝ 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 do in this thesis. 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 are “far apart”, have maximum distance. The goal is always to find the maximum 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. Relations between flags are parameterized by double cosets in the Weyl group W, and being “far apart” can be defined as having the relation corresponding to the double coset containing the longest word w0. In simple cases we succeed in classifying all maximal cocliques in our Kneser graphs (not only the ones of maximal size). See e.g. [11], where the conjecture from [82] is proved. It turns out that most of the larger examples can be derived by a simple construction from corresponding examples in an apartment of the building, [22]. In not-so-simple examples we sometimes succeed in determining the maximum coclique size, and sometimes only obtain bounds. One way to obtain bounds is via Delsarte’s LP bound, and we have to compute the eigenvalues of our Kneser graphs. These eigenvalues turn out to be prime powers, and that leads to results on p-rank and Smith normal form of the adjacency matrices. 132 ABSTRACT ACKNOWLEDGEMENTS

It is my pleasure to acknowledge the people who have contributed to this thesis. I would like to thank my advisor, Andries Brouwer. It was a very nice, interesting and enlightening experience to work with him. He supported me to find my way of being a mathematician. I would also like to thank Lex Schrijver, who is one of my thesis committee members, for financing my Ph.D. studies with the Spinoza grant. Without the support of Andries and Lex, this thesis would have never existed. I also thank the other thesis committee members, Aart Blokhuis, Arjeh Cohen, Hans Cuypers, Jan Draisma, Edwin Van Dam, and Hendrik Van Maldeghem for their valuable inputs and comments. It was a pleasure to be at the Discrete Algebra and Geometry group for four years. I thank the group members for the great time we spent together, for the nice lunch conversations and discussions. I thank my office mates Jos in ’t panhuis, Dan Roozemond, Rob Eggermont, Yael Fleischmann and Maxim Hendriks. Jos and Dan were always very kind and helpful, which helped me a lot in my first year at the group to get used to the environment. I thank Yael, for her nice company in the office and tasty cakes. I thank Jan Willem Knopper for helping me whenever there was something wrong with my computer. I thank Jan Draisma for setting an inspiring example and for the nice puzzles he came up with. I thank Hans Sterk, Peter Schwabe, Shoumin Liu, Bart Frenk, Jan Jaap Oosterwijk, Mayank Singhal, Shona Yu, Gaetan Bisson, all the PhD students and other members of the group. I thank Aart Blokhuis for all the nice mathematical discussions, Hans Cuypers, for being a good boss. I thank Anita Klooster, for helping me with the thesis procedures after I left the university. Her cheerful nature improves the quality of the working environment. In the process of writing a PhD thesis, what makes life easier is the support and encouragement of friends and family. I would like to thank Rianne van Lieshout, for being a good friend and for all our great conversations during breaks. I thank Mayla Bruso and Antonino Simone, for everything we shared, and dear Christiane Peters, for being a very good friend. I would also like to express my gratitude to my friends in the Netherlands. Ege Ünal ¸Serbetli, Semih ¸Serbetli, Mehmet Çelik, ˙Ilhan Yıldırım, Tuba Dönmez Kokal, ˙Ilkin Kokal, Barı¸sYagcı,ˇ Aylin Koca Güle¸sir, Gürcan Güle¸sir, Kamil Kiraz, Beste Ertürk Sönmez, Ozan Sönmez, Zülküf Genç, Ugurˇ Keskin, Cagda¸sAtıcı,ˇ Hilal Karatoy, ¸Sehnaz Cenani Durmazoglu,ˇ Cagda¸sDurmazoˇ glu,ˇ Zümbül Atan and Oguzˇ Atan, my good old friend Mali (Mehmet Ali Dündar), and many others. 134 ACKNOWLEDGEMENTS

I thank Atike Dicle Pekel Duhbacı ﬁrst for designing the cover of this thesis, who I ﬁnd quite alike to myself and enjoy her friendship a lot, and Koray Duhbacı for being the sun of our Turkish Eindhoven life. I thank my good friends Sandra and Tarkan, for bringing joy to our lives. Our life here would be incomplete without them.

Before coming to TU/e, an important part of my mathematical adventure was held at Koç University. I loved being there, and learned there a lot. I thank to my previous super- visor Selda Küçükçifçi, for being always supportive and positive, and all my professors at Koç University, especially, Emine ¸Sule Yazıcı, Ali Ülger, Varga Kalantarov, Halil Mete Soner, and Ali Mustafazadeh. I thank Sinan Ünver, who ended up being a good friend. I thank my new colleagues from RBS Quantitative Review team, for the warm and ﬂexible working environment, which helped me in the process of making ﬁnal adjust- ments of this thesis. I thank my parents, Hülya and Yılmaz, for supporting me and for always underlining the importance of science, reading, thinking, learning, producing and sharing information. I am very lucky to have two beloved sisters, Özge and Gökçe, and now a brother Hüseyin. I thank them for always being there for me. I thank my dearest nephew Çınar for coming into our lives and making us happy. I thank my parents-in-law, Rukiye and Ziya. Finally, I thank my husband Tanır, for being there through this process, believing in me, supporting, and encouraging me with his love. CURRICULUM VITAE

Çiçek Güven Özçelebi was born on October 6, 1982 in ˙Izmir, Turkey. She ﬁnished her pre-university education at Be¸sikta¸sAtatürk Anadolu Lisesi in 2000. She received her bachelor of science degree in mathematics under the supervision of Prof. Varga Kalantarov at Koç University in 2005. She received her master of science degree also at Koç University in 2007 with her thesis “Weak Colorings of Steiner Triple Systems”, under the supervision of Assoc. Prof. Selda Kücükçifçi. In September 2007, she started her Ph. D. studies in the Discrete Algebra and Geom- etry Group under the supervision of Prof. Andries E. Brouwer in Eindhoven University of Technology. The thesis “Buildings and Kneser Graphs” is the result of her work in this period. During her experience at TU/e, she was the secretary of the Ph.D. association Promove of the university between September 2008 and September 2009, and was the chairman of the association from then on till January 2011. She has been working as a quantitative analyst at Royal Bank of Scotland in Amster- dam from October 2011 on. She enjoys reading, traveling and experiencing art, in particular painting. 136 CURRICULUM VITAE BIBLIOGRAPHY

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J-objects, 54 color class, 4 J-shadow, 15 complex, 16 J-space, 15 connected, 3 p-rank, 84, 87, 90 connected (geometry), 15 q-Kneser graph, 51, 52, 54 convex, 4, 17 q-analogue, 8, 87 convex closure, 101 t-(v, k, λ) design, 87 coset graph, 6 Coxeter graph, 97 abstract Coxeter complex, 18 Coxeter group, 9, 20, 52, 53, 93, 97 alternating form, 24 Coxeter group of the Tits system, 12 apartments, 19, 20, 61 Coxeter matrix, 9 associate matrices, 27 Coxeter system, 9, 68, 69, 96, 101 association scheme, 26, 28, 33, 97, 100 Coxeter-Dynkin diagram, 9, 52, 96 automorphism group of a graph, 5 automorphism of a graph, 5 Delsarte’s Linear Programming Bound, viii, 29, 108 basis of minimal idempotents, 28 diameter, 4, 7, 16, 27, 33, 37, 41 bilinear form, 24, 61 dihedral group, 9 Borel subgroup, 12, 66, 99 distance, 4, 7, 27, 54 Bose-Mesner algebra, 28, 33 distance regular graph, 5, 27 Bruhat decomposition, 13, 100 distance transitive graph, 6 Buekenhout-Tits diagram, 20, 98 distribution diagram, viii, 7, 30, 94, 95, Buekenhout-Tits geometry, 14, 98 101, 107, 108, 111, 115, 120, 124 building, vii, 13, 19, 53, 54, 61, 62, 68 distribution graph, 7 distribution vector, 29 Cartan subgroup, 12, 101 double coset graph, 7, 103 chain calculus, 105 chamber, 15, 54, 71, 100 edge transitive graph, 6 chamber complex, 16–19 eigenvalue, viii, 7, 29, 30, 83–86, 88, 93, characteristic matrix, 6 108, 113, 114, 118, 122, 125, 128 Chevalley group, 13, 14, 96 elliptic quadric, 25, 77 chromatic number, 4, 33, 52, 53 classical groups, 13 face, 16, 18–20 coclique, 4, 61, 63, 71, 95, 103, 129 faithful action, 5 code, 31 far apart, vii, 34, 46, 51, 54 codeword, 31 field with one element, 8, 34 collinear, 15, 77, 100, 112 finite simple groups of Lie type, 13, 53 collinearity graph, 16, 83, 95 first eigenmatrix (P matrix), 28, 108

144 INDEX 145

flag, vii, 15, 21, 34, 46, 52, 54, 55, 71, 93, Krein parameters, 28 98, 100, 107 flag transitive graph, 6 length, 3, 9, 15, 17, 27, 40, 53, 57, 97 folding, 45, 47 linearly independent, 22, 63, 79 fundamental system of roots, 10, 97 matroid, 62 Gaussian coefficient, 22 matroid base, 62 generalized m-gon, 16 generalized hexagon, 65 nearest neighbor decoding, 31 generalized Kneser graph, vii, 34 negative root, 10, 101 generalized quadrangle, viii, 22, 23, 25, 83, non-collinearity graph, 83 84, 89 normal subgroup, 8, 13, 101 geodesic, 4 opposite folding, 18 geodetically closed, 4, 104 opposite objects, 54, 93 geometry of type X , 20 n opposite types, 54, 93 girth, 15 oppositeness graph, 54 graph of Lie type, 99 oppositeness graphs, 93 Grassmann graph, 52 orbit, 5, 27, 79, 86 grid, 23 orbital, 5, 97 group action, 5, 62, 97 orthogonal form, 24 group with a (B, N)-pair, 13, 62, 99 orthogonal vectors, 24 group with a Tits system, 11, 53, 62

Hamming bound, 32 parabolic subgroup, 11, 13, 54, 62, 100 Hamming distance, 31 parapolar spaces, viii Hamming graph, 27 partial linear space, 22 Hamming metric, 27, 31 path, 3, 37, 95 Hamming scheme, 27 permutation group, 5 Hermitian symmetric form, 24 Petersen graph, 4, 32 Hoffman bound, viii, 95, 129 point-line geometry, 15, 21, 95 hyperbolic quadric, 25 polar space, 23, 65 hyperplane, 22, 66, 71 positive root, 10, 101 projective plane, 22, 80 incidence, 14 projective space, 21, 26, 61, 64, 71 incidence graph, 15 quadratic form, 24, 65, 85 incidence system of type Xn,i , 21, 95 intersection array, 5 quotient matrix, 6, 7, 30, 108, 115, 120, intersection number, 5, 28 125 invariant factors, 83 irreducible Coxeter system, 9 radical, 24 isometry, 25 rank, 12, 15, 17 isotropic, 24 reduced root system, 9 reﬂexive form, 24, 61 Johnson graph, 27, 32, 33, 51, 55 regular partition, 6, 7, 30 Johnson metric, 27 residually connected, 15, 104 Johnson scheme, 27 residue, 15, 20, 78, 105 root system graph, 97 Kneser graph, vii, 32, 51, 61, 63, 68, 71, 77, 83, 95 Schläﬂi graph, 67 146 INDEX second eigenmatrix (Q matrix), 28, 108 sesquilinear form, 23 simplex, 16 singular vector, 24 skew symmetric form, 24 Smith Normal Form, viii, 83, 87 stabilizer, 5, 20, 54, 62, 97 standard parabolic subgroup, 13 star of a simplex, 17 Steinberg groups, 13 strongly regular, viii, 4, 27, 83, 88 subspace, 22 Suzuki-Ree groups, 13 symmetric form, 24 symplectic, 104 symplectic form, 24, 85, 91 symplecton, 67, 104 thick, 16 thin, 16, 17, 72 Tits group, 8, 13, 14 totally isotropic, 24, 65, 72, 85, 91 totally singular, 24, 65 transitive action, 5 twisted Chevalley groups, 13, 20 type-preserving, 18 unique coclique extension property, 61, 68 unitary form, 24 vertex coloring, 3 vertex transitive graph, 6

Weyl group, 11, 12, 55, 62, 66, 99 Witt index, 24, 65