Chevalley Groups and Finite Geometry

The University of Melbourne

School of Mathematics and Statistics

PhD Thesis

Chevalley Groups andFinite Geometry

PhD Student:Jon Yue Zhang Xu

(University of Melbourne)

Supervisor:Prof. Arun Ram

(University of Melbourne)

Cosupervisor:A/Prof. John Bamberg

(University of Western Australia)

May 9, 2018

An austere buildingcut into simple pieces,now we see clearly.

³WM

K;~f

Q

ikameshikiyawokizamaretesatorikeri

i

Contents

i

Acknowledgements v

Introduction 1

Chapter 1. Finite Geometry 51.1. Lattices 51.2. Incidence structures 91.3. Veblen-Young type theorems 131.4. Ovoids of projective incidence structures 141.5. Ovoids of polar incidence structures 18

Chapter 2. Chevalley Groups 232.1. Lie algebras and root systems 232.2. Chevalley bases and Chevalley groups 242.3. The Bruhat Decomposition 272.4. Decomposing double cosets into single cosets 312.5. Examples 36

Chapter 3. Twisted Chevalley groups 493.1. Definitions and basic properties 493.2. Examples 553.3. From sesquilinear forms to Steinberg endormorphisms 64

Chapter 4. Tying things together 674.1. (Generalized) flag varieties, their lattices and incidence structures 674.2. Ovoids and (twisted) Chevalley groups 714.3. The Thickness of Schubert Cells 774.4. The example G = SL3(F) 81

Appendix A. 85A.1. Finite Fields and Galois Theory 85A.2. Sesquilinear forms 86A.3. Quadratic forms 89A.4. Flag varieties and buildings 91

Appendix B. 93B.1. Hasse diagrams of subspace lattices 93B.2. Hasse diagrams of Boolean lattices 96B.3. Hasse diagrams of Schubert cells in PG(F3

2) 99

Bibliography 103

Index 109

iii

Acknowledgements

I would like to thank my supervisor Arun Ram for his honesty and determination in thePhD supervision process. He has been an inspiring figure, and he has taught me how to confrontmathematics seriously and professionally.

I would like to thank my cosupervisor John Bamberg for helping me learn finite geometry,and for his hospitality during my six month visit at the University of Western Australia.

The haiku in the opening page of the thesis was submitted to the University of Melbourne’sGraduate Students Association thesis haiku competition, where it won (joint) first place. Iwould like to thank Asako Saito and Haydn Trowell for help with the Japanese translation.

v

Introduction

The aim of this thesis is to show how aspects of representation theory can be used to studyfinite geometry, and how the finite geometric concept of ‘thickness’ can be used to study Schu-bert cells (from representation theory). It is a result of an effort to create “interdisciplinary”communication and collaboration between the finite geometry community and the represen-tation theory communities in Australia, and we hope that this thesis can help to bridge themodern language barrier between these two fields. The books of D. Taylor [Tay92], Z. Wan[Wan93], Buekenhout and Cohen [BC13], are already important contributions to this dialogue.

We chose the finite geometry question of finding and classifying ovoids as a framework forinvestigation. The goal was to shape the language of algebraic groups and Chevalley groupsto provide tools for studying ovoids. The precedent in the work of Tits [Tit61] and Steinberg[Ste67, Example (c) before Theorem 34] on the Suzuki-Tits ovoid indicated that this was afruitful research direction.

Chapter 1, Chapter 2 and Chapter 3 cover well-known definitions and results from (respec-tively) finite geometry, the theory of Chevalley groups, and the theory of twisted Chevalleygroups. Most of the theory and examples have been adapted from the literature. The followingparts do not explicitly appear in the literature, and may be useful to those interested in aspectsof Chevalley groups and/or finite geometry:

• the proof that the classical ovoid in the Hermitian spaceH(3, q) is an ovoid (Proposition1.5.5),• the worked examples after Theorem 2.4.6 of the indexing of the points of the flag

variety G/B,• the worked examples of twisted Chevalley groups in Chapter 3,• the lattice theory discussed in Section 1.1.4 used to produce the pictures of Schubert

cells in Section B.3 (these pictures provide a novel way to view Schubert cells and flagvarieties).

Chapter 4 constitutes our main contribution to the research literature, which consists of threetheorems:

Main Theorem 1. Let C be the favourite nondegenerate conic, let E be the favouriteelliptic quadric, and S the favourite Suzuki-Tits ovoid as defined in (respectively) Equation 1.1,Equation 1.2, Equation 1.3.

(1) (Section 4.2.1) Let G = PΩ3(Fq) (∼= Ω3(Fq)/Z(Ω3(Fq), where Ω3(Fq) is the commuta-tor subgroup of O3(Fq)) and B be the subgroup of upper triangular matrices. With Gacting on P2 = P(F2+1

q ) by matrix multiplication and [v+] = [1 : 0 : 0] ∈ P2, then

Φ: G/B −→ P2

gB 7−→ g[v+]is injective with Φ(G/B) = C.

(2) (Section 4.2.2) Let GF = PΩ−4 (Fq) be the orthogonal group of minus type and BF bethe subgroup of upper triangular matrices. With GF acting on P3 = P(F3+1

q ) by matrix

multiplication and [v+] = [1 : 0 : 0 : 0] ∈ P3, then

Φ: GF/BF −→ P3

gBF 7−→ g[v+]is injective with Φ(GF/BF ) = E.

1

(3) (Section 4.2.3) Let GF = Sp4(F22e+1)F be the Suzuki group and BF be the subgroup ofupper triangular matrices. With GF acting on P3 = P(F3+1

q ) by matrix multiplication

and [v+] = [1 : 0 : 0 : 0] ∈ P3, then

Φ: GF/BF −→ P3

gBF 7−→ g[v+]is injective with Φ(GF/BF ) = S.

Main Theorem 2. (Section 4.2.4) Let O be the favourite classical ovoid in the Hermitianspace H(F3+1

q2 ) as defined in Proposition 1.5.5. This means that

O is the set of totally isotropic 1-dimensional subspaces in U⊥,

where

U = [x−2 : x−1 : x1 : x2] is a choice of nondegenerate 1-dimensional subspace.

Then O has a Schubert cell decomposition given by

O = [1 : 0 : 0 : 0] | x2 = 0 t [u : 1 : 0 : 0] | u ∈ Fq2 and x2uq − x1 = 0

t [u : t : 1 : 0] | u ∈ Fq2 , t ∈ Fq and x2uq − x1t+ x−1 = 0

t [t− u′qu : −u′q : uq : 1] | u, u′ ∈ Fq2 , t ∈ Fq and x2(t− u′uq) + x1u′q + x−1u− x−2 = 0

t [t′ − u′qu : −u′q + tuq : uq : 1]

| u, u′ ∈ Fq2 , t, t′ ∈ Fq and x2(t′ − u′uq)− x1(−u′ + tu) + x−1u− x−2 = 0.

Main Theorem 3. (Section 4.3) Let G(Fq) be a Chevalley group and let Pi and Pj bethe ith and jth standard maximal parabolic subgroups of G(Fq). Let W be the Weyl group of

G(Fq), w ∈ W and let Xw = BwB be the Schubert cell corresponding to w. Then the number

of elements in BwPi incident to gPj in Xw is

q`(z), where w = uzv with u ∈ W j, zv ∈ Wj, z ∈ (Wj)i,j, v ∈ Wi,j.

We now more informally describe the results, motivations and methodology of these theo-rems.

In Section 4.1.4, we establish a relationship between incidence structures (from finite ge-ometry), and flag varieties (from representation theory). The motivation for establishing thisrelationship is the work of Tits and Steinberg on the Suzuki-Tits ovoid, and is outlined inSection 4.2.3. Main Theorem 1 explicitly describes three key examples of ovoids – the rationalnormal curve, the elliptic quadric, and the Suzuki-Tits ovoid – as flag varieties of a suitablychosen Chevalley group. It would be interesting to consider whether non-classical ovoids (see[Che04] and [Che96]) can also be realised in this way, with perhaps the role of the Chevalleygroup being played by a suitably chosen pseudo-reductive group (see [CGP15]).

Main Theorem 2 provides a Schubert cell decomposition of the classical ovoid in the Her-mitian variety H(3, q2) (as described in Proposition 1.5.5).

Before describing Main Theorem 3, let us review the definitions of ovoids (in finite geometry)and Schubert cells (in representation theory).

Ovoids. Let V be a vector space and let PG(V ) be the lattice of subspaces of V withinclusion ⊆ as the partial order. A point (respectively line, hyperplane) in PG(V ) is a subspaceS ⊆ V such that dim(S) = 1 (respectively dim(S) = 2, dim(S) = dim(V )− 1).

Let O be a set of points in PG(V ). A tangent line to O is a line in PG(V ) that containsexactly 1 point of O. Then [Tit62, §1] defines, an ovoid of PG(V ) as a set O of points of PG(V )such that

(O1) (thinness) If l is a line in PG(V ) then l contains 0, 1 or 2 points of O,(O2) (maximality) If p ∈ O then the union of the tangent lines to O through p is a hyper-

plane.

These two types of conditions, “thinness” and “maximality”, characterize the definitions ofovoids (and ovals and hyperovals) in projective spaces, projective planes, polar spaces and

2

generalized quadrangles that can be found in the finite geometry literature (see, for example,[HT15], [Bal15, §4.8], [Bro00a, §1] and [BW11, §2.1 and §4.2 and §4.4]).

Schubert cells. Let G be a Chevalley group over F and let B be a Borel subgroup. Thequotient

G/B is the (generalized) flag variety.

In the case that G = GLn(F) then

GLn(F)/B ∼= maximal chains 0 ⊆ V1 ⊆ · · · ⊆ Vn−1 ⊆ V in PG(V )where V is an F-vector space of dimension n and PG(V ) is the lattice of subspaces of V . Theflag varieties are studied with the use of the Bruhat decomposition,

G =⊔w∈W

BwB, and Xw denotes the Schubert cell BwB

viewed as subsets of the set of cosets G/B. In the case of GLn(F)/B the Xw are collections of

maximal chains in PG(V ) and thus, when F = Fq is a finite field, the Xw are natural objects infinite geometry. From the point of view of representation theory, the closures of the Schubertcells are the Schubert varieties of the projective variety G(F)/B(F), where F is the algebraicclosure of F. This makes the Schubert cell a tool in the framework of geometric representationtheory.

In Chapter 4, Section 4.3, we define an incidence structure for each Schubert cell and eachpair of maximal parabolic subgroups of the Chevalley group. This provides a way of analyzingthe Schubert cell using the viewpoint of finite projective geometry. Then, in pursuit of thequestion of what causes the “thinness” that distinguishes ovoids, we prove the main theorem(Theorem 4.3.3) which is a computation of the “thickness” of the incidence structures thatcome from Schubert cells.

This allows us to isolate basic examples of Schubert cells which are thin. We hope thatfuture work will provide a full classification of ovoids that arise from Schubert cells, Schubertvarieties and hyperplane sections of Schubert varieties.

Appendix A and Appendix B contain material whose presence would otherwise disrupt theflow of the thesis.

We have established the statements of this thesis in the context of the theory of groups ofLie type, however, we note here that the powerful Bruhat-Tits theory of spherical buildings(seen in [Tay92], [AB08], [Tit74]) is implicit in all our work. We provide a translation betweenthe language of groups of Lie type and the language of buildings in Section A.4.

3

CHAPTER 1

Finite Geometry

In this chapter, we outline the definitions and key theorems of finite geometry. We describesome key structures of finite geometry: lattices (Section 1.1), incidence structures (Section 1.2),and the (isotropic) subspaces of a vector space (Sections 1.1.1, 1.1.2, 1.1.3). We illustrate howthese structures are related via the Veblen-Young theorem (Section 1.3).

The combinatorial objects at the forefront of research in finite geometry are those whichare extremal in some sense. Ovoids are examples of such objects; conceptually, an ovoid is a‘maximally thin’ set of points. These objects arose out of the study of geometric propertiesof the classical varieties (such as conics) over finite fields, and whether these varieties can becharacterised by incidence axioms.

We define ovoids and describe the ovoids of projective and polar incidence structures inSections 1.4 and 1.5. The study of ovoids is active in the 21st century, and there are many openproblems (see [BDI15], [HT15], [DKM11], [BP09], [DM06], [Shu05]). To quote Thas [Tha01],ovoids of polar incidence structures have

“many connections with and applications to projective planes, circle geome-tries, generalised polygons, strongly regular graphs, partial geometries, semi-partial geometries, codes, designs”.

1.1. Lattices

The main references for this section are [Whi86, Chapter 3] and [Bir67, Ch. 1]. Let L bea partially ordered set with partial order ≤. The join (or supremum, or least upper bound) oftwo elements x, y ∈ L is the element

x ∨ y ∈ L such that

(V1) x ∨ y ≥ x and x ∨ y ≥ y , and(V2) if z ∈ L and z ≥ x and z ≥ y then z ≥ x ∨ y.

The meet (or infimum, or greatest lower bound) of two elements x, y ∈ L is the element

x ∧ y ∈ L such that

(W1) x ∧ y ≤ x and x ∧ y ≤ y , and(W2) if z ∈ L and z ≤ x and z ≤ y then z ≤ x ∧ y.

A join-semilattice (respectively meet-semilattice) is a poset L closed under join (respectivelymeet). A lattice is a poset L closed under both join and meet.

Let L,L′ be lattices. A lattice homomorphism [DP02, p. 2.16] is a map ϕ : L → L′ suchthat if x, y ∈ L then

f(x ∨ y) = f(x) ∨ f(y), f(x ∧ y) = f(x) ∧ f(y).

A lattice isomorphism is a bijective lattice homomorphism such that its inverse is also a latticehomomorphism.

1.1.1. The subspace lattice of a vector space. Let F be a field or division ring andlet V be a finite dimensional vector space over F. The subspace lattice of V is

PG(V ) = subspaces of V with partial order ⊆ .

5

More generally, one could consider a ring R and an R-module M and let

PG(M) = R-submodules of M with partial order ⊆ .

In the finite geometry literature, a finite projective space is the subspace lattice PG(Fn+1q ), where

Fq is the finite field with q elements. The notations PG(n,Fq) and PG(n, q) are sometimes usedfor PG(Fn+1

q ). In the algebraic geometry literature, projective space is the quotient

Pn =Fn − (0, . . . , 0)

〈(a1, . . . , an) = (ca1, . . . , can) | c ∈ F×〉.

The term ‘projective space’ is dependent on context and should, therefore, be used with care.The points of PG(V ) are the rank 1 elements, the lines of PG(V ) are the rank 2 elements,

the i-planes of PG(V ) are the rank i+ 1 elements, and the hyperplanes of PG(V ) are the rankdim(V )− 1 elements.

The joins and meets of PG(V ) are

U ∨W = U +W and U ∧W = U ∩W .

Let K1 and K2 be division rings, let V1 be a left K1-vector space and let V2 be a left K2 vectorspace. A semilinear transformation from V1 to V2 with associated isomorphism σ is a pair (f, σ)where

• σ : K1 → K2 is an isomorphism of division rings, and• f : V1 → V2 is a function satisfying

f(u+ v) = f(u) + f(v) and f(av) = σ(a)f(v)

for all u, v ∈ V1 and a ∈ K1.

We will often omit σ and refer to the function f : V1 → V2 as the semilinear transformationfrom V1 to V2.

Proposition 1.1.1. [Tay92, §3] If f : V1 → V2 is a bijective semilinear transformation thenthe induced mapping PG(f) : PG(V1)→ PG(V2) is a lattice isomorphism.

1.1.2. The subset lattice of a finite set. The subset lattice

PG(Fn+11 ) is the lattice of subsets of 1, 2, . . . n+ 1, ordered by ⊆.

The joins and meets of PG(Fn+11 ) are

U ∨W = U ∪W and U ∧W = U ∩W .

The justification for the notation PG(Fn1 ) is to relate our work to the study of the ‘field withone element’ as outlined by Tits in [Tit56, §13]. For modern research, see for example [Sou04]and [PL09]. Conceptually, we have

PG(Fn1 ) ∼= limq→1

PG(Fnq ).

1.1.3. The polar semilattice associated with a sesquilinear form. Let Fq be thefinite field with q elements, and let V = Fn+1

q , considered as a Fq-vector space. Let 〈·, ·〉 : V ×V → Fq be a σ-sesquilinear form on V (see Appendix A.2 for definitions and basic propertiesof sesquilinear forms). A vector v ∈ V is totally isotropic if 〈v, v〉 = 0. A subspace W in V istotally isotropic if the following condition is satisfied:

if v, w ∈ W then 〈v, w〉 = 0.

Note that our definition of totally isotropic follows [Tay92, pg. 56, Definition (ii)] and [Bal15,pg. 27]. The classical polar semilattice P (V, 〈·, ·〉) is the meet-semilattice of totally isotropicsubspaces in PG(V ) (often P (V, 〈·, ·〉) is called a polar space). The meet is given by

x ∧ y = x ∩ y.

Note that, in general, P (V, 〈·, ·〉) is not a lattice since it is not closed under join. Let r bethe dimension of a maximum dimension totally isotropic subspace of P (V, 〈·, ·〉), usually called

6

the rank or Witt index . The key examples appearing in the literature (see [Bal15, §4.2] and[Tha81, §1]) are:

• The symplectic polar semilattice is the semilattice W (2r − 1, q) of totally isotropicsubspaces of a nondegenerate alternating form of rank r on F2r

q .

• The Hermitian (or unitary) polar semilattices are the semilattices H(2r − 1, q2) andH(2r, q2) (respectively) of totally isotropic subspaces of a nondegenerate Hermitianform of rank r on F2r

q and F2r+1q (respectively).

• The hyperbolic orthogonal polar semilattice is the semilattice Q+(2r − 1, q) of totallyisotropic subspaces of a nonsingular quadratic form of rank r on F2r

q .• The parabolic orthogonal polar semilattice is the semilattice Q(2r, q) of totally isotropic

subspaces of a nonsingular quadratic form of rank r on F2r+1q .

• The elliptic orthogonal polar semilattice is the semilattice Q−(2r + 1, q) of totallyisotropic subspaces of a nonsingular quadratic form of rank r on F2r+2

q .

1.1.4. Lattice Theory. Let L be a lattice. A greatest element of L is a g ∈ L such thatif x ∈ L then x ≤ g. A least element of L is a g ∈ L such that if x ∈ L then x ≥ g. For theremainder of this section, we assume L is a lattice with a unique greatest element 1 and uniqueleast element 0.

The lattice L is modular if the following condition is satisfied:

if x, y, z ∈ L and x ≤ z then x ∨ (y ∧ z) = (x ∨ y) ∧ z.

The lattice L is decomposable if there exists z1, z2 ∈ L such that if x ∈ L then the followingcondition is satisfied:

there exists unique x1, x2 ∈ L such that 0 ≤ x1 ≤ z1, 0 ≤ x2 ≤ z2 and x = x1 ∨ x2.

The lattice L is indecomposable if it is not decomposable.A nonempty subset C ⊆ L is a chain if the following condition is satisfied:

if x, y ∈ L then x ≤ y or x ≥ y.

Let C ⊆ L be a chain such that C is finite. The length of C is

`(C) = |C| − 1.

The rank of an element a ∈ L is

rank(a) = sup`(C) | C is a chain in L(a)

where L(a) = supx ∈ L | x ≤ a. The rank of L is

rank(L) = suprank(a) | a ∈ L.

An element a ∈ L is an atom if rank(a) = 1. A lattice L is atomic if the following condition issatisfied: if x ∈ L then there exist atoms a1, a2, . . . , ak ∈ L with

(. . . ((a1 ∨ a2) ∨ a3) . . .) ∨ ak) = x.

A projective lattice is an indecomposable modular atomic lattice of finite rank.

Proposition 1.1.2. Let V = Fn. Then PG(V ) is a projective lattice.

Proof. Let L = PG(V ). Recall that L is a lattice with join and meet given by U ∨W =U + W and U ∧W = U ∩W . We will show that L is atomic, of finite rank, modular, andindecomposable, in turn.

If U ∈ L then there exists a basis v1, v2, . . . , vk of U . The elements

spanv1, spanv2, . . . , spanvkare atoms, and

(. . . ((spanv1+ spanv2) + spanv3) + · · ·+ spanvk) = U

Hence L is atomic.

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A chain in L is a subset C = U0, U1, . . . , Uk where

U0 ( U1 ( · · · ( Uk.

The maximal length of any chain is

`(C) = dimV − 1 = n.

So rank(L) = n. Hence L has finite rank.Assume U1, U2, U3 ∈ L with U1 ⊆ U3. To show:

(1) U1 + (U2 ∩ U3) ⊆ (U1 + U2) ∩ U3.(2) (U1 + U2) ∩ U3 ⊆ U1 + (U2 ∩ U3).

(1) Since U2 ∩ U3 ⊆ U2, we have U1 + (U2 ∩ U3) ⊆ U1 + U2. Also, since U1 ⊆ U3 andU2 ∩ U3 ⊆ U3 so that U1 + (U2 ∩ U3) ⊆ U3. Hence U1 + (U2 ∩ U3) ⊆ (U1 + U2) ∩ U3.

(2) Suppose v ∈ (U1 +U2)∩U3. Then v ∈ U1 +U2. So there exists λ1, λ2 ∈ F and v1 ∈ U1,v2 ∈ U2 such that v = λ1v1 + λ2v2.

If λ2 = 0 then v = λ1v1 ∈ U1. Hence v ∈ U1 + (U2 ∩ U3).If λ2 6= 0 then 1

λ2v = λ1

λ2v1 + v2. So v2 = 1

λ2v− λ1

λ2v1. Since v ∈ U3 and v1 ∈ U1 ⊆ U3

we have v2 ∈ U3. So v2 ∈ U2 ∩ U3. So v ∈ U1 + (U2 ∩ U3). Hence (U1 + U2) ∩ U3 ⊆U1 + (U2 ∩ U3).

Suppose for sake of contradiction that L is decomposable with V1, V2 ∈ L such that the followingcondition is satisfied: if U ∈ L then there exists a unique U1, U2 ∈ L such that 0 ⊆ U1 ⊆ V1, 0 ⊆U2 ⊆ V2, and U = U1 + U2. We first show that V1 ∩ V2 = 0 and V1 + V2 = V . Suppose, forsake of contradiction, that X = V1 ∩ V2 is nonzero. Then 0 ⊆ X ⊆ V1 and 0 ⊆ X ⊆ V2. ButX = X + 0 = 0 +X = X +X, contradicting uniqueness. Hence V1 ∩ V2 = 0.

We now show V = V1 +V2. Suppose, for sake of contradiction, that there exists v ∈ V suchthat v /∈ V1 + V2. Let U1, U2 ∈ L such that 0 ⊆ U1 ⊆ V1, 0 ⊆ U2 ⊆ V2 and Fv = U1 +U2. Weknow U1 + U2 ⊆ V1 + V2. So Fv ⊆ V1 + V2. Hence v ∈ V1 + V2, a contradiction.

Hence V1 ⊕ V2 = V . Let v1 ∈ V1 and v2 ∈ V2 such that v1 + v2 /∈ V1 and v1 + v2 /∈ V2. ThenFv1 + v2 ( V1 and Fv1 + v2 ( V2. Let U1, U2 ∈ L such that 0 ⊆ U1 ⊆ V1, 0 ⊆ U2 ⊆ V2

and Fv1 + v2 = U1 + U2. Since Fv1 + v2 is 1-dimensional, we have Fv1 + v2 = U1 orFv1 + v2 = U2. So Fv1 + v2 ⊆ V1 or Fv1 + v2 ⊆ V2, a contradiction.

A lattice L is complemented if the following condition is satisfied:

if x ∈ L then there exists an element y ∈ L such that x ∧ y = 0 and x ∨ y = 1.

A lattice L is distributive if the following condition is satisfied:

if x, y, z ∈ L then x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z).

A Boolean lattice is a complemented distributive lattice [Bir67, Ch. 1 §10], [DP02, §4.13].

Proposition 1.1.3. The subspace lattice PG(Fn+11 ) is a Boolean lattice.

Proof. Suppose x ∈ PG(Fn+11 ). Then x∪1, 2, . . . n+ 1 = 1, 2, . . . n+ 1 and x∩∅ = ∅.

So PG(Fn+11 ) is complemented. Let x, y, z ∈ PG(Fn+1

1 ). Then x ∩ (y ∪ z) = (x ∩ y) ∪ (x ∩ z).So PG(Fn+1

1 ) is distributive. Hence PG(Fn+11 ) is a Boolean lattice.

Any lattice L can be visualised using a graph called the lattice diagram or Hasse diagram([Whi86, §3.1], [Bir67, Ch. 1 §3]) as follows. Let the vertices of the Hasse diagram be theelements of L, placing y higher than x whenever y > x. We draw a straight line segment fromy to x if y > x and there does not exist z ∈ L such that y > z > x. The Hasse diagramscorresponding to subspace lattices and Boolean lattices are given in Appendix B, Sections B.1and B.2.

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1.2. Incidence structures

An incidence structure is a triple G = (P ,L, I) where P and L are sets and I ⊆ P × L.

I ⊆ P × L pr2−→ Lypr1P

• The projection maps pr1 : P ×L → P and pr2 : P ×L → L are defined by pr1(p, l) = pand pr2(p, l) = l.• A point p ∈ P is contained in a line l ∈ L if (p, l) ∈ I.• A subset S ⊆ P is collinear if there exists l ∈ L such that if p ∈ S then (p, l) ∈ I.

Often it is convenient to identify l ∈ L with the set of points pr1(pr−12 (l)). Occasionally we will

use B instead of L and call B the set of blocks.Let G = (P ,L, I) and G ′ = (P ′,L′, I ′) be incidence structures. A homomorphism from G

to G ′ is a map ϕ : P t L → P ′ t L′ such that

ϕ(P) ⊆ P ′, ϕ(L) ⊆ L′, and if (p, b) ∈ I then (ϕ(p), ϕ(b)) ∈ I.

A homomorphism of incidence structures ϕ : P t B → P ′ t B′ is an isomorphism if ϕ is abijection and ϕ−1 is a homomorphism of incidence structures.

Let L be a lattice of finite rank. The (point-line) incidence structure associated with L isG(L) = (P ,L, I) where

P = x ∈ L | rank(x) = 1,L = y ∈ L | rank(y) = 2,I = (x, y) ∈ P × L | x ≤ y.

Define

Pk = z ∈ L | rank(z) = kDefine the projection maps

pk : maximal chains in L −→ Pk(0 ⊆ x1 ⊆ x2 ⊆ · · · ⊆ xn ⊆ 1) 7−→ xk.

If a ∈ Pj then

pip−1j (a) = z ∈ Pi|a ≤ z or z ≤ a.

1.2.1. Projective incidence structures. A projective incidence structure is an incidencestructure G = (P ,L, I) such that

(1) If p1, p2 ∈ P and p1 6= p2 then there exists a unique line l(p1, p2) ∈ L containing p1

and p2;(2) (Veblen-Young axiom) Let a, b, c, d ∈ P be distinct points such that there exists a

point p ∈ P with p ∈ l(a, b) ∩ l(c, d). Then there exists a point q ∈ P such thatq ∈ l(a, c) ∩ l(b, d);

a

b

p

dc

q

(3) (thickness condition) Any line contains at least 3 points.

9

Let G1 = (P1,L1, I1) and G2 = (P2,L2, I2) be projective incidence structures. A collineationfrom G1 to G2 is a homomorphism α : P1 t L1 → P2 t L2 from G1 to G2.

Let G = (P ,L, I) be a projective incidence structure. The group of invertible collineationsof G is

PΓL(G) = α : P t L → P t L |α is an isomorphism from G to G.

Theorem 1.2.1 (Fundamental Theorem of Projective Geometry). [Tay92, Theorem 3.1]Let V1 and V2 be left vector spaces of dimension n over division rings K1 and K2, and thatn ≥ 3. Let

G1 = (P1,L1, I1) = G(PG(V1)),

G2 = (P2,L2, I2) = G(PG(V2)),

be the projective incidence structures associated with V1 and V2 respectively. If α : P1 t L1 →P2tL2 is an isomorphism of incidence structures then there exists a semilinear transformationf : V1 → V2 with associated isomorphism σ : K1 → K2 such that

α(p) = f(p) and α(l) = f(l),

for all p ∈ P1 and l ∈ L1. Furthermore, if f ′ : V1 → V2 is a semilinear transformation withassociated isomorphism σ′ : K1 → K2 such that

α(p) = f ′(p) and α(l) = f ′(l),

for all p ∈ P1 and l ∈ L1, then there exists b ∈ K2 such that for all v ∈ V1 and a ∈ K1 we havef ′(v) = bf(v) and σ′(a) = bσ(a)b−1.

Corollary 1.2.2. [Tay92, pg. 15] Let K be a division ring and let V be a left K-vectorspace such that dimKV ≥ 3. Let G = G(PG(V )) be the point line incidence structure associatedwith the subspace lattice PG(V ). Let ΓL(V ) be the group of invertible semilinear transformationsof V . Then

PΓL(V ) ∼= ΓL(V )/Z(ΓL(V ))

A projective incidence structure G is Desarguesian [BR98, §2.2] if the following statementholds: if a1, a2, a3, b1, b2, b3 ∈ P such that:

• There exists a point c distinct from a1, a2, a3, b1, b2, b3 such that

c ∈ l(a1, b1) ∩ l(a2, b2) ∩ l(a3, b3), and

• No three of the points c, a1, a2, a3 are collinear, and no three points of the pointsc, b1, b2, b3 are collinear,

then the points p12, p23, p13 are collinear, where

p12 = l(a1, a2) ∩ l(b1, b2), p23 = l(a2, a3) ∩ l(b2, b3), p13 = l(a1, a3) ∩ l(b1, b3).

c

a1

a3

a2

b1

b3

b2

p13

p23

p12

10

A projective incidence structure G is Pappian [BR98, §2.2] if for all distinct intersecting pairs oflines l1, l2 ∈ L, the following condition is satisfied: if a1, a2, a3 are distinct points on l1, b1, b2, b3

are distinct points on l2, and none of the points a1, a2, a3, b1, b2, b3 are equal to the point l1 ∩ l2,then the points

p12 = l(a1, b2) ∩ l(b1, a2), p23 = l(a2, b3) ∩ l(b2, a3), p31 = l(a3, b1) ∩ l(b3, a1)

are collinear.

l2

b1

b2

b3

p12l1

a1a2

a3

p13 p23

Theorem 1.2.3. Let V be a vector space over a division ring F, and let G(PG(V )) be thepoint-line incidence structure associated with the subspace lattice PG(V ). Then G(PG(V )) is aDesarguesian projective incidence structure. Furthermore, G(PG(V )) is Pappian if and only ifF is a field.

References for proof. See [BR98, Theorem 2.2.1] and [BR98, Theorem 2.2.2].

Assume that G = (P ,L, I) is an incidence structure such that any two points lie on a uniqueline. A subspace is a subset S ⊆ P such that S contains any line connecting two of its points,i.e.

if p1, p2 ∈ S then pr1(pr−12 (l(p1, p2))) ⊆ S.

If U ⊆ P is a subspace then G ′ = (U ,L′, I ′) with

L′ = l ∈ L | l ⊆ U,I ′ = (p, l) ∈ I | p ∈ U and l ∈ L′,

is a projective incidence structure.The span of a subset X ⊆ P is

span(X ) =⋂

X⊆S⊆P,S is a subspace

S.

A set X ⊆ P is a basis of G if the following conditions are satisfied:

(1) span(X ) = P ,(2) if X ′ ( X is a proper subset then span(X ′) 6= P .

The projective dimension or (projective) rank of G is one less than the number of elements inany basis of G. See [BR98, Theorem 1.3.8] for proof that any basis has the same number ofelements.

Let n be the projective dimension of G and write Pdim(G) = n. An i-plane is the inducedprojective geometry G ′ = (U ,L′, I ′) of a subspace U ⊆ P such that Pdim(G ′) = i. A hyperplaneis an (n− 1)-plane.

The subspace lattice of G is

PG(G) = subspaces S ⊆ P with partial order ⊆,with join and meet given by

U ∨ V = span(U ∪ V) and U ∧ V = span(U ∩ V).

11

Theorem 1.2.4. [Whi86, pg. 59 and Corollary 3.3.5] The mapprojective

lattices

−→

finite dimensional

projective incidence structures

L 7−→ G(L)

PG(G) ←−p G

is a bijection.

1.2.2. Polar incidence structures. Our definition of polar incidence structure is takenfrom [Tay92, pg. 108] (see also [Shu75, Theorem C], [Shu10, §7.1]). This definition is from thework of Buekenhout and Shult, who show in [BS74, Theorem 4] that the definition is equivalent(assuming that there does not exist an infinite chain of isotropic subspaces) to the definitionsgiven by Veldkamp [Vel59] and Tits [Tit74, §7.1]. The proof of this equivalence is also givenin [Ueb11, §2, Chapter 4]. Buekenhout and Shult’s definition, which we use below, has theadvantage of giving a complete characterisation of polar spaces in terms of points and lines.

A polar incidence structure (usually called a polar space in the literature) is an incidencestructure G = (P ,L, I)

I ⊆ P × L pr2−→ Lypr1P

such that L is a set of subsets of P and

(BS1) If l ∈ L then l contains at least 3 points,(BS2) There does not exist a point p ∈ P such that p is collinear with all points in P ,(BS3) If p ∈ P , l ∈ L with p not incident to l, then exactly one of the following hold:

(a) There exists a unique q incident with l such that q is collinear with p, or(b) If q is incident with l then q is collinear with p.

A subset U ⊆ P is a singular subspace (or subspace) [Tay92, pg. 108], [Ueb11, §4.2, pg. 125] ifthe following conditions are satisfied:

(1) If p, q ∈ U with p 6= q then p and q are collinear,(2) If l ∈ L contains two distinct points p and q in U , then l ⊆ U .

A singular subspace U is maximal if it is not strictly contained in any other singular subspace.The rank of a polar space is the projective dimension of a maximal singular subspace (see[Shu10, Section 7.3.5]).

We will sometimes refer to the point-block polar incidence structure G = (P ,B, I)

I ⊆ P × B prm−→ Bypr1P

where

B = maximal singular subspacesI = (p, b) ∈ P × B | p ⊆ b.

Theorem 1.2.5. [Tay92, pg. 107, paragraph -2], [Shu10, pg. 178]. Let V , 〈·, ·〉 andP (V, 〈·, ·〉) be as in Section 1.1.3. Let

P = 1− dimensional totally isotropic vector subspaces of P (V, 〈·, ·〉)L = 2− dimensional totally isotropic vector subspaces of P (V, 〈·, ·〉)I = (p, l) ∈ P × L | p ⊆ l.

Then G(P (V, 〈·, ·〉)) = (P ,L, I) is a polar incidence structure.

12

1.3. Veblen-Young type theorems

Let G = (P ,L, I) be a projective incidence structure. A central collineation is a collineationα : P t L → P t L such that there exists a hyperplane H (the axis of α) and a point C (thecenter of α) with the following properties:

• If p is a point on H then α(p) = p,• If l is a line incident with C then α(l) = l.

Fix a hyperplane H ⊆ P and a point O ∈ P with O not on H. Let P∗ = P\H.

Proposition 1.3.1. [BR98, Theorem 3.2.3] Let T (H) be the set of all central collineationswith axis H and center on H. For p ∈ P∗ there exists a unique τp ∈ T (H) such that τp(O) = p.Thus we can define an addition on P∗ by

p+ q = τp(τq(O)).

Theorem 1.3.2 (Baer’s theorem). [Bae42], [Bae46], [BR98, Theorem 3.1.8]. Let G be aDesarguesian projective geometry. If H is a hyperplane and c, p, p′ are distinct collinear pointsof G with p, p′ /∈ H then there exists a unique central collineation of G with axis H and centrec mapping p onto p′.

The next theorem demonstrates how to recover a division ring F from a projective incidencestructure.

Theorem 1.3.3. [BR98, Theorem 3.3.4] Let DO be the group of central collineations of Gwith axis H and center O. Let 0: P∗ → P∗ be the map which sends all points of P∗ onto O.Let F = DO ∪ 0. If σ1, σ2 ∈ F we define the map σ1 + σ2 : P → P by

(σ1 + σ2)(p) = σ1(p) + σ2(p)

for p ∈ P∗ and extending projectively to all of P (see [BR98, §3.1]) so that σ1 + σ2 ∈ F (whereaddition on P∗ is defined in Proposition 1.3.1). Furthermore, define σ1 · σ2 ∈ F by

σ1 · σ2 =

σ1 σ2, if σ1, σ2 ∈ DO,

0, if σ1 = 0 or σ2 = 0,

where means composition of functions. Then (F,+, ·) is a division ring.

Theorem 1.3.4.

(1) (Veblen-Young) If G is a projective incidence structure and PdimG ≥ 3 then G isDesarguesian and there exists a vector space V over a division ring F such that G ∼=G(PG(V )).

(2) If G is a Desarguesian projective incidence structure with PdimG = 2 then there existsa vector space V over a division ring F such that G ∼= G(PG(V )).

References for proof. (1) is [BR98, Theorem 2.7.1] [BR98, Corollary 3.4.3], [HHP94,Ch. VI, §7], the field is F = DO ∪0 as given in Theorem 1.3.3. (2) is [BR98, Theorem 3.4.2].See also [Shu10, Theorem 6.4.1].

The following theorem is an analogue of the Veblen-Young theorem for PG(n, 1).

Theorem 1.3.5. [DP02, Theorem 5.5] Let L be a finite Boolean lattice and let A(L) be theset of atoms of L. Then the map

ν : L −→ subsets of A(L) ∼= PG(F|A(L)|1 ),

x 7−→ a ∈ A(L) | a ≤ x,is a lattice isomorphism.

The following theorem is an analogous Veblen-Young theorem for classical polar semilattices.

13

Theorem 1.3.6. [Shu10, Theorem 7.9.7] If G = (P ,L, I) is a polar incidence structure ofrank at least 4 then there exists a (classical) polar semilattice P (V, 〈·, ·〉) such that

P = 1-dimensional totally isotropic subspaces of V L = 2-dimensional totally isotropic subspaces of V I = (p, l) ∈ I | p ⊆ l.

Remark 1.3.7. There is also a classification, more complicated to state, for polar incidencestructure of rank at least 3, due to Tits [Tit74] (see also [BS74, Theorem 1]). The classificationof polar incidence structures of rank 2 is, to our knowledge, still open.

1.4. Ovoids of projective incidence structures

Let G = (P ,L, I) be a projective incidence structure of projective dimension n. The keyexample is G = G(PG(Fn+1

q )). Let S ⊆ P . A line l ∈ L is secant to S if |S ∩ pr1(pr−12 (l))| = 2

(that is, if l is incident with S in exactly 2 points). Similiarly, a line l ∈ L is a tangent to S if|S ∩ pr1(pr−1

2 (l))| = 1 (that is, if l is incident with S in exactly 1 point). A hyperplane H ⊆ Pis secant to S if there exists at least 2 points in S that are incident with H. Let p ∈ S. Let

Lp = l ∈ L | p ⊆ l.

The set of tangent lines (or tangent cone) to S incident with p is

T Lp(S) = l ∈ L | l is tangent to S and p ⊆ l,and the set of secant lines to S incident with p is

SLp(S) = l ∈ L | l is secant to S and p ⊆ l.The tangent space to S at p is

Tp(S) = supl ∈ T Lp(S)

where the supremum is taken in the lattice PG(G).A cap ([HT15, §5.1], [Bar55], [Seg59b]) is a set of points S ⊆ P such that if p ∈ S then

Lp = T Lp(S) t SLp(S).

An ovoid [Bal15, §4.8], [Tit62, §1] is a set of points O ⊆ P such that

(O1) O is a cap and(O2) if p ∈ O then Tp(O) ∈ Pn and

T Lp(O) = p2p−1n (Tp(O)).

Proposition 1.4.1. If O is an ovoid in G(PG(Fn+1q )) then Card(O) = qn−1 + 1.

Proof. Let p ∈ O. Let Lp be the set of lines through p. Since O is a cap,

Lp = T Lp(O) t SLp(O)

We count the number of elements in each of the sets above. Since

Lp ↔ p1(PG(Fn+1q /p))

l 7→ l + p,

is a bijection,

Card(Lp) =Card(

Fn+1q

p− 0)

Card(F×q )=qn − 1

q − 1.

Next

T Lp ↔ l ∈ p2(PG(Tp(O))) | p ⊆ ll 7→ l,

14

is a bijection, and by the condition (O2) in the definition of an ovoid, Tp(O) ∈ Pn , so that

Card(T Lp) =qn−1 − 1

q − 1.

Then, for q ∈ O, denoting l(p, q) ∈ L for the line containing p and q, the map

O − p ↔ SLpq 7→ l(p, q),

is a bijection, giving

Card(SLp) = Card(O)− 1.

Thus Lp = T Lp t SLp gives

Card(O) = Card(Lp)− Card(T Lp) + 1 = qn−1 + 1.

Proposition 1.4.2. [Dem68, Comment after (28’), §1.4, pg. 48], [Bal15, First line ofproof of Theorem 4.37]. Let O be an ovoid in G(PG(Fn+1

q )) and let H be a secant hyperplane.

Then O′ = p1p−1n (H) ∩ O is an ovoid in G(H) ∼= G(PG(Fnq )).

O’

O

H

Corollary 1.4.3. If G(PG(Fn+1q )) has an ovoid then n ≤ 3 [Bal15, Theorem 4.37],

[Dem68, Theorem 48], [Tit62, Footnote (1)].

Proof. Let O be an ovoid in G(PG(Fn+1q )). Let

A = (x,H) ∈ O × Pn | x ⊆ H and Card(O ∩ p1p−1n (H)) ≥ 2.

Let x ∈ O. Since the only hyperplane H ∈ Pn such that x ⊆ H and p1p−1n (H) ∩ O = x is

Tx(O), and there are exactly qn−1q−1

hyperplanes H ∈ Pn with x ⊆ H, then

Card

hyperplanes incidence with a fixed pointx not tangent to O

=qn − 1

q − 1− 1,

which is independent of the choice of x ∈ O. Thus,

Card(A) = Card(O)

(qn − 1

q − 1− 1

),

= (qn−1 + 1)

(qn − 1

q − 1− 1

)(by Proposition 1.4.1)

= (qn−1 + 1)(qn−1 + qn−2 + · · ·+ q2 + q)

= q2(n−1) + q2(n−1)−1 + · · ·+ qn + qn−1 + · · ·+ q.

15

Let H ∈ Pn with Card(p1p−1n (H) ∩ O) ≥ 2. By Proposition 1.4.2, p1p

−1n (H) ∩ O is an ovoid in

G(H). By Proposition 1.4.1,

Card(p1p−1n (H) ∩ O) = qn−2 + 1,

which is independent of the choice of H. So

Card(A) = (qn−2 + 1)CardH ∈ Pn | Card(p1p−1n (H) ∩ O) ≥ 2.

So

CardH ∈ Pn | Card(p1p−1n (H) ∩ O) ≥ 2

=q2(n−1) + q2(n−1)−1 + · · ·+ qn + qn−1 + · · ·+ q

qn−2 + 1

=(qn + qn−1 + · · ·+ q3)(qn−2 + 1) + q2 + q

qn−2 + 1

= qn + qn−1 + · · ·+ q3 +q2 + q

qn−2 + 1,

but, since q ∈ Z≥1, the right hand side is not an integer unless n ∈ 2, 3.

Proposition 1.4.4. Suppose q 6= 2. Then

(1) A set of points O in PG(2,Fq) is an ovoid if and only if O is a cap and |O| = q + 1.(2) A set of points O in PG(3,Fq) is an ovoid if and only if O is a cap and |O| = q2 + 1.

References for proof. One implication is given in Proposition 1.4.1. For the converse,Tits [Tit62, pg. 37 and pg. 38] cites this as following immediately from the work of Segre[Seg59b] and Barlotti [Bar55].

Example 1.4.5. Using the Hasse diagram of the lattice PG(2, 2) in Example B.1.2, it canbe checked that O = [1 : 0 : 0] , [0 : 0 : 1] , [1 : 1 : 1] is an ovoid in PG(2, 2).

Example 1.4.6. Using the Hasse diagram of the lattice PG(2, 3) in Example B.1.3, it canbe checked that O = [1 : 0 : 0] , [0 : 0 : 1] , [1 : 1 : 1] , [1 : 2 : 1] is an ovoid in PG(2, 3).

The rational normal curve of PG(F2+1q ) is

N = [t2 : t : 1] | t ∈ Fq ∪ [1 : 0 : 0]. (1.1)

A conic in PG(F2+1q ) is

C = [x : y : z] | Q(x, y, z) = 0

where Q : F3q → Fq is a quadratic form (see Appendix, Section A.2). Equivalently, a conic is the

variety in P(F3q) whose homogeneous ideal is generated by exactly one homogeneous polynomial

(see [Har92, Example 1.20]). A conic is nondegenerate if its corresponding quadratic form Q isnondegenerate.

Proposition 1.4.7. [Bal15, Theorem 4.35]

(1) The rational normal curve N is a nondegenerate conic.(2) If C is a nondegenerate conic then there exists g ∈ PGL3(Fq) such that g ·N = C.(3) If C is a nondegenerate conic then C is an ovoid in G(PG(F2+1

q )).

Proof.

(1) Define Q : F3q → Fq by

Q(x, y, z) = xz − y2.

Then N = [x : y : z] | Q(x, y, z) = 0. So N is a conic.

16

(2) Let C be a conic with corresponding non-degenerate quadratic form Q and correspond-ing bilinear form 〈, 〉 (see Appendix B. A.3). By [Bal15, Theorem 3.26], there existsa nonzero isotropic vector e ∈ F3

q. Since 〈, 〉 is non-degenerate, there exists f ′ ∈ F3q

such that 〈e, f ′〉 6= 0. Also, since e is totally isotropic, if f ′ = λe for some nonzero λthen 〈e, f ′〉 = 〈e, λe〉 = 0, a contradiction. Hence e, f ′ is a linearly independent set.Define

f = − Q(f ′)

〈e, f ′〉2e+

1

〈e, f ′〉f ′.

Then spane, f = spane, f ′ and (e, f) is a hyperbolic pair (meaning that 〈e, e〉 = 0,〈e, f〉 = 1, 〈f, f〉 = 0). Choose a nonzero vector a ∈ spane, f⊥. Then

Q(λe+ µa+ νf) = Q((λe+ νf) + (µa))

= 〈λe+ νf, µa〉+Q(λe+ νf) +Q(µa)

= Q(λe+ νf) +Q(µa) since a ∈ spane, f⊥

= 〈λe, νf〉+Q(λe) +Q(νf) +Q(µa)

= λν +Q(λe) +Q(νf) +Q(µa)

= λν + λ2Q(e) + ν2Q(f) + µ2Q(a)

= λν + µ2Q(a) since e and f are totally isotropic

= λν + kµ2,

for some k ∈ F×q (k is nonzero since Q(a) 6= 0). Applying the basis change f 7→ kf , we

can assume k = −1. Then, in the basis e, a, f, we have Q(x, y, z) = xz − y2.(3) Since PGL3(Fq) preserves ovoids, by part (2) of this Proposition, it suffices to show

that N is an ovoid. If q = 2, N is the ovoid in Example 1.4.5.Now assume q 6= 2. Since |Fq| = q we have |N | = q + 1. By Corollary 1.4.4, it

remains to show that N is a cap. Suppose, for sake of contradiction, that p1, p2, p3

are 3 distinct points in N that are collinear. We assume none of the pi are equal to[1 : 0 : 0], if one of the pi are equal to [1 : 0 : 0] a similar proof follows. So there existsdistinct t1, t2, t3 ∈ Fq such that

p1 = [t21 : t1 : 1], p2 = [t22 : t2 : 1], p3 = [t23 : t3 : 1],

and there exists a subspace W ⊆ F3q such that

dimW = 2 and p1, p2, p3 ⊆ W.

But

det

t21 t22 t23t1 t2 t31 1 1

= (t1 − t2)(t1 − t2)(t2 − t3) 6= 0.

(This determinant is an example of a Vandermonde determinant). So dimW ≥ 3. SodimW 6= 2, a contradiction.

Theorem 1.4.8 (Segre’s theorem). If q is odd then every ovoid in PG(2, q) is a conic.

Proof. See [Bal15, Theorem 4.38], [BW11, Theorem 2.2.1], [Bro00a, §2.2], or [Seg55, The-orem 1].

Remark 1.4.9. Segre’s theorem gives a partial converse to Example 1.4.7. When q is even,some ovoids which are not normal rational curves were described in the final page of [Seg55]. Alist of known ovoids in PG(2, q) can be found in [Che04]. The ovoids in PG(2, q) are classifiedfor q = 2k where k ∈ 1, 2, 3, 4, 5, 6. The classification for k ≥ 7 is, to our knowledge, stillopen.

17

An elliptic quadric in PG(F3+1q ) is

E = [x1 : x2 : x3 : x4] | Q(x1, x2, x3, x4) = 0 (1.2)

where Q : F4q → Fq is the quadratic form (of Witt index 1) defined by

• (when q is odd) Q(x1, x2, x3, x4) = x1x2 + x23 + sx2

4 where s ∈ F×q is chosen such that

there does not exist r ∈ F×q such that r2 = −s,• (when q is even, q = 2e) Q(x1, x2, x3, x4) = x1x2 +x2

3 +ax3x4 +x24 where a ∈ Fq satisfies

a−1 + a−2 + a−22 + a−23 + · · ·+ a−2e−1

= 1.

(see Appendix A.3.5 and [Bal15, Theorem 3.28]).

Proposition 1.4.10. [Bro00a, §3], [Bal15, Theorem 4.36]. An elliptic quadric E is anovoid in PG(F3+1

q ).

Proposition 1.4.11. [Bro00a, §3.1], [Bar55], [Pan55], [Hir85, Theorem 16.1.7]. Let E bethe elliptic quadric in PG(3, q). If q is odd and O′ is an ovoid in PG(F3+1

q ) then there existsg ∈ PGL4(Fq) such that g · E = O′.

Define the field automorphism θ : F22e+1 → F22e+1 by θ(s) = s2e for s ∈ F22e+1 . The Suzuki-Tits ovoid in PG(F3+1

22e+1) is

S = [1 : 0 : 0 : 0] ∪

[t+ sθ(s2) + θ(st) : θ(t) : θ(s) : 1]∣∣s, t ∈ F22e+1

(1.3)

Note that, in the notation of [Bro00a, Theorem 3.10], [Bal15, Theorem 4.43], θ(s)2 = σ(s).

Proposition 1.4.12. [Tit61, §4], [Bro00a, Theorem 3.10], [Bal15, Theorem 4.43]. TheSuzuki-Tits ovoid S is an ovoid in PG(F3+1

22e+1).

Remark 1.4.13. The Suzuki-Tits ovoid in PG(3, 23) appears in [Seg59a, Theorem VI] asan example of an ovoid in PG(3, q) which is not an elliptic quadric.

The classification of ovoids in PG(3, q) when q is even is a longstanding open problem,leading to the following conjecture:

Conjecture 1.4.14. If O is an ovoid in PG(3, 2h) then O is an elliptic quadric or Suzuki-Tits ovoid.

See [OKe96] or [HT15, Section 5] for a survey of the work on this conjecture. MatthewBrown has made significant progress:

Theorem 1.4.15. [Bro00b] Let O be an ovoid of PG(3, 2h) and let H be a plane of PG(3, 2h).If the set of points of O incident with H form a conic in H, then O is an elliptic quadric.

1.5. Ovoids of polar incidence structures

A good general reference for this section is [Tha81]. Let G = (P ,B, I) be a polar incidencestructure as in Section 1.2.2. An ovoid [Tha81], [Bue95, pg. 330], [HT15, Definition 11.5] ina polar incidence structure G is a set of points O ⊆ P such that the following condition issatisfied:

if b ∈ B then there exists a unique point p ∈ O such that b is incident with p.

Lemma 1.5.1. [BLP09, Corollary 4.4], [PTS09, p. 1.8.4]. Let O be an ovoid in G(W (F4q)),

and let x, y ∈ P such that l(x, y) /∈ L, that is, l(x, y) is not totally isotropic. Then the linel(x, y) meets O in 0 or 2 points.

Theorem 1.5.2. [BLP09, Corollary 4.5] [Bal15, Theorem 4.40], [Tha72]. If O is an ovoidin W(F4

q) then O is an ovoid in PG(F4q).

18

Proof. Let O be an ovoid of W (F4q). There are only two types of line of PG(F4

q): totallyisotropic or non-degenerate. If l is a totally isotropic line, then it meets O in precisely onepoint. Otherwise, if l is non-degenerate, then Lemma 1.5.1 implies that l meets O in 0 or 2points. In both cases, we see that every line meets O in 0,1, or 2 points. Since |O| = q2 + 1,by Proposition 1.4.4, O is an ovoid.

Example 1.5.3. [Bal15, Theorem 4.43], [Dem68, §1.4.56(a)]. The Suzuki-Tits ovoid (seeProposition 1.4.12) in PG(3, 22e+1) is also an ovoid in W (3, 22h+1) = P (F4

22h+1 , 〈·, ·〉), where〈·, ·〉 : F4

22h+1 × F422h+1 → F4

22h+1 is the symplectic form defined by

⟨u−2

u−1

u1

u2

,v−2

v−1

v1

v2

⟩

= u−2v−1 − v−2u−1 + u1v2 − v1u2.

1.5.1. Ovoids of H(n, q2). Let Fq2 be the finite field with q2 elements, where q = pk is aprime power. The Frobenius automorphism Fr : Fq2 → Fq2 is the field automorphism definedby

Fr(a) = ap.

Let σ : Fq2 → Fq2 be the involutive field automorphism defined by

σ(a) = Frk(a).

Note that σ2(a) = a. Let V = Fn+1q2 considered as an Fq2-vector space. Let e1, e2, . . . en+1 be

a basis for V . The sesquilinear form 〈·, ·〉 : V × V → Fq2 defined by⟨ v1...

vn+1

, w1

...wn+1

⟩ = v1σ(w1) + v2σ(w2) + · · ·+ vn+1σ(wn+1)

is a nondegenerate Hermitian form (see Appendix A, Section A.2). Then 〈·, ·〉 is a Hermitianform and P (V, 〈·, ·〉) is a Hermitian polar semilattice. The Hermitian curve is H(2, q2) and theHermitian surface is H(3, q2).

There exists a nondegenerate hyperplanes in V. For example, the hyperplaneW = Fq2 e2, e3, . . . en+1is nondegenerate in V . Also, set of points in H(n, q2) is nonempty. To see this, let α ∈ Fq2 bea solution to the equation

ασα = −1.

(such a solution exists by Proposition A.1.6). Let

v = (1, α, 0, 0, . . . , 0) ∈ V.

Then

〈v, v〉 = 1 · 1 + αα + 0 · 0 + 0 · 0 + · · ·+ 0 · 0 = 1 + (−1) = 0,

so that spanv ∈ P .

Proposition 1.5.4. [Bal15, Theorem 3.11]. Let V = Fk and let β : V × V → F be anondegenerate Hermitian form. Then every maximal totally isotropic subspace has dimensionbk

2c.

Proof. Let U be a totally isotropic subspace. By Proposition A.2.10, dimU + dimU⊥ = k.Since U ⊆ U⊥, we have 2dimU ≤ k, so that dimU ≤ bk

2c.

We show that if dimU < bk2c then U can be extended to a totally isotropic subspace

U ′ = U ⊕ Fu′′. Suppose dimU < bk2c. So k − dimU⊥ < bk

2c. So k − bk

2c < dimU⊥. But

19

k − bk2c ≥ bk

2c. Hence

dimU < bk2c < dimU⊥.

Hence dimU + 2 ≤ dimU⊥.Let v ∈ U⊥\U . We know, by Proposition A.2.10, dim(F-spanv) + dimv⊥ = k so that

dimv⊥ = k − 1. Also,

dim(U⊥ + v⊥) = dimU⊥ + dimv⊥ − dim(U⊥ ∩ v⊥).

So

k − 1 + dimU⊥ = dim(U⊥ + v⊥) + dim(U⊥ ∩ v⊥).

Now dimv⊥ = k − 1 so that

dimU⊥ = dim(U⊥ ∩ v⊥) or dim(U⊥ ∩ v⊥) + 1,

so that

dim(v⊥ ∩ U⊥) = dimU⊥ or dimU⊥ − 1.

Hence there exists a nonzero w ∈ (U⊥ ∩ v⊥)\U .If β(w,w) = 0, then set u′′ = w and U ′ = U ⊕ Fu′′. If λ, λ′ ∈ F and u, u′ ∈ U then

β(u+ λu′′, u′ + λ′u′′)

= β(u, u′) + λβ(u′′, u′) + σ(λ′)β(u, u′′) + λσ(λ′)β(u′′, u′′)

= 0,

so that U ′ is a totally isotropic subspace.If β(w,w) 6= 0, then since the map F → Fσ, µ 7→ µσ(µ) is surjective (see Appendix A,

Theorem A.1.6), there exists µ ∈ F such that

µσ(µ) = − β(v, v)

β(w,w).

Then the vector u′′ = v + µw is totally isotropic, since

β(u′′, u′′) = β(v + µw, v + µw)

= β(v, v) + µβ(w, v) + σ(µ)β(v, w) + µσ(µ)β(w,w)

= β(v, v) + µσ(µ)β(w,w)

= 0.

Let U ′ = U ⊕ Fu′′. If λ, λ′ ∈ F and u, u′ ∈ U then

β(u+ λu′′, u′ + λ′u′′)

= β(u, u′′) + λβ(u′′, u′) + σ(λ′)β(u, u′′) + λσ(λ′)β(u′′, u′′)

= λβ(u′′, u′) + σ(λ′)β(u, u′′)

= 0,

since u′′ = v + λw ∈ U⊥. So U ′ is a totally isotropic subspace.

Proposition 1.5.5 (The classical ovoid in H(3, q2)). Let V = F4q2, considered as a Fq2-

vector space and let 〈·, ·〉 : V × V → Fq2 be a nondegenerate Hermitian form. Let G = (P ,B, I)be the polar incidence structure associated with the Hermitian surface H(3, q2), so that

P = 1-dimensional totally isotropic subspaces of V ,B = 2-dimensional totally isotropic subspaces of V ,I = (p, b) ∈ P × B | p ⊆ b.

20

Let W ⊆ V be a nondegenerate hyperplane and let O = Fq2 − spanv ∈ P | v ∈ W. ThenO is an ovoid in H(3, q2).

W

O

Proof. Let b ∈ B. We will show that there exists p ∈ O such that p ⊆ b. We will alsoneed to show that if p, p′ ∈ O such that p ⊆ b and p′ ⊆ b then p = p′.

Let u, v be a basis of b. By Proposition 1.5.4, dimb = 2. Also dimW = 3, and dimV = 4, sowe have dim(b ∩W ) ≥ 1. Let w ∈ b ∩W and p = Fq2 − spanw. Then p is a totally isotropic1-dimensional subspace of V with p ⊆ W and p ⊆ b and p ∈ O. But also p ⊆ b. Hence p ∈ Oand p ⊆ b as required.

Suppose p, p′ ∈ O with p ⊆ b and p′ ⊆ b. Suppose, for sake of contradiction, that p 6= p′.Then

p⊕ p′ = b.

Since p ⊆ W and p′ ⊆ W we have

b ⊆ W.

So W⊥ ⊆ b⊥. But b is totally isotropic, so b ⊆ b⊥. By Lemma A.2.8, dimV = 2 + dimb⊥, sodimb⊥ = 2. So b = b⊥. So W⊥ ⊆ b⊥ = b ⊆ W . So W ∩W⊥ 6= 0. Hence W is degenerate, acontradiction.

Remark 1.5.6. It is known that:

• There are no ovoids in H(5, 4) [DM06].• If H(n, q2) has no ovoids then H(n+ 2, q2) has no ovoids. [DKM11, Lemma 3.2].• There are no ovoids in H(2n, q2) for n ∈ Z≥0 [DKM11, Corollary 3.4].

The existence of ovoids of H(5, q2) for q > 2 is, to our knowledge, still open.

21

CHAPTER 2

Chevalley Groups

This chapter outlines the main definitions and theorems on Chevalley groups that are usedin this thesis. In particular, Theorem 2.3.7 (the Bruhat decomposition) and Theorem 2.4.6(the favourite choice of B coset representatives of a Schubert cell) are vital in setting up thetheory of flag varieties and thickness in Chapter 4. The main references are [Ste67], [PRS09,§3], [CMS95], [Ser87], [Hum72] and [Bou89].

2.1. Lie algebras and root systems

Let g be a complex Lie algebra. The Lie algebra g is simple if g has no non-trivial ideals andis not the 1-dimensional abelian Lie algebra. The Lie algebra g is semisimple if it is a directsum of simple Lie subalgebras. A representation M of a complex Lie algebra g is semisimple ifM is a direct sum of irreducible representations [Bou89, p. I.3.1]. The Lie algebra g is reductiveif its adjoint representation is semisimple [Bou89, p. I.6.4].

Suppose g is semisimple. A subalgebra a of g is nilpotent if the lower central series

a ⊇ [a, a] ⊇ [a, [a, a]] ⊇ [a, [a, [a, a]]] ⊇ . . .

is zero after finitely many terms. The normaliser of a subalgebra a in g is

Ng(a) = X ∈ g | if Y ∈ a then [X, Y ] ∈ a.A Cartan subalgebra of g is a subalgebra h such that h is nilpotent and equal to its own

normalizer [Bou04, p. VII.2.1]. Let h be a Cartan subalgebra in g. Let

h∗ = α : h→ C | α is C-linear be the dual vector space.

The root system of g is the set R ⊆ h∗−0 uniquely determined by the h-module decomposition

g = h⊕(⊕

α∈R

gα

)where gα = X ∈ g | if H ∈ h then [H,X] = α(H)X

is the α-root space of g, and the elements of R are the roots of g. The adjoint representationof g is the representation ad: g→ End(g) defined by

(ad(X)) (Y ) = [X, Y ]

for all Y ∈ g. The Killing form on g is the bilinear form

κ : g× g −→ C(X, Y ) 7−→ tr(adXadY ).

Since g is semisimple, the Killing form is non-degenerate [Hum72, Theorem 5.1], and the map

γ : g −→ g∗ defined by (γ(X))(Y ) = κ(X, Y ) for all Y ∈ g, (2.1)

is a bijection. Let

h∗R = R-span(R).

For each α ∈ R, the reflection corresponding to α is the linear transformation sα : h∗R → h∗Rgiven by

sα(v) = v − 2(v, α)

(α, α)α,

23

for v ∈ h∗R, where

(, ) : h∗R × h∗R −→ R(γH , γH′) 7−→ κ(H,H ′).

(2.2)

(see [CMS95, Ch. 1, §2.3],[Ste67, §1]). The Weyl group

W is the subgroup of GL(h∗R) generated by sα for α ∈ R.

A linearly independent subset ∆ = α1, α2, . . . , αn ⊆ R is a set of simple roots if the followingcondition is satisfied:

if β ∈ R then β ∈ Z≥0-span(∆) or β ∈ Z≤0-span(∆).

Let ∆ be a set of simple roots. The set of positive roots in R is

R+ = R ∩ Z≥0-span(∆).

The set of simple reflections or Coxeter generators is

S = sα ∈ W | α ∈ ∆ = s1, s2, . . . , snwhere si := sαi .

Theorem 2.1.1 (Coxeter). The group W has a presentation by generators s1, s2, . . . sn andrelations

s2i = 1, (sjsk)

mjk = 1,

for i, j, k ∈ 1, 2, . . . , n with j 6= k, where mjk ∈ Z≥2∪∞ is determined by W and mjk =∞means sjsk has infinite order.

A triangular decomposition of g is

g = n+ ⊕ h⊕ n−,

where

n+ =⊕α∈R+

gα and n− =⊕α∈R+

g−α.

The Borel subalgebra is

b = n+ ⊕ h.

For completeness, we state the following well known theorem which gives a classification of thecomplex simple Lie algebras up to isomorphism. For the definitions and details, see [CMS95,p. I.2.5], [Ser87, p. VI.5], [Hum72, Theorem 8.5 and its following comment], [Car89, Theorem3.5.1 and Theorem 3.5.2].

Theorem 2.1.2 (Cartan-Killing). The mapfinite dimensional complex

simple Lie algebras

−→

root systems of type

An (n≥1), Bn (n≥2), Cn (n≥3), Dn (n≥4),G2,F4,E6,E7,E8.

g 7−→ root system of g

is a bijection.

2.2. Chevalley bases and Chevalley groups

In Chevalley’s paper [Che55], a procedure is given for constructing analogues of the Liegroups over an arbitrary field F from a complex simple Lie algebra g using a special choiceof basis of g for which the structure constants are integers. The special choice of basis ofg is the Chevalley basis and the algebraic groups are now known as Chevalley groups. Thefollowing section summarises the construction of Chevalley groups following [Ste67, §1 to §3].See also [PRS09, §3], [Lus09], [Gec16], [Hum72, Chapter VII], and [Tit87]. The generalisationto the case where g is a Kac-Moody algebra can be found in [PRS09, Remark 3.2], [Tit87], and[Kum12, §VI].

24

Let g be a complex semisimple Lie algebra with a choice of Cartan subalgebra h.The rank of g (and its root system R) is dimCh.Let ∆ = α1, α2, . . . , αn be a choice of simple roots. For α ∈ R define

Hα =2

(α, α)γ−1(α)

where γ is the map in Equation (2.1). There exists a Chevalley basis

H1, H2, . . . , Hn t Xα | α ∈ R

for g, given in [Ste67, Theorem 1, §1] (see also [Car89, Theorem 4.2.1], [Che55, Theorem 1],[Bou04, Ch. VIII, §2, No. 4, Definition 3 and Ch. VIII, §4, No. 4 Corollary to Proposition 5]).A Chevalley basis is uniquely determined, up to sign changes and automorphisms of g, by theequations

[Hi, Hj] = 0,

[Hi, Xα] = 2(α, αi)

(αi, αi)Xα,

[Xα, X−α] = Hα ∈ Z-spanH1, H2, . . . , Hn[Xα, Xβ] = ±(r + 1)Xα+β if α + β ∈ R,[Xα, Xβ] = 0 if α + β 6= 0 and α + β /∈ R,

where, for α, β ∈ R, r is the greatest integer such that β − rα ∈ R. Note that the scalarsappearing in the above equations are integers, and this will help make sense of ‘exponentiation’over an arbitrary field.

Let Ug be the universal enveloping algebra of g. The Kostant Z-form for Ug is the

Z-algebra (Ug)Z generated by

Xmα

m!

∣∣∣∣ m ∈ Z≥1, α ∈ R.

Let V be a finite dimensional faithful g-module. Let M be a (Ug)Z-submodule of V such thatthere exists a Z-basis for M which is a C-basis for V . Let F be a field. Then F has the naturalZ-module structure, and we define

VF = F⊗Z M, considered as an F-vector space. (2.3)

Let α ∈ R and define xα : F −→ GL(VF) by

xα(t) = 1 + tXα +1

2!t2X2

α +1

3!t3X3

α + · · ·

for t ∈ F. This sum has finitely many terms since Xkα = 0 for k sufficiently large. The Chevalley

group [Ste67, §3] is the subgroup

G(F) of GL(VF) generated by xα(t) for t ∈ F and α ∈ R.

The adjoint group is the Chevalley group G(F) when V is the adjoint representation. Theuniversal group is the Chevalley group G(F) when V is the sum of the irreducible representa-tions having the fundamental weights as their highest weights (see [CMS95, Ch. 1, §3]). Fordiscussion on how the choice of V and M determine the Chevalley group G(F), see [Ste67,Corollary 5, pg. 44] and [Ste67, Corollary 1, pg. 64]. Following [Ste67, Lemma 19, §3], definethe Chevalley generators :

nα(t) = xα(t)x−α(−t−1)xα(t),

hα(t) = nα(t)nα(1)−1,

nα = nα(1),

25

for α ∈ R and t ∈ F×. Define

Xα = xα(t) | t ∈ F for α ∈ R (the α-root subgroups),N = 〈nα(t) ∈ G(F) | α ∈ R, t ∈ F×〉 (the monomial subgroup),T = 〈hα(t) ∈ G(F) | α ∈ R, t ∈ F×〉 (the standard maximal torus),U = 〈xα(t) ∈ G(F) | α ∈ R+, t ∈ F〉,U− = 〈xα(t) ∈ G(F) | α ∈ −R+, t ∈ F〉,B = 〈T, U〉 (the standard Borel subgroup).

Let W be the Weyl group and S the set of Coxeter generators of W , as defined in Section 2.1.Given w ∈ W , a reduced expression for w is

w = si1si2 . . . sik

where si1 , si2 , . . . , sik ∈ S and k is minimal. Define the length function ` : W → Z≥0 by

`(w) = k if there exists a reduced expression w = si1si2 . . . sik .

The group N/T is isomorphic to W via the map

N/T −→ Wnα(t)T 7−→ sα, for α ∈ R,

(see [Ste67, Lemma 22, §3]). We use the term Weyl group to refer to both N/T and W andwrite sα = nα(t)T when appropriate (in particular, when the choice of t is irrelevant). Define

nw = n−1i1n−1i2. . . n−1

i`

for each w ∈ W and choice of reduced expression w = si1si2 . . . si` .

Theorem 2.2.1. [Ste67, §3] Let G(F) be a Chevalley group. The following relations holdin G(F):

(R1) If α ∈ R and t, u ∈ F then

xα(t)xα(u) = xα(t+ u),

(R2) If rank(R) ≥ 2, α, β ∈ R with α+β 6= 0 and t, u ∈ F, then there exists unique Ci,jα,β ∈ Z

such that

xα(t)xβ(u) =xβ(u)xα(t)xα+β(C1,1α,βtu)xα+2β(C1,2

α,βtu2)x2α+β(C2,1

α,βt2u)

xα+3β(C1,3α,βtu

3)x2α+2β(C2,2α,βt

2u2)x3α+β(C3,1α,βt

3u) . . .

(see [Ste67, Lemma 15], [Car72, Theorem 5.2.2]),(R6) If α, β ∈ R and t ∈ F× then

nα(1)hβ(t)nα(1)−1 = hsα(β)(t)

(R7) If α, β ∈ R and t ∈ F then

nα(1)xβ(t)nα(1)−1 = xsαβ(ct),

where c ∈ 1,−1 is as in [Ste67, Lemma 19(a)],(R8) If α, β ∈ R and t ∈ F× and u ∈ F then

hα(t)xβ(u)hα(t)−1 = xβ(t2(β,α)(α,α) u).

26

2.3. The Bruhat Decomposition

For Q ⊆ R, let

XQ be the group generated by Xα for α ∈ Q.

Lemma 2.3.1. Let Q = α1, α2, . . . , αk ⊆ R be a subset of the roots such that the followingconditions hold:

• if α, β ∈ Q and α + β ∈ R then α + β ∈ Q, and• if α ∈ Q then −α /∈ Q.

Then the map

Fk −→ XQ

(c1, c2, . . . , ck) 7−→ xα1(c1)xα2(c2) · · ·xαk(ck),is a bijection.

Reference for proof. See [Ste67, Lemma 17].

Lemma 2.3.2. Let α ∈ ∆ be a simple root. Then sα(α) = −α and sα(R+\α) = R+\α.

Lemma 2.3.3. Let si ∈ W be a simple reflection. Then

BsiB ·BsiB = B tBsiBand B tBsiB is a subgroup of G.

Proof. [Ste67, Lemma 24] To show:

(1) If x ∈ BsiB ·BsiB then x ∈ B ∪BsiB.(2) If x ∈ B ∪BsiB then x ∈ BsiB ·BsiB.(3) B tBsiB is a subgroup of G.

(1) Let x ∈ BsiB · BsiB. Then there exists b, b′, b′′ ∈ B such that x = bnib′n−1i b′′. Now

b′ = hu for some h ∈ T and u ∈ U , so that

bnib′n−1i b′′ = bnihun

−1i b′′

= bnihn−1i niun

−1i b′′

= bh′niun−1i b′′ (where h′ ∈ T , by relation (R6)).

So by replacing b with bh′ we may assume b′ ∈ U . By Lemma 2.3.1, there exists c ∈ Fand y ∈ XR+\αi such that b′ = xαi(c)y. By Lemma 2.3.1, there exists β1, β2, . . . , βk ∈R+\αi and c1, c2, . . . , ck ∈ F such that

y = xβ1(c1)xβ2(c2) . . . xβk(ck).

Then

niyn−1i = nixβ1(c1)xβ2(c2) . . . xβk(ck)n

−1i

= nixβ1(c1)n−1i nixβ2(c2)n−1

i · · ·nixβk(ck)n−1i

= xsi(β1)(c′1)xsi(β2)(c

′2) · · ·xsi(βk)(c

′k) by relation (R7).

By Lemma 2.3.2, si(βj) ∈ R+\αi for all j. Hence niyn−1i ∈ XR+\αi. There are 2

cases:Case 1: Suppose c = 0. So nib

′n−1i = niyn

−1i ∈ B. So x = bnib

′n−1i b′′ ∈ B.

Case 2: Suppose c 6= 0. Then

x = bnixαi(c)yn−1i b′′

= bnixαi(c)n−1i niyn

−1i b′′

= bx−αi(c′)niyn

−1i b′′ (by relation (R7))

= bxαi(c−1)nαi(−c−1)xαi(c

−1)niyn−1i b′′

27

= bxαi(c−1)hαi(−c−1)nαixαi(c

−1)niyn−1i b′′

= bxαi(c−1)hαi(−c−1)nαixαi(c

−1)nαiyn−1αib′′

∈ BsiB,

since bxαi(c−1)hαi(−c−1) ∈ B and xαi(c

−1)nαiyn−1αib′′ ∈ BXR+\αiB = B.

Hence x ∈ B ∪BsiB.(2) Suppose x ∈ B ∪BsiB. If x ∈ B then

x = xni1n−1i 1,

so x ∈ BsiB ·BsiB.Suppose instead x ∈ BsiB. Then there exists b ∈ B and b′ ∈ B such that x = bnib

′.By similiar reasoning to (1), we may assume b ∈ U . So there exists c ∈ F andy ∈ XR+\αi such that b = yxi(c). There are 2 cases:Case 1: Suppose c = 0. Then

bnib′ = ynib

′

= nin−1i ynib

′

= nib′′

= xαi(1)x−αi(−1)xαi(1)b′′

= xαi(1)nixαi(±1)n−1i xαi(1)b′′

∈ BsiB ·BsiB.Case 2: Suppose c 6= 0. Then

bnib′ = yxi(c)nib

′

= ynin−1i xi(c)nib

′

= ynix−αi(c′)b′ (by relation (R7))

= ynixαi(−c′−1)nαi(c′)xαi(−c′−1)b′

∈ BsiB ·BsiB.

(3) Let S = B tBsαB. To show:(a) If x ∈ S then x−1 ∈ S.(b) If x, y ∈ S then xy ∈ S.(a) Suppose x ∈ S. If x ∈ B then x−1 ∈ B so that x ∈ S. Suppose that x ∈ BsαB.

Then there exists b, b′ ∈ B such that x = bnαb′. Then x = b′−1n−1

α b−1 ∈ BsαB.Hence x ∈ S.

(b) Suppose x, y ∈ S. If x ∈ B or y ∈ B then xy ∈ S. Suppose that x ∈ BsαBand y ∈ BsαB. Then there exists b1, b2, b3, b4 ∈ B such that x = b1nαb2 andy = b3nαb4. Then

xy = b1nαb2b3nαb4

= BnαBnαB

= BnαBn−1α B

= BnαXαXR+−αTn−1α B

= BnαXαn−1α nαXR+−αn

−1α nαTn

−1α B

= BX−αXR+−αB

= BX−αB.

But if x−α(t) ∈ X−α with t ∈ F× then

x−α(t) = xα(t−1)nα(−t−1)xα(t−1) ∈ BnαB.

28

So X−α ⊆ S and xy ∈ S.

Proposition 2.3.4. Let w ∈ W and let si ∈ W be a simple reflection. Then

(1) If `(wsi) = `(w) + 1 then

BwB ·BsiB = BwsiB.

(2) If `(wsi) = `(w)− 1 then

BwB ·BsiB = BwB ∪BwsiB.

Proof. This proof follows [Ste67, Lemma 25]. Let w ∈ W and let si ∈ W be a simplereflection.

(1) Suppose `(wsi) = `(w) + 1.To show: BwB ·BsiB = BwsiB.To show: BwB ·BsiB ⊆ BwsiB (since BwB ·BsiB is a union of double B-cosets).To show: If x ∈ BwB ·BsiB then x ∈ BwsiB.Suppose x ∈ BwB ·BsiB.To show: x ∈ BwsiB.We know there exists b, b′, b′′ ∈ B such that

x = bnwb′nib

′′.

We know that there exists t ∈ F, x′ ∈ XR+\αi, and h ∈ T such that

b′ = xi(t)x′h

So

x = bnwxi(t)x′hnib

′′.

So

x = bnwxi(t)n−1w nwnin

−1i x′nin

−1i hnib

′′ ∈ BwsiB,since

nwxi(t)n−1w ∈ Xwαi ⊆ B, n−1

i x′ni ∈ Xsi(R+\αi) ⊆ B, n−1i hni ∈ B.

Hence BwB ·BsiB = BwsiB.(2) Suppose `(wsi) = `(w)− 1.

To show: BwB ·BsiB = BwB ∪BwsiB.

LHS = BwB ·BsiB= BwsisiB ·BsiB= (BwsiB ·BsiB) ·BsiB (by Part 1)

= BwsiB · (BsiB ·BsiB)

= BwsiB · (B ∪BsiB) (by Lemma 2.3.3)

= BwsiB ∪ (BwsiB ·BsiB)

= BwsiB ∪BwB (by Part 1)

= RHS

Corollary 2.3.5. If w = si1si2 . . . si` is a reduced expression then

BwB = Bsi1B ·Bsi2B · · ·Bsi`B.

Corollary 2.3.6. If u, x ∈ W with `(ux) = `(u) + `(x) then

BuxB = BuB ·BxB.29

Theorem 2.3.7. (The Bruhat Decomposition) Let

B\G/B = BxB | x ∈ G.The map φ : W −→ B\G/B defined by

φ(w) = BwB

is a bijection.

Proof. This proof can be found in [Ste67, Theorem 4] and [Bou08, IV.§2.2.4].To show:

(1) φ is surjective.(2) φ is injective.

(1) Suppose G is a Chevalley group. By [Ste67, Lemma 26], the set ∪w∈WBwB containsa set of a generators of G. By Proposition 2.3.4, the set ∪w∈WBwB is closed undermultiplication by these generators. Since ∪w∈WBwB is also closed under inverses, itfollows that

∪w∈WBwB = G.

Hence φ is surjective.(2) (Adapted from the proof in [Ste67, Theorem 4(b)])

Suppose w,w′ ∈ W such that φ(w) = φ(w′).We show that w = w′ by induction on `(w).For the base case, we need to show that if `(w) = 0 then w = w′. Suppose `(w) = 0.

Then w = 1. Since BwB = Bw′B, we have B = Bw′B. So w′ ∈ B. So w′ = 1. Hencew = w′.

We now do the induction step: Suppose(a) `(w) > 0, and(b) If v, v′ ∈ W with `(v) < `(w) and φ(v) = φ(v′) then v = v′.

We need to show that w = w′. Let si ∈ W be a simple reflection such that`(wsi) < `(w).

We know that wsi ∈ BwB ·BsiB.So wsi ∈ Bw′B ·BsiB.By Proposition 2.3.4,

wsi ∈ Bw′B ∪Bw′siB.So wsi ∈ BwB ∪ Bw′siB. So BwsiB ⊆ BwB ∪ Bw′siB. So BwsiB = BwB orBwsiB = Bw′siB. So φ(wsi) = φ(w) or φ(wsi) = φ(w′si). So by induction,

wsi = w or wsi = w′si.

If wsi = w then si = 1, a contradiction. So wsi = w′si. Hence w = w′.

Corollary 2.3.8. [The Bruhat Decomposition]

G =⊔w∈W

BwB

Example 2.3.9. We illustrate Theorem 2.3.7 with the example G(F) = GLn(F).Assume G = GLn(F), B is the subgroup of upper triangular matrices, and T is the subgroup

of diagonal matrices. Let BxB ∈ B\G/B.To show: There exists w ∈ W such that φ(w) = BxB.To show: There exists w ∈ W such that BwB = BxB.To show: There exists w ∈ W such that x ∈ BwB.To show: There exists w ∈ W and b, b′ ∈ B such that xbw−1b′ = I.The algorithm is as follows, using column operations:

30

(1) Scale the first column so that the lowest nonzero entry is 1.(2) Use column operations to erase entries to the right of this 1.(3) Repeat on the second column, third column, and so forth.(4) The rightmost 1’s are called pivots or leading 1’s.(5) Permute the columns to make the matrix upper triangular.(6) The (1, 1)-entry will be 1. Use column operations to erase entries to the right of the

(1, 1)-entry.(7) The (2, 2)-entry will be 1. Use column operations to erase entries to the right of the

(2, 2)-entry.(8) Repeat for the remaining diagonal entries.(9) The remaining matrix will be the identity. Hence there exists b, b′ ∈ B and w ∈ W

such that

xbw−1b′ = I.

Hence x ∈ BwB.

2.4. Decomposing double cosets into single cosets

Let w ∈ W . Define

R+(w) = β ∈ R+ | w−1(β) ∈ R+R(w) = R−(w) = β ∈ R+ | w−1(β) /∈ R+

Uw,+ =∏

β∈R+(w)

Xβ

Uw,− =∏

β∈R−(w)

Xβ

Theorem 2.4.1. [Ste67, Theorem 4’] [Car89, Theorem 8.4.3] Let w ∈ W and choose arepresentative nw ∈ N of w coming from the map ϕ : W → N/T . Then

BwB = Uw,−nwB

and the map

ψ : Uw,− ×B −→ BwB(x, b) 7−→ xnwb

is a bijection.

Proof. (1) (Surjectivity of ψ) We have

BwB = TUwB

= TUw,−Uw,+wB (By Lemma 2.3.1)

= TUw,−ww−1Uw,+wB

= TUw,−wB (Since w−1Uw,+w ⊆ B)

= Uw,−TwB (by the relation 2.2.1)

= Uw,−nwB (by the relation 2.2.1)

So

ψ(Uw,− ×B) = BwB.

So ψ is surjective.

31

(2) (Injectivity) Suppose ψ(x, b) = ψ(x′, b′) so that

xnwb = x′nwb′.

Then

b′b−1 = n−1w x′−1xnw ∈ n−1

w ∈ n−1w Uw,−nw ⊆ U−.

So n−1w x′−1xnw ∈ B ∩ U− = 1. So x = x′ and b = b′. Hence ψ is injective.

Proposition 2.4.2. [Ste67, Lemma 21] The map

θ : U × T −→ B(u, h) 7−→ uh

is a bijection.

Corollary 2.4.3. Let nw|w ∈ W be a choice of coset representatives of W . Then everyg ∈ G has a unique expression g = unwhu

′ where u ∈ Uw,−, h ∈ T , u′ ∈ U .

Proposition 2.4.4. Let w ∈ W and w = si1si2 . . . si` be a reduced expression. Then

R−(w) = β1, β2, . . . , β`where

β1 = αi1 , β2 = si1αi2 , . . . , β` = si1si2 . . . si`−1αi` .

Corollary 2.4.5. Let w ∈ W with reduced expression w = si1si2 . . . si` and R−(w) =β1, β2, . . . , β` as in Proposition 2.4.4. Then

ϕw : F`(w) −→ BwB/B(c1, c2, . . . , c`) 7−→ xβ1(c1)xβ2(c2) . . . xβ`(c`)nwB,

is a bijection, where nw = n−1i1n−1i2. . . n−1

i`.

Theorem 2.4.6. Let w = si1si2 . . . si` be a reduced expression for w. Then the map

ϕ : F` −→ BwB/B

defined by

ϕ ((c1, c2, . . . , c`)) = xi1(c1)n−1i1xi2(c2)n−1

i2. . . xi`(c`)n

−1i`B

is a bijection.

Proof. By Corollary 2.4.5, the map

ϕw : F`(w) −→ BwB/B(c1, c2, . . . , c`) 7−→ xβ1(c1)xβ2(c2) . . . xβ`(c`)nwB,

is a bijection, where nw = n−1i1n−1i2. . . n−1

i`. By repeatedly using Chevalley relation (R7),

xβ1(c1)xβ2(c2) . . . xβ`(c`)nwB

= xβ1(c1)xβ2(c2) . . . xβ`(c`)n−1i1n−1i2. . . n−1

i`B

= xαi1 (c1)xsi1αi2 (c2) . . . xsi1si2 ...si`−2αi`−1

(c`−1)xsi1si2 ...si`−1αi`

(c`)n−1i1n−1i2. . . n−1

i`B


(c`−1)

n−1i1n−1i2. . . n−1

i`−1ni`−1

. . . ni2ni1xsi1si2 ...si`−1αi`

(c`)n−1i1n−1i2. . . n−1

i`B


(c`−1)

n−1i1n−1i2. . . n−1

i`−1ni`−1

. . . ni2ni1xsi1si2 ...si`−1αi`

(c`)n−1i1n−1i2. . . n−1

i`B


(c`−1)

n−1i1n−1i2. . . n−1

i`−1ni`−1

. . . ni2xsi1si1si2 ...si`−1αi`

(ε`c`)n−1i2. . . n−1

i`B

32

(for some ε` ∈ −1, 1),= xαi1 (c1)xsi1αi2 (c2) . . . xsi1si2 ...si`−2

αi`−1(c`−1)

n−1i1n−1i2. . . n−1

i`−1ni`−1

. . . ni2xsi2 ...si`−1αi`

(ε`c`)n−1i2. . . n−1

i`B


(c`−1)

n−1i1n−1i2. . . n−1

i`−1xαi` (ε

′`c`)n

−1i`B

(for some ε′` ∈ −1, 1),= xi1(c1)n−1

i1xi2(ε

′2c2)n−1

i2. . . xi`(ε

′`c`)n

−1i`B

(for some ε′2 . . . ε′` ∈ −1, 1).

Hence the map ϕ is a bijection.

Example 2.4.7. We illustrate Theorem 2.4.1 with the example G(F) = GLn(F). Let a ∈ G.Let j1 ∈ 1, 2, . . . , n be maximal such that aj1,1 6= 0. If j1 = 1 then let a(1) = a. If j1 6= 1 thenlet

a(1) = ax12(−aj1,2aj1,1

)x13(−aj1,3aj1,1

) · · ·x1n(−aj1,naj1,1

).

Let j2 ∈ 1, 2, . . . , n be maximal such that a(1)j2,26= 0. If j2 = 2 then let a(2) = a(1). If j2 6= 2

then let

a(2) = a(1)x23(−a

(1)j2,3

a(1)j2,2

)x24(−a

(1)j2,4

a(1)j2,2

) · · ·x2n(−a

(1)j2,n

a(1)j2,2

).

Continuing in this way, we produce a(n) such that every entry to the right of a(n)j1,1

is zero, every

entry to the right of a(n)j2,2

is zero, and so forth. Let

y = a(n)h1(1

a(n)j1,1

)h2(1

a(n)j2,2

) · · ·hn(1

a(n)jn,n

).

Then

yj1,1 = yj2,2 = · · · = yjn,n = 1.

Let w ∈ Sn such that

yw ∈ ULet

u =ax12(−aj1,2aj1,1

)x13(−aj1,3aj1,1

) · · ·x1n(−aj1,naj1,1

)x23(−a

(1)j2,3

a(1)j2,2

)x24(−a

(1)j2,4

a(1)j2,2

) · · ·x2n(−a

(1)j2,n

a(1)j2,2

)

· · ·h1(1

a(n)j1,1

)h2(1

a(n)j2,2

) · · ·hn(1

a(n)jn,n

)w.

Then u ∈ U and

a = uw−1hn(a(n)jn,n

) · · ·h1(a(n)j1,1

) · · · x2n(a

(1)j2,n

a(1)j2,2

) · · ·x24(a

(1)j2,4

a(1)j2,2

)x23(a

(1)j2,3

a(1)j2,2

)

x1n(aj1,naj1,1

) · · ·x13(aj1,3aj1,1

)x12(aj1,2aj1,1

)

∈ Uw−1B.

Let

b = hn(a(n)jn,n

) · · ·h1(a(n)j1,1

) · · ·x2n(a

(1)j2,n

a(1)j2,2

) · · · x24(a

(1)j2,4

a(1)j2,2

)x23(a

(1)j2,3

a(1)j2,2

)

33

x1n(aj1,naj1,1

) · · ·x13(aj1,3aj1,1

)x12(aj1,2aj1,1

),

so that

a = uw−1b.

Compute R−(w−1) and R+(w−1). There exists unique tα’s and tβ’s in F such that

u =∏

α∈R−(w−1)

xα(tα)∏

β∈R+(w−1)

xβ(tβ).

Hence

a =∏

α∈R−(w−1)

xα(tα)∏

β∈R+(w−1)

xβ(tβ)w−1b

=∏

α∈R−(w−1)

xα(tα)w−1w(∏

β∈R+(w−1)

xβ(tβ))w−1b

=∏

α∈R−(w−1)

xα(tα)w−1∏

β∈R+(w−1)

(wxβ(tβ)w−1)b

=∏

α∈R−(w−1)

xα(tα)w−1∏

β∈R+(w−1)

(xw(β)(tβ))b

∈ Uw−1,−w−1B.

since ∏β∈R+

(xw(β)(tβ)) ∈ B.

We now demonstrate this algorithm with a concrete example. Let

a =

7 6 2 41 8 7 98 6 3 50 1 1 2

so that a ∈ GL4(C). Then

a(1) = ax12(−a32

a31

)x13(−a33

a31

)x14(−a34

a31

)

= ax12(−6

8)x13(

−3

8)x14(

−5

8)

=

7 3/4 −5/8 −3/81 29/4 53/8 67/88 0 0 00 1 1 2

a(2) = a(1)x23(−1)x24(−2)

=

7 3/4 −11/8 −9/81 29/4 −5/8 9/88 0 0 00 1 0 0

a(3) = a(2)x34(9/5)

=

7 3/4 −11/8 −18/51 29/4 −5/8 08 0 0 00 1 0 0

34

a(4) = a(3)

y = a(4)h1(1/8)h2(1)h3(−8/5)h4(−5/18)

=

7/8 3/4 11/4 11/8 29/4 1 01 0 0 00 1 0 0

.

u = y

0 0 1 00 0 0 10 1 0 01 0 0 0

=

1 11/5 7/8 3/40 1 1/8 29/40 0 1 00 0 0 1

.Note that u ∈ U . Let

w =

0 0 1 00 0 0 10 1 0 01 0 0 0

= s2s3s2s1s2.

Then

R+(w−1) = ε3 − ε4,R−(w−1) = ε1 − ε2, ε1 − ε3, ε1 − ε4, ε2 − ε3, ε2 − ε4.

So write

u = x12(11/5)x13(7/8)x23(1/8)x14(7/8)x24(29/4)x34(0).

So

a =uw−1h4(−18/5)h3(−5/8)h2(1)h1(8)x34(−9/5)x24(2)x23(1)x14(5/8)x13(3/8)x12(3/4)

=x12(11/5)x13(7/8)x23(1/8)x14(7/8)x24(29/4)x34(0)w−1

h4(−18/5)h3(−5/8)h2(1)h1(8)x34(−9/5)x24(2)x23(1)x14(5/8)x13(3/8)x12(3/4)

=x12(11/5)x13(7/8)x23(1/8)x14(7/8)x24(29/4)w−1

h4(−18/5)h3(−5/8)h2(1)h1(8)x34(−9/5)x24(2)x23(1)x14(5/8)x13(3/8)x12(3/4)

∈ Uw−1,−w−1B.

Example 2.4.8. We illustrate Theorem 2.4.6 with the example G(F) = GLn(F). Let a ∈ G.Let j1 ∈ 1, 2, . . . , n be maximal such that aj1,1 6= 0. If j1 = 1 then let a(1) = a. If j1 6= 1 thenlet

a(1) = s12x12(− a1,1

aj1,1)s23x23(− a2,1

aj1,1) · · · sj1−1,j1xj1−1,1(−aj1−1,1

aj1,1)a.

Let j2 ∈ 1, 2, . . . , n be maximal such that a(1)j2,26= 0. If j2 = 2 then let a(2) = a(1). If j2 6= 2

then let

a(2) = s23x23(−a

(1)2,2

a(1)j2,2

)s34x34(−a

(1)3,2

a(1)j2,2

) · · · sj2−1,j2xj2−1,j2(−a

(1)j2−1,2

a(1)j2,2

)a(1).

Continuing in this way, we produce a(n) such that a(n) is upper triangular. Let b = a(n). Then

· · · s23x23(−a

(1)2,2

a(1)j2,2

)s34x34(−a

(1)3,2

a(1)j2,2

) · · · sj2−1,j2xj2−1,j2(−a

(1)j2−1,2

a(1)j2,2

)

s12x12(− a1,1

aj1,1)s23x23(− a2,1

aj1,1) · · · sj1−1,j1xj1−1,1(−aj1−1,1

aj1,1)a = b.

35

Then

a = xj1−1,1(aj1−1,1

aj1,1)sj1−1,j1 · · ·x23(

a2,1

aj1,1)s23x12(

a1,1

aj1,1)s12

xj2−1,j2(a

(1)j2−1,2

a(1)j2,2

)sj2−1,j2 · · ·x34(a

(1)3,2

a(1)j2,2

)s34x23(a

(1)2,2

a(1)j2,2

)s23 · · · b.

Also,

w = sj1−1,j1sj1−2,j1−1 . . . s23s12sj2−1,j2 · · · s23sj3−1,j3 · · · s34 · · ·

is a reduced expression (see [Hum92, §1.8]).

2.5. Examples

Example 2.5.1. The general linear Lie algebra is

gln = n× n matrices over C

with Lie bracket defined by [X, Y ] = XY − Y X.Let Eij be the matrix with 1 in the (i, j)-entry and 0 elsewhere. Then

Eij | i, j ∈ 1, 2, . . . , n

is a basis of gln. The Lie algebra gln is reductive but not semisimple. A standard choice ofCartan subalgebra is

h = X ∈ gln | X is diagonal

so that a basis of h is

E11, E22, . . . , Enn.

Let ε1, ε2, . . . , εn ⊆ h∗ be the dual basis so that ε(Ejj) = δij. The root system is

R = εi − εj ∈ h∗ | i, j ∈ 1, 2, . . . , n and i 6= j,

and the (εi − εj)-root space is

gεi−εj = C-spanEij.

A standard choice of simple roots in R is

∆ = ε1 − ε2, ε2 − ε3, . . . , εn−1 − εn.

and the Weyl group is

W = 〈s1, s2, . . . , sn−1|s2i = (sjsk)

mjk = 1 for i, j, k ∈ 1, 2, . . . n− 1 and j 6= k〉

where

mjk =

2 if |j − k| > 1,

3 if |j − k| = 1,

so that W ∼= Sn, the symmetric group on n letters. The set of positive roots is

R+ = εi − εj ∈ h∗ | i, j ∈ 1, 2, . . . , n and i < j,

so that

n+ = X ∈ gln | X is strictly upper triangular n− = X ∈ gln | X is strictly lower triangular ,

and

b = X ∈ gln | X is weakly upper triangular .36

Example 2.5.2. The special linear Lie algebra is

sln = X ∈ gln | Trace(X) = 0with Lie bracket inherited from gln. The Lie algebra sln is semisimple. We assume the notationused in the case of gln (Example 2.5.1). A standard choice of Cartan subalgebra is

h = X ∈ sln | X is diagonal so that a basis of h is

E11 − E22, E22 − E33, . . . , En−1,n−1 − Enn.The root system is

R = εi − εj ∈ h∗ | i, j ∈ 1, 2, . . . , n and i 6= j,and the (εi − εj)-root space is

gεi−εj = C-spanEij.A standard choice of simple roots in R is

∆ = ε1 − ε2, ε2 − ε3, . . . , εn−1 − εn.and the Weyl group is

W = 〈s1, s2, . . . , sn−1|s2i = (sjsk)

mjk = 1 for i, j, k ∈ 1, 2, . . . n− 1 and j 6= k〉where

mjk =

2 if |j − k| > 1,

3 if |j − k| = 1,

so that W ∼= Sn, the symmetric group on n letters. The set of positive roots is

R+ = εi − εj ∈ h∗ | i, j ∈ 1, 2, . . . , n and i < j,so that

n+ = X ∈ sln | X is strictly upper triangular n− = X ∈ sln | X is strictly lower triangular ,

and

b = X ∈ sln | X is weakly upper triangular .

Example 2.5.3. Let n ∈ Z≥0, let V be a finite dimensional complex vector space of dimen-sion 2n+ 1 and let 〈, 〉 be a non-degenerate symmetric bilinear form on V (see Theorem A.2.6).The special orthogonal Lie algebra so2n+1 [Bou04, Ch. VIII, §13.2] of rank n is

so2n+1 = X ∈ sl2n+1 | 〈Xv, v′〉+ 〈v,Xv′〉 = 0 for v, v′ ∈ V .By [Bou07, Chap. IX, §4, no. 2] and [Tay92, Lemma 7.5], there exists a basis

(e1, e2, . . . , en, e0, e−n, . . . , e−2, e−1)

for V such that

〈e0, e0〉 = −2 and 〈ei, e−j〉 = δij for i, j ∈ 1, 2, . . . , n.The matrix of 〈, 〉 with respect to this basis is

J =

0 0 s0 −2 0s 0 0

, where s =

0 0 · · · 0 10 0 · · · 1 0...

.... . .

......

0 1 · · · 0 01 0 · · · 0 0

,so that

so2n+1 = X ∈ sl2n+1 | X tJ + JX = 0.37

A standard choice of Cartan subalgebra is

h = X ∈ so2n+1 | X is diagonal so that a basis of h is

E11 − E−1,−1, E22 − E−2,−2, . . . , En,n − E−n,−n.Let

ε1, ε2, . . . , εnbe the dual basis in h∗ so that εi(Ejj − E−j,−j) = δij. The root system is

R = εi,−εi | i ∈ 1, 2, . . . , n t εi − εj,−εi + εj, εi + εj,−εi − εj | i, j ∈ 1, 2, . . . , n and i < j.A standard choice of simple roots in R is

∆ = ε1 − ε2, ε2 − ε3, . . . , εn−1 − εn t εn,and the Weyl group is

W = permutations and sign changes of ε1, ε2, . . . , εn= 〈s1, s2, . . . , sn−1|s2

i = (sjsk)mjk = 1 for i, j, k ∈ 1, 2, . . . n− 1 and j 6= k〉,

where

mjk =

2 if |j − k| > 1,

3 if |j − k| = 1 and (j, k) /∈ (n− 2, n− 1), (n− 2, n− 1)4 if (j, k) ∈ (n− 2, n− 1), (n− 2, n− 1).

The set of positive roots is

R+ = εi | i ∈ 1, 2, . . . , n t εi − εj, εi + εj | i, j ∈ 1, 2, . . . , n and i < j,so that

n+ = X ∈ sln | X is strictly upper triangular n− = X ∈ sln | X is strictly lower triangular ,

and

b = X ∈ sln | X is weakly upper triangular .A Chevalley basis for g is

Xεi , X−εi | i ∈ 1, 2, . . . nt Xεi−εj , Xεj−εiXεi+εj , X−εi−εj | i, j ∈ 1, 2, . . . , n and i < jt Hε1−ε2 , Hε2−ε3 , . . . , Hεn−1−εn t Hεn

where

Xεi = 2Ei,0 + E0,−i, X−εi = 2E−i,0 + E0,i,

Xεi−εj = Ei,j − E−j,−i, X−(εi−εj) = Ej,i − E−i,−j,Xεi+εj = Ei,−j − Ej,−i, X−(εi+εj) = E−j,i − E−i,j,

Hεi−εi+1= Ei,i − E−i,−i − Ei+1,i+1 + E−(i+1),−(i+1),

Hεn = 2En,n − 2E−n,−n.

We now look at so2n+1 for specific n. Let g = so3. We have

Xε1 =

0 2 00 0 10 0 0

, X−ε1 =

0 0 01 0 00 2 0

, Hε1 =

1 0 00 0 00 0 −1

.and

n = spanXε1, h = spanHε1, n− = spanX−ε1.38

Now let g = so5. We have

Xε1 =

0 0 2 0 00 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 0

, Xε2 =

0 0 0 0 00 0 2 0 00 0 0 1 00 0 0 0 00 0 0 0 0

,

X−ε1 =

0 0 0 0 00 0 0 0 01 0 0 0 00 0 0 0 00 0 2 0 0

, X−ε2 =

0 0 0 0 00 0 0 0 00 1 0 0 00 0 2 0 00 0 0 0 0

,

Xε1−ε2 =

0 1 0 0 00 0 0 0 00 0 0 0 00 0 0 0 −10 0 0 0 0

, Xε2−ε1 =

0 0 0 0 01 0 0 0 00 0 0 0 00 0 0 0 00 0 0 −1 0

,

Xε1+ε2 =

0 0 0 1 00 0 0 0 −10 0 0 0 00 0 0 0 00 0 0 0 0

, X−ε1−ε2 =

0 0 0 0 00 0 0 0 00 0 0 0 01 0 0 0 00 −1 0 0 0

,and

Hε1−ε2 =

1 0 0 0 00 −1 0 0 00 0 0 0 00 0 0 1 00 0 0 0 −1

, Hε2 =

0 0 0 0 00 1 0 0 00 0 0 0 00 0 0 −1 00 0 0 0 0

,and

n = spanXε1 , Xε2 , Xε1−ε2 , Xε1+ε2,h = spanHε1−ε2 , Hε2,

n− = spanX−ε1 , X−ε2 , X−ε1+ε2 , X−ε1−ε2.

Example 2.5.4. Let n ∈ Z≥0, let V be a finite dimensional complex vector space of dimen-sion 2n and let 〈, 〉 be a non-degenerate alternating bilinear form on V (see Theorem A.2.6).The symplectic Lie algebra sp2n [Bou04, Ch. VIII, §13.3] of rank n is

sp2n = X ∈ sl2n|〈Xv, v′〉+ 〈v,Xv′〉 = 0 for v, v′ ∈ V .

By [Bou07, Chap. IX, §4, no. 2] and [Tay92, Lemma 7.5], there exists a basis

(e1, e2, . . . , en, e−n, . . . , e−2, e−1)

for V such that

〈ei, e−j〉 = δij for i, j ∈ 1, 2, . . . , n.The matrix of 〈, 〉 with respect to this basis is

J =

[0 s−s 0

], where s =

0 0 · · · 0 10 0 · · · 1 0...

.... . .

......

0 1 · · · 0 01 0 · · · 0 0

,39

so that

sp2n = X ∈ sl2n | X tJ + JX = 0.

A standard choice of Cartan subalgebra is

h = X ∈ sp2n | X is diagonal


E11 − E−1,−1, E22 − E−2,−2, . . . , En,n − E−n,−n.

Let

ε1, ε2, . . . , εn

be the dual basis in h∗ so that εi(Ejj − E−j,−j) = δij. The root system is

R = 2εi,−2εi | i ∈ 1, 2, . . . , n t εi − εj,−εi + εj, εi + εj,−εi − εj | i, j ∈ 1, 2, . . . , n and i < j.

A standard choice of simple roots is

∆ = ε1 − ε2, ε2 − ε3, . . . , εn−1 − εn t 2εn,

and the Weyl group is

W = permutations and sign changes of ε1, ε2, . . . , εn= 〈s1, s2, . . . , sn−1|s2

i = (sjsk)mjk = 1 for i, j, k ∈ 1, 2, . . . n− 1 and j 6= k〉,

where

mjk =

2 if |j − k| > 1,

3 if |j − k| = 1 and (j, k) /∈ (n− 2, n− 1), (n− 2, n− 1)4 if (j, k) ∈ (n− 2, n− 1), (n− 2, n− 1).


R+ = 2εi | i ∈ 1, 2, . . . , n t εi − εj, εi + εj | i, j ∈ 1, 2, . . . , n and i < j,

so that

n+ = X ∈ sp2n | X is strictly upper triangular n− = X ∈ sp2n | X is strictly lower triangular ,

and

b = X ∈ sp2n | X is weakly upper triangular .

A Chevalley basis for g is

X2εi , X−2εi ∈ g | i ∈ 1, 2, . . . nt Xεi−εj , X−εi+εjXεi+εj , X−εi−εj | i, j ∈ 1, 2, . . . , n and i < jt Hε1−ε2 , Hε2−ε3 , . . . , Hεn−1−εn t H2εn

where

X2εi = Ei,−i, X−2εi = E−i,i,

Xεi−εj = Ei,j − E−j,−i, X−(εi+εj) = Ej,i − E−i,−j,Xεi+εj = Ei,−j + Ej,−i, X−(εi+εj) = E−i,j + E−j,i,


H2εn = En,n − E−n,−n.

40

We now illustrate sp2n for specific n. Let g = sp4. We have

X2ε1 =

0 0 0 10 0 0 00 0 0 00 0 0 0

, X2ε2 =

0 0 0 00 0 1 00 0 0 00 0 0 0

X−2ε1 =

0 0 0 00 0 0 00 0 0 01 0 0 0

, X−2ε2 =

0 0 0 00 0 0 00 1 0 00 0 0 0

,

Xε1−ε2 =

0 1 0 00 0 0 00 0 0 −10 0 0 0

, X−ε1+ε2 =

0 0 0 01 0 0 00 0 0 00 0 −1 0

,

Xε1+ε2 =

0 0 1 00 0 0 10 0 0 00 0 0 0

, X−ε1−ε2 =

0 0 0 00 0 0 01 0 0 00 1 0 0

,and

Hε1−ε2 =

1 0 0 00 −1 0 00 0 1 00 0 0 −1

, H2ε2 =

0 0 0 00 1 0 00 0 −1 00 0 0 0

,and

n = spanX2ε1 , X2ε2 , Xε1−ε2 , Xε1+ε2,h = spanHε1−ε2 , H2ε2,

n− = spanX−2ε1 , X−2ε2 , X−ε1+ε2 , X−ε1−ε2).

Example 2.5.5. Let n ∈ Z≥0, let V be a finite dimensional complex vector space of dimen-sion 2n and let 〈, 〉 be a non-degenerate symmetric bilinear form on V (see Theorem A.2.6).The special orthogonal Lie algebra so2n [Bou04, Ch. VIII, §13.2] of rank n is

so2n = X ∈ sl2n|〈Xv, v′〉+ 〈v,Xv′〉 = 0 for v, v′ ∈ V .By [Bou07, Chap. IX, §4, no. 2] and [Tay92, Lemma 7.5], there exists a basis

(e1, e2, . . . , en, e−n, . . . , e−2, e−1)

for V such that

〈ei, e−j〉 = δij for i, j ∈ 1, 2, . . . , n.The matrix of 〈, 〉 with respect to this basis is

J =

0 0 · · · 0 10 0 · · · 1 0...

.... . .

......

0 1 · · · 0 01 0 · · · 0 0

,so that

so2n = X ∈ sl2n | X tJ + JX = 0.A standard choice of Cartan subalgebra is

h = X ∈ so2n | X is diagonal

41


E11 − E−1,−1, E22 − E−2,−2, . . . , En,n − E−n,−n.Let

ε1, ε2, . . . , εnbe the dual basis in h∗ so that εi(Ejj − E−j,−j) = δij. The root system is

R = εi − εj,−εi + εj, εi + εj,−εi − εj | i, j ∈ 1, 2, . . . , n and i < j.A standard choice of simple roots is

∆ = ε1 − ε2, ε2 − ε3, . . . , εn−1 − εn t εn−1 + εn,and the Weyl group is

W = permutations and sign changes of an even number of ε1, ε2, . . . , εn.= 〈s1, s2, . . . , sn|s2

i = (sjsk)mjk = 1 for i, j, k ∈ 1, 2, . . . n and j 6= k〉,

where

mjk =

2 if |j − k| > 1, and j 6= n− 1 and k 6= n− 1,

3 if |j − k| = 1, and j 6= n− 1 and k 6= n− 1,

3 if j = n− 1 and k ∈ n− 2, n− 3, n− 4, or k = n− 1 and j ∈ n− 2, n− 3, n− 4,2 otherwise.


R+ = εi − εj, εi + εj | i, j ∈ 1, 2, . . . , n and i < j,so that

n+ = X ∈ so2n | X is strictly upper triangular n− = X ∈ so2n | X is strictly lower triangular ,

and

b = X ∈ so2n | X is weakly upper triangular .A Chevalley basis for g is

Xεi−εj , Xεj−εi , Xεi+εj , X−εi−εj | i, j ∈ 1, 2, . . . , n and i < jt Hε1−ε2 , Hε2−ε3 , . . . , Hεn−1−εn t Hεn−1+εn

where

Xεi−εj = Ei,j − E−j,−i, X−(εi−εj) = Ej,i − E−i,−j,Xεi+εj = Ei,−j − Ej,−i, X−(εi+εj) = E−j,i − E−i,j,


Hεn−1+εn = En−1,n−1 − E−(n−1),−(n−1) + En,n − E−n,−n.We now look at so2n for specific n. Let g = so4. We have

Xε1−ε2 =

0 1 0 00 0 0 00 0 0 −10 0 0 0

, Xε2−ε1 =

0 0 0 01 0 0 00 0 0 00 0 −1 0

,

Xε1+ε2 =

0 0 1 00 0 0 −10 0 0 00 0 0 0

, X−ε1−ε2 =

0 0 0 00 0 0 01 0 0 00 −1 0 0

,42

and

Hε1−ε2 =

1 0 0 00 −1 0 00 0 1 00 0 0 −1

, Hε1+ε2 =

1 0 0 00 1 0 00 0 −1 00 0 0 −1

,and

n = spanXε1−ε2 , Xε1+ε2,h = spanH1, H2,

n− = spanX−ε1+ε2 , X−ε1−ε2.

Example 2.5.6. Following [Ram], the Lie algebra g of type G2 is the complex simple Liealgebra given by generators E1, E2, H1, H2, F1, F2 and relations

[E1, F2] = 0, [E2, F1] = 0, [H1, H2] = 0,

[E1, F1] = H1, [E2, F2] = H2

[H1, E1] = 2E1, [H1, E2] = −3E2, [H1, F1] = −2F1, [H1, F2] = 3F2,

[H2, E1] = −E1, [H2, E2] = 2E2, [H2, F1] = F1, [H2, F2] = −2F2,

[E1, [E1, [E1, [E1, E2]]]] = 0, [E2, [E2, E1]] = 0,

[F1, [F1, [F1, [F1, F2]]]] = 0, [F2, [F2, F1]] = 0.

A choice of Cartan subalgebra is

h = C-spanH1, H2,

and a basis for h is H1, H2. Define α1, α2 ∈ h∗ by

α1(H1) = 2, α1(H2) = −1,

α2(H1) = −3, α2(H2) = 2.

The root system is

R = R+ t −R+

where

R+ = α1, α2, α1 + α2, 2α1 + α2, 3α1 + α2, 3α1 + 2α2is the set of positive roots, so that

n+ = subalgebra generated by the set E1, E2n− = subalgebra generated by the set F1, F2b = subalgebra generated by the set H1, H2, E1, E2.

A choice of simple roots is

∆ = α1, α2and the Weyl group is

W =⟨s1, s2

∣∣s21 = s2

2 = (s1s2)6 = 1⟩.

The Lie algebra g has a Chevalley basis defined as follows:

Eα1 = E1, Eα2 = E2, Eα1+α2 = [E1, E2],

E2α1+α2 =1

2![E1, [E1, E2]],

E3α1+α2 =1

3![E1, [E1, [E1, E2]]],

43

E3α1+2α2 =1

3![E2, [E1, [E1, [E1, E2]]]],

Hα1 = H1, Hα2 = H2,

Fα1 = F1, Fα2 = F2, Fα1+α2 = [F1, F2],

F2α1+α2 =1

2![F1, [F1, F2]],

F3α1+α2 =1

3![F1, [F1, [F1, F2]]],

F3α1+2α2 =1

3![F2, [F1, [F1, [F1, F2]]]].

The Lie algebra g has a faithful representation g→ gl7 which is described in [Ram].

Example 2.5.7. Let g = sl2 = CE⊕CH⊕CF where E := E12 and F := E21 as in Example2.5.2. Let B = v0, v1 be a C-basis for the 2-dimensional C-vector space V , and let the actionof E and F on V be defined on the basis B by

E =

[0 10 0

], F =

[0 01 0

].

Let M be the Z-space spanned by B. Then M is a UgZ invariant lattice in V . Let F be a field.The Chevalley group G(F) = SL2(F) is the subgroup of GL(F⊗ZM) generated by xα(t), x−α(t)where

xα(t) = exp(tE) = I + tE +t2

2!E2 +

t3

3!E3 +

t4

4!E3 + . . .

=

[1 00 1

]+ t

[0 10 0

]+t2

2!

[0 00 0

]. . . =

[1 t0 1

]and similiarly

x−α(t) =

[1 0t 1

]for t ∈ F. The elements

nα(t′) =

[0 t′

−t′−1 0

]for t′ ∈ F×, generate N

and the elements

hα(t′) =

[t′ 00 t′−1

]for t′ ∈ F×, generate H.

Example 2.5.8. Let g = sl2 = CE⊕CH⊕CF where E := E12 and F := E21 as in Example2.5.2. Let B = v0, v1, v2 be a C-basis for the 3-dimensional C-vector space V , and let theaction of E and F on V be defined on the basis B by

E =

0 2 00 0 10 0 0

, F =

0 0 01 0 00 2 0

.Let M be the Z-space spanned by B. Then M is an UgZ an invariant lattice in V (see [Ste67,Corollary 1, pg. 17]). Let F be a field. The Chevalley group G(F) is the subgroup of GL(F⊗ZM)generated by xα(t), x−α(t) where

xα(t) = exp(tE)

= I + tE +t2

2!E2 +

t3

3!E3 +

t4

4!E3 + . . .

44

=

1 0 00 1 00 0 1

+ t

0 2 00 0 10 0 0

+t2

2!

0 0 20 0 00 0 0

+t3

3!

0 0 00 0 00 0 0

=

1 2t t2

0 1 t0 0 1

and similiarly

x−α(t) =

1 0 0t 1 0t2 2t 1

for t ∈ F. The elements

nα(t) =

0 0 t2

0 −1 0t−2 0 0

for t ∈ F×, generate N . The elements

hα(t) =

t2 0 00 1 00 0 t−2

for t ∈ F×, generate T . The group G is called the projective special linear group PSL2(F) andwe have PSL2(F) ∼= SL2(F)/Z(SL2(F)).

Example 2.5.9. Let g = sl2 = CE⊕CH⊕CF where E := E12 and F := E21 as in Example2.5.2. Let B = v0, v1, v2, v3 be a C-basis for the 4-dimensional C-vector space V , and let theaction of E and F on V be defined on the basis B by

E =

0 3 0 00 0 2 00 0 0 10 0 0 0

, F =

0 0 0 01 0 0 00 2 0 00 0 3 0

.Let M be the Z-space spanned by B. Then M is an UgZ an invariant lattice in V (see [Ste67,Corollary 1, pg. 17]). Let F be a field. The Chevalley group G(F) is the subgroup of GL(F⊗ZM)generated by xα(t), x−α(t) where

xα(t) = exp(tE)

= I + tE +t2

2!E2 +

t3

3!E3 +

t4

4!E3 + . . .

=

1 0 0 00 1 0 00 0 1 00 0 0 1

+ t

0 3 0 00 0 2 00 0 0 10 0 0 0

+t2

2!

0 0 6 00 0 0 20 0 0 00 0 0 0

+t3

3!

0 0 0 60 0 0 00 0 0 00 0 0 0

+t4

4!

0 0 0 00 0 0 00 0 0 00 0 0 0

+ . . .

=

1 3t 3t2 t3

0 1 2t t2

0 0 1 t0 0 0 1

45

and similiarly

x−α(t) =

1 0 0 0t 1 0 0t2 2t 1 0t3 3t2 3t 1


nα(t) =

0 0 0 t3

0 0 −t 00 t−1 0 0−t−3 0 0 0


hα(t) =

t3 0 0 00 t 0 00 0 t−1 00 0 0 t−3

for t ∈ F×, generate H. Let G′ = SL2(F) as in Example 2.5.7. By [Ste67, Corollary 5, pg. 44],there exists an isomorphism ϕ : G′ → G such that ϕ(xα(t) = xα(t). in particular, G ∼= SL2(F).

Example 2.5.10. Continuing on from Example 2.5.4, let g = sp4, let V be the standardrepresentation of g acting on C4. Then Z4 is a UgZ submodule of V such that e1, e2, e−2, e−1is a C-basis for V . Let F be a field. The corresponding Chevalley group G(F) is the projectivesymplectic group PSp4(F). It is the subgroup of GL4(F) generated by

x2ε1(t) =

1 0 0 t0 1 0 00 0 1 00 0 0 1

, x2ε2(t) =

1 0 0 00 1 t 00 0 1 00 0 0 1

,

x−2ε1(t) =

1 0 0 00 1 0 00 0 1 0t 0 0 1

, x−2ε2(t) =

1 0 0 00 1 0 00 t 1 00 0 0 1

,

xε1−ε2(t) =

1 t 0 00 1 0 00 0 1 −t0 0 0 1

, x−ε1+ε2(t) =

1 0 0 0t 1 0 00 0 1 00 0 −t 1

,

xε1+ε2(t) =

1 0 t 00 1 0 t0 0 1 00 0 0 1

, x−ε1−ε2(t) =

1 0 0 00 1 0 0t 0 1 00 t 0 1

,for t ∈ F. The following relations hold (and are useful for computations):

xε1−ε2(t)xε1+ε2(u) = xε1+ε2(u)xε1−ε2(t)x2ε1(2tu) (2.4)

xε1−ε2(t)x2ε2(u) = x2ε2(u)xε1−ε2(t)xε1+ε2(tu)x2ε1(−t2u). (2.5)

The elements

nε1−ε2(t) =

0 t 0 0−t−1 0 0 0

0 0 0 −t0 0 t−1 0

, n2ε2(t) =

1 0 0 00 0 t 00 −t−1 0 00 0 0 1

,46

for t ∈ F× generate N . The elements

hε1−ε2(t) =

t 0 0 00 t−1 0 00 0 t 00 0 0 t−1

, h2ε2(t) =

1 0 0 00 t 0 00 0 t−1 00 0 0 1

,for t ∈ F× generate H.

Example 2.5.11. Continuing on from Example 2.5.3, let g = so3, let V be the standardrepresentation of g acting on C3. Then Z3 is a UgZ-submodule of V such that e−1, e0, e1 isa C-basis for V . Let F be a field. The corresponding Chevalley group G(F) is PΩ3(F) (see[Car72, Theorem 11.3.2]). It is generated by

xε1(t) = exp(tXε1)

= I + tE +t2

2!E2 +

t3

3!E3 +

t4

4!E3 + . . .

=

1 0 00 1 00 0 1

+ t

0 2 00 0 10 0 0

+t2

2!

0 0 20 0 00 0 0

+t3

3!

0 0 00 0 00 0 0

=

1 2t t2

0 1 t0 0 1

and similiarly

x−ε1(t) =

1 0 0t 1 0t2 2t 1


nε1(t) =

0 0 t2

0 −1 0t−2 0 0


hε1(t) =

t2 0 00 1 00 0 t−2

for t ∈ F×, generate T .

Example 2.5.12. Continuing on from Example 2.5.5, let g = so4, let V be the standardrepresentation of g acting on C4. Then Z4 is a UgZ submodule of V such that e1, e2, e−2, e−1is a C-basis for V . Let F be a field. The corresponding Chevalley group G(F) is PΩ4(F) (see[Car72, Theorem 11.3.2]). It is the subgroup of GL4(F) generated by

xε1−ε2(t) =

1 t 0 00 1 0 00 0 1 −t0 0 0 1

, xε1+ε2(t) =

1 0 t 00 1 0 −t0 0 1 00 0 0 1

,

x−(ε1−ε2)(t) =

1 0 0 0t 1 0 00 0 1 00 0 −t 1

, x−(ε1+ε2)(t) =

1 0 0 00 1 0 0t 0 1 00 −t 0 1

,47


nε1−ε2(t) =

0 t 0 0−t−1 0 0 0

0 0 0 −t0 0 t−1 0

, nε1+ε2(t) =

0 0 t 00 0 0 −t−t−1 0 0 0

0 t−1 0 0

,for t ∈ F× generate N . The elements

hε1−ε2(t) =

t 0 0 00 t−1 0 00 0 t 00 0 0 t−1

, hε1+ε2(t) =

t 0 0 00 t 0 00 0 t−1 00 0 0 t−1

,for t ∈ F× generate H.

48

CHAPTER 3

Twisted Chevalley groups

A finite twisted Chevalley group is the fixed point subgroup of a finite Chevalley groupunder an endomorphism of finite order. The machinery for constructing these groups appearin Steinberg’s paper [Ste59]. These groups also known in the literature as groups of Lie type.

In this chapter, we follow the modern treatment in [MT11, Part 3]. Some other refer-ences are [Ste67, §11], [Ste68], [Car89]. See Ramagge [Rmg] and the references therein for thegeneralisation to the case of Kac-Moody groups.

The motivation for introducing the theory of twisted Chevalley groups is their relationshipwith the classical ovoids in the Hermitian polar semilattice and the Suzuki-Tits ovoid in Sections4.2.4 and 4.2.3 of Chapter 4. We also hope in future work to generalise the thickness theory(as in Section 4.3, Chapter 4) to the twisted Chevalley groups.

3.1. Definitions and basic properties

Let Fp be the finite field with p elements, where p is a prime. Let Fp be its algebraic

closure and let G = G(Fp) be a Chevalley group over Fp, as defined in Section 2.2. Given anendomorphism F : G→ G, define

GF = g ∈ G | F (g) = g.

Recall that G is a subgroup of GL(VFp) (see Chapter 2, Equation 2.3). Choose a Fp-basis

e1, e2, . . . , en of VFp , and define the Frobenius morphism to be the group homomorphismFr: G→ G given by

(Fr(g))ij = gpij for i, j ∈ 1, 2, . . . , n.Following the terminology of [MT11, Part III], an endomorphism F : G → G is a Steinbergendomorphism if there exists k,m ∈ Z≥1, such that

Fm = Frk.

Proposition 3.1.1. If F : G→ G is a Steinberg endomorphism then F is surjective.

Proof. Let g ∈ G. Since Fp is algebraically closed, there exists fij ∈ Fp such that

fpk

ij = gij

for all i, j ∈ 1, 2, . . . , n. Let hij = (Fm−1(f))ij. Then

(F (h))ij = (Fm(f))ij = (Frk(f))ij = (f)pk

ij = gij.

Hence F (h) = g, and F is surjective.

Note that

GF ⊆ G(Fpk), since (gpk)ij = gij for g ∈ G.

In particular, GF is finite. The converse is also true.

Theorem 3.1.2. [Ste68, Theorem 10.13], [MT11, Theorem 21.5] Let F : G → G be anendomorphism of G. If GF is finite then F is a Steinberg endomorphism.

A (finite)

twisted Chevalley group or group of Lie type is GF

49

where F : G→ G is a Steinberg endomorphism. For the remainder of this section, let F : G→ Gbe a Steinberg endomorphism, and assume

F (T ) = T, F (U) = U, F (U−) = U− F (N) = N.

(This is not too strong a restriction, see [MT11, Theorem 22.5], [Ste67, Theorem 30], [Car89,Theorem 12.5.1]). Then F (B) = B, and the map

F : W −→ WnT 7−→ F (n)T

is well defined. Let

W F = w ∈ W | F (w) = w.

Define an action of T on G by

h · x = hxh−1

for h ∈ T , x ∈ G.

Proposition 3.1.3. Let X be a T -invariant subgroup of U . Then

X =∏

α∈R+,Xα⊆X

Xα.

Proof. Suppose, for sake of contradiction, that there exists x ∈ X such that x 6= 1 andx /∈

∏α∈R+,Xα⊆X Xα. Then there exists M ⊆ R+ such that x =

∏α∈M xα(cα) and there exists

β ∈M such that cβ 6= 0 and Xβ ⊆ X. Choose x such that |M | is minimal.

Suppose |M | = 1. Then there exists c ∈ Fp×

such that x = xβ(c). Let d ∈ Fp. If d = 0

then xβ(d) = 1 ∈ X. If d 6= 0 then choose t ∈ Fp×

such that t2c = d. Such a t exists since Fp isalgebraically closed. Then

xβ(d) = xβ(ct2) = hβ(t)xβ(c)hβ(t)−1 ∈ Xβ.

Hence Xβ ⊆ X.Suppose |M | > 1. Let γ ∈M with γ 6= β. Choose t2 ∈ F× such that

t2−2

(β,γ)(γ,β)(β,β)(γ,γ)

2 6= 1.

This is always possible since

(β, γ)2 6= (β, β)(γ, γ).

To see this, suppose (β, γ)2 = (β, β)(γ, γ). Since sα and sβ generate a dihedral group, β is ascalar multiple of γ. Since the underlying Lie algebra is semisimple, R is a reduced root system(see [Hum72, Section 9]), so β is not a scalar multiple of γ, a contradiction.

Define

t1 = t−(β,γ)(γ,γ)

2 .

Then

hβ(t1)hγ(t2)xhγ(t−12 )hβ(t−1

2 )x−1

=

∏α∈M,α/∈β,γ

xα(cαt2(α,β)(β,β)

1 t2(α,γ)(γ,γ)

2 )

xβ(cβt21t

2(β,γ)(γ,γ)

2 )xγ(cγt2(γ,β)(β,β)

1 t22)

(∏α∈M

xα(−cα)

)

=


xα(cα(t2(α,β)(β,β)

1 t2(α,γ)(γ,γ)

2 − 1))

xβ(cβ(t21t2(β,γ)(γ,γ)

2 − 1))xγ(cγ(t2(γ,β)(β,β)

1 t22 − 1))

50

=


xα(cα(t2(α,β)(β,β)

1 t2(α,γ)(γ,γ)

2 − 1))

xβ(cβ(t−2

(β,γ)(γ,γ)

2 t2(β,γ)(γ,γ)

2 − 1))xγ(cγ(t2−2

(β,γ)(γ,β)(β,β)(γ,γ)

2 − 1))

∈ X.

So the xβ term of hβ(t1)hγ(t2)xhγ(t−12 )hβ(t−1

2 )x−1 disappears. But the xγ term is not theidentity, hence M is not minimal, a contradiction.

Corollary 3.1.4. Let X be a minimal T -invariant subgroup of U , that is, let X be a T -invariant subgroup such that if Y is a nontrivial T -invariant subgroup X then Y = X. Thenthere exists α ∈ R+ such that X = Xα.

Proof. By Proposition 3.1.3, X =∏

α∈R+,Xα⊆X Xα. Let α ∈ R+ such that Xα ⊆ X. ButXα is a nontrivial T -invariant subgroup of X. Hence Xα = X.

Proposition 3.1.5. [Ste67, Theorem 30, Proof of (1)] There exists a bijection ρ : R+ → R+

such that

F (Xα) = Xρ(α), F (X−α) = X−ρ(α),

for all α ∈ R+.

Proof. Let α ∈ R+. To show: F (Xα) is a minimal T -invariant subgroup of U , so thatF (Xα) = Xρ(α). The proof of the result for F (X−α) is similar.

Let h ∈ T . Assume that there exists β ∈ R+ and c ∈ F such that F (hβ(c)) = h. Then

hF (Xα)h−1 = F (hβ(c))F (Xα)F (hβ(c)−1)

= F (hβ(c)Xαhβ(c−1))

⊆ F (Xα).

Hence F (Xα) is T -invariant.Let Y be a nontrivial T -invariant subgroup of F (Xα). Let y ∈ Y. Then there exists t ∈ F

such that y = F (xα(t)). Then

F (Xα) = F (hxα(t)h−1)|h ∈ T,= F (h)F (xα(t))F (h)−1|h ∈ T,= hF (xα(t))h−1|h ∈ T,= hyh−1|h ∈ T,⊆ Y.

So F (Xα) satisfies the conditions of Corollary 3.1.4. Hence

F (Xα) = Xρ(α)

for some ρ(α) ∈ R+.Since F is surjective, ρ is surjective. Since R+ is finite, ρ is bijective.

Proposition 3.1.6. Let ρ : R+ → R+ be the bijection as in Proposition 3.1.5. Then therestriction of ρ to ∆ is a bijection ∆→ ∆.

Proof. It suffices to show that if α ∈ ∆ then ρ(α) ∈ ∆. Note that B t BwB is a groupif and only if there exists α ∈ ∆ such that w = sα (see [Car72, Proposition 8.3.1]). Therefore,the map

F : B tBsαB|α ∈ ∆ −→ B tBsαB|α ∈ ∆,B tBsαB −→ B tBF (sα)B,

is a bijection. Note that (B t BsαB) ∩ U− = X−α so X−ρ(α) = F ((B t BsαB) ∩ U−) =(B tBsρ(α)B) ∩ U− and ρ(α) ∈ ∆.

51

Let W/ ∼ denote the set of equivalence classes of F -orbits of W . Define pr : R→ h∗ by

pr(α) =α + ρ(α) + ρ2(α) + · · ·+ ρm−1(α)

m.

where m is the cardinality of the ρ-orbit of α. The twisted root system is

RF = pr(α) ∈ h∗|α ∈ R.Define an equivalence class ∼ on R and R+ by

α ∼ β if there exists c ∈ R≥0 such that pr(α) = c · pr(β).

Lemma 3.1.7. [Ste67, Lemma 61] Suppose a ∈ R/ ∼. Then XFa 6= 1.

Proposition 3.1.8. [Ste67, Theorem 32(5)] For I ∈ ∆/ ∼, let sI be the longest element inWI with respect to the generators si|i ∈ I. Then W F is generated by

sI | I ∈ W/ ∼.

Proposition 3.1.9. [Ste67, Theorem 33(b)] [Car72, Proposition 13.5.2], [MT11, Propo-sition 23.2]. If w ∈ W F then there exists nw ∈ NF such that nwT = w. Furthermore, themap

θ : NF/T F −→ W F

nT F 7−→ nT

is an isomorphism.

Proof. Assume that w = sI for some I ∈ ∆/ ∼. By Lemma 3.1.7, there exists x ∈ XF−a

such that x 6= 1, where a ∈ R/ ∼ corresponds to I. We know X−a ⊆ BwB. By Corollary 2.4.3there exists unique u ∈ Uw,−, u′ ∈ U , h ∈ T such that

x = unwhu′.

where nw ∈ N is a coset representative of w. The element nw can be chosen so that h = 1, so

x = unwu′.

Since x = F (x),

F (u)F (nw)F (u′) = unwu′

Since F (w) = w, there exists h′ ∈ H such that F (nw) = nwh′, so

F (u)nwh′F (u′) = unwu

′.

Since F (u) ∈ Uw,−, h′ ∈ T , F (u′) ∈ U , applying the uniqueness of Corollary 2.4.3,

F (u) = u, F (u′) = u′, h′ = 1.

So F (nw) = nw.It has been shown that the map θ : NF → W F defined by θ(n) = nT is a surjection. The

kernel of θ is

ker(θ) = n ∈ NF | n ∈ T= T F .

Hence θ is an isomorphism.

Theorem 3.1.10. [MT11, Theorem 24.1], [Ste67, Theorem 33(c)]. For each w ∈ W F andchoose a representative nw ∈ NF of W F such that nwT = w. Let

BF\GF/BF = BFxBF | x ∈ GF.

The map ψF : W F → BF\GF/BF defined by

ψF (w) = BFnwBF

52

is a bijection. Furthermore, the map

ψ : (Uw,−)F ×BF −→ BFnwBF

(x, b) 7−→ xnwb

is a bijection.

Proof. Let x ∈ GF . By Theorem 2.4.1, there exists unique u ∈ Uw,−, b ∈ B such that

x = unwb.

Since F (x) = x,

F (u)F (nw)F (b) = unwb.

But F (nw) = nw, so

F (u)nwF (b) = unwb.

By uniqueness,

F (u) = u, and F (b) = b.

Hence u ∈ UFw,− and b ∈ BF . So x ∈ BFnwB

F . So BFxBF = BFnwBF . So ψF (nw) = BFxBF .

Hence ψF is surjective.Suppose ψF (w) = ψF (w′). Then BFnwB

F = BFnw′BF . Then BnwB = Bnw′B. Then

BwB = Bw′B. By Theorem 2.3.7, w = w′. Hence ψF is injective.To show: ψ is a bijection. Let x ∈ BFnwB

F . Then x ∈ BnwB. So by Theorem 2.4.1, thereexists unique u ∈ Uw,−, b ∈ B such that x = unwb. Since F (x) = x, F (u)F (nw)F (b) = unwb.But F (nw) = nw, so F (u)nwF (b) = unwb. By uniqueness, F (u) = u and F (b) = b. So u ∈ UF

w,−and b ∈ BF . So ψ(u, b) = unwb = x. So ψ is surjective.

Suppose ψ(u, b) = ψ(u′, b′). Then unwb = u′nwb′. By Theorem 2.4.1, u = u′ and b = b′.

Hence ψ is injective.

Proposition 3.1.11. [Ste67, Lemma 62, §11]. Let w ∈ W F . Then

(Uw,−)F =∏

a∈R+/∼,w−1(a)⊆−R+

XFa

with uniqueness of expression on the right.

Proof. Recall that

Uw,− =∏

β∈R−(w)

Xβ

where

R−(w) = β ∈ R+|w−1(β) ∈ −R+.

But

Uw,− =∏β∈R+,

w−1(β)⊆−R+

Xβ

=∏

a∈R+/∼,w−1(a)⊆−R+

Xa

Suppose

a ∈ R+/ ∼ |w−1(a) ⊆ −R+ = a1, a2, . . . , ak.

53

Note that if γ, δ ∈ R are in the same ρ-orbit then γ ∼ δ. So each ai can be partitioned intoρ-orbits. Suppose y ∈ UF

w,−. Then

y ∈

∏a∈R+/∼,

w−1(a)⊆−R+

Xa

F

.

To show:

(1)

y ∈∏

a∈R+/∼,w−1(a)⊆−R+

XFa .

(2) If

xa1xa2 . . . xak = x′a1x′a2. . . x′ak ∈

∏a∈R+/∼,

w−1(a)⊆−R+

XFa .

with xai ∈ XFai

then

xa1 = x′a1 , xa2 = x′a2 , . . . , xak = x′ak .

We know y = xa1xa2 . . . xak for some xai ∈ Xai . Since F (y) = y,

F (xa1)F (xa2) . . . F (xak) = xa1xa2 . . . xak .

Since a1 is a union of ρ-orbit of roots,

xa1 = xβ1(c1)xρ(β1)(c2) . . . xρn1−1(β1)(cn1)xβ2(cn1+1)xρ(β2)(cn1+1) . . .

∈ Xβ1Xρ(β1) . . .Xρn1−1(β1)Xβ2Xρ(β2) . . . .

Note that

F (xa1) ∈ F (Xβ1)F (Xρ(β1)) . . . F (Xρn1−1(β1))F (Xβ2)F (Xρ(β2)) . . .

= Xρ(β1)Xρ2(β1) . . .Xβ1Xρ(β2)Xρ2(β2) . . . .

Since a1 satisfies the condition

if α, β ∈ a1 and α + β ∈ R then α + β ∈ a1,

so applying Lemma 2.3.1 on the set of roots a1,

Xa1 = Xρ(β1)Xρ2(β1) . . .Xβ1Xρ(β2)Xρ2(β2) . . .

= Xβ1Xρ(β1) . . .Xρn1−1(β1)Xβ2Xρ(β2) . . . .

So there exists d1, d2, . . . ∈ F such that

F (xa1) = xβ1(d1)xρ(β1)(d2) . . . xρn1−1(β1)(dn1)xβ2(dn1+1)xρ(β2)(dn1+1) . . . .

A similiar argument applies to F (xa2), F (xa3), . . . , F (xak) so that

xβ1(c1)xρ(β1)(c2)xρ2(β1)(c3) . . . xβ`(c`) = xβ1(d1)xρ(β1)(d2)xρ2(β1)(d3) . . . xβ`(d`).

By Lemma 2.3.1, the map

F` −→ Uw,−(c1, c2, . . . , c`) 7−→ xβ1(c1) . . . xβ`(c`)

is a bijection. Hence d1 = c1, d2 = c2, . . . , d` = c` and

F (xa1) = xa1 , F (xa2) = xa2 , . . . , F (xak) = xak

and the xai ’s are unique.

54

3.2. Examples

3.2.1. The special unitary group SU3(Fq2). Let G = SL3(Fp), let T be the subgroupof diagonal matrices in G, let B the subgroup of upper triangular matrices in G, and let q = pk

be a postive integer power of p. Define F : G→ G by

F (x) = nw0(Frk(x)t)−1n−1

w0, where nw0 = n−1

1 n−12 n−1

1 =

0 0 10 −1 01 0 0

.If x ∈ G then F 2(x) = Fr2k(x) . Hence F is a Steinberg endomorphism and GF ⊆ SL3(Fq2).The special unitary group SU3(Fq2) is GF . Calculating F on the Chevalley generators in B:

F (xα(t)) = nw0(Frk(xα(t))t)−1n−1

w0

=

0 0 10 −1 01 0 0

1 0 0−tq 1 00 0 1

0 0 10 −1 01 0 0

= xβ(tq),

F (xβ(t)) = xα(tq), (by similiar reasoning)

F (xα+β(t)) =

0 0 10 −1 01 0 0

1 0 00 1 0−tq 0 1

0 0 10 −1 01 0 0

= xα+β(−tq)

for t ∈ Fp. So the permutation ρ : R+ → R+ corresponding to F is

ρ(α) = β, ρ(β) = α, ρ(α + β) = α + β.

Also,

F (s1) = F (n−11 )T = n−1

2 T = s2,

F (s2) = F (n−12 )T = n−1

1 T = s1,

so that

F (s1s2) = s2s1, F (s2s1) = s1s2, F (s1s2s1) = s2s1s2 = s1s2s1,

and F has only one orbit on S. Hence

W F = 1, s1s2s1.The twisted root system is

RF =

α + β

2, α + β,−(α + β),−α + β

2

.

By Theorem 3.1.10,

GF = BF t UFs1s2s1,−ns1s2s1B

F .

We have

R/ ∼= a,−a where a = α, β, α + β.So

UFs1s2s1,− = (XαXβXα+β)F .

If x = xα(t)xβ(v)xα+β(u) ∈ XFa then

F (x) = F (xα(t)xβ(v)xα+β(u))

= xβ(tq)xα(vq)xα+β(−uq)= xα(vq)xβ(tq)xα+β(−uq − vqtq).

55

So F (x) = x if and only if

v = tq and u = −uq − vqtq.Hence

XFa = xα(t)xβ(tq)xα+β(u) | t, u ∈ Fq2 such that tqt+ u+ uq = 0.

Hence

UFs1s2s1,−n

−11 n−1

2 n−11 BF

= xα(t)xβ(tq)xα+β(u)n−11 n−1

2 n−11 BF | t, u ∈ Fq2 such that tqt+ u+ uq = 0.

=

1 t u+ ttq

0 1 tq

0 0 1

0 0 10 −1 01 0 0

BF

∣∣∣∣∣∣t, u ∈ Fq2 such that tqt+ u+ uq = 0

=

u+ ttq −t 1

tq −1 01 0 0

BF

∣∣∣∣∣∣t, u ∈ Fq2 such that tqt+ u+ uq = 0

3.2.2. The special orthogonal group of minus type PΩ−4 (Fq). Let G = PΩ4(Fp), let

T be the subgroup of diagonal matrices in G, let B the subgroup of upper triangular matricesin G, and let q = pk (see Example 2.5.12). Recall that

G = x ∈ PSL4(Fp) | xtJx = J where J =

0 0 0 10 0 1 00 1 0 01 0 0 0

.Define F : G→ G by

F (x) = A−1(Frk(x−1)t)A where A =

0 0 0 10 1 0 00 0 1 01 0 0 0

.The function F is a well defined automorphism since if x ∈ G then

F (x)tJF (x) = (A−1Frk(x−1)tA)tJ(A−1Frk(x−1)tA)

= AFrk(x−1)AJAFrk(x−1)tA

= AFrk(x−1)JFrk(x−1)tA

= AFrk(x−1J(x−1)t)A

= AF kr ((xtJx)−1)A

= AFrk(J)A

= AJA

= J

so that F (x) ∈ G. If x ∈ G then F 2(x) = Fr2k(x). Hence F is a Steinberg endomorphism andGF ⊆ PΩ4(Fq2). Calculating F on the Chevalley generators in B:

F (xε1−ε2(t)) = xε1+ε2(tq), and F (xε1+ε2(t)) = xε1−ε2(t

q).

So the permutation ρ : R+ → R+ corresponding to F is

ρ(ε1 − ε2) = ε1 + ε2, ρ(ε1 + ε2) = ε1 − ε2.

Also,

F (sε1−ε2) = sε1+ε2 , F (sε1+ε2) = sε1−ε2 ,

56

and F has only one orbit on S. Let t0 = sε1−ε2sε1+ε2 so that

W F = 1, t0 with t20 = 1.

We have

ε1 − ε2 =1

2(ε1 − ε2 + ε1 + ε2) = ε1 = ε1 + ε2.

So the twisted root system is

RF = −ε1, ε1.

By Theorem 3.1.10,

GF/BF = BF t UFt0,−nt0B

F .

We have

R/ ∼= ±a where a = ε1 − ε2, ε1 + ε2.

By Proposition 3.1.11, UFt0,− = XF

a . If x = xε1−ε2(t)xε1+ε2(u) ∈ Xa then

F (x) = xε1−ε2(tq)xε1+ε2(u

q)

= xε1+ε2(uq)xε1−ε2(t

q).


t = uq and u = tq.

So

XFa = xε1−ε2(t)xε1+ε2(t

q) | t ∈ Fq2.

Hence

UFt0,−nt0B

F

= xε1−ε2(t)xε1+ε2(tq)n−1

ε1−ε2n−1ε1+ε2

BF | t ∈ Fq2

=

1 t 0 00 1 0 00 0 1 −t0 0 0 1

1 0 tq 00 1 0 −tq0 0 1 00 0 0 1

0 −1 0 01 0 0 00 0 0 10 0 −1 0

0 0 −1 00 0 0 11 0 0 00 −1 0 0

BF

∣∣∣∣∣∣∣∣t ∈ Fq2

=

1 t tq −tq+1

0 1 0 −tq0 0 1 −t0 0 0 1

0 0 0 −10 0 −1 00 −1 0 0−1 0 0 0

BF

∣∣∣∣∣∣∣∣t ∈ Fq2

=

tq+1 −tq −t −1tq 0 −1 0t −1 0 0−1 0 0 0

BF

∣∣∣∣∣∣∣∣t ∈ Fq2

.

By Proposition 3.3.1, we have GF = G〈,〉 where 〈, 〉 : F4q2 × F4

q2 → Fq2 is defined by

〈[u1, u2, u−2, u−1] , [v1, v2, v−2, v−1]〉

= [u1, u2, u−2, u−1]A

vq1vq2vq−2

vq−1

where A =

0 0 0 10 1 0 00 0 1 01 0 0 0

= u−1v

q1 + u2v

q2 + u−2v

q−2 + u1v

q−1.

57

Follow [Car89, Theorem 14.5.2], choose λ ∈ Fq2 such that λ generates F×q2 . Let

S =

1 0 0 00 1 −λ 00 1 −λq 00 0 0 1

.Then

StJS =

0 0 0 10 2 −λ− λq 00 −λ− λq 2λλq 01 0 0 0

Defining the quadratic form Q : F4

q → Fq by

Q([u1, u2, u−2, u−1])

= [u1, u2, u−2, u−1]

0 0 0 10 2 −λ− λq 00 −λ− λq 2λλq 01 0 0 0

u1

u2

u−2

u−1

= 2u−1u1 + 2u2

2 + 2(−λq − λ)u−2u2 + 2λλqu2−2.

Define

PΩ−4 (Fq, Q) = M ∈ Mat4×4(Fq) | if v ∈ F4q then Q(Mv) = Q(v)

= M ∈ Mat4×4(Fq) |M tStJSM = StJS.There are two things to show:

(1) The map

ϕ : PΩ4(Fp) −→ PΩ4(Fp, Q)M 7−→ S−1MS,

is an isomorphism.(2) ϕ((PΩ4(Fp)F ) = PΩ4(Fq, Q). This is the same as showing that if T ∈ PΩ4(Fp)F then

F kp (ϕ(T )) = ϕ(T )

Conceptually, showing (1) and (2) shows that the following diagram makes sense:

PΩ4(Fp)ϕ−→∼

PΩ−4 (Fp, Q)

⊆ ⊆

PΩ4(Fp)Fϕ−→∼

PΩ−4 (Fq, Q)

Note that

S−1 =

1 0 0 00 −λq

λ−λqλ

λ−λq 00 −1

λ−λq1

λ−λq 00 0 0 1

and S = AS.

(1) Let M ∈ PΩ4(Fp). Then

(S−1MS)tStJS(S−1MS) = StM t(St)−1StJSS−1MS

= StM tJMS

= StJS (since M tJM = J).

Hence S−1MS ∈ PΩ−4 (Fp, Q).

58

(2) Let M ∈ PΩ4(Fp)F . Then

ϕ(M) = S−1MS

= S−1AMA−1S

= S−1MS

= ϕ(M).

So ϕ(M) ∈ PΩ4(Fq, Q).

Now

ϕ(UFt0,−nt0B

F )

= ϕ(xε1−ε2(t)xε1+ε2(tq))ϕ(n−1

ε1−ε2n−1ε1+ε2

)ϕ(BF ) | t ∈ Fq2

=

ϕ

1 t tq −tq+1

0 1 0 −tq0 0 1 −t0 0 0 1

ϕ

0 0 0 −10 0 −1 00 −1 0 0−1 0 0 0

ϕ(BF )

∣∣∣∣∣∣∣∣t ∈ Fq2

=

1 t+ tq −tλ− tqλq −tq+1

0 1 0 tqλq−tλλ−λq

0 0 1 tq−tλ−λq

0 0 0 1

0 0 0 −10 0 −1 00 −1 0 0−1 0 0 0

ϕ(BF )

∣∣∣∣∣∣∣∣t ∈ Fq2

=

tq+1 tλ+ tqλq −t− tq −1−tqλq+tλλ−λq 0 −1 0−tq+tλ−λq −1 0 0−1 0 0 0

ϕ(BF )

∣∣∣∣∣∣∣∣t ∈ Fq2

.

3.2.3. The special unitary group SU4(Fq2). Let G = SL4(Fp), let T be the subgroup ofdiagonal matrices in G, let B the subgroup of upper triangular matrices in G, and let q = pk.Define F : G→ G by

F (x) = nw0(Frk(x)t)−1n−1

w0

where

nw0 = n−1α−1

n−1α1n−1α0n−1α−1

n−1α1n−1α0

=

0 0 0 −10 0 1 00 −1 0 01 0 0 0

.If x ∈ G then F 2(x) = Fr2k(x). Hence F is a Steinberg endomorphism, and GF ⊆ SL4(Fq2).By calculation,

F (xα−1(t)) = xα1(tq), F (xα0(t)) = xα0(t

q), F (xα1(t)) = xα−1(tq),

F (xα−1+α0(t)) = xα0+α1(−tq), F (xα0+α1(t)) = xα−1+α0(−tq),F (xα−1+α0+α1(t)) = xα−1+α0+α1(t

q).

for t ∈ Fp. So the permutation ρ : R+ → R+ corresponding to F is

ρ(α−1) = α1, ρ(α0) = α0, ρ(α1) = α−1,

ρ(α−1 + α0) = α0 + α1, ρ(α0 + α1) = α−1 + α0,

ρ(α−1 + α0 + α1) = α−1 + α0 + α1.

Also,

F (s−1) = s1, F (s0) = s0, F (s1) = s−1

59

so that S has F -orbits s−1, s1, s0. We know

Ws0 = 1, s0,Ws−1,s1 = 1, s−1, s1, s−1s1.

Let

t0 = s0 and t1 = s−1s1,

so

nt0 =

1 0 0 00 0 1 00 −1 0 00 0 0 1

, nt1 =

0 1 0 0−1 0 0 00 0 0 10 0 −1 0

(3.1)

By Proposition 3.1.8, W F is generated by t0, t1 so that

W F = 1, t0, t1, t0t1, t1t0, t0t1t0, t1t0t1, t0t1t0t1.We have

α0 = α0

α−1 =1

2(α−1 + α1) = α1

α−1 + α0 =1

2α−1 + α0 +

1

2α1 = α0 + α1

α−1 + α0 + α1 = α−1 + α0 + α1.

So the twisted root system is

RF = ±α0,1

2(α−1 + α1),

1

2α−1 + α0 +

1

2α1, α−1 + α0 + α1.

By Theorem 3.1.10,

GF/BF = BF t UFt0,−nt0B

F t UFt1,−nt1B

F t UFt0t1,−nt0t1B


F

t UFt0t1t0,−nt0t1t0B

F t UFt1t0t1,−nt1t0t1B

F t UFt0t1t0t1,−nt0t1t0t1B

F .

We have

R/ ∼= ±a, b, c, dwhere

a = α0, b = α−1, α1,c = α−1 + α0, α0 + α1, d = α−1 + α0 + α1.

By Proposition 3.1.11,

UFt0,− = XF

a , UFt1,− = XF

b , UFt0t1,− = XF

aXFc , U

Ft1t0,− = XF

b XFd

UFt0t1t0,− = XF

aXFc X

Fd , U

Ft1t0t1,− = XF

b XFc X

Fd , U

Ft0t1t0t1,− = XF

aXFb X

Fc X

Fd .

Calculating the root subgroups:

XFa = xa(t) = xα0(t) | t ∈ Fq =

1 0 0 00 1 t 00 0 1 00 0 0 1

∣∣∣∣∣∣∣∣t ∈ Fq

,

XFb = xb(u) = xα−1(u)xα1(u

q) | u ∈ Fq2 =

1 u 0 00 1 0 00 0 1 uq

0 0 0 1

∣∣∣∣∣∣∣∣u ∈ Fq2

, (3.2)

60

XFc = xc(u) = xα−1+α0(u)xα0+α1(−uq) | u ∈ Fq2 =

1 0 u 00 1 0 −uq0 0 1 00 0 0 1

∣∣∣∣∣∣∣∣u ∈ Fq2

,

XFd = xd(t) = xα−1+α0+α1(t) | t ∈ Fq =

1 0 0 t0 1 0 00 0 1 00 0 0 1

∣∣∣∣∣∣∣∣t ∈ Fq

.

Calculating the points of the Schubert cells:

BF = BF

UFt0,−nt0B

F = xα0(t)n−1α0BF | t ∈ Fq

=

1 0 0 00 t −1 00 1 0 00 0 0 1

BF

∣∣∣∣∣∣∣∣t ∈ Fq

UFt1,−nt1B

F = xα−1(u)xα1(uq)n−1

α−1n−1α1BF | u ∈ Fq2

=

u −1 0 01 0 0 00 0 uq −10 0 1 0

BF

∣∣∣∣∣∣∣∣u ∈ Fq2

UFt0t1,−nt0t1B

F = xα0(t)xα−1+α0(u)xα0+α1(−uq)n−1α0n−1α−1

n−1α1BF | t ∈ Fq, u ∈ Fq2

=

u −1 0 0t 0 −uq 11 0 0 00 0 1 0

BF

∣∣∣∣∣∣∣∣t ∈ Fq, u ∈ Fq2

UFt1t0,−nt1t0B

F = xα−1(u)xα1(uq)xα−1+α0+α1(t)n

−1α−1

n−1α1n−1α0BF | t ∈ Fq, u ∈ Fq2

=

u t 1 01 0 0 00 uq 0 −10 1 0 0

BF

∣∣∣∣∣∣∣∣t ∈ Fq, u ∈ Fq2

UFt0t1t0,−nt0t1t0B

F = xα0(t)xα−1+α0(u)xα0+α1(−uq)xα−1+α0+α1(t′) (3.3)

n−1α0n−1α−1

n−1α1n−1α0BF | t, t′ ∈ Fq, u ∈ Fq2

=

u t′ 1 0t −uq 0 11 0 0 00 1 0 0

BF

∣∣∣∣∣∣∣∣t, t′ ∈ Fq, u ∈ Fq2

UFt1t0t1,−nt1t0t1B

F = xα−1(u)xα1(uq)xα−1+α0(u

′)xα0+α1(−u′q)xα−1+α0+α1(t)

n−1α−1

n−1α1n−1α0n−1α−1

n−1α1BF | t ∈ Fq, u, u′ ∈ Fq2

=

t− u′qu −u −u′ −1−u′q −1 0 0uq 0 −1 01 0 0 0

BF

∣∣∣∣∣∣∣∣t ∈ Fq, u, u′ ∈ Fq2

UFt0t1t0t1,−nt0t1t0t1B

F = xα0(t)xα−1(u)xα1(uq)xα−1+α0(u

′)xα0+α1(−u′q)xα−1+α0+α1(t′)

n−1α0n−1α−1

n−1α1n−1α0n−1α−1

n−1α1BF | t, t′ ∈ Fq, u, u′ ∈ Fq2

61

=

t′ − u′qu −u u −1−u′q + tuq 1 0 0

uq −1 0 01 0 0 0

BF

∣∣∣∣∣∣∣∣t, t′ ∈ Fq, u, u′ ∈ Fq2

By Proposition 3.3.1, we have GF = G〈,〉 where 〈, 〉 : F4

q2 × F4q2 → Fq2 is defined by

〈u, v〉 = ut

0 0 0 −10 0 1 00 −1 0 01 0 0 0

v= u2v

q−2 − u1v

q−1 + u−1v

q1 − u−2v

q2.

Note that 〈u, v〉 = −(〈v, u〉)q. Let ν ∈ Fq2 such that ν + νq = 0. Define 〈u, v〉′ = ν〈u, v〉. Then〈u, v〉′ = (〈v, u〉′)q so that 〈, 〉′ is Hermitian, and by Proposition 3.3.1 we have

GF = G〈,〉 = G〈,〉′.

We now link GF to the ’usual’ unitary group. Define

SU4(Fq2) = x ∈ SL4(Fq2) | xtx = 1.

where : SL4(Fq2)→ SL4(Fq2) is defined by xij = xqij. Then

GF ∼= SU4(F2q)

where the isomorphism is

Φ: GF −→ SU4(Fq2)x 7−→ gxg−1

and

g =

1 0 0 λ0 1 λ 00 −µ µλq 0−µ 0 0 µλq

, and λ ∈ Fq2 , µ ∈ F×q2 ,

is defined by λ+ λq = 1 and µµq = −1 (see Appendix A.1.6).

3.2.4. The Suzuki group 2C2. Let G = Sp4(F2) with the standard maximal torus T andstandard Borel subgroup B, and let q = 22e+1. Following [Car89, Lemma 14.1.1], define

θ : F2 → F2 by θ(c) = c2e .

Define F : G→ G on generators by

F (xε1−ε2(t)) = x2ε2(θ(t2)), F (x2ε2(t)) = xε1−ε2(θ(t)),

F (xε1+ε2(t)) = x2ε1(θ(t2)), F (x2ε1(t)) = xε1+ε2(θ(t)),

and similarly for the the xα where α ∈ −R+. To show that F is a well-defined isomorphism,we use [Car89, Proposition 12.2.1] which says that it suffices to check relations (R1) and (R2)from Chapter 2, Section 2.2 and the relation hα(t1)hα(t2) = hα(t1t2) are preserved by F (seealso [Car89, Proposition 12.3.3]). The key calculations are checking that the Relations (2.4) inExample 2.5.10 are preserved by F . In other words, we need to show:

(1) F (xε1−ε2(t))F (xε1+ε2(u)) = F (xε1−ε2(u))F (xε1+ε2(t))F (xε1−ε2(2tu)).(2) F (xε1−ε2(t))F (x2ε2(u)) = F (x2ε2(u))F (xε1−ε2(t))F (xε1+ε2(tu))F (x2ε1(−t2u)).

The calculations are as follows:

62

(1)

F (xε1−ε2(t))F (xε1+ε2(u)) = x2ε2(θ(t2))x2ε1(θ(u

2))

= x2ε1(θ(u2))x2ε2(θ(t

2))

= F (xε1−ε2(u))F (xε1+ε2(t))

= F (xε1−ε2(u))F (xε1+ε2(t))F (xε1−ε2(2tu)).

(2)

F (xε1−ε2(t))F (x2ε2(u)) = xε1−ε2(θ(u))x2ε2(θ(t2))x2ε1(θ(t

2)θ(u2))xε1+ε2(θ(t2)θ(u))

= xε1−ε2(θ(u))x2ε2(θ(t2))xε1+ε2(θ(t

2)θ(u))x2ε1(θ(u2t2))

x2ε1(θ(t2)θ(u2))xε1+ε2(θ(t

2)θ(u))(using 2.2.1)

= x2ε2(θ(t2))xε1−ε2(θ(u))

= F (xε1−ε2(t))F (x2ε2(u)).

Hence F is an isomorphism. For all α ∈ R we have

F 2(xα(t)) = xα(θ(t2)) = Fre+1(xα(t))

for t ∈ F2. So F is a Steinberg endomorphism, and GF ⊆ Sp4(F22e+1). The Suzuki group is GF .The permutation ρ : R+ → R+ corresponding to F is

ρ(ε1 − ε2) = 2ε2, ρ(2ε2) = ε1 − ε2,

ρ(ε1 + ε2) = 2ε1, ρ(2ε1) = ε1 + ε2.

Letting s1 = sε1−ε2 and s2 = s2ε2 , we know

W = 1, s1, s2, s1s2, s2s1, s1s2s1, s2s1s2, s1s2s1s2.Then

F (s1) = F (n−1ε1−ε2T ) = F (n−1

ε1−ε2)T = n−12ε2T = s2,

and similiarly F (s2) = s1. So that S has only one F -orbit. By Proposition 3.1.8,

W F = 1, s1s2s1s2.Calculating α for α ∈ R+:

ε1 − ε2 =1

2(ε1 − ε2 + 2ε2) =

1

2(ε1 + ε2)

2ε2 =1

2(2ε2 + ε1 − ε2) =

1

2(ε1 + ε2)

ε1 + ε2 =1

2(ε1 + ε2 + 2ε1) =

1

2(3ε1 + ε2)

2ε1 =1

2(2ε1 + ε1 + ε2) =

1

2(3ε1 + ε2),

so the twisted root system is

RF = ±1

2(ε1 + ε2),

1

2(3ε1 + ε2).

By Theorem 3.1.10,

GF = BF t UFs1s2s1s2,−ns1s2s1s2B

F .

We have

R/ ∼= ±a, bwhere a = ε1 − ε2, 2ε2, b = ε1 + ε2, 2ε1. By Lemma 2.3.1,

UFs1s2s1s2,− = (X2ε2Xε1−ε2X2ε1Xε1+ε2)

F .

63

Let

x = x2ε2(t1)xε1−ε2(t2)x2ε1(t3)xε1+ε2(t4).

Then

F (x) = xε1−ε2(θ(t1))x2ε2(θ(t22))xε1+ε2(θ(t3))x2ε1(θ(t

24))

= x2ε2(θ(t22))xε1−ε2(θ(t1))xε1+ε2(θ(t1)θ(t22))x2ε1(θ(t

21t

22))xε1+ε2(θ(t3))x2ε1(θ(t

24))

= x2ε2(θ(t22))xε1−ε2(θ(t1))x2ε1(θ(t

21t

22 + t24))xε1+ε2(θ(t1)θ(t22) + θ(t3)).


t1 = θ(t22), t2 = θ(t1),

t3 = θ(t21t22 + t24), t4 = θ(t1)θ(t22) + θ(t3).

Letting t1 ∈ Fq we have t2 = θ(t1). Letting t3 ∈ Fq, we have t4 = t1θ(t1) + θ(t3). Hence

UFs1s2s1s2,− = x2ε2(t1)xε1−ε2(θ(t1))x2ε1(t3)xε1+ε2(t1θ(t1) + θ(t3)) | t1, t2 ∈ Fq.

Note that

ns1s2s1s2 = n−11 n−1

2 n−11 n−1

2 =

0 0 0 10 0 1 00 1 0 01 0 0 0

,and

x2ε2(t1)xε1−ε2(θ(t1))x2ε1(t3)xε1+ε2(t1θ(t1) + θ(t3))

=

1 θ(t1) t1θ(t1) + θ(t3) t3 + t1θ(t

21) + θ(t1t3)

0 1 t1 θ(t3)0 0 1 θ(t1)0 0 0 1

.Hence

UFs1s2s1s2,−ns1s2s1s2B

F

=

1 θ(t1) t1θ(t1) + θ(t3) t3 + t1θ(t21) + θ(t1t3)

0 1 t1 θ(t3)0 0 1 θ(t1)0 0 0 1

0 0 0 10 0 1 00 1 0 01 0 0 0

BF

∣∣∣∣∣∣∣∣t1, t2 ∈ Fq2

=

t3 + t1θ(t

21) + θ(t1t3) t1θ(t1) + θ(t3) θ(t1) 1θ(t3) t1 1 0θ(t1) 1 0

1 0 0 0

BF

∣∣∣∣∣∣∣∣t1, t3 ∈ Fq2

3.3. From sesquilinear forms to Steinberg endormorphisms

Let V be a vector space, G a subgroup of GL(V ), and 〈, 〉 : V × V → F be a σ-sesquilinearform. Define

G〈,〉 = x ∈ G | if u, v ∈ V then 〈xu, xv〉 = 〈u, v〉.

Proposition 3.3.1. Suppose A ∈ End(V ) is the matrix corresponding to 〈, 〉 so that

〈u, v〉 = utAσ(v),

for u, v ∈ V . Then

GF = G〈,〉.

64

where F : G→ G is defined by

F (x) = (A−1)t(σ(x)−1)t)(At).

Proof.

GF = x ∈ G | F (x) = x= x ∈ G | (A−1)t(γ(x)−1)t)(At) = x= x ∈ G | Aγ(x)−1A−1 = xt= x ∈ G | Aγ(x)A−1 = (xt)−1= x ∈ G | xtAγ(x) = A= x ∈ G | if u, v ∈ V then utxtAγ(x)σ(v) = utAσ(v)= x ∈ G | if u, v ∈ V then (xu)tAσ(xv) = utAσ(v)= x ∈ G | if u, v ∈ V then 〈xu, xv〉 = 〈u, v〉= G〈,〉.

Example 3.3.2. Let G and F : G → G be defined as in Section 3.2.1, so that GF ∼=SU3(Fq2). Then G〈,〉 = GF where 〈, 〉 : Fp

4 × Fp4 → Fp is defined by⟨ u−1

u0

u1

, v−1

v0

v1

⟩ =[u−1 u0 u1

] 0 0 10 −1 01 0 0

vq−1

vq0vq1

= u1v

q−1 − u0v

q0 + u−1v

q1

Since 〈u, v〉 = 〈v, u〉q, the form 〈, 〉 is 1-Hermitian.

Example 3.3.3. Let G and F : G→ G be defined as in Section 3.2.2 so that GF ∼= PΩ−4 (Fq).Then G〈,〉 = GF where 〈, 〉 : F4

q2 × F4q2 → Fq2 is defined by

〈[u1, u2, u−2, u−1] , [v1, v2, v−2, v−1]〉

= [u1, u2, u−2, u−1]

0 0 0 10 1 0 00 0 1 01 0 0 0

vq1vq2vq−2

vq−1

= u−1v

q1 + u2v

q2 + u−2v

q−2 + u1v

q−1.

Since 〈u, v〉 = 〈v, u〉q, the form 〈, 〉 is 1-Hermitian.

Example 3.3.4. Let G and F : G→ G be defined as in Section 3.2.3 so that GF ∼= SU4(Fq2).Then G〈,〉 = GF where 〈, 〉 : Fp

4 × Fp4 → Fp is defined by⟨

u−2

u−1

u1

u2

,v−2

v−1

v1

v2

⟩

=[u−2 u−1 u1 u2

] 0 0 0 −10 0 1 00 −1 0 01 0 0 0

vq−2

vq−1

vq1vq2

= u2v

q−2 − u1v

q−1 + u−1v

q1 − u−2v

q2

Since 〈u, v〉 = −〈v, u〉q, the form 〈, 〉 is (−1)-Hermitian.

65

CHAPTER 4

Tying things together

This chapter constitutes our main contribution to the research literature. We describe inSection 4.1.4 a relationship between lattices and incidence structures (from finite geometry),and flag varieties (from representation theory). Through this relationship, the classical ovoidin the Hermitian variety H(3, q2) (see Proposition 1.5.5) appears as points of the ‘twisted’flag variety GF/BF . In 4.2.4, we explicitly describe these points using the single BF -cosetdecomposition given in [PRS09, Theorem 4.1]. The rational normal curve, another exampleof an ovoid, appears in a similar way, and we discuss this in section 4.2.1. In each of theseexamples we determine which points of the ovoid lie in each Schubert cell of the correspondingflag variety.

In Section 4.3, we define an incidence structure for each Schubert cell and each pair ofmaximal parabolic subgroups of the Chevalley group. This provides a way of analyzing theSchubert cell using the viewpoint of projective geometry. Then, in pursuit of the question ofwhat causes the “thinness” that distinguishes ovoids, we prove the main theorem (Theorem4.3.3) which is a computation of the “thickness” of the incidence geometries that come fromSchubert cells. This allows us to isolate basic examples of Schubert cells which are thin. Wehope that future work will provide a full classification of ovoids that arise from Schubert cells,Schubert varieties and hyperplane sections of Schubert varieties.

4.1. (Generalized) flag varieties, their lattices and incidence structures

In this section, let G be a (twisted or untwisted) Chevalley group, let G ⊇ B ⊇ T where Bis a Borel subgroup of G and T is a maximal torus. Let W be the corresponding Weyl groupwith Coxeter generators s1, s2, . . . , sn.

Let k ∈ 1, 2, . . . n. The standard maximal k-parabolic subgroup of G is

Pk =⊔

w∈Wk

BwB, where Wk = 〈s1, . . . , sk−1, sk+1, . . . sn〉.

Remark 4.1.1. More generally, a parabolic subgroup is a (closed) subgroup P of G contain-ing B.

4.1.1. The flag variety of GL(V ). Let V be a F-vector space, and let G = GL(V ). TheGrassmannian of k-planes on V is

Gr(k, V ) = Vk ⊆ V | Vk is a subspace of V and dimVk = k ,and there is a G-action on Gr(k, V ) defined by g · Vk = gVk. If V = Fn we write Gr(k, n) inplace of Gr(k, V ). The full flag variety on V is

Fl(V ) = V• = (V0 ⊆ V1 ⊆ . . . ⊆ Vn) |Vk ∈ Gr(k, V ) ,with a G-action on Fl(V ) defined by

g · (V0 ⊆ V1 ⊆ . . . ⊆ Vn) = (gV0 ⊆ gV1 ⊆ . . . ⊆ gVn).

If V = Fn we write Fl(n) in place of Fl(V ). Let

e1, e2, . . . , en be a basis of V and Ek = F-spane1, e2, . . . , ek,for k ∈ 1, 2, . . . n. The standard flag in Fl(V ) is

E• = (0 ⊆ E1 ⊆ E2 ⊆ · · · ⊆ En−1 ⊆ V ).

67

Theorem 4.1.2. [FH91, §23.3] Let B be the subgroup of weakly upper triangular matricesin G = GL(V ). Then

B = StabG(E•) and Pk = StabG(Ek),

and the maps

ϕ : G/B −→ Fl(V )gB 7−→ gE•,

andϕk : G/Pk −→ Gr(k, V )

gPk 7−→ gEk,for k ∈ 1, 2, . . . , n,

are bijections.

Corollary 4.1.3. [FH91, §23.3] Let PG(Fn) be the lattice of subspaces of Fn (as outlinedin Chapter 1, Section 1.1.1). With the notation of Theorem 4.1.2, the maps

ϕ : G/B −→

maximal chainsin PG(Fn)

gB 7−→ gE•,

andϕk : G/Pk −→

rank k elements

in PG(Fn)

gPk 7−→ gEk,

for k ∈ 1, 2, . . . , n− 1 are isomorphisms of G-sets.

4.1.2. The flag variety of a classical group of Lie type. We now extend the theoryof the previous section, to the case that G is a classical group of Lie type (see [FH91, §23.3]). Aclassical group of Lie type is a Chevalley group G(F) whose underlying Lie algebra g is one ofthose in Examples 2.5.2 to 2.5.5. Let 〈·, ·〉 be the bilinear form corresponding to G(F). Recallthat a subspace W ⊆ V is isotropic if W ⊆ W⊥ (see A.2.12). The isotropic Grassmannian ofk-planes on V is

Gr〈·,·〉(k, V ) = Vk ⊆ V | Vk is an isotropic subspace of V and dimVk = k.

The full isotropic flag variety on V is

Fl〈,〉(V ) = V• = (0 ⊆ V1 ⊆ V2 ⊆ · · · ⊆ Vm) | Vk ∈ Gr〈,〉(k, V ).

where m is the maximal dimension of a totally isotropic subspace of V . Let

Ei = F− spane1, e2, . . . , ei, for i ∈ 1, 2, . . . ,m,

where the elements ek ∈ V are pairwise orthogonal isotropic vectors as defined in Examples2.5.2 to 2.5.5. Then Ei is an i-dimensional totally isotropic subspace. The standard flag inFl〈,〉(V ) is

E• = (0 ⊆ E1 ⊆ E2 ⊆ · · · ⊆ Em).

Theorem 4.1.4. [FH91, §23.3] Let B be the subgroup of weakly upper triangular matricesin G. Then

B = StabG(E•), and Pk = StabG(Ek).

Furthermore, the maps

ϕ : G/B −→ Fl〈·,·〉(V )gB 7−→ gE•,

andϕk : G/Pk −→ Gr〈·,·〉(k, V )

gPk 7−→ gEk,for k ∈ 1, 2, . . . ,m,

are bijections.

Corollary 4.1.5. [FH91, §23.3] Let P (V, 〈·, ·〉) be the semilattice of totally isotropic sub-spaces (as defined in Chapter 1, Section 1.1.3). With the notation of Theorem 4.1.4, the maps

ϕ : G/B −→

maximal chainsin P (V, 〈·, ·〉)

gB 7−→ gE•,

andϕk : G/Pk −→

rank k elements

in P (V, 〈·, ·〉)

gPk 7−→ gEk,

for k ∈ 1, 2, . . . ,m are bijections.

68

4.1.3. Generalised flag varieties. Let G be a Chevalley group, twisted or untwisted.The

generalised flag variety is G/B and the generalised Grassmannians are the G/Pk.

These are shown to be projective varieties in [CMS95, Part III, (7.4)(i) and (7.5)], [Bor91,Theorem 11.1 and Corollary 11.2], and [FH91, Claim 23.52]. The defining equations in the caseG = SLn(F) can be found in [Ful97, Chapter 9, Lemma 1].

Let G = G(F) be a Chevalley group. Then G is a subgroup of GL(VF) where VF = F⊗ZV andV is a faithful finite dimensional representation of the underlying Lie algebra g of G (see Section2.2). From the theory of highest weight representations (see [CMS95], [Ser87] or [Hum72]) Vis a highest weight representation with weight λ and highest weight vector v0. Since the groupG acts on VF,

G acts on P(VF) by g · [v] = [gv].

Furthermore,

bv0 = λ(b)v0,

for b ∈ B, so that

B ⊆ StabG([v0]),

and so StabG([v0]) is a parabolic subgroup of G.

Proposition 4.1.6. Let F : G→ G be a Steinberg endomorphism. Then

StabGF ([v0]) = StabG([v0])F .

Proof.

LHS = StabGF ([v0])

= g ∈ GF | [gv0] = [v0]= g ∈ G | [gv0] = [v0] and F (g) = g= g ∈ G | g ∈ StabG([v0]) and F (g) = g= StabG([v0])F

= RHS.

Corollary 4.1.7. BF ⊆ StabGF ([v0]).

Proof. Since B ⊆ StabG([v0]), BF ⊆ StabG([v0])F . But StabG([v0])F = StabGF ([v0]), soBF ⊆ StabGF ([v0]).

4.1.4. The incidence structures associated to (generalised) flag varieties. Let Piand Pj be parabolic subgroups. Define

pi : G/B → G/PigB 7→ gPi

andpj : G/B → G/Pj

gB 7→ gPj.

The (i, j)-incidence structure associated with G is the triple

(G)ij = (Pi,Pj, Iij)where

Pi = G/Pi,

Pj = G/Pj,

Iij =

(gPi, hPj) ∈ Pi × Pj | there exists kB∈G/B such thatpi(kB)=gPi and pj(kB)=hPj

.

The following two Propositions follow from Theorem 4.1.2 and Theorem 4.1.4.

69

Proposition 4.1.8. Let G = G(PG(Fn+1)) = (P ,L, I) be the projective incidence structureassociated with PG(Fn+1) (see Chapter 1, Section 1.2.1) and let e1, e2, . . . , en+1 be a basis ofFn+1q . Let G = GLn+1(F), let P1 and P2 be the corresponding 1st and 2nd standard maximal

parabolic subgroups, and let (G)12 = (P1,P2, I12) be the associated (1, 2)-incidence structure.Then the map ϕ : P1 t P2 → P t L defined by

ϕ(gP1) = gE1, ϕ(hP2) = hE2,

is an isomorphism of incidence structures from (G)12 to G(PG(Fn+1)).

Proof. The following statements need to be proven:

(1) ϕ is well-defined.(2) ϕ is a homomorphism of incidence structures.(3) ϕ is a bijection.(4) ϕ−1 is a homomorphism of incidence structures.

(1) Suppose gP1 = g′P1. Then g−1g′ ∈ P1. By Theorem 4.1.2, P1 = StabG(E1). Sog−1g′E1 = E1. So gE1 = g′E1. So ϕ(gP1) = ϕ(g′P1). Similiarly, if hP2 = h′P2 thenϕ(hP2) = ϕ(hP2). Hence ϕ is well-defined.

(2) Suppose (gP1, hP2) ∈ I12. Then there exists kB ∈ G/B such that p1(kB) = gP1

and p2(kB) = hP2. So kP1 = gP1 and kP2 = hP2. So k−1g ∈ P1 and k−1h ∈ P2.By Theorem 4.1.2, P1 = StabG(E1) and P2 = StabG(E2). So k−1gE1 = E1 andk−1hE2 = E2. So gE1 = kE1 and hE2 = kE2. But kE1 ⊆ kE2, so gE1 ⊆ hE2. Soϕ(gP1) ⊆ ϕ(hP2). Hence (ϕ(gP1), ϕ(hP2)) ∈ I.

(3) Suppose ϕ(gP1) = ϕ(g′P1). Then gE1 = g′E1. Then g−1g′E1 = E1. But by Theorem4.1.2, P1 = StabG(E1). So g−1g′ ∈ P1. So gP1 = g′P1. Similiarly, if ϕ(hP2) = ϕ(h′P2)then hP2 = h′P2. Hence ϕ is injective.

Suppose p ∈ P . Since G is transitive on P , there exists g ∈ G such that gE1 = p.So ϕ(gP1) = p. Similiarly, if l ∈ L then there exists h ∈ G such that ϕ(hP2) = l.Hence ϕ is surjective.

(4) Suppose (p, l) ∈ I. So p ⊆ l. Then there exists g, h ∈ G such that ϕ−1(p) = gP1 andϕ−1(l) = hP2. So p = gE1 and l = hE2. So gE1 ⊆ hE2. Extend this to an element ofFl(V ):

gE1 ⊆ hE2 ⊆ V3 ⊆ V4 ⊆ . . . ⊆ V.

By Theorem 4.1.2, there exists k ∈ G such that

kE• = (gE1 ⊆ hE2 ⊆ V3 ⊆ V4 ⊆ . . . ⊆ V ).

So kE1 = gE1 and kE2 = hE2. So k−1g ∈ StabG(E1) and k−1h ∈ StabG(E2). ByTheorem 4.1.2, k−1g ∈ P1 and k−1h ∈ P2. So kP1 = gP1 and kP2 = hP2. Sop1(kB) = gP1 and p2(kB) = hP2. So (gP1, hP2) ∈ I12. So (ϕ−1(p), ϕ−1(l)) ∈ I12.

The claim that G = G(PG(Fn+1)) is a projective incidence structure appears as Theorem 1.2.3,and is proven in [BR98, Theorem 2.2.1] and [BR98, Theorem 2.2.2].

Proposition 4.1.9. [Tay92, pg. 107], [Shu10, pg. 178].

(1) Let G = (P ,L, I) be the polar incidence structure associated with the classical polarsemilattice P (V, 〈·, ·〉) (see Chapter 1, Section 1.2.2). Let G = G〈,〉, and let P1 and P2

be the corresponding 1st and 2nd standard maximal parabolic subgroups, and let (G)12 =(P1,P2, I12) be the associated (1, 2)-incidence structure. Then the map ϕ : P1 t P2 →P t L defined by

ϕ(gP1) = gE1, ϕ(hP2) = hE2,

is an isomorphism from (G)12 to G.

70

(2) Let G = (P ,B, I) be the polar incidence structure associated with the classical polarsemilattice P (V, 〈·, ·〉) (see Chapter 1, Section 1.2.2). Let G = G〈,〉, and let P1 and Pmbe the corresponding 1st and mth standard maximal parabolic subgroups, where m isthe dimension of a maximal totally isotropic subspace. Let (G)1m = (P1,Pm, I1m) bethe associated (1,m)-incidence structure. Then the map ϕ : P1 t Pm → P t B definedby

ϕ(gP1) = gE1, ϕ(hPm) = hEm,

is an isomorphism from (G)1m to G.

Proof. The proof is the same as Proposition 4.1.8, except Theorem 4.1.4 is used in placeof Theorem 4.1.2. The claim that G is a polar incidence structure appears as Theorem 1.2.5,and is stated and proven in [Tay92, pg. 107, paragraph -2].

4.1.5. Schubert cells. The Schubert cells are

Xw = BwB ⊆ G/B

Xkv = BvPk ⊆ G/Pk

where w ∈ W and v ∈ W k.

Remark 4.1.10. The closures of the Schubert cells are the Schubert varieties of the projec-tive variety G(F)/B and this makes them tools in the framework of geometric representationtheory (see [BL09], [EW16], [Wil16], [Ful97, Part III], [KL72], [BL00], [Man01]).

Theorem 4.1.11. (The geometric picture for Schubert cells). Let G = GLn(C) so thatW = Sn. Let ϕ : G/B → Fl(V ), ϕk : G/Pk → Gr(k, V ) be defined as in Theorem 4.1.2. Forw ∈ W define

rp,q(w) = #i ∈ Z≥0 | 1 ≤ i ≤ p and 1 ≤ w(i) ≤ q.

Then

ϕ(Xw) = V• ∈ Fl(V )|dim(Vp ∩ Eq) = rp,q(w) for p, q ∈ 1, 2, . . . n .

Let v ∈ W k. Then

ϕk(Xkv ) = Vk ∈ Gr(k, V )|dim(Vk ∩ Eq) = rk,q(v) for q ∈ 1, 2, . . . n .

References for proof. See [Ful97, §10.2] and [Ful97, §10.5, Corollary to Proposition7]. See also [Man01] and [GH14].

The (i, j)-Schubert incidence structure is (Xw)ij = (Pi,Pj, Iij) where

Pi = pi(Xw) = BwPi,

Pj = pj(Xw) = BwPj,

Iij = (gPi, hPj) ∈ Pi × Pj | there exists kB ∈ Xw such that pi(kB) = gPi and pj(kB) = hPj.

4.2. Ovoids and (twisted) Chevalley groups

In Sections 4.2.1, 4.2.2, 4.2.3, we display (respectively) the rational normal curve, the ellipticquadric, and the Suzuki-Tits ovoid as orbits of (twisted or untwisted) Chevalley groups. Section4.2.3 builds on the work of Tits [Tit61]. Given that these key examples of ovoids can be realisedas orbits of a suitably chosen Chevalley group, it would be interesting to consider whether non-classical ovoids (see [Che04] and [Che96]) can also be realised in this way, with perhaps the roleof the Chevalley group being played by a suitably chosen pseudo-reductive group (see [CGP15]).

In Section 4.2.4 we identify the points of the classical ovoid in the Hermitian variety H(3, q2)with the points of each Schubert cell of GF/BF where GF is the unitary group, providing apowerful way of indexing the points of the ovoid.

71

4.2.1. The special orthogonal group PΩ3(F) and the rational normal curve. TheChevalley group G = PΩ3(F) is the exponential of the Lie algebra so3 on VF = F⊗Z V whereV is the standard representation with basis e−1, e0, e1 (see 2.5.11). The Chevalley generatorsof G are

xε1(t) =

1 2t t2

0 1 t0 0 1

, x−ε1(t) =

1 0 0t 1 0t2 2t 1

,nε1(t) =

0 0 t2

0 −1 0t−2 0 0

, hε1(t) =

t2 0 00 1 00 0 t−2

for t ∈ F. By Corollary 2.3.8 and Theorem 2.4.6, the Bruhat decomposition is

G/B = B tBs1B with Bs1B = xα(t)n−1ε1B ∈ G/B | t ∈ F.

The group G acts on the set P(V ) and the vector [v0] is stabilized by B, and the orbit of G on[v0] in P(V ) is

G · [v0] = G · [1 : 0 : 0] = B · [1 : 0 : 0] tBs1B · [1 : 0 : 0]

= [1 : 0 : 0] t xε1(t)n−1ε1· [1 : 0 : 0] | t ∈ F

= [1 : 0 : 0] t [t2 : t : 1] | t ∈ F

since

xε1(t)n−1ε1

=

t2 −2t 1t −1 01 0 0

.When F = Fq, the orbit G · [v0] is the rational normal curve in P(F2+1

q ) as defined in Equation(1.1), Section 1.4.

4.2.2. The special orthogonal group of minus type PΩ−4 (Fq) and the ellipticquadric. Many portions of this section is explained in more detail in Section 3.2.2. TheChevalley group G = PΩ−4 (Fq) is the exponential of the Lie algebra so4 on VFp = Fp ⊗Z V

where V is the standard representation with basis e−2, e−1, e1, e2 (see Example 2.5.12) wherev0 = e−2 is a highest weight vector. The Chevalley generators of G are

xε1−ε2(t) =

1 t 0 00 1 0 00 0 1 −t0 0 0 1

, xε1+ε2(t) =

1 0 t 00 1 0 −t0 0 1 00 0 0 1

,

x−(ε1−ε2)(t) =

1 0 0 0t 1 0 00 0 1 00 0 −t 1

, x−(ε1+ε2)(t) =

1 0 0 00 1 0 0t 0 1 00 −t 0 1

,

nε1−ε2(t) =

0 t 0 0−t−1 0 0 0

0 0 0 −t0 0 t−1 0

, nε1+ε2(t) =

0 0 t 00 0 0 −t−t−1 0 0 0

0 t−1 0 0

,

hε1−ε2(t) =

t 0 0 00 t−1 0 00 0 t 00 0 0 t−1

, hε1+ε2(t) =

t 0 0 00 t 0 00 0 t−1 00 0 0 t−1

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Let q = pk be a prime power. Define F : G→ G by

F (x) = A−1((x−1)t)A where A =

0 0 0 10 1 0 00 0 1 01 0 0 0

,so that F is an automorphism and

F (xε1−ε2(t)) = xε1+ε2(t).

Then F : R+ → R+ and F : W → W with

F (ε1 − ε2) = ε1 + ε2 and F (sε1−ε2) = sε1+ε2 ,

giving

W F = 1, t0 where t0 = sε1−ε2sε1+ε2 ,

and Bruhat decomposition

GF = BF t UFt0,−nt0B

F

where

UFt0,− = xε1−ε2(t)xε1+ε2(t) | t ∈ Fq2.

Let 〈, 〉 : F4q2 × F4

q2 → Fq2 be defined by

〈u, v〉 = utAv.

Then

GF = g ∈ PSL4(Fq2) | if u, v ∈ F4q2 then 〈gu, gv〉 = 〈u, v〉.

Let λ ∈ Fp be a generator of F×q2 (see Appendix A.1.7). Let

K = StJS =

0 0 0 10 2 −λ− λ 00 −λ− λ 2λλ 01 0 0 0

where S =

1 0 0 00 1 −λ 00 1 −λ 00 0 0 1

.Then K ∈ GL(F4

q). Let Q : F4q → Fq be defined by

Q(v) = vtKv.

Let

PΩ−4 (Fq) = x ∈ PSL4(Fq) | xtKx = K= x ∈ PSL4(Fq) | if v ∈ F4

q then Q(xv) = Q(v).Then

ϕ : GF −→ PΩ−4 (Fq),M 7−→ S−1MS

is an isomorphism. The group PΩ−4 (Fq) acts on the set P(VFp) and

ϕ(BF ) ⊆ StabPΩ−4 (Fq)([v0]).

Then

PΩ−4 (Fq) · [v0] = ϕ(GF ) · [1 : 0 : 0 : 0]

= ϕ(BF ) · [1 : 0 : 0 : 0] t ϕ(UFt0,−nt0B

F ) · [1 : 0 : 0 : 0]

= [1 : 0 : 0 : 0] t[

tt :−tλ+ tλ

λ− λ:−t+ t

λ− λ: −1

]∣∣∣∣t ∈ Fq2

73

since

ϕ(xε1−ε2(t)xε1+ε2(t)nt0) =

tt tλ+ tλ −t− t −1

−tλ+tλλ−λ 0 −1 0−t+tλ−λ −1 0 0

−1 0 0 0

.If t ∈ Fq2 then t ∈ Fq if and only if t = t. So the coordinates of the points of PΩ−4 (Fq) · [v0] areelements of Fq and PΩ−4 (Fq) · [v0] ⊆ P(F3+1

q ). The orbit PΩ−4 (Fq) · [v0] is the elliptic quadric in

P(F3+1q ) as defined in Section 1.4.

4.2.3. The Suzuki group 2C2(F22e+1) and the Suzuki-Tits ovoid. Many portions ofthis section is explained in more detail in Section 3.2.4. The Chevalley group G = Sp4(F2) isthe exponential of the Lie algebra sp4 on VF2

= F2⊗Z V where V is the standard representationof sp4 with basis e−2, e−1, e1, e2 (see Example 2.5.10) where v0 = e−2 is a highest weightvector. The Chevalley generators of G are

x2ε1(t) =

1 0 0 t0 1 0 00 0 1 00 0 0 1

, x2ε2(t) =

1 0 0 00 1 t 00 0 1 00 0 0 1

,

x−2ε1(t) =

1 0 0 00 1 0 00 0 1 0t 0 0 1

, x−2ε2(t) =

1 0 0 00 1 0 00 t 1 00 0 0 1

,

xε1−ε2(t) =

1 t 0 00 1 0 00 0 1 −t0 0 0 1

, x−ε1+ε2(t) =

1 0 0 0t 1 0 00 0 1 00 0 −t 1

,

xε1+ε2(t) =

1 0 t 00 1 0 t0 0 1 00 0 0 1

, x−ε1−ε2(t) =

1 0 0 00 1 0 0t 0 1 00 t 0 1

,

nε1−ε2(t) =

0 t 0 0−t−1 0 0 0

0 0 0 −t0 0 t−1 0

, n2ε2(t) =

1 0 0 00 0 t 00 −t−1 0 00 0 0 1

,

hε1−ε2(t) =

t 0 0 00 t−1 0 00 0 t 00 0 0 t−1

, h2ε2(t) =

1 0 0 00 t 0 00 0 t−1 00 0 0 1

.Define

θ : F2 → F2 by θ(c) = c2e .

Define F : G→ G on generators by

F (xε1−ε2(t)) = x2ε2(θ(t2)), F (x2ε2(t)) = xε1−ε2(θ(t)),

F (xε1+ε2(t)) = x2ε1(θ(t2)), F (x2ε1(t)) = xε1+ε2(θ(t)),

and similarly for the the xα where α ∈ −R+. Then GF is the Suzuki group and F : R+ → R+

with

F (ε1 − ε2) = 2ε2, F (2ε2) = ε1 − ε2,

74

F (ε1 + ε2) = 2ε1, F (2ε1) = ε1 + ε2,

and F : W → W with

F (s1) = s2 F (s2) = s1,

where s1 = sε1−ε2 and s2 = s2ε2 . So

W F = 1, s1s2s1s2

with Bruhat decomposition

GF = BF t UFs1s2s1s2,−ns1s2s1s2B

F

where

UFs1s2s1s2,− = x2ε2(t1)xε1−ε2(θ(t1))x2ε1(t3)xε1+ε2(t1θ(t1) + θ(t3)|t1, t2 ∈ F22e+1.

The group GF acts on the set P(VF2), the point [v0] is stabilised by BF , and the orbit GF on

[v0] in P(VF2) is

GF · [v0] = BF · [v0] t UFs1s2s1s2,−ns1s2s1s2B

F · [v0]

= [1 : 0 : 0 : 0] t [t3 + t1θ(t21) + θ(t1t3) : θ(t3) : θ(t1) : 1] ∈ P(V ) | t1, t3 ∈ F22e+1

since

UFs1s2s1s2,−ns1s2s1s2 =

t3 + t1θ(t

21) + θ(t1t3) t1θ(t1) + θ(t3) θ(t1) 1θ(t3) t1 1 0θ(t1) 1 0 0

1 0 0 0

BF

∣∣∣∣∣∣∣∣t1, t3 ∈ F22e+1

.

The coordinates of the points of G · [v0] are elements of F22e+1 so that GF · [v0] ⊆ P(F422e+1). The

orbit GF · [v0] is the Suzuki-Tits ovoid in P(F3+122e+1) as defined in Section 1.4.

4.2.4. The Schubert cell decomposition of the classical ovoid in H(3, q2). Manyportions of this section is explained in more detail in Section 3.2.3. The Chevalley groupG = PSL4(Fp) is the exponential of the Lie algebra sl4 on VFp = Fp⊗ZV where V is the standard

representation of sl4 with basis e−2, e−1, e1, e2 (see Example 2.5.10) where v0 = e−2 is a highestweight vector. The set of simple roots of G is ∆ = α−1 = ε−2−ε−1, α0 = ε−1−ε1, α1 = ε1−ε2.Define F : G→ G by

F (x) = nw0(xt)−1n−1

w0

where

nw0 = n−1α−1

n−1α1n−1α0n−1α−1

n−1α1n−1α0

=

0 0 0 −10 0 1 00 −1 0 01 0 0 0

.Then F : R+ → R+ with

F (α−1) = α1, F (α0) = α0, F (α1) = α−1,

F (α−1 + α0) = α0 + α1, F (α0 + α1) = α−1 + α0,

F (α−1 + α0 + α1) = α−1 + α0 + α1.

and F : W → W with

F (s−1) = s1, F (s0) = s0, F (s1) = s−1.

75

The Chevalley generators of GF corresponding to the positive roots are

xa(t) =

1 0 0 00 1 t 00 0 1 00 0 0 1

xb(u) =

1 u 0 00 1 0 00 0 1 u0 0 0 1

xc(u) =

1 0 u 00 1 0 −u0 0 1 00 0 0 1

xd(t) =

1 0 0 t0 1 0 00 0 1 00 0 0 1

for t ∈ Fq and u ∈ Fq2 . The coset representatives of the Coxeter generators of W F in N are

nt0 =

1 0 0 00 0 1 00 −1 0 00 0 0 1

, nt1 =

0 1 0 0−1 0 0 00 0 0 10 0 −1 0

where t0 = s0 and t1 = s−1s1. The group GF acts on the set P(VFp) with BF ⊆ StabGF ([v0])

and the orbit of [v0] in P(VFp) is the set of points in the Hermitian polar semilattice H(3, q2)

(see 1.1.3). Explicitly calculating the orbit,

GF · [v0] = BF t UFt0,−nt0B

F t UFt1,−nt1B



F

t UFt0t1t0,−nt0t1t0B

F t UFt1t0t1,−nt1t0t1B

F t UFt0t1t0t1,−nt0t1t0t1B

F [v0]

= (BF t xa(t)nt0BF | t ∈ Fq t xb(u)nt1BF | u ∈ Fq2

t xa(t)xc(u)nt0t1BF | t ∈ Fq, u ∈ Fq2 t xb(u)xd(t)nt1t0B

F | t ∈ Fq, u ∈ Fq2t xa(t)xc(u)xd(t

′)nt0t1t0BF | t, t ∈ Fq, u ∈ Fq2

t xb(u)xc(u′)xd(t)nt1t0t1B

F | t ∈ Fq, u, u′ ∈ Fq2t xa(t)xb(u)xc(u

′)xd(t′)nt0t1t0t1B

F | t, t′ ∈ Fq, u, u′ ∈ Fq2) · [1 : 0 : 0 : 0]

= [1 : 0 : 0 : 0] t [u : 1 : 0 : 0] | u ∈ Fq2 t [u : t : 1 : 0] | t ∈ Fq, u ∈ Fq2t [t− u′u : −u′ : u : 1] | t ∈ Fq, u, u′ ∈ Fq2t [t′ − u′u : −u′ + tu : u : 1] || t, t′ ∈ Fq, u, u′ ∈ Fq2.

Let 〈, 〉 : F4q2 × F4

q2 → Fq2 be the nondegenerate form defined by

〈u, v〉 = ut

0 0 0 −10 0 1 00 −1 0 01 0 0 0

v = u2v−2 − u1v−1 + u−1v1 − u−2v2.

It is shown in Section 3.2.3 that

GF = g ∈ G | if u, v ∈ VFq2 then〈gu, gv〉 = 〈u, v〉.

76

Let

U = Fq2

x−2

x−1

x1

x2

be a choice of nondegenerate 1-dimensional subspace (see Appendix A.2). Then U⊥ is nonde-generate and

U⊥ =

y−2

y−1

y1

y2

∈ V∣∣∣∣∣∣∣∣x2y−2 − x1y−1 + x−1y1 − x−2y2 = 0

Let

O be the set of totally isotropic 1-dimensional subspaces in U⊥.

It was shown in Proposition 1.5.5 that O is an ovoid in the polar incidence structure of H(3, q2).Another way to express O is

O = [v] ∈ P(VF) | v ∈ U⊥ ∩ (GF · [v0]).Then O has a decomposition into its points of each Schubert cell given by

O = [1 : 0 : 0 : 0] | x2 = 0 t [u : 1 : 0 : 0] | u ∈ Fq2 and x2uq − x1 = 0

t [u : t : 1 : 0] | u ∈ Fq2 , t ∈ Fq and x2uq − x1t+ x−1 = 0

t [t− u′qu : −u′q : uq : 1] | u, u′ ∈ Fq2 , t ∈ Fq and x2(t− u′uq) + x1u′q + x−1u− x−2 = 0

t [t′ − u′qu : −u′q + tuq : uq : 1]

| u, u′ ∈ Fq2 , t, t′ ∈ Fq and x2(t′ − u′uq)− x1(−u′ + tu) + x−1u− x−2 = 0.

4.3. The Thickness of Schubert Cells

Let G be a Chevalley group, let G ⊇ B ⊇ T where B is a Borel subgroup of G and T is amaximal torus, let W be the corresponding Weyl groups with Coxeter generators s1, s2, . . . , sn.Let w ∈ W and let

Xw = BwB

be the corresponding Schubert cell. Recall the projection maps are

pi : G/B → G/PigB 7→ gPi

andpj : G/B → G/Pj

gB 7→ gPj,

and the (i, j)-Schubert incidence structure is (Xw)ij = (Pi,Pj, Iij) where

Pi = pi(Xw) = BwPi,

Pj = pj(Xw) = BwPj,

Iij = (gPi, hPj) ∈ Pi × Pj | there exists kB ∈ Xw such that pi(kB) = gPi and pj(kB) = hPj.Let

R+i = α ∈ R+ | sα ∈ Wi,

R+j = α ∈ R+ | sα ∈ Wj,

and R+i,j = R+

i ∩R+j .

Recall that

R(z) = α ∈ R+ | z−1(α) /∈ R+,Wi = subgroup of W generated by s1, s2, . . . , si−1, si+1, . . . , sn,

W i = minimal length representatives of cosets in W/Wi.

77

Let

Wi,j = Wi ∩Wj,

W i,j = minimal length representatives of cosets in W/Wi,j.Then

Wi = z ∈ W | R(z) ⊆ R+i ,

W i = z ∈ W | R(z) ∩R+i = ∅,

(Wj)i,j = z ∈ W | R(z) ⊆ R+

j and R(z) ∩R+i,j = ∅.

Recall the ‘favourite’ set of coset representatives for BwB and BwPi.

Proposition 4.3.1. (1) For each w ∈ W fix a reduced decomposition w = si1 · · · si`.Then

G/B =⊔w∈W

BwB with BwB = xi1(c1)n−1i1· · ·xi`(c`)n−1

i`B | c1, . . . , c` ∈ F,

and xi1(c1)n−1i1· · ·xi`(c`)n−1

i`| c1, . . . , c` ∈ F a set of representatives of the cosets of

B in BwB.(2) For each z ∈ W i fix a reduced decomposition z = si1 · · · sik . Then

G/Pi =⊔z∈W i

BzPi with BzPi = xi1(c1)n−1i1· · ·xik(ck)n−1

ikPi | c1, . . . , ck ∈ F,

and xi1(c1)n−1i1· · ·xik(ck)n−1

ik| c1, . . . , ck ∈ F a set of representatives of the cosets of

Pi in BzPi.

Proof. The proof for BwB is given in Theorem 2.4.6, and the proof for BwPi is similiar.See also [Ste67, Theorem 4′], [PRS09, (7.3)].

We will also make good use of the following lemma in proving our main theorem.

Lemma 4.3.2. If z ∈ W i,jj then z ∈ W i.

Proof. Suppose z ∈ W i,jj . Then R(z) ⊆ R+

j and R(z)∩R+i,j = ∅. So R(z)∩(R+

i ∩R+j ) = ∅.

So (R(z) ∩R+j ) ∩R+

i = ∅. So R(z) ∩R+i = ∅. Hence z ∈ W i.

The following theorem is the main theorem of the thesis. It determines the “thickness” ofthe Schubert cells Xw = BwB or, more precisely, of the incidence structures (Xw)ij.

Theorem 4.3.3. Let G(F) be a Chevalley group and let Pi and Pj be standard maximal

parabolic subgroups of G(F). Let W be the Weyl group of G(F), w ∈ W and let Xw = BwB

be the Schubert cell corresponding to w. Let (Xw)ij be the incidence structure associated to Xw

and let gPj ∈ BwPj. Then the number of elements of pi(Xw) incident to gPj in (Xw)ij is

q`(z), where w = uzv with u ∈ W j, zv ∈ Wj, z ∈ (Wj)i,j and v ∈ Wi,j.

Proof. Note that

pip−1j (gPj) = hPi ∈ G/Pi | there exists kB ∈ G/B such that pi(kB) = hPi and pj(kB) = gPj

In other words, the set of elements of G/Pi incident to gPj is the set pip−1j (gPj). So the set of

elements of pi(Xw) incident to gPj is the set

pi(Xw) ∩ pip−1j (gPj).

Theorem 4.3.3 follows from the fact that

ψ : F`(z) −→ pi(Xw) ∩ pip−1j (gPj)

(d1, d2, . . . , d`) 7−→ gxk1(d1)n−1k1. . . xk`(d`)n

−1k`Pi,

78

is a bijection, where sk1 . . . sk` is a reduced expression for z, and g = xi1(c1)n−1i1. . . xik(cik)n

−1ik

is a favourite coset representative of gPj, where si1 . . . sik ∈ W j is a reduced expression for u.To show that ψ is a bijection, there are 3 things to show:

(1) ψ(d1, . . . , d`) ∈ pip−1j (gPj) ∩ pi(Xw).

(2) ψ is injective.(3) ψ is surjective.

(1) To show:

(a) ψ(d1, . . . , d`) ∈ pi(Xw).(b) ψ(d1, . . . , d`) ∈ pip−1

j (gPj).(a) We have

ψ(d1, . . . , d`) = gxk1(d1)n−1k1. . . xk`(d`)n

−1k`Pi

∈ BuzPi= BuzvPi

= BwPi

= pi(Xw).

(b) We have

ψ(d1, . . . , d`) = gxk1(d1)n−1k1. . . xk`(d`)n

−1k`Pi

= pi(gxk1(d1)n−1k1. . . xk`(d`)n

−1k`B).

We wish to show that

gxk1(d1)n−1k1. . . xk`(d`)n

−1k`B ∈ p−1

j (gPj).

But

pj(gxk1(d1)n−1k1. . . xk`(d`)n

−1k`B) = gxk1(d1)n−1

k1. . . xk`(d`)n

−1k`Pj

= gPj,

since sk1 . . . sk` ∈ Wj.(2) Suppose

ψ(d1, . . . , d`) = ψ(d′1, . . . , d′`).

Then

gxk1(d1)n−1k1. . . xk`(d`)n

−1k`Pi = gxk1(d

′1)n−1

k1. . . xk`(d

′`)n−1k`Pi.

Then

xk1(d1)n−1k1. . . xk`(d`)n

−1k`Pi = xk1(d

′1)n−1

k1. . . xk`(d

′`)n−1k`Pi.

By Proposition 4.3.2, z ∈ W i. By Proposition 4.3.1,

d1 = d′1, d2 = d′2, . . . , dr = d′r.

So ψ is injective.(3) Let kPi ∈ pi(p−1

j (gPj)) ∩ pi(Xw). We know that

p−1j (gPj) = g ∪w∈Wj

BwB,

so that

pip−1j (gPj) = g ∪w∈Wj

BwPi.

So there exists w ∈ Wj such that

kPi ∈ gBwPi.

Let z′ ∈ W i,jj such that z′Wi,j = wWi,j. Then z′Wi = wWi. So kPi ∈ gBz′Pi. So

kPi ∈ Buz′Pi.79

Since kPi ∈ pi(Xw), we have kPi ∈ BuzPi. So uz′Wi = uzWi. So z′Wi = zWi. ByProposition 4.3.2, we know z, z′ ∈ W i. So z = z′ (See [Hum92, Proposition 1.10(c)]or [Bou08, Chapter 4, §1, Exercise 3, 2nd paragraph]). So there exists d1, . . . , d` ∈ Fsuch that

kPi = gxk1(d1)n−1k1. . . xk`(d`)n

−1k`Pi.

So

ψ(d1, d2, . . . , d`) = kPi.

Hence ψ is surjective.

Example 4.3.4. Take G(F) = GL4(C) so that

W = S4, W1 = S1 × S3, W2 = S2 × S2, and W1,2 = S1 × S1 × S2.

Then W 1 = 1, s1, s2s1, s3s2s1,

W 2 = 1, s2, s1s2, s3s2, s1s3s2, s2s1s3s2 and (W2)1,2 = 1, s1.

Let w = uzy = (s1s3s2)(s1)(s3). If (Xw)12 is the (1, 2)-incidence structure associated to Xw

and g = x1(c1)n−11 x3(c2)n−1

3 x2(c3)n−12 then

p1(p−12 (gP2)) = p1(p−1

2 (x1(c1)n−11 x3(c2)n−1

3 x2(c3)n−12 P2))

= x1(c1)n−11 x3(c2)n−1

3 x2(c3)n−12 x1(d1)n−1

1 P1 | d1 ∈ F.

Note that this illustrates that p1(p−12 (gP2)) ∼= F even though the elements

x1(c1)n−11 x3(c2)n−1

3 x2(c3)n−12 x1(d1)n−1

1 P1 | d1 ∈ F

are not the “favourite” coset representatives of the cosets in G/P1. This provides a conceptualexplanation of why Theorem 4.3.3 is nontrivial (one needs to find the right coordinatization tosucceed in displaying p1(p−1

2 (gP2)) as an affine space).

Using Theorem 4.3.3 to determine the Schubert incidence structures that are “thin” pro-duces the following result.

Corollary 4.3.5. Let G(Fq) be a Chevalley group over a finite field Fq. Then the Schubert

incidence structures (Xw)ij such that pi(Xw) is a cap (see Chapter 1, Section 1.4) correspondto

triples (w, i, j) such that

w ∈ W jWi,j, if q > 2,

w ∈ W jWi,j ∪W jsiWi,j, if q = 2.

Proof. Assume w = uzy with u ∈ W j, z ∈ (Wj)i,j, y ∈ Wi,j. By Theorem 4.3.3, if

gPj ∈ BwPj then gPj is incident with q`(z) elements of pi(Xw). Note that pi(Xw) is an arc ifand only if q`(z) ≤ 2. But `(z) = 0 only when z = 1 and `(z) = 1 only when z = si, and theresult follows.

Example 4.3.6. Suppose G(F) = SL4(Fq) with q > 2. By Corollary 4.3.5, the number of

Schubert incidence structures (Xw)ij such that pi(Xw) is a cap is∑i,j

|W jWi,j| = |W 1W1,1|+ |W 2W1,2|+ |W 3W1,3|+ |W 1W2,1|+ |W 2W2,2|

+ |W 3W2,3|+ |W 1W3,1|+ |W 2W3,2|+ |W 3W3,3|= 4 · 6 + 6 · 2 + 4 · 2 + 4 · 2 + 6 · 4 + 4 · 2 + 4 · 2 + 6 · 2 + 4 · 6= 128.

80

4.4. The example G = SL3(F)

We illustrate the theory of the previous section (as well as much of the theory of Chapter2) with the example SL3(F). Let F be a field and let G = SL3(F). The standard Cartandecomposition is

G

⊆

B =

x ∈ G∣∣∣∣∣∣ x =

∗ ∗ ∗0 ∗ ∗0 0 ∗

⊆

T =

x ∈ G∣∣∣∣∣∣ x =

∗ 0 00 ∗ 00 0 ∗

.

The Weyl group of G is W = S3 and

W =⟨s1, s2 | s2

1 = s22 = (s1s2)3 = 1

⟩= 1, s1, s2, s1s2, s2s1, s1s2s1.

Then

W1 = 〈s2〉 = 1, s2, W2 = 〈s1〉 = 1, s1,

and

P1 = B tBs2B, P2 = B tBs1B.

The sets

W 1 = 1, s1, s2s1, and W 2 = 1, s2, s1s2,

are minimal length coset representatives for W/W1 and W/W2 respectively. The Bruhat de-compositions of G are

G/B = B tBs1B tBs2B tBs1s2B tBs2s1B tBs1s2s1B,

G/P1 = P1 tBs1P1 tBs2s1P1, and G/P2 = P2 tBs2P2 tBs1s2P2.

Th Schubert varieties in G/B are

X1 = B, Xs1 = B tBs1B, Xs2 = B tBs2B,

Xs1s2 = B tBs1B tBs2B tBs1s2B, Xs2s1 = B tBs2B tBs1B tBs2s1B,

Xs1s2s1 = Xs2s1s2 = B tBs1B tBs2B tBs1s2B tBs2s1B tBs1s2s1B = G/B.

The Schubert varieties in G/P1 are

X11 = P1, X1

s1= P1 tBs1P1,

X1s2s1

= P1 tBs1P1 tBs2s1P1 = G/P1.

The Schubert varieties in G/P2 are

X21 = P2, X2

s2= P2 tBs2P2,

X2s1s2

= P2 tBs2P2 tBs1s2P2 = G/P2.

The matrices

xα1(c) =

1 c 00 1 00 0 1

, xα2(c) =

1 0 00 1 c0 0 1

, xα1+α2(c) =

1 0 c0 1 00 0 1

,81

nα1 =

0 1 0−1 0 00 0 1

, nα2 =

1 0 00 0 10 −1 0

, nα1+α2 =

0 0 10 1 0−1 0 0

.generate G. By Theorem 2.4.6, the Schubert cells of G/B have single B-coset decompositions

B = B

Bs1B = xα1(c1)n−1α1B | c1 ∈ F =

c1 −1 0

1 0 00 0 1

B∣∣∣∣∣∣c1 ∈ F

,

Bs2B = xα2(c1)n−1α2B | c1 ∈ F =

1 0 0

0 c1 −10 1 0

B∣∣∣∣∣∣c1 ∈ F

,

Bs1s2B = xα1(c1)n−1α1xα2(c2)n−1

α2B | c1, c2 ∈ F =

c1 −c2 1

1 0 00 1 0

B∣∣∣∣∣∣c1, c2 ∈ F

,

Bs2s1B = xα2(c1)n−1α2xα1(c2)n−1

α1B | c1, c2 ∈ F =

c2 −1 0c1 0 −11 0 0

B∣∣∣∣∣∣c1, c2 ∈ F

,

Bs1s2s1B = xα1(c1)n−1α1xα2(c2)n−1

α2xα1(c3)n−1

α1B | c1, c2, c3 ∈ F

=

c1c3 − c2 −c1 1

c3 −1 01 0 0

B∣∣∣∣∣∣c1, c2, c3 ∈ F

,

the Schubert cells of G/P1 have single P1-coset decompositions

P1 = P1

Bs1P1 = xα1(c1)n−1α1P1 | c1 ∈ F =

c1 −1 0

1 0 00 0 1

P1

∣∣∣∣∣∣c1 ∈ F

,

Bs2s1P1 = xα2(c1)n−1α2xα1(c2)n−1

α1P1 | c1, c2 ∈ F =

c2 −1 0c1 0 −11 0 0

B∣∣∣∣∣∣c1, c2 ∈ F

,

and the Schubert cells of G/P2 have single P2-coset decompositions

P2 = P2

Bs2P2 = xα2(c1)n−1α2P2 | c1 ∈ F =

1 0 0

0 c1 −10 1 0

P2

∣∣∣∣∣∣c1 ∈ F

,

Bs1s2P2 = xα1(c1)n−1α1xα2(c2)n−1

α2P2 | c1, c2 ∈ F =

c1 −c2 1

1 0 00 1 0

P2

∣∣∣∣∣∣c1, c2 ∈ F

.

The map

ψ : G/B −→ Fl: gB 7−→ (0 ⊆ gFe1 ⊆ gFe1, e2 ⊆ F3

2)

is a bijection. The image of ψ of the favourite choices of coset representatives are:

ψ(B) =

0 ⊆

⟨100

⟩⊆

⟨1 00 10 0

⟩82

ψ(x1(c1)s1B) =

0 ⊆

⟨c1

10

⟩⊆

⟨c1 11 00 0

⟩ψ(x2(c1)s2B) =

0 ⊆

⟨100

⟩⊆

⟨1 00 c1

0 1

⟩ψ(x1(c1)s1x2(c2)s2B) =

0 ⊆

⟨c1

10

⟩⊆

⟨c1 c2

1 00 1

⟩ψ(x2(c1)s2x1(c2)s1B) =

0 ⊆

⟨c2

c1

1

⟩⊆

⟨c2 1c1 01 0

⟩ψ(x1(c1)s1x2(c2)s2x1(c3)s1B) =

0 ⊆

⟨c1c3 + c2

c3

1

⟩⊆

⟨c1c3 + c2 c1

c3 11 0

⟩ψ(x2(c1)s2x1(c2)s1x2(c3)s2B) =

0 ⊆

⟨c2

c1

1

⟩⊆

⟨c2 c3

c1 11 0

⟩The map

ψ1 : G/P1 −→ Gr(1, 3): gP1 7−→ (0 ⊆ gFe1 ⊆ F3)

is a bijection. The image under the favourite choice of coset reprentatives is

ψ1(P1) =

0 ⊆

⟨100

⟩ψ1(x1(c1)s1P1) =

0 ⊆

⟨c1

10

⟩ψ(x2(c1)s2x1(c2)s1P1) =

0 ⊆

⟨c2

c1

1

⟩The map

ψ2 : G/P2 −→ Gr(2, 3): gP2 7−→ (0 ⊆ gFe1, e2 ⊆ F3)

is a bijection. The image of the favourite choice of coset representatives is

ψ(P2) =

0 ⊆

⟨1 00 10 0

⟩ψ(x2(c1)s2P2) =

0 ⊆

⟨1 00 c1

0 1

⟩ψ(x1(c1)s1x2(c2)s2P2) =

0 ⊆

⟨c1 c2

1 00 1

⟩83

We have established the following isomorphisms of G-sets

G/B ∼= maximal chains in PG(2, q),G/P1

∼= rank 1 elements in PG(2, q),G/P2

∼= rank 2 elements in PG(2, q),

and can therefore illustrate the Schubert cells Xw ⊆ G/B on the Hasse diagram of PG(2, q). We

do this for PG(2, 2) in Appendix B.3. The thickness of the (1, 2)-incidence structures (Xw)12

can be calculated using Theorem 4.3.3. Recall that

W 2 = 1, s2, s1s2W1,2 = W1 ∩W2 = 1

(W2)1,2 = 1, s1.We write each w ∈ W as w = uzv with u ∈ W 2, z ∈ (W2)1,2 and v ∈ W1,2 :

1 = 1 · 1 · 1,s1 = 1 · s1 · 1,s2 = s2 · 1 · 1,

s1s2 = s1s2 · 1 · 1,s2s1 = s2 · s1 · 1,

s1s2s1 = s1s2 · s1 · 1,

so that the ‘thickness’ of X1, Xs1 , Xs2 , Xs1s2 , Xs2s1 , Xs1s2s1 are, respectively,

1, q, 1, 1, q, q.

This can be manually checked using the Hasse diagrams in Appendix B.3.

84

APPENDIX A

A.1. Finite Fields and Galois Theory

A reference for this section is [DF04, Part IV].Let p ∈ Z>0 be a prime, let Fp = Z/pZ be the finite field with p elements, and let Fp be the

algebraic closure of Fp. The Frobenius endomorphism is the field endomorphism

F : Fp −→ Fp defined by F (x) = xp.

Let k ∈ Z≥0. The finite field with pk elements is

Fpk = x ∈ Fp | F k(x) = x.

Proposition A.1.1. The set Fpk is a subfield of Fp with pk elements.

Proof. The fact that Fpk has pk elements follows from the fact that xpk

= x is separablei.e has distinct linear factors. For complete details, see [DF04, Example: (Existence andUniqueness of Finite Fields), Section 13.5].

Proposition A.1.2.

Gal(Fpk : Fp) = 1, F, F 2, . . . , F k−1

Proposition A.1.3.

Fp =∞⋃k=1

Fpk .

Proposition A.1.4. [DF04, §14.3, Proposition 15] The function

m ∈ Z≥0 | m divides k −→ subfields of Fpkm 7−→ Fpm ,

is a bijection.

Corollary A.1.5. Suppose F is a finite field of even characteristic. Then F has an auto-morphism τ such that τ 2(x) = x2 if and only if F ∼= F2k for some odd k.

Proof. We know F ∼= F2k for some k ∈ Z≥1. Suppose k is even. By Proposition A.1.4,there exists a unique subfield E ∼= F2k such that E ∼= F22 . By uniqueness, τ(E) = E. We knowE ∼= 0, 1, α, 1 + α where α2 + α + 1 = 0. There are exactly 2 automorphisms of τ , uniquelydetermined by τ(α) = α or τ(α) = α + 1. If τ(α) = α then τ 2(α) = α, a contradiction. Ifτ(α) = α + 1 then τ 2(α) = τ(α + 1) = α, a contradiction. Hence k is not even. Hence k isodd.

Proposition A.1.6. Let Fq2 be the finite field with q2 elements, where q = pk is a prime

power. Define : Fq2 → Fq2 by x = xq.

(1) Let c ∈ F×q . Then the equation

xx = c

has exactly q + 1 solutions in F×q2.

85

(2) Let d ∈ Fq. Then the equation

x+ x = d

has exactly q solutions in Fq2.

Proof. [Wan93, Lemma 5.1]. Define

φ : F×q2 → F×q ,x 7→ xx,

ψ : Fq2 → Fq,x 7→ x+ x,

We consider first the map φ. If x ∈ F×q2 we have (xx) = xx so that xx ∈ Fq. Considering the

set F×q2 as a group under multiplication, φ is a group homomorphism. Since

Ker(φ) = x ∈ F×q2 | xq+1 = 1,

we have |Ker(φ)| ≤ q + 1. Also, |Im(φ)| ≤ |F×q | = q − 1. But |F×q2| = |Ker(φ)||Im(φ)| and

|F×q2 | = q2 − 1. So |Ker(φ)| = q + 1 so that |φ−1(c)| = q + 1 for c ∈ F×q2 . Hence (1) is true.

A similar argument for ψ shows that (2) is true.

Proposition A.1.7. [Art11, Theorem 15.7.3(c)], [DF04, Proposition 9.18] The group F×qis a cyclic group of order q − 1.

A.2. Sesquilinear forms

The reference for this section is [Tay92, pg. 52] (see also [Ueb11, Chapter 4, Section 5] and[Bou07]). Let F be a field and let

V be a F-vector space and σ : F→ F be a field automorphism.

A σ-sesquilinear form (or form) on V is a function 〈, 〉 : V × V → F such that

(1) If λ, µ ∈ F and u, v ∈ V then

〈λu, µv〉 = λ(σ(µ))〈u, v〉,(2) If u, v, w ∈ V then

〈u+ v, w〉 = 〈u,w〉+ 〈v, w〉 and 〈u, v + w〉 = 〈u, v〉+ 〈u,w〉.A bilinear form is a σ-sesquilinear form such that σ = id.

Theorem A.2.1. Assume that V is n-dimensional and e1, e2, . . . , en is a basis of V .Define σ : V → V by

σ(λ1e1 + λ2e2 + · · ·+ λnen) = σ(λ1)e1 + σ(λ2)e2 + · · ·+ σ(λn)en.

Define the map

ϕ :

σ − sesquilinear forms

on V

−→

n× n matriceswith entries in F

〈·, ·〉 7−→ A

by Aij = 〈ei, ej〉. Then ϕ is a bijection, and the σ-sesquilinear form corresponding to A is givenby

〈v, w〉 = vtAσ(w).

Two forms 〈, 〉 and 〈, 〉′ on V are isometric (or equivalent) if there exists T ∈ GL(V ) suchthat

if u, v ∈ V then 〈Tu, Tv〉 = 〈u, v〉′.

Proposition A.2.2. If A is the matrix corresponding to the form 〈, 〉, then 〈, 〉 is equivalentto 〈, 〉′ if and only if there exists a T ∈ GL(V ) such that the matrix corresponding to 〈, 〉′ is

T tAσ(T ),

where (σ(T ))ij = σ(Tij).

86

Define

θ− : V → V ∗

v 7→ θvby θv(u) = 〈u, v〉, and −θ : V → V ∗

v 7→ vθby vθ(u) = 〈v, u〉.

The form 〈, 〉 is nondegenerate if θ− and −θ are bijections [Bou07, Chapter IX, §1, No. 6,Proposition 6]. The form 〈, 〉 is degenerate if it is not nondegenerate. A subspace W ⊆ V isnondegenerate if the restricted form 〈, 〉|W : W ×W → F is nondegenerate.

Proposition A.2.3. A form 〈, 〉 : V × V → F is nondegenerate [Ueb11, Chapter 4, Section5], [Bou07, Chapter IX, §1, No. 1, Definition 3] if and only if the following conditions aresatisfied:

(1) If v ∈ V and v 6= 0 then there exists x ∈ V such that 〈v, x〉 6= 0,(2) If v′ ∈ V and v′ 6= 0 then there exists x′ ∈ V such that 〈x′, v′〉 6= 0.

Theorem A.2.4. Let 〈, 〉 : V ×V → F be a σ-sesquilinear form defined by 〈v, w〉 = vtAσ(w).Then 〈, 〉 is nondegenerate if and only if A is invertible.

Proof. Suppose A is invertible. Let v ∈ V such that v 6= 0. Then vtA 6= 0, since A isinvertible. So there exists i ∈ 1, 2, . . . , n such that vtAσ(ei) 6= 0. So 〈v, ei〉 6= 0. Similiarly,there exists j ∈ 1, 2, . . . , n such that 〈ej, v〉 6= 0. Hence 〈, 〉 is nondegenerate.

Suppose 〈, 〉 is nondegenerate, and suppose for the sake of contradiction that A is notinvertible. Then there exists a nonzero v ∈ V such that Av = 0. Hence utAσ(v) = 0 for allu ∈ V . Hence 〈u, v〉 = 0 for all u ∈ V . Hence 〈, 〉 is degenerate, a contradiction.

Proposition A.2.5. [Bou07, Chapter IX, §1, No. 6, Corollary to Proposition 6] Let V be afinite dimensional vector space, and let 〈, 〉 be a form on V . Then the form 〈, 〉 is nondegenerateif and only if for every basis e1, e2, . . . , en there exists a basis f1, f2, . . . , fn such that

〈ei, fj〉 = δij.

A form 〈, 〉 : V × V → F is reflexive if the following condition is satisfied:

if u, v ∈ V and 〈u, v〉 = 0 then 〈v, u〉 = 0.

Theorem A.2.6. [The Birkhoff Von-Neumann Theorem] [Bal15, Theorem 3.6], [Tay92,Theorem 7.1], [BV36]. Let 〈, 〉 : V × V → F be a nondegenerate reflexive σ-sesquilinear form.Up to a scalar, 〈, 〉 is exactly one of the following types:

(1) 〈, 〉 is an alternating form, that is,

〈u, v〉 = −〈v, u〉

for u, v ∈ V if the characteristic of F is 6= 2, and 〈u, u〉 = 0 for u ∈ V if the charac-teristic of F is 2.

(2) 〈, 〉 is a symmetric form, that is,

〈u, v〉 = 〈v, u〉,for u, v ∈ V ,

(3) 〈, 〉 is a hermitian form, that is,

〈u, v〉 = σ(〈v, u〉),

for u, v ∈ V , where σ2 = 1 and σ 6= 1.

Assume that 〈, 〉 : V × V → F is a reflexive form. The orthogonal component of a subspaceW ⊆ V is

W⊥ = x ∈ V | if w ∈ W then 〈w, x〉 = 0

Proposition A.2.7. Let V be a finite dimensional vector space and let 〈, 〉 : V × V → F bea reflexive form. A subspace W ⊆ V is nondegenerate if any only if W ∩W⊥ = 0.

87

Proof. Suppose W ⊆ V is nondegenerate. Let x ∈ W ∩W⊥. Then 〈x, v〉 = 0 = 〈v, x〉 forall v ∈ W . Since W is nondegenerate, x = 0. Hence W ∩W⊥ = 0.

Suppose W ∩W⊥ = 0. Let v ∈ W with v 6= 0. Suppose, for sake of contradiction, that〈v, x〉 = 0 for all x ∈ W . Then v ∈ W⊥. So v = 0, a contradiction. Hence there existsx ∈ W such that 〈v, x〉 6= 0. Similarly, there exists x ∈ W such that 〈x, v〉 6= 0. So W isnondegenerate.

Proposition A.2.8. [Kah08, Lemma 1.1.5] Let V be a finite-dimensional vector space, and〈, 〉 : V × V → F be a reflexive nondegenerate form. Let W ⊆ V be a nondegenerate subspace.Then

V = W ⊕W⊥

Proof. By Proposition A.2.7, we know W ∩W⊥ = 0. Suppose e1, e2, . . . , em is a basisfor W . By Proposition A.2.5, there exists a basis f1, f2, . . . , fm of W such that 〈ei, fj〉 = δij.Let x ∈ V . Then

x−m∑i=1

〈x, fi〉ei ∈ W⊥,

so x ∈ W +W⊥. So V = W +W⊥. Hence V = W ⊕W⊥.

Corollary A.2.9. If 〈, 〉 : V × V → F is a reflexive nondegenerate form and W is anondegenerate subspace of V then W⊥ is a nondegenerate subspace of V .

Proof. Assume 〈, 〉 : V × V → F is a reflexive nondegenerate form and W is a nondegen-erate subspace of V .

Suppose v ∈ W⊥ with v 6= 0. Furthermore, suppose for sake of contradiction that 〈v, x〉 = 0for all x ∈ W⊥. Then 〈v, x′ + x〉 = 0 for all x ∈ W⊥ and x′ ∈ W . So by Proposition A.2.8,〈v, y〉 = 0 for all y ∈ V . Hence 〈, 〉 is degenerate, a contradiction. So there exists x ∈ W⊥

such that 〈v, x〉 6= 0. Similarly, there exists x ∈ W⊥ such that 〈x, v〉 6= 0. Hence W⊥ isnondegenerate.

Proposition A.2.10. [Bal15, Lemma 3.1] Let W be a subspace of a finite dimensionalvector space V , and let 〈, 〉 : V × V → F be a reflexive nondegenerate form. Then

dimW + dimW⊥ = dimV.

Proof. Suppose that dimV = n, and let e1, e2, . . . , er be a basis for W . Define αi ∈ V ∗for i ∈ 1, 2, . . . , r by

αi(v) = 〈ei, v〉.Suppose

∑ri=1 λiαi = 0. Then

∑ri=1 λiαi(v) = 0 for all v ∈ V . Therefore,

〈r∑i=1

λiei, v〉 =r∑i=1

λiαi(v) = 0

for all v ∈ V . Hence∑r

i=1 λiei. So λ1 = λ2 = · · · = λr = 0. Hence α1, α2, . . . , αr is a linearlyindependent set.

So

W⊥ = v ∈ V | if i ∈ 1, 2, . . . , r then 〈ei, v〉 = 0= v ∈ V | if i ∈ 1, 2, . . . , r then αi(v) = 0.

The linearly independent set α1, . . . , αr can be extended to a basis α1, . . . , αr, αr+1, . . . , αnof V ∗. Let e1, e2, . . . , er, er+1, . . . , en be the basis in V dual to α1, . . . , αr, αr+1, . . . , αn.Then

W⊥ = Spaner+1, . . . , en.

So dimW⊥ = n− r.

88

Proposition A.2.11. Let U,W ⊆ V be vector subspaces with U ⊆ W . Then W⊥ ⊆ U⊥.

Proof. Let v ∈ W⊥. If w ∈ W then 〈w, v〉 = 0. In particular, if w ∈ U then 〈w, v〉 = 0.So v ∈ U⊥. Hence W⊥ ⊆ U⊥.

Following [Tay92, pg. 56, Definition (ii)] and [Bal15, pg. 27], a subspace W ⊆ V is totallyisotropic if the following condition is satisfied:

if u, v ∈ W then 〈u, v〉 = 0.

The rank or Witt index of V is the maximum dimension of a totally isotropic subspace.

Proposition A.2.12. A subspace W ⊆ V is totally isotropic if and only if W ⊆ W⊥.

Proof. Suppose W is totally isotropic. Let v ∈ W . If w ∈ W then 〈v, w〉 = 0. So v ∈ W⊥.Hence W ⊆ W⊥.

Suppose W ⊆ W⊥. Let v, w ∈ W . Then 〈v, w〉 = 0. Hence W is totally isotropic.

A.3. Quadratic forms

References for this section include [Kah08] and [Lam]. A quadratic form on V is a functionQ : V → F such that the following conditions are satisfied:

(1) If λ ∈ F and u ∈ V then Q(λu) = λ2Q(u),(2) The map 〈, 〉 : V × V → F defined by

〈u, v〉 = Q(u+ v)−Q(u)−Q(v)

is a bilinear form.

Proposition A.3.1. Define

ψ :

quadratic formson V

−→

symmetric bilinear

forms on V

Q 7−→ 〈, 〉Q,

where 〈u, v〉Q = Q(u+ v)−Q(u)−Q(v). If F has characteristic 6= 2 then ψ is a bijection.

Let Q : V → F be a quadratic form. Suppose the characteristic of F is 6= 2. Then Q is degenerateif its corresponding bilinear form is degenerate. Also, Q is non-degenerate if its correspondingbilinear form is non-degenerate.

Suppose the characteristic of F is 2. Then Q is non-degenerate if the following condition issatisfied:

If v 6= 0 and 〈x, v〉 = 0 for all x ∈ V then Q(v) 6= 0.

A nonzero vector x ∈ V is singular if Q(x) = 0. A subspace W ⊆ V is singular if thereexists a singular vector x ∈ W such that x is orthogonal to W (with respect to the bilinearform associated with Q). A subspace W ⊆ V is totally singular if Q(x) = 0 for all x ∈ W .

A quadratic space is a tuple (V,Q) where V is a finite dimensional vector space over afield F (characteristic 6= 2) and Q : V → F is a quadratic form. An isometry between twoquadratic spaces (V,Q) and (V ′, Q′) is a linear isomorphism τ : V → V ′ such that if v ∈ Vthen Q′(τv) = Q(v). Two quadratic spaces (V,Q) and (V ′, Q′) are isometric if there exists anisometry τ : V → V ′, in this case we write (V,Q) ∼= (V ′, Q′). A quadratic space (V,Q) is

• isotropic if there exists a nonzero v ∈ V such that Q(v) = 0,• anisotropic if there does not exist a nonzero v ∈ V such that Q(v) = 0,• totally isotropic if Q(v) = 0 for all nonzero v ∈ V .• hyperbolic if it is isometric to (V ′, Q′) where dimV ′ = 2m is even and

Q′(x1, x2, . . . , x2m) = x1x2 + x3x4 + · · ·+ x2m−1x2m.

Let (V1, Q1) and (V2, Q2) be quadratic spaces. The orthogonal direct sum is the quadratic space(V1 ⊥ V2, Q1 ⊥ Q2) where

V1 ⊥ V2 = V1 ⊕ V2, and (Q1 ⊥ Q2)(v1 + v2) = Q1(v1) +Q2(v2)

89

for all v1 ∈ V1 and v2 ∈ V2. The tensor product is (V1 ⊗ V2, Q1 ⊗ Q2) where V1 ⊗ V2 is theusual tensor product of vector spaces and Q1 ⊗Q2 is the quadratic form whose correspondingsymmetric bilinear form corresponds to (via Proposition A.3)

〈u1 ⊗ u2, v1 ⊗ v2〉Q1⊗Q2 = 〈u1, v1〉Q1〈u2, v2〉Q2 .

Lemma A.3.2. [Kah08, Lemma 1.2.8] Let (V,Q) be a nondegenerate quadratic space. Sup-pose dimV = 2 and V contains an isotropic vector. Then (V,Q) is hyperbolic, that is, thereexists a basis u, v of V such that

Q(x1u+ x2v) = x1x2.

Proof. Let v ∈ V be an isotropic vector. Let w ∈ V \spanv. Since (V,Q) is nondegen-erate, 〈v, w〉 6= 0. Let

u = − Q(w)

2〈v, w〉2v +

1

〈v, w〉w.

Then Q(u) = 0 and 〈u, v〉 = 1. So Q is hyperbolic with respect to the basis u, v.

Theorem A.3.3. [Lam, Chapter I, Theorem 4.1] Suppose (V,Q) is a quadratic space. Then

(V,Q) ∼= (Vt, Qt) ⊥ (Vh, Qh) ⊥ (Va, Qa)

where

(Vt, Qt) is totally isotropic,

(Vh, Qh) is a hyperbolic space,

(Va, Qa) is anisotropic.

Furthermore, the quadratic spaces (Vt, Qt), (Vh, Qh), (Va, Qa) are uniquely determined up toisometry.

Two quadratic spaces (V,Q) and (V ′, Q′) are Witt equivalent and we write

(V,Q) ' (V ′, Q′) if (Va, Qa) ∼= (V ′a, Q′a).

Given the decomposition in Theorem A.3.3, the Witt index or rank is equal to 12dim(Vh) if Q

is nondegenerate (see [Lam, Corollary 4.4]).The Witt ring of F, denoted W (F), is the set of Witt equivalence classes of quadratic spaces

(V,Q) over F, with addition given by ⊥ and multiplication given by ⊗.

Lemma A.3.4. Suppose q ∈ Z≥1 with q odd. Then −1 ∈ Fq is a square if and only ifq ≡ 1(mod4).

Proof. We know that F×q is a cyclic group of order q− 1 (see [Art11, Theorem 15.7.3(c)]).So we may write

F×q = 1, x, x2, x3, . . . , xq−3, xq−2.

Let S be the subgroup of nonzero squares in F×q . Then

S = 1, x2, x4, . . . , xq−3.

Suppose q = 1(mod4) so that we may write q = 4m+ 1 where m ∈ Z. Then

(x2m)2 = x4m = xq−1 = 1.

So x2m is a square root of 1 not equal to 1. So x2m = −1. Hence xm is a square root of −1.Suppose q = 3(mod4) and write q = 4m+ 3 where m ∈ Z. Then

(x2m+1)2 = x4m+2 = xq−1 = 1.

90

So x2m+1 = −1. Suppose for sake of contradiction that −1 has a square root. Then there existsk ∈ Z≥0 such that

(xk)2 = x2m+1.

So x2(m−k)+1 = 1, a contradiction. Hence −1 is not a square root.

Proposition A.3.5. [Lam, Chapter II, Corollary 3.6] Suppose F = Fq with q odd.

• Suppose q ≡ 1(mod4). Then a full set of representatives for the equivalence classes ofW (F) are the four quadratic spaces (V,Q) where(1) V = 0 and Q = 0,(2) V = Fq and Q(x1) = x2

1,(3) V = Fq and Q(x1) = sx2

1, where s is any nonsquare element of Fq,(4) V = F2

q and Q(x1, x2) = x21 + sx2

2, where s is any nonsquare element of Fq.Furthermore, W (F) ∼= F2[Z/2Z].• Suppose q ≡ 3(mod4). Then a full set of representatives for the equivalence classes ofW (F) are the four quadratic spaces (V,Q) where(1) V = 0 and Q = 0,(2) V = Fq and Q(x1) = x2

1,(3) V = F2

q and Q(x1, x2) = x21 + x2

2,

(4) V = Fq and Q(x1) = −x21.

Furthermore, W (F) ∼= Z/4Z.

A.4. Flag varieties and buildings

Here we provide a translation between flag varieties and the theory of buildings. In partic-ular, every flag variety G/B has the structure of a building. The main references are [AB08,§6.1.4], [Car89, §15.5], [Tit74, Theorem 5.2] and [Tay92].

Let W be a Coxeter group and S its set of simple reflections. Following [AB08, §5.1.1], abuilding of type (W,S) is a pair (C, δ) consisting of a nonempty set C, whose elements are calledchambers, together with a map δ : C × C→W , called the Weyl distance function, such that forall c, d ∈ C, the following three conditions hold:

(B1) δ(c, d) = 1 if and only if c = d(B2) If δ(c, d) = w and d′ ∈ C satisfies δ(d, d′) = s ∈ S, then δ(c, d′) = ws or w. If, in

addition, `(ws) = `(w) + 1, then δ(c, d′) = ws.(B3) If δ(c, d) = w, then for any s ∈ S there is a chamber d′ ∈ C such that δ(d, d′) = s and

δ(c, d′) = ws.

Intuitively, a building of type (W,S) is a ‘W -metric space’, where the distance between anytwo chambers is given by an element of the Coxeter group.

A building is thin if for every chamber c ∈ C and s ∈ S, there exists exactly two chambersd ∈ C such that δ(c, d) ∈ 1, s. A building is thick if for every chamber c ∈ C and s ∈ S, thereexists three or more chambers d ∈ C such that δ(c, d) ∈ 1, s.

Theorem A.4.1. Let G be a Chevalley group, B a Borel subgroup, and T ⊆ B a maximaltorus. Let W = NG(T )/T be the corresponding Weyl group with simple reflections S. Define afunction δ :G/B ×G/B→W by

δ(gB, hB) = w if and only if g−1hB ⊆ BwB.

Then (G/B, δ) is a thick building of type (W,S).

Proof. First we check that he function δ is well defined. If gB, hB ∈ G/B then by theBruhat decomposition (Theorem 2.3.7), there exists a unique w ∈ W such that g−1hB ⊆ BwB.It remains to show that δ(gB, hB) does not depend on the choice of coset representatives of gBand hB. Suppose that gB = g′B, hB = h′B and δ(gB, hB) = w. Then g′−1gB = h′−1hB = Band so

g′−1h′B = g′−1h′h′−1hB = g′−1hB = g′−1gg−1hB ⊆ g′−1gBwB = BwB

91

So δ(g′B, h′B) = w. Hence δ is well defined.We have δ(gB, hB) = 1 if and only if gh−1B = B if and only if gB = hB, so (B1) is

satisfied.Suppose δ(gB, hB) = w and δ(hB, h′B) = s. Then g−1hB ⊆ BwB and h−1h′B ⊆ BsB.

Using Proposition 2.3.4 we have

g−1h′B = g−1hh−1h′B ⊆ g−1hB · h−1h′B ⊆ BwB ·BsB ⊆ BwB ∪BwsB.Hence g−1h′B ⊆ BwB or g−1h′B ⊆ BwsB, therefore δ(gB, h′B) = w or δ(gB, h′B) = ws.Furthermore, if `(w) = `(ws) + 1 then from Proposition 2.3.4 we have BwB ·BsB = BwsB sothat g−1h′B ⊆ BwsB, therefore δ(gB, h′B) = ws. So (B2) is satisfied.

We divide the proof of (B3) into two cases.(Case 1) Suppose `(ws) > `(w) and δ(gB, hB) = w. Then g−1hB ⊆ BwB so that

g−1hB ·BsB ⊆ BwB ·BsBThen Proposition 2.3.4 implies that BwB · BsB = BwsB. Let h′ = hn−1, where n ∈ N ischosen so that n−1T = s. Then h−1h′B ∈ BsB. So

g−1h′B = g−1hh−1h′B ⊆ g−1hB ·BsB ⊆ BwB ·BsB = BwsB

so that δ(gB, h′B) = ws. Hence (B3) holds in this case.(Case 2) Suppose `(ws) < `(w). We know that

g−1hB ⊆ BwB = BwssB ⊆ BwsB ·BsBHence g−1h = xy for some x ∈ BwsB, y ∈ BsB. Define h′ = hy−1. Then g−1h′ = g−1hy−1 =x ∈ BwsB. Therefore δ(gB, h′B) = ws, so that (B3) is satisfied.

To show that we have a thick building, let gB ∈ G/B and s ∈ S. We need to find hB, h′B ∈G/B with hB 6= h′B, δ(gB, hB) = s and δ(gB, h′B) = s. By the Bruhat decomposition(Theorem 2.3.7), there exists w ∈ W such that g ∈ BwB. Define h = gn where n ∈ N ischosen so that nT = s. Then g−1h ∈ BsB so that δ(gB, hB) = s. By Proposition 2.3.4, wehave BsB · BsB = B ∪ BsB. So we can choose a k ∈ BsB such that g−1hk ∈ BsB. Defineh′ = hk, so that g−1h′ = g−1hk ∈ BsB. Thus δ(gB, h′B) = s. Also, h−1h′ = k ∈ BsB so thathB 6= h′B, as required.

92

APPENDIX B

B.1. Hasse diagrams of subspace lattices

Example B.1.1. The subspace lattice PG(F22) has lattice diagram

0

⟨01

⟩⟨10

⟩ ⟨11

⟩V


0

⟨100

⟩ ⟨010

⟩ ⟨110

⟩ ⟨001

⟩ ⟨101

⟩ ⟨011

⟩ ⟨111

⟩

⟨1 00 10 0

⟩ ⟨1 00 00 1

⟩ ⟨1 00 10 1

⟩ ⟨0 01 00 1

⟩ ⟨1 01 00 1

⟩ ⟨0 11 00 1

⟩ ⟨1 11 00 1

⟩

V

Another way to visualise PG(F32) is the Fano plane:

93

⟨001

⟩

⟨100

⟩⟨

111

⟩⟨

010

⟩

⟨101

⟩ ⟨110

⟩ ⟨011

⟩

94


0

⟨00

1

⟩ ⟨01

0

⟩ ⟨01

1

⟩ ⟨01

2

⟩ ⟨10

0

⟩ ⟨10

1

⟩ ⟨10

2

⟩ ⟨11

0

⟩ ⟨11

1

⟩ ⟨11

2

⟩ ⟨12

0

⟩ ⟨12

1

⟩ ⟨12

2

⟩

⟨0 0

0 1

1 0

⟩ ⟨0 1

0 0

1 0

⟩ ⟨0 1

0 1

1 0

⟩ ⟨0 1

0 2

1 0

⟩ ⟨0 1

1 0

0 0

⟩ ⟨0 1

1 0

0 1

⟩ ⟨0 1

1 0

0 2

⟩ ⟨0 1

1 0

1 0

⟩ ⟨0 1

1 0

1 1

⟩ ⟨0 1

1 0

1 2

⟩ ⟨0 1

1 0

2 0

⟩ ⟨0 1

1 0

2 1

⟩ ⟨0 1

1 0

2 2

⟩

V

95

B.2. Hasse diagrams of Boolean lattices

Example B.2.1. The Boolean lattice PG(F11) has lattice diagram

∅

1


∅

1 2

1, 2


∅

1 2 3

1, 2 1, 3 2, 3

1, 2, 3

96


∅

1 2 3 4

1, 2 1, 3 1, 4 2, 3 2, 4 3, 4

1, 2, 3 1, 2, 4 1, 3, 4 2, 3, 4

1, 2, 3, 4

97


∅

1 2 3 4 5

1, 2 1, 3 1, 4 1, 5 2, 3 2, 4 2, 5 3, 4 3, 5 4, 5

1, 2, 3 1, 2, 4 1, 2, 5 1, 3, 4 1, 3, 5 1, 4, 5 2, 3, 4 2, 3, 5 2, 4, 5 3, 4, 5

1, 2, 3, 4 1, 2, 3, 5 1, 2, 4, 5 1, 3, 4, 5 2, 3, 4, 5

1, 2, 3, 4, 5

98

B.3. Hasse diagrams of Schubert cells in PG(F32)

The Schubert cell Xe = B in PG(F32) is labeled with thick lines.

0

⟨100

⟩ ⟨010

⟩ ⟨110

⟩ ⟨001

⟩ ⟨101

⟩ ⟨011

⟩ ⟨111

⟩

⟨1 00 10 0

⟩ ⟨1 00 00 1

⟩ ⟨1 00 10 1

⟩ ⟨0 01 00 1

⟩ ⟨1 01 00 1

⟩ ⟨0 11 00 1

⟩ ⟨1 11 00 1

⟩

V

The Schubert cell Xs1 = Bs1B in PG(F32) is labeled with thick lines.

0

⟨100

⟩ ⟨010

⟩ ⟨110

⟩ ⟨001

⟩ ⟨101

⟩ ⟨011

⟩ ⟨111

⟩

⟨1 00 10 0

⟩ ⟨1 00 00 1

⟩ ⟨1 00 10 1

⟩ ⟨0 01 00 1

⟩ ⟨1 01 00 1

⟩ ⟨0 11 00 1

⟩ ⟨1 11 00 1

⟩

V

99

The Schubert cell Xs2 = Bs2B in PG(F32) is labeled with thick lines.

0

⟨100

⟩ ⟨010

⟩ ⟨110

⟩ ⟨001

⟩ ⟨101

⟩ ⟨011

⟩ ⟨111

⟩

⟨1 00 10 0

⟩ ⟨1 00 00 1

⟩ ⟨1 00 10 1

⟩ ⟨0 01 00 1

⟩ ⟨1 01 00 1

⟩ ⟨0 11 00 1

⟩ ⟨1 11 00 1

⟩

V

The Schubert cell Xs1s2 = Bs1s2B in PG(F32) is labeled with thick lines.

0

⟨100

⟩ ⟨010

⟩ ⟨110

⟩ ⟨001

⟩ ⟨101

⟩ ⟨011

⟩ ⟨111

⟩

⟨1 00 10 0

⟩ ⟨1 00 00 1

⟩ ⟨1 00 10 1

⟩ ⟨0 01 00 1

⟩ ⟨1 01 00 1

⟩ ⟨0 11 00 1

⟩ ⟨1 11 00 1

⟩

V

100

The Schubert cell Xs2s1 = Bs2s1B in PG(F32) is labeled with thick lines.

0

⟨100

⟩ ⟨010

⟩ ⟨110

⟩ ⟨001

⟩ ⟨101

⟩ ⟨011

⟩ ⟨111

⟩

⟨1 00 10 0

⟩ ⟨1 00 00 1

⟩ ⟨1 00 10 1

⟩ ⟨0 01 00 1

⟩ ⟨1 01 00 1

⟩ ⟨0 11 00 1

⟩ ⟨1 11 00 1

⟩

V

The Schubert cell Xs1s2s1 = Bs1s2s1B = Bs2s1s2B = Xs2s1s2 in PG(F32) is labeled with thick

lines.

0

⟨100

⟩ ⟨010

⟩ ⟨110

⟩ ⟨001

⟩ ⟨101

⟩ ⟨011

⟩ ⟨111

⟩

⟨1 00 10 0

⟩ ⟨1 00 00 1

⟩ ⟨1 00 10 1

⟩ ⟨0 01 00 1

⟩ ⟨1 01 00 1

⟩ ⟨0 11 00 1

⟩ ⟨1 11 00 1

⟩

V

101

Bibliography

[AB08] P. Abramenko and K.S. Brown. Buildings: Theory and Applications. Graduate Textsin Mathematics. Springer New York, 2008. isbn: 9780387788357. url: https://books.google.com.au/books?id=7AJf5J55h28C.

[Art11] M. Artin. Algebra. Pearson Prentice Hall, 2011. isbn: 9780132413770. url: https://books.google.com.au/books?id=QsOfPwAACAAJ.

[Bae42] Reinhold Baer. “Homogeneity of projective planes”. In: Amer. J. Math. 64 (1942),pp. 137–152. issn: 0002-9327.

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Index

adjoint group, 25

Baer’s Theorem, 13basis, 11Borel subgroup, 26Bruhat decomposition, 30Building, 91

cap, 14Cartan-Killing theorem, 24Chevalley, 24

basis, 24, 25generators, 25group, 25relations, 26

collineation, 10conic, 16Coxeter generators, 24Coxeter’s theorem, 24

Desarguesian, 10dimension

projective, 11

flag variety, 67Frobenius automorphism, 19Frobenius morphism, 49

group of Lie type, 49

Hasse diagram, 8

incidence structure, 9polar, 12projective, 9

Kostant Z-form, 25

lattice, 5Boolean, 8diagram, 8projective, 7subset, 6subspace, 5, 6, 11

Lie algebra, 23Borel subalgebra, 24Cartan subalgebra, 23examples, 36Killing form, 23positive roots, 24rank, 25

reductive, 23root system, 23semisimple, 23simple, 23simple roots, 24triangular decomposition, 24

maximal torus, 26

ovoid, 14, 18and (twisted) Chevalley groups, 71classical ovoid in the Hermitian surface, 20Suzuki-Tits, 18

Pappian, 11polar

semilattice, 6space, 6, 12

projective space, 6

rank, 7rational normal curve, 16reflection, 23root subgroups, 26

Schubert cell, 71Segre’s theorem, 17semilinear transformation, 6Shult’s theorem, 14simple reflections, 24singular

subspace, 12span, 11Steinberg endomorphism, 49Suzuki group, 62

Thickness, 77totally isotropic

subspace, 6vector, 6

Twisted Chevalley group, 49

universal group, 25

Veblen-Youngaxiom, 9theorem, 13

Weyl group, 24, 26length function, 26reduced expression, 26

109

Witt index, 7

110

Minerva Access is the Institutional Repository of The University of Melbourne

Author/s:

Xu, Jon

Title:

Chevalley groups and finite geometry

Date:

2018

Persistent Link:

http://hdl.handle.net/11343/212228

File Description:

Chevalley Groups and Finite Geometry (PhD Thesis)

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Chevalley Groups and Finite Geometry

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