Chevalley Groups and Finite Geometry

Home , Chevalley basis, Galois geometry

The University of Melbourne

School of Mathematics and Statistics PhD Thesis

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 building cut into simple pieces, now we see clearly.

³WM K;~f Q ikameshiki yawokizamarete satorikeri i

Contents

i Acknowledgements v Introduction 1 Chapter 1. Finite Geometry 5 1.1. Lattices 5 1.2. Incidence structures 9 1.3. Veblen-Young type theorems 13 1.4. Ovoids of projective incidence structures 14 1.5. Ovoids of polar incidence structures 18 Chapter 2. Chevalley Groups 23 2.1. Lie algebras and root systems 23 2.2. Chevalley bases and Chevalley groups 24 2.3. The Bruhat Decomposition 27 2.4. Decomposing double cosets into single cosets 31 2.5. Examples 36 Chapter 3. Twisted Chevalley groups 49 3.1. Deﬁnitions and basic properties 49 3.2. Examples 55 3.3. From sesquilinear forms to Steinberg endormorphisms 64 Chapter 4. Tying things together 67 4.1. (Generalized) ﬂag varieties, their lattices and incidence structures 67 4.2. Ovoids and (twisted) Chevalley groups 71 4.3. The Thickness of Schubert Cells 77 4.4. The example G = SL3(F) 81 Appendix A. 85 A.1. Finite Fields and Galois Theory 85 A.2. Sesquilinear forms 86 A.3. Quadratic forms 89 A.4. Flag varieties and buildings 91 Appendix B. 93 B.1. Hasse diagrams of subspace lattices 93 B.2. Hasse diagrams of Boolean lattices 96 3 B.3. Hasse diagrams of Schubert cells in PG(F2) 99 Bibliography 103 Index 109

iii

Acknowledgements

I would like to thank my supervisor Arun Ram for his honesty and determination in the PhD supervision process. He has been an inspiring figure, and he has taught me how to confront mathematics 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’s Graduate Students Association thesis haiku competition, where it won (joint) first place. I would like to thank Asako Saito and Haydn Trowell for help with the Japanese translation.

Introduction

The aim of this thesis is to show how aspects of representation theory can be used to study finite 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 representation theory communities in Australia, and we hope that this thesis can help to bridge the modern 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 for investigation. The goal was to shape the language of algebraic groups and Chevalley groups to 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 a fruitful research direction. Chapter 1, Chapter 2 and Chapter 3 cover well-known definitions and results from (respectively) finite geometry, the theory of Chevalley groups, and the theory of twisted Chevalley groups. Most of the theory and examples have been adapted from the literature. The following parts do not explicitly appear in the literature, and may be useful to those interested in aspects of Chevalley groups and/or finite geometry: • the proof that the classical ovoid in the Hermitian space H(3, q) is an ovoid (Proposition 1.5.5), • 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 flag varieties). Chapter 4 constitutes our main contribution to the research literature, which consists of three theorems:

Main Theorem 1. Let C be the favourite nondegenerate conic, let E be the favourite elliptic 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 = P Ω3(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 G 2 2+1 + 2 acting on P = P(Fq ) by matrix multiplication and [v ] = [1 : 0 : 0] ∈ P , then Φ: G/B −→ 2 P is injective with Φ(G/B) = C. gB 7−→ g[v+]

F − F (2) (Section 4.2.2) Let G = P Ω4 (Fq) be the orthogonal group of minus type and B be F 3 3+1 the subgroup of upper triangular matrices. With G acting on P = P(Fq ) by matrix multiplication and [v+] = [1 : 0 : 0 : 0] ∈ P3, then Φ: GF /BF −→ 3 P is injective with Φ(GF /BF ) = E. gBF 7−→ g[v+]

1 F F F (3) (Section 4.2.3) Let G = Sp4(F22e+1 ) be the Suzuki group and B be the subgroup of F 3 3+1 upper triangular matrices. With G acting on P = P(Fq ) by matrix multiplication and [v+] = [1 : 0 : 0 : 0] ∈ P3, then Φ: GF /BF −→ 3 P is injective with Φ(GF /BF ) = S. gBF 7−→ g[v+] Main Theorem 2. (Section 4.2.4) Let O be the favourite classical ovoid in the Hermitian 3+1 space H(Fq2 ) 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 q O = {[1 : 0 : 0 : 0] | x2 = 0} t {[u : 1 : 0 : 0] | u ∈ Fq2 and x2u − x1 = 0} q t {[u : t : 1 : 0] | u ∈ Fq2 , t ∈ Fq and x2u − x1t + x−1 = 0} 0q 0q q 0 0 q 0q t {[t − u u : −u : u : 1] | u, u ∈ Fq2 , t ∈ Fq and x2(t − u u ) + x1u + x−1u − x−2 = 0} t {[t0 − u0qu : −u0q + tuq : uq : 1] 0 0 0 0 q 0 | u, u ∈ Fq2 , t, t ∈ Fq and x2(t − u u ) − 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 be the 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 `(z) j i,j q , where w = uzv with u ∈ W , zv ∈ Wj, z ∈ (Wj) , v ∈ Wi,j. We now more informally describe the results, motivations and methodology of these theorems. In Section 4.1.4, we establish a relationship between incidence structures (from finite geometry), and flag varieties (from representation theory). The motivation for establishing this relationship is the work of Tits and Steinberg on the Suzuki-Tits ovoid, and is outlined in Section 4.2.3. Main Theorem 1 explicitly describes three key examples of ovoids – the rational normal curve, the elliptic quadric, and the Suzuki-Tits ovoid – as flag varieties 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 role of the Chevalley group 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 with inclusion ⊆ as the partial order. A point (respectively line, hyperplane) in PG(V ) is a subspace S ⊆ 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 contains exactly 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 hyperplane. These two types of conditions, “thinness” and “maximality”, characterize the definitions of ovoids (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. The quotient 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 . The flag varieties are studied with the use of the Bruhat decomposition, G ˚ G = BwB, and Xw denotes the Schubert cell BwB w∈W ˚ 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 in finite geometry. From the point of view of representation theory, the closures of the Schubert cells are the Schubert varieties of the projective variety G(F)/B(F), where F is the algebraic closure of F. This makes the Schubert cell a tool in the framework of geometric representation theory. In Chapter 4, Section 4.3, we define an incidence structure for each Schubert cell and each pair of maximal parabolic subgroups of the Chevalley group. This provides a way of analyzing the Schubert cell using the viewpoint of finite projective geometry. Then, in pursuit of the question 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 that come from Schubert cells. This allows us to isolate basic examples of Schubert cells which are thin. We hope that future work will provide a full classification of ovoids that arise from Schubert cells, Schubert varieties and hyperplane sections of Schubert varieties. Appendix A and Appendix B contain material whose presence would otherwise disrupt the flow of the thesis. We have established the statements of this thesis in the context of the theory of groups of Lie 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 between the language of groups of Lie type and the language of buildings in Section A.4.

CHAPTER 1

Finite Geometry

In this chapter, we outline the definitions and key theorems of finite geometry. We describe some 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 how these structures are related via the Veblen-Young theorem (Section 1.3). The combinatorial objects at the forefront of research in finite geometry are those which are 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 properties of the classical varieties (such as conics) over finite fields, and whether these varieties can be characterised by incidence axioms. We define ovoids and describe the ovoids of projective and polar incidence structures in Sections 1.4 and 1.5. The study of ovoids is active in the 21st century, and there are many open problems (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 geometries, 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 be a partially ordered set with partial order ≤. The join (or supremum, or least upper bound) of two 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 (respectively meet). A lattice is a poset L closed under both join and meet. Let L, L0 be lattices. A lattice homomorphism [DP02, p. 2.16] is a map ϕ: L → L0 such that 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 lattice homomorphism.

1.1.1. The subspace lattice of a vector space. Let F be a ﬁeld or division ring and let 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 ⊆ .

n+1 In the finite geometry literature, a finite projective space is the subspace lattice PG(Fq ), where Fq is the finite field with q elements. The notations PG(n, Fq) and PG(n, q) are sometimes used n+1 for PG(Fq ). In the algebraic geometry literature, projective space is the quotient n n F − {(0,..., 0)} P = × . h(a1, . . . , an) = (ca1, . . . , can) | c ∈ F i 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 rank dim(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 vector space. 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 transformation from V1 to V2.

Proposition 1.1.1. [Tay92, §3] If f : V1 → V2 is a bijective semilinear transformation then the 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 n+1 PG(F1 ) is the lattice of subsets of {1, 2, . . . n + 1}, ordered by ⊆. n+1 The joins and meets of PG(F1 ) are U ∨ W = U ∪ W and U ∧ W = U ∩ W . n The justification for the notation PG(F1 ) is to relate our work to the study of the ‘field with one element’ as outlined by Tits in [Tit56, §13]. For modern research, see for example [Sou04] and [PL09]. Conceptually, we have n ∼ n PG( 1 ) = lim PG( q ). F q→1 F

1.1.3. The polar semilattice associated with a sesquilinear form. Let Fq be the n+1 finite field with q elements, and let V = Fq , considered as a Fq-vector space. Let h·, ·i: V × V → Fq be a σ-sesquilinear form on V (see Appendix A.2 for definitions and basic properties of sesquilinear forms). A vector v ∈ V is totally isotropic if hv, vi = 0. A subspace W in V is totally isotropic if the following condition is satisfied: if v, w ∈ W then hv, wi = 0. Note that our definition of totally isotropic follows [Tay92, pg. 56, Definition (ii)] and [Bal15, pg. 27]. The classical polar semilattice P (V, h·, ·i) is the meet-semilattice of totally isotropic subspaces in PG(V ) (often P (V, h·, ·i) is called a polar space). The meet is given by x ∧ y = x ∩ y. Note that, in general, P (V, h·, ·i) is not a lattice since it is not closed under join. Let r be the dimension of a maximum dimension totally isotropic subspace of P (V, h·, ·i), 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 isotropic 2r subspaces of a nondegenerate alternating form of rank r on Fq . • The Hermitian (or unitary) polar semilattices are the semilattices H(2r − 1, q2) and H(2r, q2) (respectively) of totally isotropic subspaces of a nondegenerate Hermitian 2r 2r+1 form of rank r on Fq and Fq (respectively). • The hyperbolic orthogonal polar semilattice is the semilattice Q+(2r − 1, q) of totally 2r isotropic subspaces of a nonsingular quadratic form of rank r on Fq . • The parabolic orthogonal polar semilattice is the semilattice Q(2r, q) of totally isotropic 2r+1 subspaces of a nonsingular quadratic form of rank r on Fq . • The elliptic orthogonal polar semilattice is the semilattice Q−(2r + 1, q) of totally 2r+2 isotropic subspaces of a nonsingular quadratic form of rank r on Fq . 1.1.4. Lattice Theory. Let L be a lattice. A greatest element of L is a g ∈ L such that if x ∈ L then x ≤ g.A least element of L is a g ∈ L such that if x ∈ L then x ≥ g. For the remainder of this section, we assume L is a lattice with a unique greatest element 1 and unique least 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 following condition 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) = sup{x ∈ L | x ≤ a}. The rank of L is rank(L) = sup{rank(a) | a ∈ L}. An element a ∈ L is an atom if rank(a) = 1. A lattice L is atomic if the following condition is satisfied: 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, and indecomposable, in turn. If U ∈ L then there exists a basis {v1, v2, . . . , vk} of U. The elements

span{v1}, span{v2},..., span{vk} are atoms, and

(... ((span{v1} + span{v2}) + span{v3}) + ··· + span{vk}) = U Hence L is atomic. 7 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 and U2 ∩ 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). 1 λ1 1 λ1 If λ2 6= 0 then v = v1 + v2. So v2 = v − v1. Since v ∈ U3 and v1 ∈ U1 ⊆ U3 λ2 λ2 λ2 λ2 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 following condition 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, for sake of contradiction, that X = V1 ∩ V2 is nonzero. Then 0 ⊆ X ⊆ V1 and 0 ⊆ X ⊆ V2. But X = 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 such that v∈ / V1 + V2. Let U1,U2 ∈ L such that 0 ⊆ U1 ⊆ V1, 0 ⊆ U2 ⊆ V2 and F{v} = U1 + U2. We know U1 + U2 ⊆ V1 + V2. So F{v} ⊆ 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. Then F{v1 + v2} ( V1 and F{v1 + v2} ( V2. Let U1,U2 ∈ L such that 0 ⊆ U1 ⊆ V1, 0 ⊆ U2 ⊆ V2 and F{v1 + v2} = U1 + U2. Since F{v1 + v2} is 1-dimensional, we have F{v1 + v2} = U1 or F{v1 + v2} = U2. So F{v1 + v2} ⊆ V1 or F{v1 + 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].

n+1 Proposition 1.1.3. The subspace lattice PG(F1 ) is a Boolean lattice.

n+1 Proof. Suppose x ∈ PG(F1 ). Then x ∪ {1, 2, . . . n + 1} = {1, 2, . . . n + 1} and x ∩ ∅ = ∅. n+1 n+1 So PG(F1 ) is complemented. Let x, y, z ∈ PG(F1 ). Then x ∩ (y ∪ z) = (x ∩ y) ∪ (x ∩ z). n+1 n+1 So PG(F1 ) is distributive. Hence PG(F1 ) 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 the elements of L, placing y higher than x whenever y > x. We draw a straight line segment from y to x if y > x and there does not exist z ∈ L such that y > z > x. The Hasse diagrams corresponding to subspace lattices and Boolean lattices are given in Appendix B, Sections B.1 and B.2. 8 1.2. Incidence structures An incidence structure is a triple G = (P, L, I) where P and L are sets and I ⊆ P × L. pr I ⊆ P × L −→2 L  pr1 y P

• The projection maps pr1 : P × L → P and pr2 : P × L → L are defined by pr1(p, l) = p and 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. −1 Often it is convenient to identify l ∈ L with the set of points pr1(pr2 (l)). Occasionally we will use B instead of L and call B the set of blocks. Let G = (P, L, I) and G0 = (P0, L0, I0) be incidence structures. A homomorphism from G to G0 is a map ϕ: P t L → P0 t L0 such that ϕ(P) ⊆ P0, ϕ(L) ⊆ L0, and if (p, b) ∈ I then (ϕ(p), ϕ(b)) ∈ I. A homomorphism of incidence structures ϕ: P t B → P0 t B0 is an isomorphism if ϕ is a bijection 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 is G(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) = k} Deﬁne the projection maps

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

If a ∈ Pj then −1 pipj (a) = {z ∈ Pi|a ≤ z or z ≤ a}. 1.2.1. Projective incidence structures. A projective incidence structure is an incidence structure 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 that q ∈ l(a, c) ∩ l(b, d);

b p

d c

(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 collineation from 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 collineations of 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 that n ≥ 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 → P2 tL2 is an isomorphism of incidence structures then there exists a semilinear transformation f : V1 → V2 with associated isomorphism σ : K1 → K2 such that α(p) = f(p) and α(l) = f(l),

0 for all p ∈ P1 and l ∈ L1. Furthermore, if f : V1 → V2 is a semilinear transformation with 0 associated isomorphism σ : K1 → K2 such that α(p) = f 0(p) and α(l) = f 0(l), for all p ∈ P1 and l ∈ L1, then there exists b ∈ K2 such that for all v ∈ V1 and a ∈ K1 we have f 0(v) = bf(v) and σ0(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-vector space such that dimK V ≥ 3. Let G = G(PG(V )) be the point line incidence structure associated with the subspace lattice PG(V ). Let ΓL(V ) be the group of invertible semilinear transformations of V . Then PΓL(V ) ∼= ΓL(V )/Z(ΓL(V )) A projective incidence structure G is Desarguesian [BR98, §2.2] if the following statement holds: 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 points c, 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).

p13

p12 c a2 b2

a 1 p23

10 A projective incidence structure G is Pappian [BR98, §2.2] if for all distinct intersecting pairs of lines 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.

a1 l2 a2

p p 13 23 l p12 1

Theorem 1.2.3. Let V be a vector space over a division ring F, and let G(PG(V )) be the point-line incidence structure associated with the subspace lattice PG(V ). Then G(PG(V )) is a Desarguesian projective incidence structure. Furthermore, G(PG(V )) is Pappian if and only if F 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 unique line. A subspace is a subset S ⊆ P such that S contains any line connecting two of its points, i.e. −1 if p1, p2 ∈ S then pr1(pr2 (l(p1, p2))) ⊆ S. If U ⊆ P is a subspace then G0 = (U, L0, I0) with L0 = {l ∈ L | l ⊆ U}, I0 = {(p, l) ∈ I | p ∈ U and l ∈ L0}, is a projective incidence structure. The span of a subset X ⊆ P is \ span(X ) = S. X ⊆S⊆P, S is a subspace A set X ⊆ P is a basis of G if the following conditions are satisﬁed: (1) span(X ) = P, (2) if X 0 ( X is a proper subset then span(X 0) 6= P. The projective dimension or (projective) rank of G is one less than the number of elements in any basis of G. See [BR98, Theorem 1.3.8] for proof that any basis has the same number of elements. Let n be the projective dimension of G and write Pdim(G) = n. An i-plane is the induced projective geometry G0 = (U, L0, I0) of a subspace U ⊆ P such that Pdim(G0) = i.A hyperplane is 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 map n o n o projective −→ finite dimensional lattices 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 taken from [Tay92, pg. 108] (see also [Shu75, Theorem C], [Shu10, §7.1]). This definition is from the work 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 definitions given by Veldkamp [Vel59] and Tits [Tit74, §7.1]. The proof of this equivalence is also given in [Ueb11, §2, Chapter 4]. Buekenhout and Shult’s definition, which we use below, has the advantage 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 incidence structure G = (P, L, I) pr I ⊆ P × L −→2 L  pr1 y P 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] if the 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) pr I ⊆ P × B −→m B  pr1 y P where B = {maximal singular subspaces} I = {(p, b) ∈ P × B | p ⊆ b}. Theorem 1.2.5. [Tay92, pg. 107, paragraph -2], [Shu10, pg. 178]. Let V , h·, ·i and P (V, h·, ·i) be as in Section 1.1.3. Let P = {1 − dimensional totally isotropic vector subspaces of P (V, h·, ·i)} L = {2 − dimensional totally isotropic vector subspaces of P (V, h·, ·i)} I = {(p, l) ∈ P × L | p ⊆ l}. Then G(P (V, h·, ·i)) = (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 collineation α: P t L → P t L such that there exists a hyperplane H (the axis of α) and a point C (the center 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 collineations ∗ with 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 a Desarguesian projective geometry. If H is a hyperplane and c, p, p0 are distinct collinear points of G with p, p0 ∈/ H then there exists a unique central collineation of G with axis H and centre c mapping p onto p0.

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

Theorem 1.3.3. [BR98, Theorem 3.3.4] Let DO be the group of central collineations of G with 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 (where ∗ addition on P is defined in Proposition 1.3.1). Furthermore, define σ1 · σ2 ∈ F by ( σ1 ◦ σ2, if σ1, σ2 ∈ DO, σ1 · σ2 = 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 is ∼ Desarguesian 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 exists ∼ a 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 the set of atoms of L. Then the map ∼ |A(L)| ν : L −→ {subsets of A(L)} = PG(F1 ), 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 of rank at least 4 then there exists a (classical) polar semilattice P (V, h·, ·i) 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 incidence structure of rank at least 3, due to Tits [Tit74] (see also [BS74, Theorem 1]). The classification of 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 key n+1 −1 example is G = G(PG(Fq )). Let S ⊆ P. A line l ∈ L is secant to S if |S ∩ pr1(pr2 (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 −1 |S ∩ pr1(pr2 (l))| = 1 (that is, if l is incident with S in exactly 1 point). A hyperplane H ⊆ P is 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

TLp(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) = sup{l ∈ 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 = TLp(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 −1 TLp(O) = p2pn (Tp(O)). n+1 n−1 Proposition 1.4.1. If O is an ovoid in G(PG(Fq )) then Card(O) = q + 1.

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

Lp = TLp(O) t SLp(O) We count the number of elements in each of the sets above. Since n+1 Lp ↔ p1(PG(Fq /p)) l 7→ l + p, is a bijection, n+1 Fq n Card( p − {0}) q − 1 Card(Lp) = × = . Card(Fq ) q − 1 Next

TLp ↔ {l ∈ p2(PG(Tp(O))) | p ⊆ l} l 7→ l,

14 is a bijection, and by the condition (O2) in the deﬁnition of an ovoid, Tp(O) ∈ Pn , so that qn−1 − 1 Card(TL ) = . p q − 1 Then, for q ∈ O, denoting l(p, q) ∈ L for the line containing p and q, the map

O − {p} ↔ SLp q 7→ l(p, q), is a bijection, giving

Card(SLp) = Card(O) − 1.

Thus Lp = TLp t SLp gives n−1 Card(O) = Card(Lp) − Card(TLp) + 1 = q + 1. Proposition 1.4.2. [Dem68, Comment after (28’), §1.4, pg. 48], [Bal15, First line of n+1 proof of Theorem 4.37]. Let O be an ovoid in G(PG(Fq )) and let H be a secant hyperplane. 0 −1 ∼ n Then O = p1pn (H) ∩ O is an ovoid in G(H) = G(PG(Fq )).

O’

n+1 Corollary 1.4.3. If G(PG(Fq )) has an ovoid then n ≤ 3 [Bal15, Theorem 4.37], [Dem68, Theorem 48], [Tit62, Footnote (1)].

n+1 Proof. Let O be an ovoid in G(PG(Fq )). Let −1 A = {(x, H) ∈ O × Pn | x ⊆ H and Card(O ∩ p1pn (H)) ≥ 2}. −1 Let x ∈ O. Since the only hyperplane H ∈ Pn such that x ⊆ H and p1pn (H) ∩ O = {x} is qn−1 Tx(O), and there are exactly q−1 hyperplanes H ∈ Pn with x ⊆ H, then n o qn − 1 Card hyperplanes incidence with a ﬁxed point = − 1, x not tangent to O q − 1 which is independent of the choice of x ∈ O. Thus, qn − 1 Card(A) = Card(O) − 1 , q − 1 qn − 1 = (qn−1 + 1) − 1 (by Proposition 1.4.1) q − 1 = (qn−1 + 1)(qn−1 + qn−2 + ··· + q2 + q) = q2(n−1) + q2(n−1)−1 + ··· + qn + qn−1 + ··· + q.

15 −1 −1 Let H ∈ Pn with Card(p1pn (H) ∩ O) ≥ 2. By Proposition 1.4.2, p1pn (H) ∩ O is an ovoid in G(H). By Proposition 1.4.1, −1 n−2 Card(p1pn (H) ∩ O) = q + 1, which is independent of the choice of H. So n−2 −1 Card(A) = (q + 1)Card{H ∈ Pn | Card(p1pn (H) ∩ O) ≥ 2}. So −1 Card{H ∈ Pn | Card(p1pn (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 q2 + q = qn + qn−1 + ··· + q3 + , 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 (2) A set of points O in PG(3, Fq) is an ovoid if and only if O is a cap and |O| = q + 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 can be 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 can be checked that O = {[1 : 0 : 0] , [0 : 0 : 1] , [1 : 1 : 1] , [1 : 2 : 1]} is an ovoid in PG(2, 3).

2+1 The rational normal curve of PG(Fq ) is 2 N = {[t : t : 1] | t ∈ Fq} ∪ {[1 : 0 : 0]}. (1.1) 2+1 A conic in PG(Fq ) is C = {[x : y : z] | Q(x, y, z) = 0}

3 where Q: Fq → Fq is a quadratic form (see Appendix, Section A.2). Equivalently, a conic is the 3 variety in P(Fq) 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 is nondegenerate. 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 ∈ P GL3(Fq) such that g · N = C. 2+1 (3) If C is a nondegenerate conic then C is an ovoid in G(PG(Fq )). Proof. 3 (1) Define Q: Fq → 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 corresponding bilinear form h, i (see Appendix B. A.3). By [Bal15, Theorem 3.26], there exists 3 0 3 a nonzero isotropic vector e ∈ Fq. Since h, i is non-degenerate, there exists f ∈ Fq such that he, f 0i= 6 0. Also, since e is totally isotropic, if f 0 = λe for some nonzero λ then he, f 0i = he, λei = 0, a contradiction. Hence {e, f 0} is a linearly independent set. Define Q(f 0) 1 f = − e + f 0. he, f 0i2 he, f 0i Then span{e, f} = span{e, f 0} and (e, f) is a hyperbolic pair (meaning that he, ei = 0, he, fi = 1, hf, fi = 0). Choose a nonzero vector a ∈ span{e, f}⊥. Then Q(λe + µa + νf) = Q((λe + νf) + (µa)) = hλe + νf, µai + Q(λe + νf) + Q(µa) = Q(λe + νf) + Q(µa) since a ∈ span{e, f}⊥ = hλe, νfi + 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 ∈ Fq (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 exists distinct t1, t2, t3 ∈ Fq such that 2 2 2 p1 = [t1 : t1 : 1], p2 = [t2 : t2 : 1], p3 = [t3 : t3 : 1], 3 and there exists a subspace W ⊆ Fq such that

dimW = 2 and p1, p2, p3 ⊆ W. But  2 2 2  t1 t2 t3 det  t1 t2 t3  = (t1 − t2)(t1 − t2)(t2 − t3) 6= 0. 1 1 1 (This determinant is an example of a Vandermonde determinant). So dimW ≥ 3. So dimW 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]. A list of known ovoids in PG(2, q) can be found in [Che04]. The ovoids in PG(2, q) are classified for q = 2k where k ∈ {1, 2, 3, 4, 5, 6}. The classification for k ≥ 7 is, to our knowledge, still open. 17 3+1 An elliptic quadric in PG(Fq ) is

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

4 where Q: Fq → Fq is the quadratic form (of Witt index 1) defined by 2 2 × • (when q is odd) Q(x1, x2, x3, x4) = x1x2 + x3 + sx4 where s ∈ Fq is chosen such that × 2 there does not exist r ∈ Fq such that r = −s, e 2 2 • (when q is even, q = 2 ) Q(x1, x2, x3, x4) = x1x2 +x3 +ax3x4 +x4 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 an 3+1 ovoid in PG(Fq ). Proposition 1.4.11. [Bro00a, §3.1], [Bar55], [Pan55], [Hir85, Theorem 16.1.7]. Let E be 0 3+1 the elliptic quadric in PG(3, q). If q is odd and O is an ovoid in PG(Fq ) then there exists 0 g ∈ P GL4(Fq) such that g · E = O . 2e Define the field automorphism θ : F22e+1 → F22e+1 by θ(s) = s for s ∈ F22e+1 . The Suzuki- 3+1 Tits ovoid in PG(F22e+1 ) is 2 S = {[1 : 0 : 0 : 0]} ∪ [t + sθ(s ) + 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]. The 3+1 Suzuki-Tits ovoid S is an ovoid in PG(F22e+1 ). Remark 1.4.13. The Suzuki-Tits ovoid in PG(3, 23) appears in [Seg59a, Theorem VI] as an 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. Matthew Brown 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 incidence structure as in Section 1.2.2. An ovoid [Tha81], [Bue95, pg. 330], [HT15, Deﬁnition 11.5] in a polar incidence structure G is a set of points O ⊆ P such that the following condition is satisﬁed: if b ∈ B then there exists a unique point p ∈ O such that b is incident with p.

4 Lemma 1.5.1. [BLP09, Corollary 4.4], [PTS09, p. 1.8.4]. Let O be an ovoid in G(W (Fq)), and let x, y ∈ P such that l(x, y) ∈/ L, that is, l(x, y) is not totally isotropic. Then the line l(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 ovoid 4 4 in W(Fq) then O is an ovoid in PG(Fq). 18 4 4 Proof. Let O be an ovoid of W (Fq). There are only two types of line of PG(Fq): totally isotropic or non-degenerate. If l is a totally isotropic line, then it meets O in precisely one point. Otherwise, if l is non-degenerate, then Lemma 1.5.1 implies that l meets O in 0 or 2 points. 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 (see 2e+1 2h+1 4 Proposition 1.4.12) in PG(3, 2 ) is also an ovoid in W (3, 2 ) = P (F22h+1 , h·, ·i), where 4 4 4 h·, ·i: F22h+1 × F22h+1 → F22h+1 is the symplectic form deﬁned by     * u−2 v−2 +  u−1   v−1    ,   = u−2v−1 − v−2u−1 + u1v2 − v1u2.  u1   v1  u2 v2

2 2 k 1.5.1. Ovoids of H(n, q ). Let Fq2 be the finite field with q elements, where q = p is a prime power. The Frobenius automorphism F r : Fq2 → Fq2 is the field automorphism defined by F r(a) = ap.

Let σ : Fq2 → Fq2 be the involutive ﬁeld automorphism deﬁned by σ(a) = F rk(a).

2 n+1 Note that σ (a) = a. Let V = Fq2 considered as an Fq2 -vector space. Let {e1, e2, . . . en+1} be a basis for V . The sesquilinear form h·, ·i: V × V → Fq2 deﬁned by

* v1   w1 + . .  .  ,  .  = v1σ(w1) + v2σ(w2) + ··· + vn+1σ(wn+1) vn+1 wn+1 is a nondegenerate Hermitian form (see Appendix A, Section A.2). Then h·, ·i is a Hermitian form and P (V, h·, ·i) is a Hermitian polar semilattice. The Hermitian curve is H(2, q2) and the Hermitian surface is H(3, q2). There exists a nondegenerate hyperplanes in V. For example, the hyperplane W = Fq2 {e2, e3, . . . en+1} 2 is nondegenerate in V . Also, set of points in H(n, q ) is nonempty. To see this, let α ∈ Fq2 be a solution to the equation ασα = −1. (such a solution exists by Proposition A.1.6). Let v = (1, α, 0, 0,..., 0) ∈ V. Then hv, vi = 1 · 1 + αα + 0 · 0 + 0 · 0 + ··· + 0 · 0 = 1 + (−1) = 0, so that span{v} ∈ P.

Proposition 1.5.4. [Bal15, Theorem 3.11]. Let V = Fk and let β : V × V → F be a nondegenerate Hermitian form. Then every maximal totally isotropic subspace has dimension k b 2 c. Proof. Let U be a totally isotropic subspace. By Proposition A.2.10, dimU + dimU ⊥ = k. ⊥ k Since U ⊆ U , we have 2dimU ≤ k, so that dimU ≤ b 2 c. k We show that if dimU < b 2 c then U can be extended to a totally isotropic subspace 0 00 k ⊥ k k ⊥ U = U ⊕ F{u }. Suppose dimU < b 2 c. So k − dimU < b 2 c. So k − b 2 c < dimU . But 19 k k k − b 2 c ≥ b 2 c. Hence k dimU < b c < dimU ⊥. 2 Hence dimU + 2 ≤ dimU ⊥. Let v ∈ U ⊥\U. We know, by Proposition A.2.10, dim(F-span{v}) + 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 u00 = w and U 0 = U ⊕ F{u00}. If λ, λ0 ∈ F and u, u0 ∈ U then β(u + λu00, u0 + λ0u00) = β(u, u0) + λβ(u00, u0) + σ(λ0)β(u, u00) + λσ(λ0)β(u00, u00) = 0, so that U 0 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 β(v, v) µσ(µ) = − . β(w, w) Then the vector u00 = v + µw is totally isotropic, since β(u00, u00) = 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 0 = U ⊕ F{u00}. If λ, λ0 ∈ F and u, u0 ∈ U then β(u + λu00, u0 + λ0u00) = β(u, u00) + λβ(u00, u0) + σ(λ0)β(u, u00) + λσ(λ0)β(u00, u00) = λβ(u00, u0) + σ(λ0)β(u, u00) = 0,

00 ⊥ 0 since u = v + λw ∈ U . So U is a totally isotropic subspace. 2 4 Proposition 1.5.5 (The classical ovoid in H(3, q )). Let V = Fq2 , considered as a Fq2 - vector space and let h·, ·i: 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 − span{v} ∈ P | v ∈ W }. Then O is an ovoid in H(3, q2).

Proof. Let b ∈ B. We will show that there exists p ∈ O such that p ⊆ b. We will also need to show that if p, p0 ∈ O such that p ⊆ b and p0 ⊆ b then p = p0. Let u, v be a basis of b. By Proposition 1.5.4, dimb = 2. Also dimW = 3, and dimV = 4, so we have dim(b ∩ W ) ≥ 1. Let w ∈ b ∩ W and p = Fq2 − span{w}. Then p is a totally isotropic 1-dimensional subspace of V with p ⊆ W and p ⊆ b and p ∈ O. But also p ⊆ b. Hence p ∈ O and p ⊆ b as required. Suppose p, p0 ∈ O with p ⊆ b and p0 ⊆ b. Suppose, for sake of contradiction, that p 6= p0. Then p ⊕ p0 = b. Since p ⊆ W and p0 ⊆ W we have b ⊆ W. So W ⊥ ⊆ b⊥. But b is totally isotropic, so b ⊆ b⊥. By Lemma A.2.8, dimV = 2 + dimb⊥, so dimb⊥ = 2. So b = b⊥. So W ⊥ ⊆ b⊥ = b ⊆ W . So W ∩ W ⊥ 6= 0. Hence W is degenerate, a contradiction. 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]. 2 • There are no ovoids in H(2n, q ) 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.

CHAPTER 2

Chevalley Groups

This chapter outlines the main deﬁnitions and theorems on Chevalley groups that are used in 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 the theory 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 and is not the 1-dimensional abelian Lie algebra. The Lie algebra g is semisimple if it is a direct sum of simple Lie subalgebras. A representation M of a complex Lie algebra g is semisimple if M is a direct sum of irreducible representations [Bou89, p. I.3.1]. The Lie algebra g is reductive if 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 ! M M g = h gα where gα = {X ∈ g | if H ∈ h then [H,X] = α(H)X} α∈R is the α-root space of g, and the elements of R are the roots of g. The adjoint representation of 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 s : h∗ → h∗ α R R given by 2(v, α) s (v) = v − α, α (α, α)

23 for v ∈ h∗ , where R (, ): h∗ × h∗ −→ R R R 0 (2.2) (γH , γH0 ) 7−→ κ(H,H ). (see [CMS95, Ch. 1, §2.3],[Ste67, §1]). The Weyl group W is the subgroup of GL(h∗ ) generated by s for α ∈ R. R α A linearly independent subset ∆ = {α1, α2, . . . , αn} ⊆ R is a set of simple roots if the following condition 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, . . . , sn} where si := sαi .

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

2 mjk si = 1, (sjsk) = 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 + M − M n = gα and n = g−α. α∈R+ α∈R+ The Borel subalgebra is b = n+ ⊕ h. For completeness, we state the following well known theorem which gives a classification of the complex 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, Theorem 3.5.1 and Theorem 3.5.2]. Theorem 2.1.2 (Cartan-Killing). The map n o n o finite dimensional complex −→ root systems of type simple Lie algebras 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 Lie groups over an arbitrary ﬁeld F from a complex simple Lie algebra g using a special choice of basis of g for which the structure constants are integers. The special choice of basis of g is the Chevalley basis and the algebraic groups are now known as Chevalley groups. The following section summarises the construction of Chevalley groups following [Ste67, §1 to §3]. See also [PRS09, §3], [Lus09], [Gec16], [Hum72, Chapter VII], and [Tit87]. The generalisation to 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 2 H = γ−1(α) α (α, 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 the equations

[Hi,Hj] = 0,

(α, αi) [Hi,Xα] = 2 Xα, (αi, αi) [Xα,X−α] = Hα ∈ Z-span{H1,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 scalars appearing 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 m Xα Z-algebra (Ug) generated by m ∈ Z≥1, α ∈ R . Z m!

Let V be a finite dimensional faithful g-module. Let M be a (Ug)Z-submodule of V such that there exists a Z-basis for M which is a C-basis for V . Let F be a field. Then F has the natural Z-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 1 1 x (t) = 1 + tX + t2X2 + t3X3 + ··· α α 2! α 3! α k for t ∈ F. This sum has finitely many terms since Xα = 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. The universal group is the Chevalley group G(F) when V is the sum of the irreducible representations having the fundamental weights as their highest weights (see [CMS95, Ch. 1, §3]). For discussion 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], deﬁne the Chevalley generators:

−1 nα(t) = xα(t)x−α(−t )xα(t), −1 hα(t) = nα(t)nα(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 = hnα(t) ∈ G(F) | α ∈ R, t ∈ F i (the monomial subgroup), × T = hhα(t) ∈ G(F) | α ∈ R, t ∈ F i (the standard maximal torus), + U = hxα(t) ∈ G(F) | α ∈ R , t ∈ Fi, + U− = hxα(t) ∈ G(F) | α ∈ −R , t ∈ Fi, B = hT,Ui (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 = si1 si2 . . . 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 = si1 si2 . . . sik . The group N/T is isomorphic to W via the map

N/T −→˜ W nα(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 and write sα = nα(t)T when appropriate (in particular, when the choice of t is irrelevant). Deﬁne

n = n−1n−1 . . . n−1 w i1 i2 i` for each w ∈ W and choice of reduced expression w = si1 si2 . . . si` .

Theorem 2.2.1. [Ste67, §3] Let G(F) be a Chevalley group. The following relations hold in G(F): (R1) If α ∈ R and t, u ∈ F then

xα(t)xα(u) = xα(t + u),

i,j (R2) If rank(R) ≥ 2, α, β ∈ R with α+β 6= 0 and t, u ∈ F, then there exists unique Cα,β ∈ Z such that

1,1 1,2 2 2,1 2 xα(t)xβ(u) =xβ(u)xα(t)xα+β(Cα,βtu)xα+2β(Cα,βtu )x2α+β(Cα,βt u) 1,3 3 2,2 2 2 3,1 3 xα+3β(Cα,βtu )x2α+2β(Cα,βt u )x3α+β(Cα,βt u) ... (see [Ste67, Lemma 15], [Car72, Theorem 5.2.2]), (R6) If α, β ∈ R and t ∈ F× then

−1 nα(1)hβ(t)nα(1) = hsα(β)(t)

(R7) If α, β ∈ R and t ∈ F then

−1 nα(1)xβ(t)nα(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

2(β,α) −1 hα(t)xβ(u)hα(t) = xβ(t (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 following conditions hold: • if α, β ∈ Q and α + β ∈ R then α + β ∈ Q, and • if α ∈ Q then −α∈ / Q. Then the map k F −→ 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 t BsiB and B t BsiB 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 t BsiB is a subgroup of G. 0 00 0 −1 00 (1) Let x ∈ BsiB · BsiB. Then there exists b, b , b ∈ B such that x = bnib ni b . Now b0 = hu for some h ∈ T and u ∈ U, so that 0 −1 00 −1 00 bnib ni b = bnihuni b −1 −1 00 = bnihni niuni b 0 −1 00 0 = bh niuni b (where h ∈ T , by relation (R6)). So by replacing b with bh0 we may assume b0 ∈ U. By Lemma 2.3.1, there exists c ∈ F 0 + and 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 −1 −1 niyni = nixβ1 (c1)xβ2 (c2) . . . xβk (ck)ni −1 −1 −1 = nixβ1 (c1)ni nixβ2 (c2)ni ··· nixβk (ck)ni 0 0 0 = xsi(β1)(c1)xsi(β2)(c2) ··· xsi(βk)(ck) by relation (R7). + −1 + By Lemma 2.3.2, si(βj) ∈ R \{αi} for all j. Hence niyni ∈ XR \{αi}. There are 2 cases: 0 −1 −1 0 −1 00 Case 1: Suppose c = 0. So nib ni = niyni ∈ B. So x = bnib ni b ∈ B. Case 2: Suppose c 6= 0. Then −1 00 x = bnixαi (c)yni b −1 −1 00 = bnixαi (c)ni niyni b 0 −1 00 = bx−αi (c )niyni b (by relation (R7)) −1 −1 −1 −1 00 = bxαi (c )nαi (−c )xαi (c )niyni b 27 −1 −1 −1 −1 00 = bxαi (c )hαi (−c )nαi xαi (c )niyni b = bx (c−1)h (−c−1)n x (c−1)n yn−1b00 αi αi αi αi αi αi ∈ BsiB, −1 −1 −1 −1 00 since bx (c )h (−c ) ∈ B and x (c )n yn b ∈ BX + B = B. αi αi αi αi αi R \{αi} Hence x ∈ B ∪ BsiB. (2) Suppose x ∈ B ∪ BsiB. If x ∈ B then −1 x = xni1ni 1,

so x ∈ BsiB · BsiB. 0 0 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 and + y ∈ XR \{αi} such that b = yxi(c). There are 2 cases: Case 1: Suppose c = 0. Then 0 0 bnib = ynib −1 0 = nini ynib 00 = nib 00 = xαi (1)x−αi (−1)xαi (1)b −1 00 = xαi (1)nixαi (±1)ni xαi (1)b

∈ BsiB · BsiB. Case 2: Suppose c 6= 0. Then 0 0 bnib = yxi(c)nib −1 0 = ynini xi(c)nib 0 0 = ynix−αi (c )b (by relation (R7)) 0−1 0 0−1 0 = ynixαi (−c )nαi (c )xαi (−c )b

∈ BsiB · BsiB.

(3) Let S = B t BsαB. To show: (a) If x ∈ S then x−1 ∈ S. (b) If x, y ∈ S then xy ∈ S. −1 (a) Suppose x ∈ S. If x ∈ B then x ∈ B so that x ∈ S. Suppose that x ∈ BsαB. 0 0 0−1 −1 −1 Then there exists b, b ∈ B such that x = bnαb . Then x = b nα b ∈ BsαB. Hence x ∈ S. (b) Suppose x, y ∈ S. If x ∈ B or y ∈ B then xy ∈ S. Suppose that x ∈ BsαB and y ∈ BsαB. Then there exists b1, b2, b3, b4 ∈ B such that x = b1nαb2 and y = b3nαb4. Then

xy = b1nαb2b3nαb4

= BnαBnαB −1 = BnαBnα B −1 = BnαXαXR+−{α}T nα B −1 −1 −1 = BnαXαnα nαXR+−{α}nα nαT nα B

= BX−αXR+−{α}B

= BX−αB. × But if x−α(t) ∈ X−α with t ∈ F then −1 −1 −1 x−α(t) = xα(t )nα(−t )xα(t ) ∈ 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 simple reﬂection.

(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, b0, b00 ∈ B such that 0 00 x = bnwb nib . 0 + We know that there exists t ∈ F, x ∈ XR \{αi}, and h ∈ T such that 0 0 b = xi(t)x h So 0 00 x = bnwxi(t)x hnib . So −1 −1 0 −1 00 x = bnwxi(t)nw nwnini x nini hnib ∈ BwsiB, since −1 −1 0 −1 + nwxi(t)nw ∈ Xwαi ⊆ B, ni x ni ∈ Xsi(R \{αi}) ⊆ B, ni 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 = si1 si2 . . . si` is a reduced expression then

BwB = Bsi1 B · Bsi2 B ··· 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∈W BwB contains a set of a generators of G. By Proposition 2.3.4, the set ∪w∈W BwB is closed under multiplication by these generators. Since ∪w∈W BwB is also closed under inverses, it follows that

∪w∈W BwB = G. Hence φ is surjective. (2) (Adapted from the proof in [Ste67, Theorem 4(b)]) Suppose w, w0 ∈ W such that φ(w) = φ(w0). We show that w = w0 by induction on `(w). For the base case, we need to show that if `(w) = 0 then w = w0. Suppose `(w) = 0. Then w = 1. Since BwB = Bw0B, we have B = Bw0B. So w0 ∈ B. So w0 = 1. Hence w = w0. We now do the induction step: Suppose (a) `(w) > 0, and (b) If v, v0 ∈ W with `(v) < `(w) and φ(v) = φ(v0) then v = v0. 0 We need to show that w = w . Let si ∈ W be a simple reﬂection such that `(wsi) < `(w). We know that wsi ∈ BwB · BsiB. 0 So wsi ∈ Bw B · BsiB. By Proposition 2.3.4, 0 0 wsi ∈ Bw B ∪ Bw siB. 0 0 So wsi ∈ BwB ∪ Bw siB. So BwsiB ⊆ BwB ∪ Bw siB. So BwsiB = BwB or 0 0 BwsiB = Bw siB. So φ(wsi) = φ(w) or φ(wsi) = φ(w si). So by induction, 0 wsi = w or wsi = w si. 0 0 If wsi = w then si = 1, a contradiction. So wsi = w si. Hence w = w . Corollary 2.3.8. [The Bruhat Decomposition] G G = BwB w∈W

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, b0 ∈ B such that xbw−1b0 = 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, b0 ∈ B and w ∈ W such that xbw−1b0 = I. Hence x ∈ BwB.

2.4. Decomposing double cosets into single cosets Let w ∈ W . Deﬁne

+ −1 + R+(w) = {β ∈ R | w (β) ∈ R } + −1 + R(w) = R−(w) = {β ∈ R | w (β) ∈/ R } Y Uw,+ = Xβ

β∈R+(w) Y Uw,− = Xβ

β∈R−(w)

Theorem 2.4.1. [Ste67, Theorem 4’] [Car89, Theorem 8.4.3] Let w ∈ W and choose a representative 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 = T UwB

= TUw,−Uw,+wB (By Lemma 2.3.1) −1 = TUw,−ww Uw,+wB −1 = TUw,−wB (Since w Uw,+w ⊆ B)

= Uw,−T wB (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(x0, b0) so that 0 0 xnwb = x nwb . Then 0 −1 −1 0−1 −1 −1 − b b = nw x xnw ∈ nw ∈ nw Uw,−nw ⊆ U . −1 0−1 − 0 0 So nw x xnw ∈ 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 every 0 0 g ∈ G has a unique expression g = unwhu where u ∈ Uw,−, h ∈ T , u ∈ U.

Proposition 2.4.4. Let w ∈ W and w = si1 si2 . . . si` be a reduced expression. Then

R−(w) = {β1, β2, . . . , β`} where

β1 = αi1 , β2 = si1 αi2 , . . . , β` = si1 si2 . . . si`−1 αi` .

Corollary 2.4.5. Let w ∈ W with reduced expression w = si1 si2 . . . si` and R−(w) = {β1, β2, . . . , β`} as in Proposition 2.4.4. Then `(w) ϕw : F −→ BwB/B

(c1, c2, . . . , c`) 7−→ xβ1 (c1)xβ2 (c2) . . . xβ` (c`)nwB, is a bijection, where n = n−1n−1 . . . n−1. w i1 i2 i`

Theorem 2.4.6. Let w = si1 si2 . . . si` be a reduced expression for w. Then the map ` ϕ: F −→ BwB/B deﬁned by ϕ ((c , c , . . . , c )) = x (c )n−1x (c )n−1 . . . x (c )n−1B 1 2 ` i1 1 i1 i2 2 i2 i` ` i` is a bijection. Proof. By Corollary 2.4.5, the map `(w) ϕw : F −→ BwB/B

(c1, c2, . . . , c`) 7−→ xβ1 (c1)xβ2 (c2) . . . xβ` (c`)nwB, is a bijection, where n = n−1n−1 . . . n−1. By repeatedly using Chevalley relation (R7), w i1 i2 i`

xβ1 (c1)xβ2 (c2) . . . xβ` (c`)nwB = x (c )x (c ) . . . x (c )n−1n−1 . . . n−1B β1 1 β2 2 β` ` i1 i2 i` = x (c )x (c ) . . . x (c )x (c )n−1n−1 . . . n−1B αi1 1 si1 αi2 2 si1 si2 ...si`−2 αi`−1 `−1 si1 si2 ...si`−1 αi` ` i1 i2 i` = x (c )x (c ) . . . x (c ) αi1 1 si1 αi2 2 si1 si2 ...si`−2 αi`−1 `−1 n−1n−1 . . . n−1 n . . . n n x (c )n−1n−1 . . . n−1B i1 i2 i`−1 i`−1 i2 i1 si1 si2 ...si`−1 αi` ` i1 i2 i` = x (c )x (c ) . . . x (c ) αi1 1 si1 αi2 2 si1 si2 ...si`−2 αi`−1 `−1 n−1n−1 . . . n−1 n . . . n n x (c )n−1n−1 . . . n−1B i1 i2 i`−1 i`−1 i2 i1 si1 si2 ...si`−1 αi` ` i1 i2 i` = x (c )x (c ) . . . x (c ) αi1 1 si1 αi2 2 si1 si2 ...si`−2 αi`−1 `−1 n−1n−1 . . . n−1 n . . . n x ( c )n−1 . . . n−1B i1 i2 i`−1 i`−1 i2 si1 si1 si2 ...si`−1 αi` ` ` i2 i` 32 (for some ` ∈ {−1, 1}), = x (c )x (c ) . . . x (c ) αi1 1 si1 αi2 2 si1 si2 ...si`−2 αi`−1 `−1 n−1n−1 . . . n−1 n . . . n x ( c )n−1 . . . n−1B i1 i2 i`−1 i`−1 i2 si2 ...si`−1 αi` ` ` i2 i` = x (c )x (c ) . . . x (c ) αi1 1 si1 αi2 2 si1 si2 ...si`−2 αi`−1 `−1 n−1n−1 . . . n−1 x (0 c )n−1B i1 i2 i`−1 αi` ` ` i` 0 (for some ` ∈ {−1, 1}), = x (c )n−1x (0 c )n−1 . . . x (0 c )n−1B i1 1 i1 i2 2 2 i2 i` ` ` i` 0 0 (for some 2 . . . ` ∈ {−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. (1) Let j1 ∈ {1, 2, . . . , n} be maximal such that aj1,1 6= 0. If j1 = 1 then let a = a. If j1 6= 1 then let

(1) aj1,2 aj1,3 aj1,n a = ax12(− )x13(− ) ··· x1n(− ). aj1,1 aj1,1 aj1,1 Let j ∈ {1, 2, . . . , n} be maximal such that a(1) 6= 0. If j = 2 then let a(2) = a(1). If j 6= 2 2 j2,2 2 2 then let a(1) a(1) a(1) (2) (1) j2,3 j2,4 j2,n a = a x23(− )x24(− ) ··· x2n(− ). a(1) a(1) a(1) j2,2 j2,2 j2,2 Continuing in this way, we produce a(n) such that every entry to the right of a(n) is zero, every j1,1 entry to the right of a(n) is zero, and so forth. Let j2,2

(n) 1 1 1 y = a h1( )h2( ) ··· hn( ). a(n) a(n) a(n) j1,1 j2,2 jn,n Then

yj1,1 = yj2,2 = ··· = yjn,n = 1.

Let w ∈ Sn such that yw ∈ U Let a(1) a(1) a(1) aj1,2 aj1,3 aj1,n j2,3 j2,4 j2,n u =ax12(− )x13(− ) ··· x1n(− )x23(− (1) )x24(− (1) ) ··· x2n(− (1) ) aj1,1 aj1,1 aj1,1 a a a j2,2 j2,2 j2,2 1 1 1 ··· h1( )h2( ) ··· hn( )w. a(n) a(n) a(n) j1,1 j2,2 jn,n Then u ∈ U and a(1) a(1) a(1) −1 (n) (n) j2,n j2,4 j2,3 a = uw hn(aj ,n) ··· h1(aj ,1) ··· x2n( ) ··· x24( )x23( ) n 1 a(1) a(1) a(1) j2,2 j2,2 j2,2

aj1,n aj1,3 aj1,2 x1n( ) ··· x13( )x12( ) aj1,1 aj1,1 aj1,1 ∈ Uw−1B. Let a(1) a(1) a(1) (n) (n) j2,n j2,4 j2,3 b = hn(aj ,n) ··· h1(aj ,1) ··· x2n( ) ··· x24( )x23( ) n 1 a(1) a(1) a(1) j2,2 j2,2 j2,2 33 aj1,n aj1,3 aj1,2 x1n( ) ··· x13( )x12( ), aj1,1 aj1,1 aj1,1 so that a = uw−1b. −1 −1 Compute R−(w ) and R+(w ). There exists unique tα’s and tβ’s in F such that Y Y u = xα(tα) xβ(tβ). −1 −1 α∈R−(w ) β∈R+(w ) Hence Y Y −1 a = xα(tα) xβ(tβ)w b −1 −1 α∈R−(w ) β∈R+(w ) Y −1 Y −1 = xα(tα)w w( xβ(tβ))w b −1 −1 α∈R−(w ) β∈R+(w ) Y −1 Y −1 = xα(tα)w (wxβ(tβ)w )b −1 −1 α∈R−(w ) β∈R+(w ) Y −1 Y = xα(tα)w (xw(β)(tβ))b −1 −1 α∈R−(w ) β∈R+(w ) −1 ∈ Uw−1,−w B. since Y (xw(β)(tβ)) ∈ B.

β∈R+ We now demonstrate this algorithm with a concrete example. Let  7 6 2 4   1 8 7 9  a =    8 6 3 5  0 1 1 2 so that a ∈ GL4(C). Then

(1) −a32 −a33 −a34 a = ax12( )x13( )x14( ) a31 a31 a31 −6 −3 −5 = ax ( )x ( )x ( ) 12 8 13 8 14 8  7 3/4 −5/8 −3/8   1 29/4 53/8 67/8  =    8 0 0 0  0 1 1 2 (2) (1) a = a x23(−1)x24(−2)  7 3/4 −11/8 −9/8   1 29/4 −5/8 9/8  =    8 0 0 0  0 1 0 0 (3) (2) a = a x34(9/5)  7 3/4 −11/8 −18/5   1 29/4 −5/8 0  =    8 0 0 0  0 1 0 0

34 a(4) = a(3) (4) y = a h1(1/8)h2(1)h3(−8/5)h4(−5/18)  7/8 3/4 11/4 1   1/8 29/4 1 0  =   .  1 0 0 0  0 1 0 0  0 0 1 0   1 11/5 7/8 3/4   0 0 0 1   0 1 1/8 29/4  u = y   =   .  0 1 0 0   0 0 1 0  1 0 0 0 0 0 0 1 Note that u ∈ U. Let  0 0 1 0   0 0 0 1  w =   = s2s3s2s1s2.  0 1 0 0  1 0 0 0 Then −1 R+(w ) = {ε3 − ε4}, −1 R−(w ) = {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 −1 a =uw 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) −1 =x12(11/5)x13(7/8)x23(1/8)x14(7/8)x24(29/4)x34(0)w

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

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) −1 ∈ Uw−1,−w B.

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

(1) a1,1 a2,1 aj1−1,1 a = s12x12(− )s23x23(− ) ··· sj1−1,j1 xj1−1,1(− )a. aj1,1 aj1,1 aj1,1 Let j ∈ {1, 2, . . . , n} be maximal such that a(1) 6= 0. If j = 2 then let a(2) = a(1). If j 6= 2 2 j2,2 2 2 then let a(1) a(1) a(1) (2) 2,2 3,2 j2−1,2 (1) a = s23x23(− )s34x34(− ) ··· sj2−1,j2 xj2−1,j2 (− )a . a(1) a(1) a(1) j2,2 j2,2 j2,2 Continuing in this way, we produce a(n) such that a(n) is upper triangular. Let b = a(n). Then (1) (1) a(1) a2,2 a3,2 j2−1,2 ··· s23x23(− )s34x34(− ) ··· sj2−1,j2 xj2−1,j2 (− ) a(1) a(1) a(1) j2,2 j2,2 j2,2

a1,1 a2,1 aj1−1,1 s12x12(− )s23x23(− ) ··· sj1−1,j1 xj1−1,1(− )a = b. aj1,1 aj1,1 aj1,1

35 Then

aj1−1,1 a2,1 a1,1 a = xj1−1,1( )sj1−1,j1 ··· x23( )s23x12( )s12 aj1,1 aj1,1 aj1,1 a(1) (1) (1) j2−1,2 a3,2 a2,2 xj2−1,j2 ( )sj2−1,j2 ··· x34( )s34x23( )s23 ··· b. a(1) a(1) a(1) j2,2 j2,2 j2,2 Also,

w = sj1−1,j1 sj1−2,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 − YX. 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 of Cartan 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-span{Eij}. A standard choice of simple roots in R is

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

2 mjk W = hs1, s2, . . . , sn−1|si = (sjsk) = 1 for i, j, k ∈ {1, 2, . . . n − 1} and j 6= ki where ( 2 if |j − k| > 1, m = jk 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) = 0} with Lie bracket inherited from gln. The Lie algebra sln is semisimple. We assume the notation used 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-span{Eij}. A standard choice of simple roots in R is

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

2 mjk W = hs1, s2, . . . , sn−1|si = (sjsk) = 1 for i, j, k ∈ {1, 2, . . . n − 1} and j 6= ki where ( 2 if |j − k| > 1, m = jk 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 dimension 2n + 1 and let h, i 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 0 0 0 so2n+1 = {X ∈ sl2n+1 | hXv, v i + hv, Xv i = 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

he0, e0i = −2 and hei, e−ji = δij for i, j ∈ {1, 2, . . . , n}. The matrix of h, i with respect to this basis is  0 0 ··· 0 1    0 0 s  0 0 ··· 1 0   . . . . .  J =  0 −2 0  , where s =  ......  ,   s 0 0  0 1 ··· 0 0  1 0 ··· 0 0 so that t so2n+1 = {X ∈ sl2n+1 | X J + 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, . . . , εn} ∗ be 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}

2 mjk = hs1, s2, . . . , sn−1|si = (sjsk) = 1 for i, j, k ∈ {1, 2, . . . n − 1} and j 6= ki, where  2 if |j − k| > 1,  mjk = 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, . . . n}}

t {Xεi−εj ,Xεj −εi Xεi+εj ,X−εi−εj | i, j ∈ {1, 2, . . . , n} and i < j}

t {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  0 2 0   0 0 0   1 0 0 

Xε1 =  0 0 1  ,X−ε1 =  1 0 0  ,Hε1 =  0 0 0  . 0 0 0 0 2 0 0 0 −1 and − n = span{Xε1 }, h = span{Hε1 }, n = span{X−ε1 }. 38 Now let g = so5. We have  0 0 2 0 0   0 0 0 0 0   0 0 0 0 0   0 0 2 0 0      Xε1 =  0 0 0 0 1 ,Xε2 =  0 0 0 1 0  ,      0 0 0 0 0   0 0 0 0 0  0 0 0 0 0 0 0 0 0 0  0 0 0 0 0   0 0 0 0 0   0 0 0 0 0   0 0 0 0 0      X−ε1 =  1 0 0 0 0 ,X−ε2 =  0 1 0 0 0  ,      0 0 0 0 0   0 0 2 0 0  0 0 2 0 0 0 0 0 0 0  0 1 0 0 0   0 0 0 0 0   0 0 0 0 0   1 0 0 0 0      Xε1−ε2 =  0 0 0 0 0 ,Xε2−ε1 =  0 0 0 0 0  ,      0 0 0 0 −1   0 0 0 0 0  0 0 0 0 0 0 0 0 −1 0  0 0 0 1 0   0 0 0 0 0   0 0 0 0 −1   0 0 0 0 0      Xε1+ε2 =  0 0 0 0 0 ,X−ε1−ε2 =  0 0 0 0 0  ,      0 0 0 0 0   1 0 0 0 0  0 0 0 0 0 0 −1 0 0 0 and  1 0 0 0 0   0 0 0 0 0   0 −1 0 0 0   0 1 0 0 0      Hε1−ε2 =  0 0 0 0 0  ,Hε2 =  0 0 0 0 0  ,      0 0 0 1 0   0 0 0 −1 0  0 0 0 0 −1 0 0 0 0 0 and

n = span{Xε1 ,Xε2 ,Xε1−ε2 ,Xε1+ε2 },

h = span{Hε1−ε2 ,Hε2 }, − n = span{X−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 dimension 2n and let h, i 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 0 0 0 sp2n = {X ∈ sl2n|hXv, v i + hv, Xv i = 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

hei, e−ji = δij for i, j ∈ {1, 2, . . . , n}. The matrix of h, i with respect to this basis is  0 0 ··· 0 1   0 0 ··· 1 0  0 s  . . . . .  J = , where s =  ......  , −s 0    0 1 ··· 0 0  1 0 ··· 0 0

39 so that t sp2n = {X ∈ sl2n | X J + JX = 0}. A standard choice of Cartan subalgebra is

h = {X ∈ sp2n | 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, . . . , 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}

2 mjk = hs1, s2, . . . , sn−1|si = (sjsk) = 1 for i, j, k ∈ {1, 2, . . . n − 1} and j 6= ki, where  2 if |j − k| > 1,  mjk = 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 = {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, . . . n}}

t {Xεi−εj ,X−εi+εj Xεi+εj ,X−εi−εj | i, j ∈ {1, 2, . . . , n} and i < j}

t {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,

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

H2εn = En,n − E−n,−n.

40 We now illustrate sp2n for speciﬁc n. Let g = sp4. We have  0 0 0 1   0 0 0 0   0 0 0 0   0 0 1 0  X2ε1 =  ,X2ε2 =    0 0 0 0   0 0 0 0  0 0 0 0 0 0 0 0  0 0 0 0   0 0 0 0   0 0 0 0   0 0 0 0  X−2ε1 =  ,X−2ε2 =   ,  0 0 0 0   0 1 0 0  1 0 0 0 0 0 0 0  0 1 0 0   0 0 0 0   0 0 0 0   1 0 0 0  Xε1−ε2 =  ,X−ε1+ε2 =   ,  0 0 0 −1   0 0 0 0  0 0 0 0 0 0 −1 0  0 0 1 0   0 0 0 0   0 0 0 1   0 0 0 0  Xε1+ε2 =  ,X−ε1−ε2 =   ,  0 0 0 0   1 0 0 0  0 0 0 0 0 1 0 0 and  1 0 0 0   0 0 0 0   0 −1 0 0   0 1 0 0  Hε1−ε2 =   ,H2ε2 =   ,  0 0 1 0   0 0 −1 0  0 0 0 −1 0 0 0 0 and

n = span{X2ε1 ,X2ε2 ,Xε1−ε2 ,Xε1+ε2 },

h = span{Hε1−ε2 ,H2ε2 }, − n = span{X−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 dimension 2n and let h, i 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 0 0 0 so2n = {X ∈ sl2n|hXv, v i + hv, Xv i = 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

hei, e−ji = δij for i, j ∈ {1, 2, . . . , n}. The matrix of h, i with respect to this basis is  0 0 ··· 0 1   0 0 ··· 1 0   . . . . .  J =  ......  ,    0 1 ··· 0 0  1 0 ··· 0 0 so that t so2n = {X ∈ sl2n | X J + JX = 0}. A standard choice of Cartan subalgebra is

h = {X ∈ so2n | X is diagonal }

41 so that a basis of h is

{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 = {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}.

2 mjk = hs1, s2, . . . , sn|si = (sjsk) = 1 for i, j, k ∈ {1, 2, . . . n} and j 6= ki, where 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, m = jk 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. The set of positive roots is + 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 < j}

t {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,

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

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  0 1 0 0   0 0 0 0   0 0 0 0   1 0 0 0  Xε1−ε2 =  ,Xε2−ε1 =   ,  0 0 0 −1   0 0 0 0  0 0 0 0 0 0 −1 0  0 0 1 0   0 0 0 0   0 0 0 −1   0 0 0 0  Xε1+ε2 =  ,X−ε1−ε2 =   ,  0 0 0 0   1 0 0 0  0 0 0 0 0 −1 0 0

42 and  1 0 0 0   1 0 0 0   0 −1 0 0   0 1 0 0  Hε1−ε2 =   ,Hε1+ε2 =   ,  0 0 1 0   0 0 −1 0  0 0 0 −1 0 0 0 −1 and

n = span{Xε1−ε2 ,Xε1+ε2 },

h = span{H1,H2}, − n = span{X−ε1+ε2 ,X−ε1−ε2 }.

Example 2.5.6. Following [Ram], the Lie algebra g of type G2 is the complex simple Lie algebra 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-span{H1,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 + 2α2} is the set of positive roots, so that + n = subalgebra generated by the set {E1,E2} − n = subalgebra generated by the set {F1,F2}

b = subalgebra generated by the set {H1,H2,E1,E2}. A choice of simple roots is

∆ = {α1, α2} and the Weyl group is 2 2 6 W = s1, s2 s1 = s2 = (s1s2) = 1 . The Lie algebra g has a Chevalley basis deﬁned as follows:

Eα1 = E1,Eα2 = E2,Eα1+α2 = [E1,E2], 1 E = [E , [E ,E ]], 2α1+α2 2! 1 1 2 1 E = [E , [E , [E ,E ]]], 3α1+α2 3! 1 1 1 2 43 1 E = [E , [E , [E , [E ,E ]]]], 3α1+2α2 3! 2 1 1 1 2

Hα1 = H1,Hα2 = H2,

Fα1 = F1,Fα2 = F2,Fα1+α2 = [F1,F2], 1 F = [F , [F ,F ]], 2α1+α2 2! 1 1 2 1 F = [F , [F , [F ,F ]]], 3α1+α2 3! 1 1 1 2 1 F = [F , [F , [F , [F ,F ]]]]. 3α1+2α2 3! 2 1 1 1 2

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 Example 2.5.2. Let B = {v0, v1} be a C-basis for the 2-dimensional C-vector space V , and let the action of E and F on V be deﬁned on the basis B by 0 1 0 0 E = ,F = . 0 0 1 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⊗Z M) generated by xα(t), x−α(t) where t2 t3 t4 x (t) = exp(tE) = I + tE + E2 + E3 + E3 + ... α 2! 3! 4! 1 0 0 1 t2 0 0 1 t = + t + ... = 0 1 0 0 2! 0 0 0 1 and similiarly 1 0 x (t) = −α t 1 for t ∈ F. The elements 0 t0 n (t0) = for t0 ∈ ×, generate N α −t0−1 0 F and the elements t0 0 h (t0) = for t0 ∈ ×, generate H. α 0 t0−1 F

Example 2.5.8. Let g = sl2 = CE ⊕CH ⊕CF where E := E12 and F := E21 as in Example 2.5.2. Let B = {v0, v1, v2} be a C-basis for the 3-dimensional C-vector space V , and let the action of E and F on V be deﬁned on the basis B by  0 2 0   0 0 0  E =  0 0 1  ,F =  1 0 0  . 0 0 0 0 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) t2 t3 t4 = I + tE + E2 + E3 + E3 + ... 2! 3! 4! 44  1 0 0   0 2 0   0 0 2   0 0 0  t2 t3 =  0 1 0  + t  0 0 1  +  0 0 0  +  0 0 0  0 0 1 0 0 0 2! 0 0 0 3! 0 0 0  1 2t t2  =  0 1 t  0 0 1 and similiarly  1 0 0  x−α(t) =  t 1 0  t2 2t 1 for t ∈ F. The elements  0 0 t2  nα(t) =  0 −1 0  t−2 0 0 for t ∈ F×, generate N. The elements  t2 0 0  hα(t) =  0 1 0  0 0 t−2

× for t ∈ F , generate T . The group G is called the projective special linear group PSL2(F) and ∼ we 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 Example 2.5.2. Let B = {v0, v1, v2, v3} be a C-basis for the 4-dimensional C-vector space V , and let the action of E and F on V be deﬁned on the basis B by  0 3 0 0   0 0 0 0   0 0 2 0   1 0 0 0  E =   ,F =   .  0 0 0 1   0 2 0 0  0 0 0 0 0 0 3 0

xα(t) = exp(tE) t2 t3 t4 = I + tE + E2 + E3 + E3 + ... 2! 3! 4!  1 0 0 0   0 3 0 0   0 0 6 0   0 0 0 6   0 1 0 0   0 0 2 0  t2  0 0 0 2  t3  0 0 0 0  =   + t   +   +    0 0 1 0   0 0 0 1  2!  0 0 0 0  3!  0 0 0 0  0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0  t4  0 0 0 0  +   + ... 4!  0 0 0 0  0 0 0 0  1 3t 3t2 t3   0 1 2t t2  =    0 0 1 t  0 0 0 1

45 and similiarly  1 0 0 0   t 1 0 0  x−α(t) =  2   t 2t 1 0  t3 3t2 3t 1 for t ∈ F. The elements  0 0 0 t3   0 0 −t 0  nα(t) =  −1   0 t 0 0  −t−3 0 0 0 for t ∈ F×, generate N. The elements  t3 0 0 0   0 t 0 0  hα(t) =  −1   0 0 t 0  0 0 0 t−3

× 0 for t ∈ F , generate H. Let G = SL2(F) as in Example 2.5.7. By [Ste67, Corollary 5, pg. 44], 0 ∼ 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 standard 4 4 representation of g acting on C . Then Z is a UgZ submodule of V such that {e1, e2, e−2, e−1} is a C-basis for V . Let F be a ﬁeld. The corresponding Chevalley group G(F) is the projective symplectic group PSp4(F). It is the subgroup of GL4(F) generated by  1 0 0 t   1 0 0 0   0 1 0 0   0 1 t 0  x2ε1 (t) =   , x2ε2 (t) =   ,  0 0 1 0   0 0 1 0  0 0 0 1 0 0 0 1  1 0 0 0   1 0 0 0   0 1 0 0   0 1 0 0  x−2ε1 (t) =   , x−2ε2 (t) =   ,  0 0 1 0   0 t 1 0  t 0 0 1 0 0 0 1  1 t 0 0   1 0 0 0   0 1 0 0   t 1 0 0  xε1−ε2 (t) =   , x−ε1+ε2 (t) =   ,  0 0 1 −t   0 0 1 0  0 0 0 1 0 0 −t 1  1 0 t 0   1 0 0 0   0 1 0 t   0 1 0 0  xε1+ε2 (t) =   , x−ε1−ε2 (t) =   ,  0 0 1 0   t 0 1 0  0 0 0 1 0 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) 2 xε1−ε2 (t)x2ε2 (u) = x2ε2 (u)xε1−ε2 (t)xε1+ε2 (tu)x2ε1 (−t u). (2.5) The elements  0 t 0 0   1 0 0 0   −t−1 0 0 0   0 0 t 0  nε1−ε2 (t) =   , n2ε2 (t) =  −1  ,  0 0 0 −t   0 −t 0 0  0 0 t−1 0 0 0 0 1

46 for t ∈ F× generate N. The elements  t 0 0 0   1 0 0 0   0 t−1 0 0   0 t 0 0  hε1−ε2 (t) =   , h2ε2 (t) =  −1  ,  0 0 t 0   0 0 t 0  0 0 0 t−1 0 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 standard 3 3 representation of g acting on C . Then Z is a UgZ-submodule of V such that {e−1, e0, e1} is a 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 ) t2 t3 t4 = I + tE + E2 + E3 + E3 + ... 2! 3! 4!  1 0 0   0 2 0   0 0 2   0 0 0  t2 t3 =  0 1 0  + t  0 0 1  +  0 0 0  +  0 0 0  0 0 1 0 0 0 2! 0 0 0 3! 0 0 0  1 2t t2  =  0 1 t  0 0 1 and similiarly  1 0 0 

x−ε1 (t) =  t 1 0  t2 2t 1 for t ∈ F. The elements  0 0 t2 

nε1 (t) =  0 −1 0  t−2 0 0 for t ∈ F×, generate N. The elements  t2 0 0 

hε1 (t) =  0 1 0  0 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 standard 4 4 representation of g acting on C . Then Z is a UgZ submodule of V such that {e1, e2, e−2, e−1} is 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  1 t 0 0   1 0 t 0   0 1 0 0   0 1 0 −t  xε1−ε2 (t) =   , xε1+ε2 (t) =   ,  0 0 1 −t   0 0 1 0  0 0 0 1 0 0 0 1  1 0 0 0   1 0 0 0   t 1 0 0   0 1 0 0  x−(ε1−ε2)(t) =   , x−(ε1+ε2)(t) =   ,  0 0 1 0   t 0 1 0  0 0 −t 1 0 −t 0 1

47 for t ∈ F. The elements  0 t 0 0   0 0 t 0   −t−1 0 0 0   0 0 0 −t  nε1−ε2 (t) =   , nε1+ε2 (t) =  −1  ,  0 0 0 −t   −t 0 0 0  0 0 t−1 0 0 t−1 0 0 for t ∈ F× generate N. The elements  t 0 0 0   t 0 0 0   0 t−1 0 0   0 t 0 0  hε1−ε2 (t) =   , hε1+ε2 (t) =  −1  ,  0 0 t 0   0 0 t 0  0 0 0 t−1 0 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 group under an endomorphism of finite order. The machinery for constructing these groups appear in 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 references are [Ste67, §11], [Ste68], [Car89]. See Ramagge [Rmg] and the references therein for the generalisation to the case of Kac-Moody groups. The motivation for introducing the theory of twisted Chevalley groups is their relationship with the classical ovoids in the Hermitian polar semilattice and the Suzuki-Tits ovoid in Sections 4.2.4 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 an endomorphism F : G → G, define GF = {g ∈ G | F (g) = g}. Recall that G is a subgroup of GL(V ) (see Chapter 2, Equation 2.3). Choose a -basis Fp Fp {e , e , . . . , e } of V , and define the Frobenius morphism to be the group homomorphism 1 2 n Fp Fr: G → G given by p (F r(g))ij = gij for i, j ∈ {1, 2, . . . , n}. Following the terminology of [MT11, Part III], an endomorphism F : G → G is a Steinberg endomorphism if there exists k, m ∈ Z≥1, such that F m = F rk. 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 pk fij = gij m−1 for all i, j ∈ {1, 2, . . . , n}. Let hij = (F (f))ij. Then m k pk (F (h))ij = (F (f))ij = (F r (f))ij = (f)ij = gij. Hence F (h) = g, and F is surjective. Note that F pk G ⊆ G(Fpk ), since (g )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 an endomorphism 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 → G be 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 −→ W nT 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 Y X = Xα. + α∈R ,Xα⊆X Proof. Suppose, for sake of contradiction, that there exists x ∈ X such that x 6= 1 and Q + Q x∈ / + X . Then there exists M ⊆ R such that x = x (c ) and there exists α∈R ,Xα⊆X α α∈M α α 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 × 2 then xβ(d) = 1 ∈ X. If d 6= 0 then choose t ∈ Fp such that t c = d. Such a t exists since Fp is algebraically closed. Then 2 −1 xβ(d) = xβ(ct ) = hβ(t)xβ(c)hβ(t) ∈ Xβ.

Hence Xβ ⊆ X. × Suppose |M| > 1. Let γ ∈ M with γ 6= β. Choose t2 ∈ F such that (β,γ)(γ,β) 2−2 (β,β)(γ,γ) t2 6= 1. This is always possible since (β, γ)2 6= (β, β)(γ, γ).

2 To see this, suppose (β, γ) = (β, β)(γ, γ). Since sα and sβ generate a dihedral group, β is a scalar 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 −(β,γ) (γ,γ) t1 = t2 . Then −1 −1 −1 hβ(t1)hγ(t2)xhγ(t2 )hβ(t2 )x   ! 2 (α,β) 2 (α,γ) 2 (β,γ) 2 (γ,β) Y (β,β) (γ,γ) 2 (γ,γ) (β,β) 2 Y =  xα(cαt1 t2 ) xβ(cβt1t2 )xγ(cγt1 t2) xα(−cα) α∈M,α/∈{β,γ} α∈M   2 (α,β) 2 (α,γ) 2 (β,γ) 2 (γ,β) Y (β,β) (γ,γ) 2 (γ,γ) (β,β) 2 =  xα(cα(t1 t2 − 1)) xβ(cβ(t1t2 − 1))xγ(cγ(t1 t2 − 1)) α∈M,α/∈{β,γ}

50   (α,β) (α,γ) (β,γ) (β,γ) (β,γ)(γ,β) Y 2 (β,β) 2 (γ,γ) −2 (γ,γ) 2 (γ,γ) 2−2 (β,β)(γ,γ) =  xα(cα(t1 t2 − 1)) xβ(cβ(t2 t2 − 1))xγ(cγ(t2 − 1)) α∈M,α/∈{β,γ} ∈ X.

−1 −1 −1 So the xβ term of hβ(t1)hγ(t2)xhγ(t2 )hβ(t2 )x disappears. But the xγ term is not the identity, 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. Then + there exists α ∈ R such that X = Xα. Q + By Proposition 3.1.3, X = + X . Let α ∈ R such that X ⊆ X. But Proof. α∈R ,Xα⊆X α α Xα 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 that F (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 −1 −1 hF (Xα)h = F (hβ(c))F (Xα)F (hβ(c) )

= 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 −1 F (Xα) = {F (hxα(t)h )|h ∈ T }, −1 = {F (h)F (xα(t))F (h) |h ∈ T }, −1 = {hF (xα(t))h |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 the restriction of ρ to ∆ is a bijection ∆ → ∆. Proof. It suﬃces to show that if α ∈ ∆ then ρ(α) ∈ ∆. Note that B t BwB is a group if and only if there exists α ∈ ∆ such that w = sα (see [Car72, Proposition 8.3.1]). Therefore, the map

F : {B t BsαB|α ∈ ∆} −→ {B t BsαB|α ∈ ∆}, B t BsαB −→ B t BF (sα)B, is a bijection. Note that (B t BsαB) ∩ U− = X−α so X−ρ(α) = F ((B t BsαB) ∩ U−) = (B t Bsρ(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 α + ρ(α) + ρ2(α) + ··· + ρm−1(α) pr(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). F Lemma 3.1.7. [Ste67, Lemma 61] Suppose a ∈ R/ ∼. Then Xa 6= 1.

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

{sI | I ∈ W/ ∼}. Proposition 3.1.9. [Ste67, Theorem 33(b)] [Car72, Proposition 13.5.2], [MT11, Propo- F F sition 23.2]. If w ∈ W then there exists nw ∈ N such that nwT = w. Furthermore, the map θ : N F /T F −→ W F nT F 7−→ nT is an isomorphism.

F Proof. Assume that w = sI for some I ∈ ∆/ ∼. By Lemma 3.1.7, there exists x ∈ X−a such that x 6= 1, where a ∈ R/ ∼ corresponds to I. We know X−a ⊆ BwB. By Corollary 2.4.3 0 there exists unique u ∈ Uw,−, u ∈ U, h ∈ T such that 0 x = unwhu . where nw ∈ N is a coset representative of w. The element nw can be chosen so that h = 1, so 0 x = unwu . Since x = F (x), 0 0 F (u)F (nw)F (u ) = unwu 0 0 Since F (w) = w, there exists h ∈ H such that F (nw) = nwh , so 0 0 0 F (u)nwh F (u ) = unwu . 0 0 Since F (u) ∈ Uw,−, h ∈ T , F (u ) ∈ U, applying the uniqueness of Corollary 2.4.3, F (u) = u, F (u0) = u0, h0 = 1.

So F (nw) = nw. It has been shown that the map θ : N F → W F deﬁned by θ(n) = nT is a surjection. The kernel of θ is ker(θ) = {n ∈ N F | 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 and F F choose a representative nw ∈ N of W such that nwT = w. Let BF \GF /BF = {BF xBF | x ∈ GF }. The map ψF : W F → BF \GF /BF deﬁned by F F F ψ (w) = B nwB

52 is a bijection. Furthermore, the map F F F F ψ :(Uw,−) × B −→ B nwB (x, b) 7−→ xnwb is a bijection.

F Proof. Let x ∈ G . 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.

F F F F F F F F F F F Hence u ∈ Uw,− and b ∈ B . So x ∈ B nwB . So B xB = B nwB . So ψ (nw) = B xB . Hence ψF is surjective. F F 0 F F F F Suppose ψ (w) = ψ (w ). Then B nwB = B nw0 B . Then BnwB = Bnw0 B. Then BwB = Bw0B. By Theorem 2.3.7, w = w0. Hence ψF is injective. F F To show: ψ is a bijection. Let x ∈ B nwB . Then x ∈ BnwB. So 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. F But F (nw) = nw, so F (u)nwF (b) = unwb. By uniqueness, F (u) = u and F (b) = b. So u ∈ Uw,− F and b ∈ B . So ψ(u, b) = unwb = x. So ψ is surjective. 0 0 0 0 0 0 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

F Y F (Uw,−) = Xa a∈R+/∼, w−1(a)⊆−R+ with uniqueness of expression on the right. Proof. Recall that Y Uw,− = Xβ

β∈R−(w) where + −1 + R−(w) = {β ∈ R |w (β) ∈ −R }. But Y Uw,− = Xβ β∈R+, w−1(β)⊆−R+ Y = Xa a∈R+/∼, w−1(a)⊆−R+ Suppose + −1 + {a ∈ R / ∼ |w (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 into F ρ-orbits. Suppose y ∈ Uw,−. Then  F  Y  y ∈  Xa .   a∈R+/∼, w−1(a)⊆−R+ To show: (1)

Y F y ∈ Xa . a∈R+/∼, w−1(a)⊆−R+ (2) If Y x x . . . x = x0 x0 . . . x0 ∈ XF . a1 a2 ak a1 a2 ak a a∈R+/∼, w−1(a)⊆−R+ with x ∈ XF then ai ai x = x0 , x = x0 , . . . , x = x0 . a1 a1 a2 a2 ak ak

We know y = xa1 xa2 . . . xak for some xai ∈ Xai . Since F (y) = y,

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

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

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

n −1 ∈ Xβ1 Xρ(β1) ... Xρ 1 (β1)Xβ2 Xρ(β2) .... Note that

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

2 2 = Xρ(β1)Xρ (β1) ... Xβ1 Xρ(β2)Xρ (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,

2 2 Xa1 = Xρ(β1)Xρ (β1) ... Xβ1 Xρ(β2)Xρ (β2) ...

n −1 = Xβ1 Xρ(β1) ... Xρ 1 (β1)Xβ2 Xρ(β2) ....

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

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

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

2 2 xβ1 (c1)xρ(β1)(c2)xρ (β1)(c3) . . . xβ` (c`) = xβ1 (d1)xρ(β1)(d2)xρ (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 subgroup of 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  0 0 1  k t −1 −1 −1 −1 −1 0 −1 0 F (x) = nw0 (F r (x) ) nw0 , where nw0 = n1 n2 n1 =   . 1 0 0 2 2k F If x ∈ G then F (x) = F r (x) . Hence F is a Steinberg endomorphism and G ⊆ SL3(Fq2 ). F The special unitary group SU3(Fq2 ) is G . Calculating F on the Chevalley generators in B: k t −1 −1 F (xα(t)) = nw0 (F r (xα(t)) ) nw0  0 0 1   1 0 0   0 0 1  =  0 −1 0   −tq 1 0   0 −1 0  1 0 0 0 0 1 1 0 0 q = xβ(t ), q F (xβ(t)) = xα(t ), (by similiar reasoning)  0 0 1   1 0 0   0 0 1  F (xα+β(t)) =  0 −1 0   0 1 0   0 −1 0  1 0 0 −tq 0 1 1 0 0 q = xα+β(−t ) + + 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, −1 −1 F (s1) = F (n1 )T = n2 T = s2, −1 −1 F (s2) = F (n2 )T = n1 T = s1, so that

F (s1s2) = s2s1,F (s2s1) = s1s2,F (s1s2s1) = s2s1s2 = s1s2s1, and F has only one orbit on S. Hence F W = {1, s1s2s1}. The twisted root system is α + β α + β RF = , α + β, −(α + β), − . 2 2 By Theorem 3.1.10, F F F F G = B t Us1s2s1,−ns1s2s1 B . We have R/ ∼= {a, −a} where a = {α, β, α + β}. So F F Us1s2s1,− = (XαXβXα+β) . F If x = xα(t)xβ(v)xα+β(u) ∈ Xa then

F (x) = F (xα(t)xβ(v)xα+β(u)) q q q = xβ(t )xα(v )xα+β(−u ) q q q q q = xα(v )xβ(t )xα+β(−u − v t ).

55 So F (x) = x if and only if v = tq and u = −uq − vqtq. Hence F q q q Xa = {xα(t)xβ(t )xα+β(u) | t, u ∈ Fq2 such that t t + u + u = 0}. Hence F −1 −1 −1 F Us1s2s1,−n1 n2 n1 B q −1 −1 −1 F q q = {xα(t)xβ(t )xα+β(u)n1 n2 n1 B | t, u ∈ Fq2 such that t t + u + u = 0}.  q     1 t u + tt 0 0 1  q F q q  =  0 1 t   0 −1 0  B t, u ∈ Fq2 such that t t + u + u = 0

 0 0 1 1 0 0   q   u + tt −t 1  q F q q  =  t −1 0  B t, u ∈ Fq2 such that t t + u + u = 0

 1 0 0 

− 3.2.2. The special orthogonal group of minus type P Ω4 (Fq). Let G = P Ω4(Fp), let T be the subgroup of diagonal matrices in G, let B the subgroup of upper triangular matrices in G, and let q = pk (see Example 2.5.12). Recall that 0 0 0 1 t 0 0 1 0 G = {x ∈ PSL4(Fp) | x Jx = J} where J =   . 0 1 0 0 1 0 0 0 Deﬁne F : G → G by 0 0 0 1 −1 k −1 t 0 1 0 0 F (x) = A (F r (x ) )A where A =   . 0 0 1 0 1 0 0 0 The function F is a well deﬁned automorphism since if x ∈ G then F (x)tJF (x) = (A−1F rk(x−1)tA)tJ(A−1F rk(x−1)tA) = AF rk(x−1)AJAF rk(x−1)tA = AF rk(x−1)JF rk(x−1)tA = AF rk(x−1J(x−1)t)A k t −1 = AFr ((x Jx) )A = AF rk(J)A = AJA = J so that F (x) ∈ G. If x ∈ G then F 2(x) = F r2k(x). Hence F is a Steinberg endomorphism and F G ⊆ P Ω4(Fq2 ). Calculating F on the Chevalley generators in B: q q F (xε1−ε2 (t)) = xε1+ε2 (t ), and F (xε1+ε2 (t)) = xε1−ε2 (t ). 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−ε2 sε1+ε2 so that F 2 W = {1, t0} with t0 = 1. We have 1 ε − ε = (ε − ε + ε + ε ) = ε = ε + ε . 1 2 2 1 2 1 2 1 1 2 So the twisted root system is F R = {−ε1, ε1}. By Theorem 3.1.10, F F F F F G /B = B t Ut0,−nt0 B . We have

R/ ∼= ±{a} where a = {ε1 − ε2, ε1 + ε2}.

F F By Proposition 3.1.11, Ut0,− = Xa . If x = xε1−ε2 (t)xε1+ε2 (u) ∈ Xa then q q F (x) = xε1−ε2 (t )xε1+ε2 (u ) q q = xε1+ε2 (u )xε1−ε2 (t ). So F (x) = x if and only if t = uq and u = tq. So F q 2 Xa = {xε1−ε2 (t)xε1+ε2 (t ) | t ∈ Fq }. Hence F F Ut0,−nt0 B q −1 −1 F 2 = {xε1−ε2 (t)xε1+ε2 (t )nε1−ε2 nε1+ε2 B | t ∈ Fq }    q       1 t 0 0 1 0 t 0 0 −1 0 0 0 0 −1 0  q   0 1 0 0   0 1 0 −t   1 0 0 0   0 0 0 1  F  =         B t ∈ Fq2  0 0 1 −t   0 0 1 0   0 0 0 1   1 0 0 0     0 0 0 1 0 0 0 1 0 0 −1 0 0 −1 0 0   q q+1     1 t t −t 0 0 0 −1  q   0 1 0 −t   0 0 −1 0  F  =     B t ∈ Fq2  0 0 1 −t   0 −1 0 0     0 0 0 1 −1 0 0 0     tq+1 −tq −t −1  q   t 0 −1 0  F  =   B t ∈ Fq2 .  t −1 0 0     −1 0 0 0  F h,i 4 4 By Proposition 3.3.1, we have G = G where h, i: Fq2 × Fq2 → Fq2 is deﬁned by

h[u1, u2, u−2, u−1] , [v1, v2, v−2, v−1]i  q    v1 0 0 0 1 q  v2  0 1 0 0 = [u1, u2, u−2, u−1] A  q  where A =   v−2 0 0 1 0 q v−1 1 0 0 0 q q q q = u−1v1 + u2v2 + u−2v−2 + u1v−1.

57 × Follow [Car89, Theorem 14.5.2], choose λ ∈ Fq2 such that λ generates Fq2 . Let 1 0 0 0 0 1 −λ 0 S =  q  . 0 1 −λ 0 0 0 0 1 Then 0 0 0 1 q t 0 2 −λ − λ 0 S JS =  q q  0 −λ − λ 2λλ 0 1 0 0 0

4 Deﬁning the quadratic form Q: Fq → Fq by

Q([u1, u2, u−2, u−1])     0 0 0 1 u1 q 0 2 −λ − λ 0  u2  = [u1, u2, u−2, u−1]  q q    0 −λ − λ 2λλ 0 u−2 1 0 0 0 u−1 2 q q 2 = 2u−1u1 + 2u2 + 2(−λ − λ)u−2u2 + 2λλ u−2. Deﬁne − 4 P Ω4 (Fq,Q) = {M ∈ Mat4×4(Fq) | if v ∈ Fq then Q(Mv) = Q(v)} t t t = {M ∈ Mat4×4(Fq) | M S JSM = S JS}. There are two things to show: (1) The map

ϕ: P Ω4(Fp) −→ P Ω4(Fp,Q) M 7−→ S−1MS, is an isomorphism. F F (2) ϕ((P Ω4(Fp) ) = P Ω4(Fq,Q). This is the same as showing that if T ∈ P Ω4(Fp) then k Fp (ϕ(T )) = ϕ(T ) Conceptually, showing (1) and (2) shows that the following diagram makes sense:

ϕ − P Ω4( p) −→ P Ω4 ( p,Q) F ∼ F ⊆ ⊆ F ϕ − P Ω4( p) −→ P Ω4 ( q,Q) F ∼ F Note that 1 0 0 0 −λq λ −1 0 λ−λq λ−λq 0 S =  −1 1  and S = AS. 0 λ−λq λ−λq 0 0 0 0 1

(1) Let M ∈ P Ω4(Fp). Then (S−1MS)tStJS(S−1MS) = StM t(St)−1StJSS−1MS = StM tJMS = StJS (since M tJM = J).

−1 − Hence S MS ∈ P Ω4 (Fp,Q). 58 F (2) Let M ∈ P Ω4(Fp) . Then ϕ(M) = S−1MS = S−1AMA−1S = S−1MS = ϕ(M).

So ϕ(M) ∈ P Ω4(Fq,Q). Now F F ϕ(Ut0,−nt0 B ) q −1 −1 F 2 = {ϕ(xε1−ε2 (t)xε1+ε2 (t ))ϕ(nε1−ε2 nε1+ε2 )ϕ(B ) | t ∈ Fq }   q q+1     1 t t −t 0 0 0 −1  q    0 1 0 −t   0 0 −1 0  F  = ϕ   ϕ   ϕ(B ) t ∈ Fq2  0 0 1 −t   0 −1 0 0     0 0 0 1 −1 0 0 0   q q q q+1     1 t + t −tλ − t λ −t 0 0 0 −1  q q   t λ −tλ   0 1 0 λ−λq   0 0 −1 0  F =  tq−t    ϕ(B ) t ∈ Fq2  0 0 1 q   0 −1 0 0   λ−λ   0 0 0 1 −1 0 0 0   q+1 q q q   t tλ + t λ −t − t −1  q q   −t λ +tλ   λ−λq 0 −1 0  F =  −tq+t  ϕ(B ) t ∈ Fq2 .  q −1 0 0   λ−λ   −1 0 0 0 

3.2.3. The special unitary group SU4(Fq2 ). Let G = SL4(Fp), let T be the subgroup of diagonal matrices in G, let B the subgroup of upper triangular matrices in G, and let q = pk. Deﬁne F : G → G by k t −1 −1 F (x) = nw0 (F r (x) ) nw0 where  0 0 0 −1  −1 −1 −1 −1 −1 −1  0 0 1 0  nw0 = n n n n n n =   . α−1 α1 α0 α−1 α1 α0  0 −1 0 0  1 0 0 0

2 2k F If x ∈ G then F (x) = F r (x). Hence F is a Steinberg endomorphism, and G ⊆ SL4(Fq2 ). By calculation, q q q F (xα−1 (t)) = xα1 (t ),F (xα0 (t)) = xα0 (t ),F (xα1 (t)) = xα−1 (t ), q q F (xα−1+α0 (t)) = xα0+α1 (−t ),F (xα0+α1 (t)) = xα−1+α0 (−t ), q F (xα−1+α0+α1 (t)) = xα−1+α0+α1 (t ). + + 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

W{s0} = {1, s0},

W{s−1,s1} = {1, s−1, s1, s−1s1}. Let

t0 = s0 and t1 = s−1s1, so 1 0 0 0  0 1 0 0 0 0 1 0 −1 0 0 0 nt0 =   , nt1 =   (3.1) 0 −1 0 0  0 0 0 1 0 0 0 1 0 0 −1 0

F By Proposition 3.1.8, W is generated by t0, t1 so that F W = {1, t0, t1, t0t1, t1t0, t0t1t0, t1t0t1, t0t1t0t1}. We have

α0 = α0 1 α = (α + α ) = α −1 2 −1 1 1 1 1 α + α = α + α + α = α + α −1 0 2 −1 0 2 1 0 1 α−1 + α0 + α1 = α−1 + α0 + α1. So the twisted root system is 1 1 1 RF = ±{α , (α + α ), α + α + α , α + α + α }. 0 2 −1 1 2 −1 0 2 1 −1 0 1 By Theorem 3.1.10, F F F F F F F F F F F G /B = B t Ut0,−nt0 B t Ut1,−nt1 B t Ut0t1,−nt0t1 B t Ut1t0,−nt1t0 B F F F F F F t Ut0t1t0,−nt0t1t0 B t Ut1t0t1,−nt1t0t1 B t Ut0t1t0t1,−nt0t1t0t1 B . We have R/ ∼= ±{a, b, c, d} where

a = {α0}, b = {α−1, α1},

c = {α−1 + α0, α0 + α1}, d = {α−1 + α0 + α1}. By Proposition 3.1.11, F F F F F F F F F F Ut0,− = Xa ,Ut1,− = Xb ,Ut0t1,− = Xa Xc ,Ut1t0,− = Xb Xd F F F F F F F F F F F F F Ut0t1t0,− = Xa Xc Xd ,Ut1t0t1,− = Xb Xc Xd ,Ut0t1t0t1,− = Xa Xb Xc Xd . Calculating the root subgroups:    1 0 0 0   F  0 1 t 0   Xa = {xa(t) = xα0 (t) | t ∈ Fq} =   t ∈ Fq ,  0 0 1 0     0 0 0 1     1 u 0 0   F q  0 1 0 0   2 2 Xb = {xb(u) = xα−1 (u)xα1 (u ) | u ∈ Fq } =  q  u ∈ Fq , (3.2)  0 0 1 u     0 0 0 1  60    1 0 u 0  q  F q  0 1 0 −u   2 2 Xc = {xc(u) = xα−1+α0 (u)xα0+α1 (−u ) | u ∈ Fq } =   u ∈ Fq ,  0 0 1 0     0 0 0 1     1 0 0 t   F  0 1 0 0   Xd = {xd(t) = xα−1+α0+α1 (t) | t ∈ Fq} =   t ∈ Fq .  0 0 1 0     0 0 0 1  Calculating the points of the Schubert cells: BF = BF F F −1 F Ut0,−nt0 B = {xα0 (t)nα0 B | t ∈ Fq}    1 0 0 0    0 t −1 0  F  =   B t ∈ Fq  0 1 0 0     0 0 0 1  F F q −1 −1 F 2 Ut1,−nt1 B = {xα−1 (u)xα1 (u )nα−1 nα1 B | u ∈ Fq }    u −1 0 0    1 0 0 0  F  =  q  B u ∈ Fq2  0 0 u −1     0 0 1 0  F F q −1 −1 −1 F 2 Ut0t1,−nt0t1 B = {xα0 (t)xα−1+α0 (u)xα0+α1 (−u )nα0 nα−1 nα1 B | t ∈ Fq, u ∈ Fq }    u −1 0 0  q   t 0 −u 1  F  =   B t ∈ Fq, u ∈ Fq2  1 0 0 0     0 0 1 0  F F q −1 −1 −1 F 2 Ut1t0,−nt1t0 B = {xα−1 (u)xα1 (u )xα−1+α0+α1 (t)nα−1 nα1 nα0 B | t ∈ Fq, u ∈ Fq }    u t 1 0    1 0 0 0  F  =  q  B t ∈ Fq, u ∈ Fq2  0 u 0 −1     0 1 0 0  F F q 0 Ut0t1t0,−nt0t1t0 B = {xα0 (t)xα−1+α0 (u)xα0+α1 (−u )xα−1+α0+α1 (t ) (3.3) −1 −1 −1 −1 F 0 2 nα0 nα−1 nα1 nα0 B | t, t ∈ Fq, u ∈ Fq }    u t0 1 0  q   t −u 0 1  F 0  =   B t, t ∈ Fq, u ∈ Fq2  1 0 0 0     0 1 0 0  F F q 0 0q Ut1t0t1,−nt1t0t1 B = {xα−1 (u)xα1 (u )xα−1+α0 (u )xα0+α1 (−u )xα−1+α0+α1 (t) −1 −1 −1 −1 −1 F 0 2 nα−1 nα1 nα0 nα−1 nα1 B | t ∈ Fq, u, u ∈ Fq }    t − u0qu −u −u0 −1  0q   −u −1 0 0  F 0  =  q  B t ∈ Fq, u, u ∈ Fq2  u 0 −1 0     1 0 0 0  F F q 0 0q 0 Ut0t1t0t1,−nt0t1t0t1 B = {xα0 (t)xα−1 (u)xα1 (u )xα−1+α0 (u )xα0+α1 (−u )xα−1+α0+α1 (t ) −1 −1 −1 −1 −1 −1 F 0 0 2 nα0 nα−1 nα1 nα0 nα−1 nα1 B | t, t ∈ Fq, u, u ∈ Fq } 61    t0 − u0qu −u u −1  0q q   −u + tu 1 0 0  F 0 0  =  q  B t, t ∈ Fq, u, u ∈ Fq2  u −1 0 0     1 0 0 0 

F h,i 4 4 By Proposition 3.3.1, we have G = G where h, i: Fq2 × Fq2 → Fq2 is defined by  0 0 0 −1  t  0 0 1 0  hu, vi = u   v  0 −1 0 0  1 0 0 0 q q q q = u2v−2 − u1v−1 + u−1v1 − u−2v2. q q 0 Note that hu, vi = −(hv, ui) . Let ν ∈ Fq2 such that ν + ν = 0. Define hu, vi = νhu, vi. Then hu, vi0 = (hv, ui0)q so that h, i0 is Hermitian, and by Proposition 3.3.1 we have GF = Gh,i = Gh,i0 . We now link GF to the ’usual’ unitary group. Define

t SU4(Fq2 ) = {x ∈ SL4(Fq2 ) | x x = 1}. q where : SL4(Fq2 ) → SL4(Fq2 ) is deﬁned by xij = xij. Then F ∼ 2 G = SU4(Fq) where the isomorphism is F Φ: G −→ SU4(Fq2 ) x 7−→ gxg−1 and  1 0 0 λ   0 1 λ 0  × g =  q  , and λ ∈ Fq2 , µ ∈ F 2 ,  0 −µ µλ 0  q −µ 0 0 µλq is deﬁned by λ + λq = 1 and µµq = −1 (see Appendix A.1.6).

2 3.2.4. The Suzuki group C2. Let G = Sp4(F2) with the standard maximal torus T and standard Borel subgroup B, and let q = 22e+1. Following [Car89, Lemma 14.1.1], deﬁne

2e θ : F2 → F2 by θ(c) = c . Deﬁne F : G → G on generators by

2 F (xε1−ε2 (t)) = x2ε2 (θ(t )),F (x2ε2 (t)) = xε1−ε2 (θ(t)), 2 F (xε1+ε2 (t)) = x2ε1 (θ(t )),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 (see also [Car89, Proposition 12.3.3]). The key calculations are checking that the Relations (2.4) in Example 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 (2) F (xε1−ε2 (t))F (x2ε2 (u)) = F (x2ε2 (u))F (xε1−ε2 (t))F (xε1+ε2 (tu))F (x2ε1 (−t u)). The calculations are as follows: 62 (1) 2 2 F (xε1−ε2 (t))F (xε1+ε2 (u)) = x2ε2 (θ(t ))x2ε1 (θ(u )) 2 2 = x2ε1 (θ(u ))x2ε2 (θ(t ))

= F (xε1−ε2 (u))F (xε1+ε2 (t))

= F (xε1−ε2 (u))F (xε1+ε2 (t))F (xε1−ε2 (2tu)). (2) 2 2 2 2 F (xε1−ε2 (t))F (x2ε2 (u)) = xε1−ε2 (θ(u))x2ε2 (θ(t ))x2ε1 (θ(t )θ(u ))xε1+ε2 (θ(t )θ(u)) 2 2 2 2 = xε1−ε2 (θ(u))x2ε2 (θ(t ))xε1+ε2 (θ(t )θ(u))x2ε1 (θ(u t )) 2 2 2 x2ε1 (θ(t )θ(u ))xε1+ε2 (θ(t )θ(u))(using 2.2.1) 2 = x2ε2 (θ(t ))xε1−ε2 (θ(u))

= F (xε1−ε2 (t))F (x2ε2 (u)). Hence F is an isomorphism. For all α ∈ R we have 2 2 e+1 F (xα(t)) = xα(θ(t )) = F r (xα(t)) F F for t ∈ F2. So F is a Steinberg endomorphism, and G ⊆ Sp4(F22e+1 ). The Suzuki group is G . 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 −1 −1 −1 F (s1) = F (nε1−ε2 T ) = F (nε1−ε2 )T = n2ε2 T = s2, and similiarly F (s2) = s1. So that S has only one F -orbit. By Proposition 3.1.8, F W = {1, s1s2s1s2}. Calculating α for α ∈ R+: 1 1 ε − ε = (ε − ε + 2ε ) = (ε + ε ) 1 2 2 1 2 2 2 1 2 1 1 2ε = (2ε + ε − ε ) = (ε + ε ) 2 2 2 1 2 2 1 2 1 1 ε + ε = (ε + ε + 2ε ) = (3ε + ε ) 1 2 2 1 2 1 2 1 2 1 1 2ε = (2ε + ε + ε ) = (3ε + ε ), 1 2 1 1 2 2 1 2 so the twisted root system is 1 1 RF = ±{ (ε + ε ), (3ε + ε )}. 2 1 2 2 1 2 By Theorem 3.1.10, F F F F G = B t Us1s2s1s2,−ns1s2s1s2 B . We have R/ ∼= ±{a, b} where a = {ε1 − ε2, 2ε2}, b = {ε1 + ε2, 2ε1}. By Lemma 2.3.1, F F Us1s2s1s2,− = (X2ε2 Xε1−ε2 X2ε1 Xε1+ε2 ) .

63 Let

x = x2ε2 (t1)xε1−ε2 (t2)x2ε1 (t3)xε1+ε2 (t4). Then 2 2 F (x) = xε1−ε2 (θ(t1))x2ε2 (θ(t2))xε1+ε2 (θ(t3))x2ε1 (θ(t4)) 2 2 2 2 2 = x2ε2 (θ(t2))xε1−ε2 (θ(t1))xε1+ε2 (θ(t1)θ(t2))x2ε1 (θ(t1t2))xε1+ε2 (θ(t3))x2ε1 (θ(t4)) 2 2 2 2 2 = x2ε2 (θ(t2))xε1−ε2 (θ(t1))x2ε1 (θ(t1t2 + t4))xε1+ε2 (θ(t1)θ(t2) + θ(t3)). So F (x) = x if and only if 2 t1 = θ(t2), t2 = θ(t1), 2 2 2 2 t3 = θ(t1t2 + t4), t4 = θ(t1)θ(t2) + θ(t3).

Letting t1 ∈ Fq we have t2 = θ(t1). Letting t3 ∈ Fq, we have t4 = t1θ(t1) + θ(t3). Hence F Us1s2s1s2,− = {x2ε2 (t1)xε1−ε2 (θ(t1))x2ε1 (t3)xε1+ε2 (t1θ(t1) + θ(t3)) | t1, t2 ∈ Fq}. Note that  0 0 0 1  −1 −1 −1 −1  0 0 1 0  ns1s2s1s2 = n n n n =   , 1 2 1 2  0 1 0 0  1 0 0 0 and

x2ε2 (t1)xε1−ε2 (θ(t1))x2ε1 (t3)xε1+ε2 (t1θ(t1) + θ(t3))  2  1 θ(t1) t1θ(t1) + θ(t3) t3 + t1θ(t1) + θ(t1t3)  0 1 t1 θ(t3)  =   .  0 0 1 θ(t1)  0 0 0 1 Hence F F Us1s2s1s2,−ns1s2s1s2 B  2     1 θ(t1) t1θ(t1) + θ(t3) t3 + t1θ(t ) + θ(t1t3) 0 0 0 1  1   0 1 t1 θ(t3)   0 0 1 0  F  =     B t1, t2 ∈ Fq2  0 0 1 θ(t1)   0 1 0 0     0 0 0 1 1 0 0 0   2   t3 + t1θ(t ) + θ(t1t3) t1θ(t1) + θ(t3) θ(t1) 1  1   θ(t3) t1 1 0  F  =   B t1, t3 ∈ Fq2  θ(t1) 1 0     1 0 0 0 

3.3. From sesquilinear forms to Steinberg endormorphisms Let V be a vector space, G a subgroup of GL(V ), and h, i: V × V → F be a σ-sesquilinear form. Deﬁne Gh,i = {x ∈ G | if u, v ∈ V then hxu, xvi = hu, vi}. Proposition 3.3.1. Suppose A ∈ End(V ) is the matrix corresponding to h, i so that hu, vi = utAσ(v), for u, v ∈ V . Then GF = Gh,i.

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 hxu, xvi = hu, vi} = Gh,i. Example 3.3.2. Let G and F : G → G be defined as in Section 3.2.1, so that GF ∼= h,i F 4 4 SU3(Fq2 ). Then G = G where h, i: Fp × Fp → Fp is defined by        q  * u−1 v−1 + 0 0 1 v−1 q  u0  ,  v0  = u−1 u0 u1  0 −1 0   v0  q u1 v1 1 0 0 v1 q q q = u1v−1 − u0v0 + u−1v1 Since hu, vi = hv, uiq, the form h, i is 1-Hermitian. F ∼ − Example 3.3.3. Let G and F : G → G be defined as in Section 3.2.2 so that G = P Ω4 (Fq). h,i F 4 4 Then G = G where h, i: Fq2 × Fq2 → Fq2 is defined by

h[u1, u2, u−2, u−1] , [v1, v2, v−2, v−1]i    q  0 0 0 1 v1 q 0 1 0 0  v2  = [u1, u2, u−2, u−1]    q  0 0 1 0 v−2 q 1 0 0 0 v−1 q q q q = u−1v1 + u2v2 + u−2v−2 + u1v−1. Since hu, vi = hv, uiq, the form h, i is 1-Hermitian. F ∼ Example 3.3.4. Let G and F : G → G be deﬁned as in Section 3.2.3 so that G = SU4(Fq2 ). h,i F 4 4 Then G = G where h, i: Fp × Fp → Fp is deﬁned by        q  u−2 v−2 0 0 0 −1 v−2 * + q  u−1   v−1   0 0 1 0   v−1    ,   = u−2 u−1 u1 u2    q   u1   v1   0 −1 0 0   v1  q u2 v2 1 0 0 0 v2 q q q q = u2v−2 − u1v−1 + u−1v1 − u−2v2 Since hu, vi = −hv, uiq, the form h, i is (−1)-Hermitian.

CHAPTER 4

Tying things together

This chapter constitutes our main contribution to the research literature. We describe in Section 4.1.4 a relationship between lattices and incidence structures (from finite geometry), and flag varieties (from representation theory). Through this relationship, the classical ovoid in 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 -coset decomposition given in [PRS09, Theorem 4.1]. The rational normal curve, another example of an ovoid, appears in a similar way, and we discuss this in section 4.2.1. In each of these examples we determine which points of the ovoid lie in each Schubert cell of the corresponding flag variety. In Section 4.3, we define an incidence structure for each Schubert cell and each pair of maximal parabolic subgroups of the Chevalley group. This provides a way of analyzing the Schubert cell using the viewpoint of projective geometry. Then, in pursuit of the question 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 geometries that come from Schubert cells. This allows us to isolate basic examples of Schubert cells which are thin. We hope 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 B is a Borel subgroup of G and T is a maximal torus. Let W be the corresponding Weyl group with Coxeter generators s1, s2, . . . , sn. Let k ∈ {1, 2, . . . n}. The standard maximal k-parabolic subgroup of G is G Pk = BwB, where Wk = hs1, . . . , sk−1, sk+1, . . . sni.

w∈Wk Remark 4.1.1. More generally, a parabolic subgroup is a (closed) subgroup P of G containing B. 4.1.1. The ﬂag variety of GL(V ). Let V be a F-vector space, and let G = GL(V ). The Grassmannian of k-planes on V is

Gr(k, V ) = {Vk ⊆ V | Vk is a subspace of V and dimVk = k} , n and there is a G-action on Gr(k, V ) deﬁned by g · Vk = gVk. If V = F we write Gr(k, n) in place of Gr(k, V ). The full ﬂag variety on V is

F l(V ) = {V• = (V0 ⊆ V1 ⊆ ... ⊆ Vn) | Vk ∈ Gr(k, V )} , with a G-action on F l(V ) deﬁned by

g · (V0 ⊆ V1 ⊆ ... ⊆ Vn) = (gV0 ⊆ gV1 ⊆ ... ⊆ gVn). If V = Fn we write F l(n) in place of F l(V ). Let {e1, e2, . . . , en} be a basis of V and Ek = F-span{e1, e2, . . . , ek}, for k ∈ {1, 2, . . . n}. The standard ﬂag in F l(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 matrices in G = GL(V ). Then

B = StabG(E•) and Pk = StabG(Ek), and the maps ϕ: G/B −→ F l(V ) ϕ : G/P −→ Gr(k, V ) and k k for k ∈ {1, 2, . . . , n}, gB 7−→ gE•, gPk 7−→ gEk, are bijections.

Corollary 4.1.3. [FH91, §23.3] Let PG(Fn) be the lattice of subspaces of Fn (as outlined in Chapter 1, Section 1.1.1). With the notation of Theorem 4.1.2, the maps maximal chains rank k elements ϕ: G/B −→ ϕk : G/Pk −→ in PG(Fn) and in PG(Fn) gB 7−→ gE•, 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 theory of the previous section, to the case that G is a classical group of Lie type (see [FH91, §23.3]). A classical group of Lie type is a Chevalley group G(F) whose underlying Lie algebra g is one of those in Examples 2.5.2 to 2.5.5. Let h·, ·i be the bilinear form corresponding to G(F). Recall that a subspace W ⊆ V is isotropic if W ⊆ W ⊥ (see A.2.12). The isotropic Grassmannian of k-planes on V is h·,·i Gr (k, V ) = {Vk ⊆ V | Vk is an isotropic subspace of V and dimVk = k}. The full isotropic ﬂag variety on V is h,i h,i F l (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 − span{e1, e2, . . . , ei}, for i ∈ {1, 2, . . . , m}, where the elements ek ∈ V are pairwise orthogonal isotropic vectors as deﬁned in Examples 2.5.2 to 2.5.5. Then Ei is an i-dimensional totally isotropic subspace. The standard ﬂag in F lh,i(V ) is

E• = (0 ⊆ E1 ⊆ E2 ⊆ · · · ⊆ Em). Theorem 4.1.4. [FH91, §23.3] Let B be the subgroup of weakly upper triangular matrices in G. Then

B = StabG(E•), and Pk = StabG(Ek). Furthermore, the maps ϕ: G/B −→ F lh·,·i(V ) ϕ : G/P −→ Grh·,·i(k, V ) and k k for k ∈ {1, 2, . . . , m}, gB 7−→ gE•, gPk 7−→ gEk, are bijections. Corollary 4.1.5. [FH91, §23.3] Let P (V, h·, ·i) be the semilattice of totally isotropic subspaces (as deﬁned in Chapter 1, Section 1.1.3). With the notation of Theorem 4.1.4, the maps maximal chains rank k elements ϕ: G/B −→ ϕ : G/P −→ in P (V, h·, ·i) and k k in P (V, h·, ·i) gB 7−→ gE•, 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 case G = 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 and V is a faithful finite dimensional representation of the underlying Lie algebra g of G (see Section 2.2). From the theory of highest weight representations (see [CMS95], [Ser87] or [Hum72]) V is a highest weight representation with weight λ and highest weight vector v0. Since the group G 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 F StabGF ([v0]) = StabG([v0]) . Proof.

LHS = StabGF ([v0]) F = {g ∈ G | [gv0] = [v0]}

= {g ∈ G | [gv0] = [v0] and F (g) = g}

= {g ∈ G | g ∈ StabG([v0]) and F (g) = g} F = StabG([v0]) = RHS. F Corollary 4.1.7. B ⊆ StabGF ([v0]). F F F Proof. Since B ⊆ StabG([v0]), B ⊆ StabG([v0]) . But StabG([v0]) = StabGF ([v0]), so F B ⊆ StabGF ([v0]).

4.1.4. The incidence structures associated to (generalised) ﬂag varieties. Let Pi and Pj be parabolic subgroups. Deﬁne p : G/B → G/P p : G/B → G/P i i and j j gB 7→ gPi 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, n there exists kB∈G/B such that o Iij = (gPi, hPj) ∈ Pi × Pj | . pi(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 structure n+1 associated with PG(F ) (see Chapter 1, Section 1.2.1) and let {e1, e2, . . . , en+1} be a basis of n+1 Fq . 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,

n+1 is an isomorphism of incidence structures from (G)12 to G(PG(F )). 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. 0 −1 0 (1) Suppose gP1 = g P1. Then g g ∈ P1. By Theorem 4.1.2, P1 = StabG(E1). So −1 0 0 0 0 g g E1 = E1. So gE1 = g E1. So ϕ(gP1) = ϕ(g P1). Similiarly, if hP2 = h P2 then ϕ(hP2) = ϕ(hP2). Hence ϕ is well-deﬁned. (2) Suppose (gP1, hP2) ∈ I12. Then there exists kB ∈ G/B such that p1(kB) = gP1 −1 −1 and p2(kB) = hP2. So kP1 = gP1 and kP2 = hP2. So k g ∈ P1 and k h ∈ P2. −1 By Theorem 4.1.2, P1 = StabG(E1) and P2 = StabG(E2). So k gE1 = E1 and −1 k hE2 = E2. So gE1 = kE1 and hE2 = kE2. But kE1 ⊆ kE2, so gE1 ⊆ hE2. So ϕ(gP1) ⊆ ϕ(hP2). Hence (ϕ(gP1), ϕ(hP2)) ∈ I. 0 0 −1 0 (3) Suppose ϕ(gP1) = ϕ(g P1). Then gE1 = g E1. Then g g E1 = E1. But by Theorem −1 0 0 0 4.1.2, P1 = StabG(E1). So g g ∈ P1. So gP1 = g P1. Similiarly, if ϕ(hP2) = ϕ(h P2) 0 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. −1 (4) Suppose (p, l) ∈ I. So p ⊆ l. Then there exists g, h ∈ G such that ϕ (p) = gP1 and −1 ϕ (l) = hP2. So p = gE1 and l = hE2. So gE1 ⊆ hE2. Extend this to an element of F l(V ):

gE1 ⊆ hE2 ⊆ V3 ⊆ V4 ⊆ ... ⊆ V. By Theorem 4.1.2, there exists k ∈ G such that

kE• = (gE1 ⊆ hE2 ⊆ V3 ⊆ V4 ⊆ ... ⊆ V ).

−1 −1 So kE1 = gE1 and kE2 = hE2. So k g ∈ StabG(E1) and k h ∈ StabG(E2). By −1 −1 Theorem 4.1.2, k g ∈ P1 and k h ∈ P2. So kP1 = gP1 and kP2 = hP2. So −1 −1 p1(kB) = gP1 and p2(kB) = hP2. So (gP1, hP2) ∈ I12. So (ϕ (p), ϕ (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 polar h,i semilattice P (V, h·, ·i) (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 polar h,i semilattice P (V, h·, ·i) (see Chapter 1, Section 1.2.2). Let G = G , and let P1 and Pm be the corresponding 1st and mth standard maximal parabolic subgroups, where m is the dimension of a maximal totally isotropic subspace. Let (G)1m = (P1, Pm, I1m) be the associated (1, m)-incidence structure. Then the map ϕ: P1 t Pm → P t B deﬁned by

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 place of 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 X˚ = BwB ⊆ G/B w where w ∈ W and v ∈ W k. ˚k Xv = BvPk ⊆ G/Pk Remark 4.1.10. The closures of the Schubert cells are the Schubert varieties of the projective variety G(F)/B and this makes them tools in the framework of geometric representation theory (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 that W = Sn. Let ϕ: G/B → F l(V ), ϕk : G/Pk → Gr(k, V ) be deﬁned as in Theorem 4.1.2. For w ∈ W deﬁne

rp,q(w) = #{i ∈ Z≥0 | 1 ≤ i ≤ p and 1 ≤ w(i) ≤ q}. Then ˚ ϕ(Xw) = {V• ∈ F l(V )|dim(Vp ∩ Eq) = rp,q(w) for p, q ∈ {1, 2, . . . n}} . Let v ∈ W k. Then ˚k ϕk(Xv ) = {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 Proposition 7]. 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 elliptic quadric, and the Suzuki-Tits ovoid as orbits of (twisted or untwisted) Chevalley groups. Section 4.2.3 builds on the work of Tits [Tit61]. Given that these key examples of ovoids can be realised as 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 role of 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 a powerful way of indexing the points of the ovoid. 71 4.2.1. The special orthogonal group P Ω3(F) and the rational normal curve. The Chevalley group G = P Ω3(F) is the exponential of the Lie algebra so3 on VF = F ⊗Z V where V is the standard representation with basis {e−1, e0, e1} (see 2.5.11). The Chevalley generators of G are  1 2t t2   1 0 0 

xε1 (t) =  0 1 t  , x−ε1 (t) =  t 1 0  , 0 0 1 t2 2t 1  0 0 t2   t2 0 0 

nε1 (t) =  0 −1 0  , hε1 (t) =  0 1 0  t−2 0 0 0 0 t−2 for t ∈ F. By Corollary 2.3.8 and Theorem 2.4.6, the Bruhat decomposition is −1 G/B = B t Bs1B with Bs1B = {xα(t)nε1 B ∈ 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] t Bs1B · [1 : 0 : 0] −1 = {[1 : 0 : 0]} t {xε1 (t)nε1 · [1 : 0 : 0] | t ∈ F} 2 = {[1 : 0 : 0]} t {[t : t : 1] | t ∈ F} since  t2 −2t 1  −1 t −1 0 xε1 (t)nε1 =   . 1 0 0

2+1 When F = Fq, the orbit G · [v0] is the rational normal curve in P(Fq ) as deﬁned in Equation (1.1), Section 1.4.

− 4.2.2. The special orthogonal group of minus type P Ω4 (Fq) and the elliptic quadric. Many portions of this section is explained in more detail in Section 3.2.2. The Chevalley group G = P Ω−( ) is the exponential of the Lie algebra so on V = ⊗ V 4 Fq 4 Fp Fp Z where V is the standard representation with basis {e−2, e−1, e1, e2} (see Example 2.5.12) where v0 = e−2 is a highest weight vector. The Chevalley generators of G are  1 t 0 0   1 0 t 0   0 1 0 0   0 1 0 −t  xε1−ε2 (t) =   , xε1+ε2 (t) =   ,  0 0 1 −t   0 0 1 0  0 0 0 1 0 0 0 1  1 0 0 0   1 0 0 0   t 1 0 0   0 1 0 0  x−(ε1−ε2)(t) =   , x−(ε1+ε2)(t) =   ,  0 0 1 0   t 0 1 0  0 0 −t 1 0 −t 0 1  0 t 0 0   0 0 t 0   −t−1 0 0 0   0 0 0 −t  nε1−ε2 (t) =   , nε1+ε2 (t) =  −1  ,  0 0 0 −t   −t 0 0 0  0 0 t−1 0 0 t−1 0 0  t 0 0 0   t 0 0 0   0 t−1 0 0   0 t 0 0  hε1−ε2 (t) =   , hε1+ε2 (t) =  −1  .  0 0 t 0   0 0 t 0  0 0 0 t−1 0 0 0 t−1

72 Let q = pk be a prime power. Deﬁne F : G → G by 0 0 0 1 −1 −1 t 0 1 0 0 F (x) = A ((x ) )A where A =   , 0 0 1 0 1 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 F W = {1, t0} where t0 = sε1−ε2 sε1+ε2 , and Bruhat decomposition F F F F G = B t Ut0,−nt0 B where F 2 Ut0,− = {xε1−ε2 (t)xε1+ε2 (t) | t ∈ Fq }. 4 4 Let h, i: Fq2 × Fq2 → Fq2 be deﬁned by hu, vi = utAv. Then F 4 G = {g ∈ PSL4(Fq2 ) | if u, v ∈ Fq2 then hgu, gvi = hu, vi}. × Let λ ∈ Fp be a generator of Fq2 (see Appendix A.1.7). Let 0 0 0 1 1 0 0 0 t 0 2 −λ − λ 0 0 1 −λ 0 K = S JS =   where S =   . 0 −λ − λ 2λλ 0 0 1 −λ 0 1 0 0 0 0 0 0 1 4 4 Then K ∈ GL(Fq). Let Q: Fq → Fq be deﬁned by Q(v) = vtKv. Let − t P Ω4 (Fq) = {x ∈ PSL4(Fq) | x Kx = K} 4 = {x ∈ PSL4(Fq) | if v ∈ Fq then Q(xv) = Q(v)}. Then F − ϕ: G −→ P Ω4 (Fq), M 7−→ S−1MS is an isomorphism. The group P Ω−( ) acts on the set (V ) and 4 Fq P Fp F ϕ(B ) ⊆ Stab − ([v0]). P Ω4 (Fq) Then − F P Ω4 (Fq) · [v0] = ϕ(G ) · [1 : 0 : 0 : 0] F F F = ϕ(B ) · [1 : 0 : 0 : 0] t ϕ(Ut0,−nt0 B ) · [1 : 0 : 0 : 0] −tλ + tλ −t + t = {[1 : 0 : 0 : 0]} t tt : : : −1 t ∈ Fq2 λ − λ λ − λ

73 since  tt tλ + tλ −t − t −1   −tλ+tλ 0 −1 0  λ−λ ϕ(xε −ε (t)xε +ε (t)nt ) =   . 1 2 1 2 0  −t+t −1 0 0   λ−λ  −1 0 0 0 − If t ∈ Fq2 then t ∈ Fq if and only if t = t. So the coordinates of the points of P Ω4 (Fq) · [v0] are − 3+1 − elements of Fq and P Ω4 (Fq) · [v0] ⊆ P(Fq ). The orbit P Ω4 (Fq) · [v0] is the elliptic quadric in 3+1 P(Fq ) as defined in Section 1.4. 2 4.2.3. The Suzuki group C2(F22e+1 ) and the Suzuki-Tits ovoid. Many portions of this section is explained in more detail in Section 3.2.4. The Chevalley group G = Sp4(F2) is the exponential of the Lie algebra sp on V = ⊗ V where V is the standard representation 4 F2 F2 Z of sp4 with basis {e−2, e−1, e1, e2} (see Example 2.5.10) where v0 = e−2 is a highest weight vector. The Chevalley generators of G are  1 0 0 t   1 0 0 0   0 1 0 0   0 1 t 0  x2ε1 (t) =   , x2ε2 (t) =   ,  0 0 1 0   0 0 1 0  0 0 0 1 0 0 0 1  1 0 0 0   1 0 0 0   0 1 0 0   0 1 0 0  x−2ε1 (t) =   , x−2ε2 (t) =   ,  0 0 1 0   0 t 1 0  t 0 0 1 0 0 0 1  1 t 0 0   1 0 0 0   0 1 0 0   t 1 0 0  xε1−ε2 (t) =   , x−ε1+ε2 (t) =   ,  0 0 1 −t   0 0 1 0  0 0 0 1 0 0 −t 1  1 0 t 0   1 0 0 0   0 1 0 t   0 1 0 0  xε1+ε2 (t) =   , x−ε1−ε2 (t) =   ,  0 0 1 0   t 0 1 0  0 0 0 1 0 t 0 1  0 t 0 0   1 0 0 0   −t−1 0 0 0   0 0 t 0  nε1−ε2 (t) =   , n2ε2 (t) =  −1  ,  0 0 0 −t   0 −t 0 0  0 0 t−1 0 0 0 0 1  t 0 0 0   1 0 0 0   0 t−1 0 0   0 t 0 0  hε1−ε2 (t) =   , h2ε2 (t) =  −1  .  0 0 t 0   0 0 t 0  0 0 0 t−1 0 0 0 1 Define 2e θ : F2 → F2 by θ(c) = c . Define F : G → G on generators by 2 F (xε1−ε2 (t)) = x2ε2 (θ(t )),F (x2ε2 (t)) = xε1−ε2 (θ(t)), 2 F (xε1+ε2 (t)) = x2ε1 (θ(t )),F (x2ε1 (t)) = xε1+ε2 (θ(t)), + F + + and similarly for the the xα where α ∈ −R . Then G 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 F W = {1, s1s2s1s2} with Bruhat decomposition

F F F F G = B t Us1s2s1s2,−ns1s2s1s2 B where

F 2e+1 Us1s2s1s2,− = {x2ε2 (t1)xε1−ε2 (θ(t1))x2ε1 (t3)xε1+ε2 (t1θ(t1) + θ(t3)|t1, t2 ∈ F2 }. The group GF acts on the set (V ), the point [v ] is stabilised by BF , and the orbit GF on P F2 0 [v ] in (V ) is 0 P F2 F F F F G · [v0] = B · [v0] t Us1s2s1s2,−ns1s2s1s2 B · [v0] 2 = {[1 : 0 : 0 : 0]} t {[t3 + t1θ(t1) + θ(t1t3): θ(t3): θ(t1) : 1] ∈ P(V ) | t1, t3 ∈ F22e+1 } since  2   t3 + t1θ(t ) + θ(t1t3) t1θ(t1) + θ(t3) θ(t1) 1  1  F  θ(t3) t1 1 0  F  2e+1 Us s s s ,−ns1s2s1s2 =   B t1, t3 ∈ F2 . 1 2 1 2  θ(t1) 1 0 0     1 0 0 0 

F 4 The coordinates of the points of G · [v0] are elements of F22e+1 so that G · [v0] ⊆ P(F22e+1 ). The F 3+1 orbit G · [v0] is the Suzuki-Tits ovoid in P(F22e+1 ) as deﬁned in Section 1.4.

4.2.4. The Schubert cell decomposition of the classical ovoid in H(3, q2). Many portions of this section is explained in more detail in Section 3.2.3. The Chevalley group G = PSL ( ) is the exponential of the Lie algebra sl on V = ⊗ V where V is the standard 4 Fp 4 Fp Fp Z representation of sl4 with basis {e−2, e−1, e1, e2} (see Example 2.5.10) where v0 = e−2 is a highest weight 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

t −1 −1 F (x) = nw0 (x ) nw0 where  0 0 0 −1  −1 −1 −1 −1 −1 −1  0 0 1 0  nw0 = n n n n n n =   . α−1 α1 α0 α−1 α1 α0  0 −1 0 0  1 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  1 0 0 0   0 1 t 0  xa(t) =    0 0 1 0  0 0 0 1  1 u 0 0   0 1 0 0  xb(u) =    0 0 1 u  0 0 0 1  1 0 u 0   0 1 0 −u  xc(u) =    0 0 1 0  0 0 0 1  1 0 0 t   0 1 0 0  xd(t) =    0 0 1 0  0 0 0 1

F for t ∈ Fq and u ∈ Fq2 . The coset representatives of the Coxeter generators of W in N are 1 0 0 0  0 1 0 0 0 0 1 0 −1 0 0 0 nt0 =   , nt1 =   0 −1 0 0  0 0 0 1 0 0 0 1 0 0 −1 0

F F where t = s and t = s s . The group G acts on the set (V ) with B ⊆ Stab F ([v ]) 0 0 1 −1 1 P Fp G 0 and the orbit of [v ] in (V ) is the set of points in the Hermitian polar semilattice H(3, q2) 0 P Fp (see 1.1.3). Explicitly calculating the orbit, F F F F F F F F F F G · [v0] = B t Ut0,−nt0 B t Ut1,−nt1 B t Ut0t1,−nt0t1 B t Ut1t0,−nt1t0 B F F F F F F t Ut0t1t0,−nt0t1t0 B t Ut1t0t1,−nt1t0t1 B t Ut0t1t0t1,−nt0t1t0t1 B [v0] F F F 2 = (B t {xa(t)nt0 B | t ∈ Fq} t {xb(u)nt1 B | u ∈ Fq } F F 2 2 t {xa(t)xc(u)nt0t1 B | t ∈ Fq, u ∈ Fq } t {xb(u)xd(t)nt1t0 B | t ∈ Fq, u ∈ Fq } 0 F 2 t {xa(t)xc(u)xd(t )nt0t1t0 B | t, t ∈ Fq, u ∈ Fq } 0 F 0 2 t {xb(u)xc(u )xd(t)nt1t0t1 B | t ∈ Fq, u, u ∈ Fq } 0 0 F 0 0 2 t {xa(t)xb(u)xc(u )xd(t )nt0t1t0t1 B | t, t ∈ Fq, u, u ∈ Fq }) · [1 : 0 : 0 : 0] = {[1 : 0 : 0 : 0]} t {[u : 1 : 0 : 0] | u ∈ Fq2 } t {[u : t : 1 : 0] | t ∈ Fq, u ∈ Fq2 } 0 0 0 t {[t − u u : −u : u : 1] | t ∈ Fq, u, u ∈ Fq2 } 0 0 0 0 0 t {[t − u u : −u + tu : u : 1] || t, t ∈ Fq, u, u ∈ Fq2 }. 4 4 Let h, i: Fq2 × Fq2 → Fq2 be the nondegenerate form deﬁned by  0 0 0 −1  t  0 0 1 0  hu, vi = u   v = u2v−2 − u1v−1 + u−1v1 − u−2v2.  0 −1 0 0  1 0 0 0 It is shown in Section 3.2.3 that GF = {g ∈ G | if u, v ∈ V thenhgu, gvi = hu, vi}. Fq2

76 Let  x   −2   x−1  U = Fq2    x1     x2  be a choice of nondegenerate 1-dimensional subspace (see Appendix A.2). Then U ⊥ is nondegenerate and    y−2   ⊥  y−1   U =   ∈ V x2y−2 − x1y−1 + x−1y1 − x−2y2 = 0  y1     y2  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 ⊥ F O = {[v] ∈ P(VF) | v ∈ U ∩ (G · [v0])}. Then O has a decomposition into its points of each Schubert cell given by q O = {[1 : 0 : 0 : 0] | x2 = 0} t {[u : 1 : 0 : 0] | u ∈ Fq2 and x2u − x1 = 0} q t {[u : t : 1 : 0] | u ∈ Fq2 , t ∈ Fq and x2u − x1t + x−1 = 0} 0q 0q q 0 0 q 0q t {[t − u u : −u : u : 1] | u, u ∈ Fq2 , t ∈ Fq and x2(t − u u ) + x1u + x−1u − x−2 = 0} t {[t0 − u0qu : −u0q + tuq : uq : 1] 0 0 0 0 q 0 | u, u ∈ Fq2 , t, t ∈ Fq and x2(t − u u ) − 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 a maximal 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 p : G/B → G/P p : G/B → G/P i i and j j gB 7→ gPi 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 + + Ri = {α ∈ R | sα ∈ Wi}, + + + + + and Ri,j = Ri ∩ Rj . Rj = {α ∈ R | sα ∈ Wj}, Recall that R(z) = {α ∈ R+ | z−1(α) ∈/ R+},

Wi = subgroup of W generated by s1, s2, . . . , si−1, si+1, . . . , sn, i W = {minimal length representatives of cosets in W/Wi}.

77 Let

Wi,j = Wi ∩ Wj, i,j W = {minimal length representatives of cosets in W/Wi,j}. Then + Wi = {z ∈ W | R(z) ⊆ Ri }, i + W = {z ∈ W | R(z) ∩ Ri = ∅}, i,j + + (Wj) = {z ∈ W | R(z) ⊆ Rj and R(z) ∩ Ri,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 G/B = BwB with BwB = {x (c )n−1 ··· x (c )n−1B | c , . . . , c ∈ }, i1 1 i1 i` ` i` 1 ` F w∈W and {x (c )n−1 ··· x (c )n−1 | c , . . . , c ∈ } a set of representatives of the cosets of i1 1 i1 i` ` i` 1 ` F B in BwB. i (2) For each z ∈ W ﬁx a reduced decomposition z = si1 ··· sik . Then G G/P = BzP with BzP = {x (c )n−1 ··· x (c )n−1P | c , . . . , c ∈ }, i i i i1 1 i1 ik k ik i 1 k F z∈W i and {x (c )n−1 ··· x (c )n−1 | c , . . . , c ∈ } a set of representatives of the cosets of i1 1 i1 ik k ik 1 k F Pi in BzPi.

Proof. The proof for BwB is given in Theorem 2.4.6, and the proof for BwPi is similiar. 0 See also [Ste67, Theorem 4 ], [PRS09, (7.3)]. We will also make good use of the following lemma in proving our main theorem.

i,j i Lemma 4.3.2. If z ∈ Wj then z ∈ W . i,j + + + + Proof. Suppose z ∈ Wj . Then R(z) ⊆ Rj and R(z)∩Ri,j = ∅. So R(z)∩(Ri ∩Rj ) = ∅. + + + i So (R(z) ∩ Rj ) ∩ Ri = ∅. So R(z) ∩ Ri = ∅. Hence z ∈ W . The following theorem is the main theorem of the thesis. It determines the “thickness” of ˚ ˚ the 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 `(z) j i,j q , where w = uzv with u ∈ W , zv ∈ Wj, z ∈ (Wj) and v ∈ Wi,j. Proof. Note that −1 pipj (gPj) = {hPi ∈ G/Pi | there exists kB ∈ G/B such that pi(kB) = hPi and pj(kB) = gPj} −1 In other words, the set of elements of G/Pi incident to gPj is the set pipj (gPj). So the set of ˚ elements of pi(Xw) incident to gPj is the set ˚ −1 pi(Xw) ∩ pipj (gPj). Theorem 4.3.3 follows from the fact that `(z) ˚ −1 ψ : F −→ pi(Xw) ∩ pipj (gPj) (d , d , . . . , d ) 7−→ gx (d )n−1 . . . x (d )n−1P , 1 2 ` k1 1 k1 k` ` k` i

78 is a bijection, where s . . . s is a reduced expression for z, and g = x (c )n−1 . . . x (c )n−1 k1 k` i1 1 i1 ik ik ik j is a favourite coset representative of gPj, where si1 . . . sik ∈ W is a reduced expression for u. To show that ψ is a bijection, there are 3 things to show: −1 ˚ (1) ψ(d1, . . . , d`) ∈ pipj (gPj) ∩ pi(Xw). (2) ψ is injective. (3) ψ is surjective. (1) To show: ˚ (a) ψ(d1, . . . , d`) ∈ pi(Xw). −1 (b) ψ(d1, . . . , d`) ∈ pipj (gPj). (a) We have ψ(d , . . . , d ) = gx (d )n−1 . . . x (d )n−1P 1 ` k1 1 k1 k` ` k` i ∈ BuzPi

= BuzvPi

= BwPi ˚ = pi(Xw). (b) We have ψ(d , . . . , d ) = gx (d )n−1 . . . x (d )n−1P 1 ` k1 1 k1 k` ` k` i = p (gx (d )n−1 . . . x (d )n−1B). i k1 1 k1 k` ` k` We wish to show that gx (d )n−1 . . . x (d )n−1B ∈ p−1(gP ). k1 1 k1 k` ` k` j j But p (gx (d )n−1 . . . x (d )n−1B) = gx (d )n−1 . . . x (d )n−1P j k1 1 k1 k` ` k` k1 1 k1 k` ` k` j = gPj,

since sk1 . . . sk` ∈ Wj. (2) Suppose 0 0 ψ(d1, . . . , d`) = ψ(d1, . . . , d`). Then gx (d )n−1 . . . x (d )n−1P = gx (d0 )n−1 . . . x (d0 )n−1P . k1 1 k1 k` ` k` i k1 1 k1 k` ` k` i Then x (d )n−1 . . . x (d )n−1P = x (d0 )n−1 . . . x (d0 )n−1P . k1 1 k1 k` ` k` i k1 1 k1 k` ` k` i By Proposition 4.3.2, z ∈ W i. By Proposition 4.3.1, 0 0 0 d1 = d1, d2 = d2, . . . , dr = dr. So ψ is injective. −1 ˚ (3) Let kPi ∈ pi(pj (gPj)) ∩ pi(Xw). We know that −1 pj (gPj) = g ∪w∈Wj BwB, so that −1 pipj (gPj) = g ∪w∈Wj BwPi.

So there exists w ∈ Wj such that

kPi ∈ gBwPi. 0 i,j 0 0 0 Let z ∈ Wj such that z Wi,j = wWi,j. Then z Wi = wWi. So kPi ∈ gBz Pi. So 0 kPi ∈ Buz Pi. 79 ˚ 0 0 Since kPi ∈ pi(Xw), we have kPi ∈ BuzPi. So uz Wi = uzWi. So z Wi = zWi. By Proposition 4.3.2, we know z, z0 ∈ W i. So z = z0 (See [Hum92, Proposition 1.10(c)] § or [Bou08, Chapter 4, 1, Exercise 3, 2nd paragraph]). So there exists d1, . . . , d` ∈ F such that kP = gx (d )n−1 . . . x (d )n−1P . i k1 1 k1 k` ` k` i 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. 1 Then W = {1, s1, s2s1, s3s2s1}, 2 1,2 W = {1, s2, s1s2, s3s2, s1s3s2, s2s1s3s2} and (W2) = {1, s1}. ˚ ˚ Let w = uzy = (s1s3s2)(s1)(s3). If (Xw)12 is the (1, 2)-incidence structure associated to Xw −1 −1 −1 and g = x1(c1)n1 x3(c2)n3 x2(c3)n2 then −1 −1 −1 −1 −1 p1(p2 (gP2)) = p1(p2 (x1(c1)n1 x3(c2)n3 x2(c3)n2 P2)) −1 −1 −1 −1 = {x1(c1)n1 x3(c2)n3 x2(c3)n2 x1(d1)n1 P1 | d1 ∈ F}. −1 ∼ Note that this illustrates that p1(p2 (gP2)) = F even though the elements −1 −1 −1 −1 {x1(c1)n1 x3(c2)n3 x2(c3)n2 x1(d1)n1 P1 | d1 ∈ F} are not the “favourite” coset representatives of the cosets in G/P1. This provides a conceptual explanation of why Theorem 4.3.3 is nontrivial (one needs to ﬁnd the right coordinatization to −1 succeed in displaying p1(p2 (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) correspond to ( j w ∈ W Wi,j, if q > 2, triples (w, i, j) such that j j w ∈ W Wi,j ∪ W siWi,j, if q = 2.

j i,j Proof. Assume w = uzy with u ∈ W , z ∈ (Wj) , y ∈ Wi,j. By Theorem 4.3.3, if `(z) ˚ ˚ gPj ∈ BwPj then gPj is incident with q elements of pi(Xw). Note that pi(Xw) is an arc if `(z) and only if q ≤ 2. But `(z) = 0 only when z = 1 and `(z) = 1 only when z = si, and the result 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

X j 1 2 3 1 2 |W Wi,j| = |W W1,1| + |W W1,2| + |W W1,3| + |W W2,1| + |W W2,2| i,j 3 1 2 3 + |W W2,3| + |W W3,1| + |W W3,2| + |W W3,3| = 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 Chapter 2) with the example SL3(F). Let F be a ﬁeld and let G = SL3(F). The standard Cartan decomposition is G ⊆    ∗ ∗ ∗   B = x ∈ G x =  0 ∗ ∗ 

 0 0 ∗  ⊆    ∗ 0 0   T = x ∈ G x =  0 ∗ 0  .

 0 0 ∗ 

The Weyl group of G is W = S3 and 2 2 3 W = s1, s2 | s1 = s2 = (s1s2) = 1 = {1, s1, s2, s1s2, s2s1, s1s2s1}. Then

W1 = hs2i = {1, s2},W2 = hs1i = {1, s1}, and

P1 = B t Bs2B,P2 = B t Bs1B. The sets 1 2 W = {1, s1, s2s1}, and W = {1, s2, s1s2}, are minimal length coset representatives for W/W1 and W/W2 respectively. The Bruhat decompositions of G are

G/B = B t Bs1B t Bs2B t Bs1s2B t Bs2s1B t Bs1s2s1B,

G/P1 = P1 t Bs1P1 t Bs2s1P1, and G/P2 = P2 t Bs2P2 t Bs1s2P2. Th Schubert varieties in G/B are

X1 = B,Xs1 = B t Bs1B,Xs2 = B t Bs2B,

Xs1s2 = B t Bs1B t Bs2B t Bs1s2B,Xs2s1 = B t Bs2B t Bs1B t Bs2s1B,

Xs1s2s1 = Xs2s1s2 = B t Bs1B t Bs2B t Bs1s2B t Bs2s1B t Bs1s2s1B = G/B.

The Schubert varieties in G/P1 are 1 1 X1 = P1,Xs1 = P1 t Bs1P1, 1 Xs2s1 = P1 t Bs1P1 t Bs2s1P1 = G/P1.

The Schubert varieties in G/P2 are 2 2 X1 = P2,Xs2 = P2 t Bs2P2, 2 Xs1s2 = P2 t Bs2P2 t Bs1s2P2 = G/P2. The matrices 1 c 0 1 0 0 1 0 c

xα1 (c) = 0 1 0 , xα2 (c) = 0 1 c , xα1+α2 (c) = 0 1 0 , 0 0 1 0 0 1 0 0 1

81  0 1 0 1 0 0  0 0 1

nα1 = −1 0 0 , nα2 = 0 0 1 , nα1+α2 =  0 1 0 . 0 0 1 0 −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    c1 −1 0 −1   Bs1B = {xα1 (c1)nα B | c1 ∈ F} =  1 0 0 B c1 ∈ F , 1  0 0 1     1 0 0 −1   Bs2B = {xα2 (c1)nα B | c1 ∈ F} = 0 c1 −1 B c1 ∈ F , 2  0 1 0     c1 −c2 1 −1 −1   Bs1s2B = {xα1 (c1)nα xα2 (c2)nα B | c1, c2 ∈ F} =  1 0 0 B c1, c2 ∈ F , 1 2  0 1 0     c2 −1 0 −1 −1   Bs2s1B = {xα2 (c1)nα xα1 (c2)nα B | c1, c2 ∈ F} = c1 0 −1 B c1, c2 ∈ F , 2 1  1 0 0  −1 −1 −1 Bs1s2s1B = {xα1 (c1)nα1 xα2 (c2)nα2 xα1 (c3)nα1 B | c1, c2, c3 ∈ F}    c1c3 − c2 −c1 1   =  c3 −1 0 B c1, c2, c3 ∈ F ,

 1 0 0 

the Schubert cells of G/P1 have single P1-coset decompositions

P1 = P1    c1 −1 0 −1   Bs1P1 = {xα1 (c1)nα P1 | c1 ∈ F} =  1 0 0 P1 c1 ∈ F , 1  0 0 1     c2 −1 0 −1 −1   Bs2s1P1 = {xα2 (c1)nα xα1 (c2)nα P1 | c1, c2 ∈ F} = c1 0 −1 B c1, c2 ∈ F , 2 1  1 0 0 

and the Schubert cells of G/P2 have single P2-coset decompositions

P2 = P2    1 0 0 −1   Bs2P2 = {xα2 (c1)nα P2 | c1 ∈ F} = 0 c1 −1 P2 c1 ∈ F , 2  0 1 0     c1 −c2 1 −1 −1   Bs1s2P2 = {xα1 (c1)nα xα2 (c2)nα P2 | c1, c2 ∈ F} =  1 0 0 P2 c1, c2 ∈ F . 1 2  0 1 0  The map ψ : G/B −→ F l 3 : gB 7−→ (0 ⊆ gF{e1} ⊆ gF{e1, e2} ⊆ F2) is a bijection. The image of ψ of the favourite choices of coset representatives are:  * 1 + * 1 0 + ψ(B) = 0 ⊆ 0 ⊆ 0 1  0 0 0

82   * c1 + * c1 1 + ψ(x1(c1)s1B) = 0 ⊆ 1 ⊆ 1 0  0 0 0  * 1 + * 1 0 + ψ(x2(c1)s2B) = 0 ⊆ 0 ⊆ 0 c1  0 0 1   * c1 + * c1 c2 + ψ(x1(c1)s1x2(c2)s2B) = 0 ⊆ 1 ⊆ 1 0  0 0 1   * c2 + * c2 1 + ψ(x2(c1)s2x1(c2)s1B) = 0 ⊆ c1 ⊆ c1 0  1 1 0   * c1c3 + c2 + * c1c3 + c2 c1 + ψ(x1(c1)s1x2(c2)s2x1(c3)s1B) = 0 ⊆ c3 ⊆ c3 1  1 1 0   * c2 + * c2 c3 + ψ(x2(c1)s2x1(c2)s1x2(c3)s2B) = 0 ⊆ c1 ⊆ c1 1  1 1 0 The map

ψ1 : G/P1 −→ Gr(1, 3) 3 : gP1 7−→ (0 ⊆ gF{e1} ⊆ F ) is a bijection. The image under the favourite choice of coset reprentatives is  * 1 + ψ1(P1) = 0 ⊆ 0  0   * c1 + ψ1(x1(c1)s1P1) = 0 ⊆ 1  0   * c2 + ψ(x2(c1)s2x1(c2)s1P1) = 0 ⊆ c1  1 The map

ψ2 : G/P2 −→ Gr(2, 3) 3 : gP2 7−→ (0 ⊆ gF{e1, e2} ⊆ F ) is a bijection. The image of the favourite choice of coset representatives is  * 1 0 + ψ(P2) = 0 ⊆ 0 1  0 0  * 1 0 + ψ(x2(c1)s2P2) = 0 ⊆ 0 c1  0 1   * c1 c2 + ψ(x1(c1)s1x2(c2)s2P2) = 0 ⊆ 1 0  0 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 2 W = {1, s2, s1s2}

W1,2 = W1 ∩ W2 = {1} 1,2 (W2) = {1, s1}. 2 1,2 We write each w ∈ W as w = uzv with u ∈ W , z ∈ (W2) 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 p F : Fp −→ Fp defined by F (x) = x . k Let k ∈ Z≥0. The finite field with p elements is k Fpk = {x ∈ Fp | F (x) = x}.

k Proposition A.1.1. The set Fpk is a subﬁeld of Fp with p elements.

k pk Proof. The fact that Fpk has p elements follows from the fact that x = x is separable i.e has distinct linear factors. For complete details, see [DF04, Example: (Existence and Uniqueness of Finite Fields), Section 13.5]. Proposition A.1.2. 2 k−1 Gal(Fpk : Fp) = {1,F,F ,...,F } Proposition A.1.3. ∞ [ Fp = Fpk . k=1 Proposition A.1.4. [DF04, §14.3, Proposition 15] The function

{m ∈ Z≥0 | m divides k} −→ {subﬁelds of Fpk } m 7−→ Fpm , is a bijection.

Corollary A.1.5. Suppose F is a finite field of even characteristic. Then F has an auto- 2 2 ∼ morphism τ such that τ (x) = x 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 know E ∼= {0, 1, α, 1 + α} where α2 + α + 1 = 0. There are exactly 2 automorphisms of τ, uniquely determined by τ(α) = α or τ(α) = α + 1. If τ(α) = α then τ 2(α) = α, a contradiction. If τ(α) = α + 1 then τ 2(α) = τ(α + 1) = α, a contradiction. Hence k is not even. Hence k is odd. 2 k Proposition A.1.6. Let Fq2 be the finite field with q elements, where q = p is a prime q power. Define : Fq2 → Fq2 by x = x . × (1) Let c ∈ Fq . Then the equation xx = c

× has exactly q + 1 solutions in Fq2 . 85 (2) Let d ∈ Fq. Then the equation x + x = d

has exactly q solutions in Fq2 . Proof. [Wan93, Lemma 5.1]. Define × × φ: Fq2 → Fq , ψ : Fq2 → Fq, x 7→ xx, x 7→ x + x, × We consider first the map φ. If x ∈ Fq2 we have (xx) = xx so that xx ∈ Fq. Considering the × set Fq2 as a group under multiplication, φ is a group homomorphism. Since × q+1 Ker(φ) = {x ∈ Fq2 | x = 1}, × × we have |Ker(φ)| ≤ q + 1. Also, |Im(φ)| ≤ |Fq | = q − 1. But |Fq2 | = |Ker(φ)||Im(φ)| and × 2 −1 × |Fq2 | = q − 1. So |Ker(φ)| = q + 1 so that |φ (c)| = q + 1 for c ∈ Fq2 . 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 Fq is 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 h, i: V × V → F such that (1) If λ, µ ∈ F and u, v ∈ V then hλu, µvi = λ(σ(µ))hu, vi, (2) If u, v, w ∈ V then hu + v, wi = hu, wi + hv, wi and hu, v + wi = hu, vi + hu, wi. 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 σ − sesquilinear forms n × n matrices ϕ: −→ on V with entries in F h·, ·i 7−→ A by Aij = hei, eji. Then ϕ is a bijection, and the σ-sesquilinear form corresponding to A is given by hv, wi = vtAσ(w). Two forms h, i and h, i0 on V are isometric (or equivalent) if there exists T ∈ GL(V ) such that if u, v ∈ V then hT u, T vi = hu, vi0. Proposition A.2.2. If A is the matrix corresponding to the form h, i, then h, i is equivalent to h, i0 if and only if there exists a T ∈ GL(V ) such that the matrix corresponding to h, i0 is T tAσ(T ), where (σ(T ))ij = σ(Tij). 86 Deﬁne ∗ ∗ θ− : V → V −θ : V → V by θv(u) = hu, vi, and by vθ(u) = hv, ui. v 7→ θv v 7→ vθ

The form h, i is nondegenerate if θ− and −θ are bijections [Bou07, Chapter IX, §1, No. 6, Proposition 6]. The form h, i is degenerate if it is not nondegenerate. A subspace W ⊆ V is nondegenerate if the restricted form h, i|W : W × W → F is nondegenerate. Proposition A.2.3. A form h, i: V × V → F is nondegenerate [Ueb11, Chapter 4, Section 5], [Bou07, Chapter IX, §1, No. 1, Deﬁnition 3] if and only if the following conditions are satisﬁed: (1) If v ∈ V and v 6= 0 then there exists x ∈ V such that hv, xi= 6 0, (2) If v0 ∈ V and v0 6= 0 then there exists x0 ∈ V such that hx0, v0i= 6 0.

Theorem A.2.4. Let h, i: V ×V → F be a σ-sesquilinear form deﬁned by hv, wi = vtAσ(w). Then h, i 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 is t invertible. So there exists i ∈ {1, 2, . . . , n} such that v Aσ(ei) 6= 0. So hv, eii= 6 0. Similiarly, there exists j ∈ {1, 2, . . . , n} such that hej, vi= 6 0. Hence h, i is nondegenerate. Suppose h, i is nondegenerate, and suppose for the sake of contradiction that A is not invertible. Then there exists a nonzero v ∈ V such that Av = 0. Hence utAσ(v) = 0 for all u ∈ V . Hence hu, vi = 0 for all u ∈ V . Hence h, i is degenerate, a contradiction. Proposition A.2.5. [Bou07, Chapter IX, §1, No. 6, Corollary to Proposition 6] Let V be a ﬁnite dimensional vector space, and let h, i be a form on V . Then the form h, i is nondegenerate if and only if for every basis {e1, e2, . . . , en} there exists a basis {f1, f2, . . . , fn} such that

hei, fji = δij. A form h, i: V × V → F is reflexive if the following condition is satisfied: if u, v ∈ V and hu, vi = 0 then hv, ui = 0. Theorem A.2.6. [The Birkhoff Von-Neumann Theorem] [Bal15, Theorem 3.6], [Tay92, Theorem 7.1], [BV36]. Let h, i: V × V → F be a nondegenerate reflexive σ-sesquilinear form. Up to a scalar, h, i is exactly one of the following types: (1) h, i is an alternating form, that is, hu, vi = −hv, ui for u, v ∈ V if the characteristic of F is 6= 2, and hu, ui = 0 for u ∈ V if the characteristic of F is 2. (2) h, i is a symmetric form, that is, hu, vi = hv, ui, for u, v ∈ V , (3) h, i is a hermitian form, that is, hu, vi = σ(hv, ui), for u, v ∈ V , where σ2 = 1 and σ 6= 1.

Assume that h, i: V × V → F is a reﬂexive form. The orthogonal component of a subspace W ⊆ V is W ⊥ = {x ∈ V | if w ∈ W then hw, xi = 0}

Proposition A.2.7. Let V be a finite dimensional vector space and let h, i : V × V → F be a 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 hx, vi = 0 = hv, xi for all 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 hv, xi = 0 for all x ∈ W . Then v ∈ W ⊥. So v = 0, a contradiction. Hence there exists x ∈ W such that hv, xi 6= 0. Similarly, there exists x ∈ W such that hx, vi 6= 0. So W is nondegenerate. Proposition A.2.8. [Kah08, Lemma 1.1.5] Let V be a finite-dimensional vector space, and h, i : 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 basis for W . By Proposition A.2.5, there exists a basis {f1, f2, . . . , fm} of W such that hei, fji = δij. Let x ∈ V . Then m X ⊥ x − hx, fiiei ∈ W , i=1 ⊥ ⊥ ⊥ so x ∈ W + W . So V = W + W . Hence V = W ⊕ W . Corollary A.2.9. If h, i : V × V → F is a reflexive nondegenerate form and W is a nondegenerate subspace of V then W ⊥ is a nondegenerate subspace of V . Proof. Assume h, i : V × V → F is a reflexive nondegenerate form and W is a nondegenerate subspace of V . Suppose v ∈ W ⊥ with v 6= 0. Furthermore, suppose for sake of contradiction that hv, xi = 0 for all x ∈ W ⊥. Then hv, x0 + xi = 0 for all x ∈ W ⊥ and x0 ∈ W . So by Proposition A.2.8, hv, yi = 0 for all y ∈ V . Hence h, i is degenerate, a contradiction. So there exists x ∈ W ⊥ such that hv, xi 6= 0. Similarly, there exists x ∈ W ⊥ such that hx, vi 6= 0. Hence W ⊥ is nondegenerate. Proposition A.2.10. [Bal15, Lemma 3.1] Let W be a subspace of a finite dimensional vector space V , and let h, i : 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) = hei, vi. Pr Pr Suppose i=1 λiαi = 0. Then i=1 λiαi(v) = 0 for all v ∈ V . Therefore, r r X X h λiei, vi = λiαi(v) = 0 i=1 i=1 Pr for all v ∈ V . Hence i=1 λiei. So λ1 = λ2 = ··· = λr = 0. Hence {α1, α2, . . . , αr} is a linearly independent set. So ⊥ W = {v ∈ V | if i ∈ {1, 2, . . . , r} then hei, vi = 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, . . . , αn} ∗ of V . Let {e1, e2, . . . , er, er+1, . . . , en} be the basis in V dual to {α1, . . . , αr, αr+1, . . . , αn}. Then ⊥ W = Span{er+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 hw, vi = 0. In particular, if w ∈ U then hw, vi = 0. ⊥ ⊥ ⊥ So v ∈ U . Hence W ⊆ U . Following [Tay92, pg. 56, Definition (ii)] and [Bal15, pg. 27], a subspace W ⊆ V is totally isotropic if the following condition is satisfied: if u, v ∈ W then hu, vi = 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 hv, wi = 0. So v ∈ W ⊥. Hence W ⊆ W ⊥. ⊥ Suppose W ⊆ W . Let v, w ∈ W . Then hv, wi = 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 function Q: V → F such that the following conditions are satisfied: (1) If λ ∈ F and u ∈ V then Q(λu) = λ2Q(u), (2) The map h, i: V × V → F defined by hu, vi = Q(u + v) − Q(u) − Q(v) is a bilinear form. Proposition A.3.1. Define n o n o ψ : quadratic forms −→ symmetric bilinear on V forms on V Q 7−→ h, iQ, where hu, viQ = 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 degenerate if its corresponding bilinear form is degenerate. Also, Q is non-degenerate if its corresponding bilinear form is non-degenerate. Suppose the characteristic of F is 2. Then Q is non-degenerate if the following condition is satisfied: If v 6= 0 and hx, vi = 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 there exists a singular vector x ∈ W such that x is orthogonal to W (with respect to the bilinear form 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 a field F (characteristic 6= 2) and Q: V → F is a quadratic form. An isometry between two quadratic spaces (V,Q) and (V 0,Q0) is a linear isomorphism τ : V → V 0 such that if v ∈ V then Q0(τv) = Q(v). Two quadratic spaces (V,Q) and (V 0,Q0) are isometric if there exists an isometry τ : V → V 0, in this case we write (V,Q) ∼= (V 0,Q0). 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 0,Q0) where dimV 0 = 2m is even and 0 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 the usual tensor product of vector spaces and Q1 ⊗ Q2 is the quadratic form whose corresponding symmetric bilinear form corresponds to (via Proposition A.3)

hu1 ⊗ u2, v1 ⊗ v2iQ1⊗Q2 = hu1, v1iQ1 hu2, v2iQ2 . 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, there exists a basis {u, v} of V such that

Q(x1u + x2v) = x1x2. Proof. Let v ∈ V be an isotropic vector. Let w ∈ V \span{v}. Since (V,Q) is nondegenerate, hv, wi= 6 0. Let Q(w) 1 u = − v + w. 2hv, wi2 hv, wi Then Q(u) = 0 and hu, vi = 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 to isometry. Two quadratic spaces (V,Q) and (V 0,Q0) are Witt equivalent and we write 0 0 ∼ 0 0 (V,Q) ' (V ,Q ) if (Va,Qa) = (Va,Qa). 1 Given the decomposition in Theorem A.3.3, the Witt index or rank is equal to 2 dim(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 if q ≡ 1(mod4).

× Proof. We know that Fq is a cyclic group of order q − 1 (see [Art11, Theorem 15.7.3(c)]). So we may write × 2 3 q−3 q−2 Fq = {1, x, x , x , . . . , x , x }. × Let S be the subgroup of nonzero squares in Fq . 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 exists k ∈ Z≥0 such that (xk)2 = x2m+1. 2(m−k)+1 So x = 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 of W (F) are the four quadratic spaces (V,Q) where (1) V = 0 and Q = 0, 2 (2) V = Fq and Q(x1) = x1, 2 (3) V = Fq and Q(x1) = sx1, where s is any nonsquare element of Fq, 2 2 2 (4) V = Fq and Q(x1, x2) = x1 + sx2, 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 of W (F) are the four quadratic spaces (V,Q) where (1) V = 0 and Q = 0, 2 (2) V = Fq and Q(x1) = x1, 2 2 2 (3) V = Fq and Q(x1, x2) = x1 + x2, 2 (4) V = Fq and Q(x1) = −x1. ∼ 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 particular, 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], a building of type (W, S) is a pair (C, δ) consisting of a nonempty set C, whose elements are called chambers, together with a map δ : C × C →W , called the Weyl distance function, such that for all 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 d0 ∈ C satisfies δ(d, d0) = s ∈ S, then δ(c, d0) = ws or w. If, in addition, `(ws) = `(w) + 1, then δ(c, d0) = ws. (B3) If δ(c, d) = w, then for any s ∈ S there is a chamber d0 ∈ C such that δ(d, d0) = s and δ(c, d0) = ws. Intuitively, a building of type (W, S) is a ‘W -metric space’, where the distance between any two 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 chambers d ∈ C such that δ(c, d) ∈ {1, s}. A building is thick if for every chamber c ∈ C and s ∈ S, there exists 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 maximal torus. Let W = NG(T )/T be the corresponding Weyl group with simple reflections S. Define a function 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 the Bruhat 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 gB and hB. Suppose that gB = g0B, hB = h0B and δ(gB, hB) = w. Then g0−1gB = h0−1hB = B and so g0−1h0B = g0−1h0h0−1hB = g0−1hB = g0−1gg−1hB ⊆ g0−1gBwB = BwB 91 So δ(g0B, h0B) = 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, h0B) = s. Then g−1hB ⊆ BwB and h−1h0B ⊆ BsB. Using Proposition 2.3.4 we have g−1h0B = g−1hh−1h0B ⊆ g−1hB · h−1h0B ⊆ BwB · BsB ⊆ BwB ∪ BwsB. Hence g−1h0B ⊆ BwB or g−1h0B ⊆ BwsB, therefore δ(gB, h0B) = w or δ(gB, h0B) = ws. Furthermore, if `(w) = `(ws) + 1 then from Proposition 2.3.4 we have BwB · BsB = BwsB so that g−1h0B ⊆ BwsB, therefore δ(gB, h0B) = 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 · BsB Then Proposition 2.3.4 implies that BwB · BsB = BwsB. Let h0 = hn−1, where n ∈ N is chosen so that n−1T = s. Then h−1h0B ∈ BsB. So g−1h0B = g−1hh−1h0B ⊆ g−1hB · BsB ⊆ BwB · BsB = BwsB so that δ(gB, h0B) = ws. Hence (B3) holds in this case. (Case 2) Suppose `(ws) < `(w). We know that g−1hB ⊆ BwB = BwssB ⊆ BwsB · BsB Hence g−1h = xy for some x ∈ BwsB, y ∈ BsB. Define h0 = hy−1. Then g−1h0 = g−1hy−1 = x ∈ BwsB. Therefore δ(gB, h0B) = 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, h0B ∈ G/B with hB 6= h0B, δ(gB, hB) = s and δ(gB, h0B) = s. By the Bruhat decomposition (Theorem 2.3.7), there exists w ∈ W such that g ∈ BwB. Define h = gn where n ∈ N is chosen so that nT = s. Then g−1h ∈ BsB so that δ(gB, hB) = s. By Proposition 2.3.4, we have BsB · BsB = B ∪ BsB. So we can choose a k ∈ BsB such that g−1hk ∈ BsB. Define h0 = hk, so that g−1h0 = g−1hk ∈ BsB. Thus δ(gB, h0B) = s. Also, h−1h0 = k ∈ BsB so that 0 hB 6= h B, as required.

92 APPENDIX B

B.1. Hasse diagrams of subspace lattices 2 Example B.1.1. The subspace lattice PG(F2) has lattice diagram

1 0 1 0 1 1

3 Example B.1.2. The subspace lattice PG(F2) has lattice diagram

*1 0+ *1 0+ *1 0+ *0 0+ *1 0+ *0 1+ *1 1+ 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1

*1+ *0+ *1+ *0+ *1+ *0+ *1+ 0 1 1 0 0 1 1 0 0 0 1 1 1 1

3 Another way to visualise PG(F2) is the Fano plane: 93 *0+ 0 1

*1+ *0+ 0 1 0 *1+ 0 1 1

*1+ *1+ *0+ 0 1 1 1 0 1

94 3 Example B.1.3. The subspace lattice PG(F3) has lattice diagram

*0 0+ *0 1+ *0 1+ *0 1+ *0 1+ *0 1+ *0 1+ *0 1+ *0 1+ *0 1+ *0 1+ *0 1+ *0 1+ 0 1 0 0 0 1 0 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 2 1 0 1 1 1 2 2 0 2 1 2 2 95

*0+ *0+ *0+ *0+ *1+ *1+ *1+ *1+ *1+ *1+ *1+ *1+ *1+ 0 1 1 1 0 0 0 1 1 1 2 2 2 1 0 1 2 0 1 2 0 1 2 0 1 2

0 B.2. Hasse diagrams of Boolean lattices 1 Example B.2.1. The Boolean lattice PG(F1) has lattice diagram

{1}

∅

2 Example B.2.2. The Boolean lattice PG(F1) has lattice diagram

{1, 2}

{1} {2}

∅

3 Example B.2.3. The Boolean lattice PG(F1) has lattice diagram

{1, 2, 3}

{1, 2} {1, 3} {2, 3}

{1} {2} {3}

∅

96 4 Example B.2.4. The Boolean lattice PG(F1) has lattice diagram {1, 2, 3, 4}

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

{1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} 97

{1} {2} {3} {4}

∅ 5 Example B.2.5. The Boolean lattice PG(F1) has lattice diagram {1, 2, 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} {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} 98

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

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

∅ 3 B.3. Hasse diagrams of Schubert cells in PG(F2) ˚ 3 The Schubert cell Xe = B in PG(F2) is labeled with thick lines. V

*1 0+ *1 0+ *1 0+ *0 0+ *1 0+ *0 1+ *1 1+ 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1

*1+ *0+ *1+ *0+ *1+ *0+ *1+ 0 1 1 0 0 1 1 0 0 0 1 1 1 1

0 ˚ 3 The Schubert cell Xs1 = Bs1B in PG(F2) is labeled with thick lines. V

*1 0+ *1 0+ *1 0+ *0 0+ *1 0+ *0 1+ *1 1+ 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1

*1+ *0+ *1+ *0+ *1+ *0+ *1+ 0 1 1 0 0 1 1 0 0 0 1 1 1 1

99 ˚ 3 The Schubert cell Xs2 = Bs2B in PG(F2) is labeled with thick lines. V

*1 0+ *1 0+ *1 0+ *0 0+ *1 0+ *0 1+ *1 1+ 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1

*1+ *0+ *1+ *0+ *1+ *0+ *1+ 0 1 1 0 0 1 1 0 0 0 1 1 1 1

0 ˚ 3 The Schubert cell Xs1s2 = Bs1s2B in PG(F2) is labeled with thick lines. V

*1 0+ *1 0+ *1 0+ *0 0+ *1 0+ *0 1+ *1 1+ 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1

*1+ *0+ *1+ *0+ *1+ *0+ *1+ 0 1 1 0 0 1 1 0 0 0 1 1 1 1

100 ˚ 3 The Schubert cell Xs2s1 = Bs2s1B in PG(F2) is labeled with thick lines. V

*1 0+ *1 0+ *1 0+ *0 0+ *1 0+ *0 1+ *1 1+ 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1

*1+ *0+ *1+ *0+ *1+ *0+ *1+ 0 1 1 0 0 1 1 0 0 0 1 1 1 1

0 ˚ ˚ 3 The Schubert cell Xs1s2s1 = Bs1s2s1B = Bs2s1s2B = Xs2s1s2 in PG(F2) is labeled with thick lines. V

*1 0+ *1 0+ *1 0+ *0 0+ *1 0+ *0 1+ *1 1+ 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1

*1+ *0+ *1+ *0+ *1+ *0+ *1+ 0 1 1 0 0 1 1 0 0 0 1 1 1 1

101

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107

Index

adjoint group, 25 reductive, 23 root system, 23 Baer’s Theorem, 13 semisimple, 23 basis, 11 simple, 23 Borel subgroup, 26 simple roots, 24 Bruhat decomposition, 30 triangular decomposition, 24 Building, 91 maximal torus, 26 cap, 14 Cartan-Killing theorem, 24 ovoid, 14, 18 Chevalley, 24 and (twisted) Chevalley groups, 71 basis, 24, 25 classical ovoid in the Hermitian surface, 20 generators, 25 Suzuki-Tits, 18 group, 25 relations, 26 Pappian, 11 collineation, 10 polar conic, 16 semilattice, 6 Coxeter generators, 24 space, 6, 12 Coxeter’s theorem, 24 projective space, 6

Desarguesian, 10 rank, 7 dimension rational normal curve, 16 projective, 11 reflection, 23 root subgroups, 26 flag variety, 67 Frobenius automorphism, 19 Schubert cell, 71 Frobenius morphism, 49 Segre’s theorem, 17 semilinear transformation, 6 group of Lie type, 49 Shult’s theorem, 14 simple reflections, 24 Hasse diagram, 8 singular subspace, 12 incidence structure, 9 span, 11 polar, 12 Steinberg endomorphism, 49 projective, 9 Suzuki group, 62 Kostant -form, 25 Z Thickness, 77 lattice, 5 totally isotropic Boolean, 8 subspace, 6 diagram, 8 vector, 6 projective, 7 Twisted Chevalley group, 49 subset, 6 universal group, 25 subspace, 5, 6, 11 Lie algebra, 23 Veblen-Young Borel subalgebra, 24 axiom, 9 Cartan subalgebra, 23 theorem, 13 examples, 36 Killing form, 23 Weyl group, 24, 26 positive roots, 24 length function, 26 rank, 25 reduced 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

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