The Underlying Geometry of the Graphs CD(K, Q)

The underlying geometry of the graphs CD(k, q)

Andrew Woldar Villanova University

Algebraic and Extremal Graph Theory Conference University of Delaware, August 7-10, 2017

To Ron Solomon, who gave me the gift of knowledge and the rewards of friendship, “Semper gratiam habebo.”

Andrew Woldar The underlying geometry of the graphs CD(k, q) They have the greatest number of edges among all known graphs of a ﬁxed order and girth at least g for g ≥ 5, g 6= 11, 12. (For g = 11, 12, they are outperformed by the generalized hexagon of type G2.)

They are very close to being Ramanujan, with a conjectured bound √ of λ2 ≤ 2 q where λ2 is the second largest eigenvalue. p Ramanujan ⇐⇒ λ2 ≤ 2 q − 1

Introduction

The graphs CD(k, q) exhibit many interesting extremal properties, and have sundry applications to coding theory, cryptography and network topology.

Andrew Woldar The underlying geometry of the graphs CD(k, q) They are very close to being Ramanujan, with a conjectured bound √ of λ2 ≤ 2 q where λ2 is the second largest eigenvalue. p Ramanujan ⇐⇒ λ2 ≤ 2 q − 1

Introduction

The graphs CD(k, q) exhibit many interesting extremal properties, and have sundry applications to coding theory, cryptography and network topology.

They have the greatest number of edges among all known graphs of a ﬁxed order and girth at least g for g ≥ 5, g 6= 11, 12. (For g = 11, 12, they are outperformed by the generalized hexagon of type G2.)

Andrew Woldar The underlying geometry of the graphs CD(k, q) p Ramanujan ⇐⇒ λ2 ≤ 2 q − 1

Introduction

The graphs CD(k, q) exhibit many interesting extremal properties, and have sundry applications to coding theory, cryptography and network topology.

They are very close to being Ramanujan, with a conjectured bound √ of λ2 ≤ 2 q where λ2 is the second largest eigenvalue.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Introduction

The graphs CD(k, q) exhibit many interesting extremal properties, and have sundry applications to coding theory, cryptography and network topology.

They are very close to being Ramanujan, with a conjectured bound √ of λ2 ≤ 2 q where λ2 is the second largest eigenvalue. p Ramanujan ⇐⇒ λ2 ≤ 2 q − 1

Andrew Woldar The underlying geometry of the graphs CD(k, q) More explicitly, each CD(k, q) is a connected component of the graph D(k, q).

They exist for every positive integer k and every prime power q.

Introduction

Graphs CD(k, q) are derived from the graphs D(k, q), which are examples of algebraically deﬁned graphs, that is, graphs whose vertices are coordinate vectors, with two vertices adjacent provided their coordinates satisfy a prescribed set of equations (adjacency relations).

Andrew Woldar The underlying geometry of the graphs CD(k, q) They exist for every positive integer k and every prime power q.

Introduction

More explicitly, each CD(k, q) is a connected component of the graph D(k, q).

Andrew Woldar The underlying geometry of the graphs CD(k, q) Introduction

More explicitly, each CD(k, q) is a connected component of the graph D(k, q).

They exist for every positive integer k and every prime power q.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Adjacency relations:

p11 − `11 = p1`1 p12 − `12 = p1`11 p21 − `21 = p11`1 p22 − `22 = p12`1

Introduction

— CD(5, q) —

5 Points: (p) = (p1, p11, p12, p21, p22) ⊂ Fq 5 Lines: [`] = [`1, `11, `12, `21, `22] ⊂ Fq

Andrew Woldar The underlying geometry of the graphs CD(k, q) Introduction

— CD(5, q) —

5 Points: (p) = (p1, p11, p12, p21, p22) ⊂ Fq 5 Lines: [`] = [`1, `11, `12, `21, `22] ⊂ Fq

Adjacency relations:

p11 − `11 = p1`1 p12 − `12 = p1`11 p21 − `21 = p11`1 p22 − `22 = p12`1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Introduction

— CD(4, q) —

4 Points: (p) = (p1, p11, p12, p21, p22) ⊂ Fq 4 Lines: [`] = [`1, `11, `12, `21, `22] ⊂ Fq

Adjacency relations:

p11 − `11 = p1`1 p12 − `12 = p1`11 p21 − `21 = p11`1 p22 − `22 = p12`1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Introduction

— CD(3, q) —

3 Points: (p) = (p1, p11, p12, p21, p22) ⊂ Fq 3 Lines: [`] = [`1, `11, `12, `21, `22] ⊂ Fq

Adjacency relations:

p11 − `11 = p1`1 p12 − `12 = p1`11 p21 − `21 = p11`1 p22 − `22 = p12`1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Introduction

— CD(2, q) —

2 Points: (p) = (p1, p11, p12, p21, p22) ⊂ Fq 2 Lines: [`] = [`1, `11, `12, `21, `22] ⊂ Fq

Adjacency relations:

p11 − `11 = p1`1 p12 − `12 = p1`11 p21 − `21 = p11`1 p22 − `22 = p12`1

Andrew Woldar The underlying geometry of the graphs CD(k, q) This bound is known to be sharp for only three values of k, namely k = 2, 3, 5.

Problem: How close to this bound can we come for the remaining values of k?

Motivation

Even Circuit Theorem (Erd˝os): Let Γ be a graph with v vertices and e edges, and assume Γ contains no 2k-cycle. Then

1+ 1 e ≤ O v k

Andrew Woldar The underlying geometry of the graphs CD(k, q) Problem: How close to this bound can we come for the remaining values of k?

Motivation

Even Circuit Theorem (Erd˝os): Let Γ be a graph with v vertices and e edges, and assume Γ contains no 2k-cycle. Then

1+ 1 e ≤ O v k

This bound is known to be sharp for only three values of k, namely k = 2, 3, 5.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Motivation

Even Circuit Theorem (Erd˝os): Let Γ be a graph with v vertices and e edges, and assume Γ contains no 2k-cycle. Then

1+ 1 e ≤ O v k

This bound is known to be sharp for only three values of k, namely k = 2, 3, 5.

Problem: How close to this bound can we come for the remaining values of k?

Andrew Woldar The underlying geometry of the graphs CD(k, q) The cases k = 2, 3, 5 are realized by the respective incidence graphs of the generalized triangle of type A2, the generalized quadrangle of type B2, and the generalized hexagon of type G2.

The BAD news: Theorem (Feit-Higman): A ﬁnite thick generalized m-gon exists only for m ∈ {2, 3, 4, 6, 8}.

The cases k = 2, 3, 5 correspond to m = 3, 4, 6 respectively. No other value of m can contribute anything meaningful to our extremal graph theory problem.

Motivation

The GOOD news:

Andrew Woldar The underlying geometry of the graphs CD(k, q) The BAD news: Theorem (Feit-Higman): A ﬁnite thick generalized m-gon exists only for m ∈ {2, 3, 4, 6, 8}.

The cases k = 2, 3, 5 correspond to m = 3, 4, 6 respectively. No other value of m can contribute anything meaningful to our extremal graph theory problem.

Motivation

The GOOD news: The cases k = 2, 3, 5 are realized by the respective incidence graphs of the generalized triangle of type A2, the generalized quadrangle of type B2, and the generalized hexagon of type G2.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Theorem (Feit-Higman): A ﬁnite thick generalized m-gon exists only for m ∈ {2, 3, 4, 6, 8}.

The cases k = 2, 3, 5 correspond to m = 3, 4, 6 respectively. No other value of m can contribute anything meaningful to our extremal graph theory problem.

Motivation

The BAD news:

Andrew Woldar The underlying geometry of the graphs CD(k, q) The cases k = 2, 3, 5 correspond to m = 3, 4, 6 respectively. No other value of m can contribute anything meaningful to our extremal graph theory problem.

Motivation

The BAD news: Theorem (Feit-Higman): A ﬁnite thick generalized m-gon exists only for m ∈ {2, 3, 4, 6, 8}.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Motivation

The BAD news: Theorem (Feit-Higman): A ﬁnite thick generalized m-gon exists only for m ∈ {2, 3, 4, 6, 8}.

The cases k = 2, 3, 5 correspond to m = 3, 4, 6 respectively. No other value of m can contribute anything meaningful to our extremal graph theory problem.

Andrew Woldar The underlying geometry of the graphs CD(k, q) To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

1 Root systems 2 The Weyl group 3 Dynkin diagrams and the Cartan matrix 4 Lie algebras and the upper Borel subalgebra 5 Lie groups and ﬁnite groups of Lie type 6 Rank 2 buildings 7 Embedding buildings into Lie algebras

8 Objects of type Ae1

Motivation

What was the goal of our research?

Andrew Woldar The underlying geometry of the graphs CD(k, q) (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously

Andrew Woldar The underlying geometry of the graphs CD(k, q) (ii) exist for inﬁnitely many values of m.

Outline:

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons,

Andrew Woldar The underlying geometry of the graphs CD(k, q) Outline:

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Andrew Woldar The underlying geometry of the graphs CD(k, q) 1 Root systems 2 The Weyl group 3 Dynkin diagrams and the Cartan matrix 4 Lie algebras and the upper Borel subalgebra 5 Lie groups and ﬁnite groups of Lie type 6 Rank 2 buildings 7 Embedding buildings into Lie algebras

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

Andrew Woldar The underlying geometry of the graphs CD(k, q) 2 The Weyl group 3 Dynkin diagrams and the Cartan matrix 4 Lie algebras and the upper Borel subalgebra 5 Lie groups and ﬁnite groups of Lie type 6 Rank 2 buildings 7 Embedding buildings into Lie algebras

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

1 Root systems

Andrew Woldar The underlying geometry of the graphs CD(k, q) 3 Dynkin diagrams and the Cartan matrix 4 Lie algebras and the upper Borel subalgebra 5 Lie groups and ﬁnite groups of Lie type 6 Rank 2 buildings 7 Embedding buildings into Lie algebras

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

1 Root systems 2 The Weyl group

Andrew Woldar The underlying geometry of the graphs CD(k, q) 4 Lie algebras and the upper Borel subalgebra 5 Lie groups and ﬁnite groups of Lie type 6 Rank 2 buildings 7 Embedding buildings into Lie algebras

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

1 Root systems 2 The Weyl group 3 Dynkin diagrams and the Cartan matrix

Andrew Woldar The underlying geometry of the graphs CD(k, q) 5 Lie groups and ﬁnite groups of Lie type 6 Rank 2 buildings 7 Embedding buildings into Lie algebras

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

1 Root systems 2 The Weyl group 3 Dynkin diagrams and the Cartan matrix 4 Lie algebras and the upper Borel subalgebra

Andrew Woldar The underlying geometry of the graphs CD(k, q) 6 Rank 2 buildings 7 Embedding buildings into Lie algebras

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

1 Root systems 2 The Weyl group 3 Dynkin diagrams and the Cartan matrix 4 Lie algebras and the upper Borel subalgebra 5 Lie groups and ﬁnite groups of Lie type

Andrew Woldar The underlying geometry of the graphs CD(k, q) 7 Embedding buildings into Lie algebras

8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

1 Root systems 2 The Weyl group 3 Dynkin diagrams and the Cartan matrix 4 Lie algebras and the upper Borel subalgebra 5 Lie groups and ﬁnite groups of Lie type 6 Rank 2 buildings

Andrew Woldar The underlying geometry of the graphs CD(k, q) 8 Objects of type Ae1

Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

Andrew Woldar The underlying geometry of the graphs CD(k, q) Motivation

What was the goal of our research? To derive families of graphs that simultaneously (i) mimic the behavior of incidence graphs of generalized m-gons, (ii) exist for inﬁnitely many values of m.

Outline:

8 Objects of type Ae1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Simple Lie Thin Weyl group algebras building embedding

Groups of Borel Thick Lie type subgroup building

Schematic

Irreducible root systems

Andrew Woldar The underlying geometry of the graphs CD(k, q) Thin Weyl group building embedding

Groups of Borel Thick Lie type subgroup building

Schematic

Irreducible root systems

Simple Lie algebras

Andrew Woldar The underlying geometry of the graphs CD(k, q) Thin Weyl group building embedding

Borel Thick subgroup building

Schematic

Irreducible root systems

Simple Lie algebras

Groups of Lie type

Andrew Woldar The underlying geometry of the graphs CD(k, q) Thin building embedding

Borel Thick subgroup building

Schematic

Irreducible root systems

Simple Lie Weyl group algebras

Groups of Lie type

Andrew Woldar The underlying geometry of the graphs CD(k, q) Thin building embedding

Borel Thick subgroup building

Schematic

Irreducible root systems

Simple Lie Weyl group algebras

Groups of Lie type

Andrew Woldar The underlying geometry of the graphs CD(k, q) Thin building embedding

Thick building

Schematic

Irreducible root systems

Simple Lie Weyl group algebras

Groups of Borel Lie type subgroup

Andrew Woldar The underlying geometry of the graphs CD(k, q) embedding

Thick building

Schematic

Irreducible root systems

Simple Lie Thin Weyl group algebras building

Groups of Borel Lie type subgroup

Andrew Woldar The underlying geometry of the graphs CD(k, q) embedding

Schematic

Irreducible root systems

Simple Lie Thin Weyl group algebras building

Groups of Borel Thick Lie type subgroup building

Andrew Woldar The underlying geometry of the graphs CD(k, q) embedding

Schematic

Irreducible root systems

Simple Lie Thin Weyl group algebras building

Groups of Borel Thick Lie type subgroup building

Andrew Woldar The underlying geometry of the graphs CD(k, q) Schematic

Irreducible root systems

Simple Lie Thin Weyl group algebras building embedding

Groups of Borel Thick Lie type subgroup building

Andrew Woldar The underlying geometry of the graphs CD(k, q) A root system Φ is a set of vectors in V such that

1 |Φ| < ∞

2 Φ spans V (r,s) 3 For every r, s ∈ Φ, one has 2 (r,r) ∈ Z (r,s) 4 For every r, s ∈ Φ, one has s − 2 (r,r) r ∈ Φ 5 For every r ∈ Φ, one has {αr | α ∈ R} ∩ Φ = {r, −r} (Hence Φ = Φ+ ∪ Φ− where Φ− = −Φ+.)

Root systems deﬁnition

Let V be a ﬁnite-dimensional Euclidean space with standard inner product (·, ·).

Andrew Woldar The underlying geometry of the graphs CD(k, q) 1 |Φ| < ∞

Root systems deﬁnition

Let V be a ﬁnite-dimensional Euclidean space with standard inner product (·, ·).

A root system Φ is a set of vectors in V such that

Andrew Woldar The underlying geometry of the graphs CD(k, q) 2 Φ spans V (r,s) 3 For every r, s ∈ Φ, one has 2 (r,r) ∈ Z (r,s) 4 For every r, s ∈ Φ, one has s − 2 (r,r) r ∈ Φ 5 For every r ∈ Φ, one has {αr | α ∈ R} ∩ Φ = {r, −r} (Hence Φ = Φ+ ∪ Φ− where Φ− = −Φ+.)

Root systems deﬁnition

Let V be a ﬁnite-dimensional Euclidean space with standard inner product (·, ·).

A root system Φ is a set of vectors in V such that

1 |Φ| < ∞

Andrew Woldar The underlying geometry of the graphs CD(k, q) (r,s) 3 For every r, s ∈ Φ, one has 2 (r,r) ∈ Z (r,s) 4 For every r, s ∈ Φ, one has s − 2 (r,r) r ∈ Φ 5 For every r ∈ Φ, one has {αr | α ∈ R} ∩ Φ = {r, −r} (Hence Φ = Φ+ ∪ Φ− where Φ− = −Φ+.)

Root systems deﬁnition

Let V be a ﬁnite-dimensional Euclidean space with standard inner product (·, ·).

A root system Φ is a set of vectors in V such that

1 |Φ| < ∞

2 Φ spans V

Andrew Woldar The underlying geometry of the graphs CD(k, q) (r,s) 4 For every r, s ∈ Φ, one has s − 2 (r,r) r ∈ Φ 5 For every r ∈ Φ, one has {αr | α ∈ R} ∩ Φ = {r, −r} (Hence Φ = Φ+ ∪ Φ− where Φ− = −Φ+.)

Root systems deﬁnition

Let V be a ﬁnite-dimensional Euclidean space with standard inner product (·, ·).

A root system Φ is a set of vectors in V such that

1 |Φ| < ∞

2 Φ spans V (r,s) 3 For every r, s ∈ Φ, one has 2 (r,r) ∈ Z

Andrew Woldar The underlying geometry of the graphs CD(k, q) 5 For every r ∈ Φ, one has {αr | α ∈ R} ∩ Φ = {r, −r} (Hence Φ = Φ+ ∪ Φ− where Φ− = −Φ+.)

Root systems deﬁnition

Let V be a ﬁnite-dimensional Euclidean space with standard inner product (·, ·).

A root system Φ is a set of vectors in V such that

1 |Φ| < ∞

2 Φ spans V (r,s) 3 For every r, s ∈ Φ, one has 2 (r,r) ∈ Z (r,s) 4 For every r, s ∈ Φ, one has s − 2 (r,r) r ∈ Φ

Andrew Woldar The underlying geometry of the graphs CD(k, q) (Hence Φ = Φ+ ∪ Φ− where Φ− = −Φ+.)

Root systems deﬁnition

Let V be a ﬁnite-dimensional Euclidean space with standard inner product (·, ·).

A root system Φ is a set of vectors in V such that

1 |Φ| < ∞

2 Φ spans V (r,s) 3 For every r, s ∈ Φ, one has 2 (r,r) ∈ Z (r,s) 4 For every r, s ∈ Φ, one has s − 2 (r,r) r ∈ Φ 5 For every r ∈ Φ, one has {αr | α ∈ R} ∩ Φ = {r, −r}

Andrew Woldar The underlying geometry of the graphs CD(k, q) Root systems deﬁnition

Let V be a ﬁnite-dimensional Euclidean space with standard inner product (·, ·).

A root system Φ is a set of vectors in V such that

1 |Φ| < ∞

Andrew Woldar The underlying geometry of the graphs CD(k, q) Root systems rank 2 case

A1 × A1

π/2 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Root systems rank 2 case

A2 r2

2π/3 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Root systems rank 2 case

r 2 B2

3π/4 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Root systems rank 2 case

5π/6 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Set Π = {r1, r2, . . . , rn} (fundamental basis).

For each ri ∈ Π, denote by wi the reﬂection in the hyperplane in n V = R orthogonal to ri, that is,

(ri,s) wi(s) = s − 2 ri ∈ Φ (Axiom 4 of a root system) (ri,ri)

Thus each wi permutes the roots of Φ.

We call wi a fundamental reﬂection.

The Weyl group is generated by all fundamental reﬂections, i.e.,

W = hw1, w2, . . . , wni

The Weyl group deﬁnition

Let Φ be a root system with fundamental roots r1, r2, . . . , rn.

Andrew Woldar The underlying geometry of the graphs CD(k, q) For each ri ∈ Π, denote by wi the reﬂection in the hyperplane in n V = R orthogonal to ri, that is,

(ri,s) wi(s) = s − 2 ri ∈ Φ (Axiom 4 of a root system) (ri,ri)

Thus each wi permutes the roots of Φ.

We call wi a fundamental reﬂection.

The Weyl group is generated by all fundamental reﬂections, i.e.,

W = hw1, w2, . . . , wni

The Weyl group deﬁnition

Let Φ be a root system with fundamental roots r1, r2, . . . , rn.

Set Π = {r1, r2, . . . , rn} (fundamental basis).

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∈ Φ (Axiom 4 of a root system)

Thus each wi permutes the roots of Φ.

We call wi a fundamental reﬂection.

The Weyl group is generated by all fundamental reﬂections, i.e.,

W = hw1, w2, . . . , wni

The Weyl group deﬁnition

Let Φ be a root system with fundamental roots r1, r2, . . . , rn.

Set Π = {r1, r2, . . . , rn} (fundamental basis).

For each ri ∈ Π, denote by wi the reﬂection in the hyperplane in n V = R orthogonal to ri, that is,

(ri,s) wi(s) = s − 2 ri (ri,ri)

Andrew Woldar The underlying geometry of the graphs CD(k, q) Thus each wi permutes the roots of Φ.

We call wi a fundamental reﬂection.

The Weyl group is generated by all fundamental reﬂections, i.e.,

W = hw1, w2, . . . , wni

The Weyl group deﬁnition

Let Φ be a root system with fundamental roots r1, r2, . . . , rn.

Set Π = {r1, r2, . . . , rn} (fundamental basis).

For each ri ∈ Π, denote by wi the reﬂection in the hyperplane in n V = R orthogonal to ri, that is,

(ri,s) wi(s) = s − 2 ri ∈ Φ (Axiom 4 of a root system) (ri,ri)

Andrew Woldar The underlying geometry of the graphs CD(k, q) We call wi a fundamental reﬂection.

The Weyl group is generated by all fundamental reﬂections, i.e.,

W = hw1, w2, . . . , wni

The Weyl group deﬁnition

Let Φ be a root system with fundamental roots r1, r2, . . . , rn.

Set Π = {r1, r2, . . . , rn} (fundamental basis).

For each ri ∈ Π, denote by wi the reﬂection in the hyperplane in n V = R orthogonal to ri, that is,

(ri,s) wi(s) = s − 2 ri ∈ Φ (Axiom 4 of a root system) (ri,ri)

Thus each wi permutes the roots of Φ.

Andrew Woldar The underlying geometry of the graphs CD(k, q) The Weyl group is generated by all fundamental reﬂections, i.e.,

W = hw1, w2, . . . , wni

The Weyl group deﬁnition

Let Φ be a root system with fundamental roots r1, r2, . . . , rn.

Set Π = {r1, r2, . . . , rn} (fundamental basis).

For each ri ∈ Π, denote by wi the reﬂection in the hyperplane in n V = R orthogonal to ri, that is,

(ri,s) wi(s) = s − 2 ri ∈ Φ (Axiom 4 of a root system) (ri,ri)

Thus each wi permutes the roots of Φ.

We call wi a fundamental reﬂection.

Andrew Woldar The underlying geometry of the graphs CD(k, q) The Weyl group deﬁnition

Let Φ be a root system with fundamental roots r1, r2, . . . , rn.

Set Π = {r1, r2, . . . , rn} (fundamental basis).

For each ri ∈ Π, denote by wi the reﬂection in the hyperplane in n V = R orthogonal to ri, that is,

(ri,s) wi(s) = s − 2 ri ∈ Φ (Axiom 4 of a root system) (ri,ri)

Thus each wi permutes the roots of Φ.

We call wi a fundamental reﬂection.

The Weyl group is generated by all fundamental reﬂections, i.e.,

W = hw1, w2, . . . , wni

Andrew Woldar The underlying geometry of the graphs CD(k, q) m−1 Then ]rirj = m π where m ∈ {2, 3, 4, 6}.

In fact, m is the order o(wiwj) of the rotation wiwj in the plane determined by ri, rj.

This implies that the Weyl group of every rank 2 root system is a dihedral group of order 2m: ∼ W (A1 × A1) = D4 (m = 2) ∼ W (A2) = D6 (m = 3) ∼ W (B2) = D8 (m = 4) ∼ W (G2) = D12 (m = 6)

The Weyl group product of two reﬂections

Let ]rirj denote the angle between ri, rj ∈ Π.

Andrew Woldar The underlying geometry of the graphs CD(k, q) In fact, m is the order o(wiwj) of the rotation wiwj in the plane determined by ri, rj.

This implies that the Weyl group of every rank 2 root system is a dihedral group of order 2m: ∼ W (A1 × A1) = D4 (m = 2) ∼ W (A2) = D6 (m = 3) ∼ W (B2) = D8 (m = 4) ∼ W (G2) = D12 (m = 6)

The Weyl group product of two reﬂections

Let ]rirj denote the angle between ri, rj ∈ Π.

m−1 Then ]rirj = m π where m ∈ {2, 3, 4, 6}.

Andrew Woldar The underlying geometry of the graphs CD(k, q) This implies that the Weyl group of every rank 2 root system is a dihedral group of order 2m: ∼ W (A1 × A1) = D4 (m = 2) ∼ W (A2) = D6 (m = 3) ∼ W (B2) = D8 (m = 4) ∼ W (G2) = D12 (m = 6)

The Weyl group product of two reﬂections

Let ]rirj denote the angle between ri, rj ∈ Π.

m−1 Then ]rirj = m π where m ∈ {2, 3, 4, 6}.

In fact, m is the order o(wiwj) of the rotation wiwj in the plane determined by ri, rj.

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∼ W (A1 × A1) = D4 (m = 2) ∼ W (A2) = D6 (m = 3) ∼ W (B2) = D8 (m = 4) ∼ W (G2) = D12 (m = 6)

The Weyl group product of two reﬂections

Let ]rirj denote the angle between ri, rj ∈ Π.

m−1 Then ]rirj = m π where m ∈ {2, 3, 4, 6}.

In fact, m is the order o(wiwj) of the rotation wiwj in the plane determined by ri, rj.

This implies that the Weyl group of every rank 2 root system is a dihedral group of order 2m:

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∼ W (A2) = D6 (m = 3) ∼ W (B2) = D8 (m = 4) ∼ W (G2) = D12 (m = 6)

The Weyl group product of two reﬂections

Let ]rirj denote the angle between ri, rj ∈ Π.

m−1 Then ]rirj = m π where m ∈ {2, 3, 4, 6}.

In fact, m is the order o(wiwj) of the rotation wiwj in the plane determined by ri, rj.

This implies that the Weyl group of every rank 2 root system is a dihedral group of order 2m: ∼ W (A1 × A1) = D4 (m = 2)

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∼ W (B2) = D8 (m = 4) ∼ W (G2) = D12 (m = 6)

The Weyl group product of two reﬂections

Let ]rirj denote the angle between ri, rj ∈ Π.

m−1 Then ]rirj = m π where m ∈ {2, 3, 4, 6}.

In fact, m is the order o(wiwj) of the rotation wiwj in the plane determined by ri, rj.

This implies that the Weyl group of every rank 2 root system is a dihedral group of order 2m: ∼ W (A1 × A1) = D4 (m = 2) ∼ W (A2) = D6 (m = 3)

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∼ W (G2) = D12 (m = 6)

The Weyl group product of two reﬂections

Let ]rirj denote the angle between ri, rj ∈ Π.

m−1 Then ]rirj = m π where m ∈ {2, 3, 4, 6}.

In fact, m is the order o(wiwj) of the rotation wiwj in the plane determined by ri, rj.

This implies that the Weyl group of every rank 2 root system is a dihedral group of order 2m: ∼ W (A1 × A1) = D4 (m = 2) ∼ W (A2) = D6 (m = 3) ∼ W (B2) = D8 (m = 4)

Andrew Woldar The underlying geometry of the graphs CD(k, q) The Weyl group product of two reﬂections

Let ]rirj denote the angle between ri, rj ∈ Π.

m−1 Then ]rirj = m π where m ∈ {2, 3, 4, 6}.

In fact, m is the order o(wiwj) of the rotation wiwj in the plane determined by ri, rj.

This implies that the Weyl group of every rank 2 root system is a dihedral group of order 2m: ∼ W (A1 × A1) = D4 (m = 2) ∼ W (A2) = D6 (m = 3) ∼ W (B2) = D8 (m = 4) ∼ W (G2) = D12 (m = 6)

Andrew Woldar The underlying geometry of the graphs CD(k, q) It is a useful device for codifying a root system:

Nodes of diagram ←→ fundamental roots edge weights determine angles between pairs of roots

It also codiﬁes the action of the Weyl group on a root system:

Nodes of diagram ←→ fundamental reﬂections edge weights determine orders of products of pairs of reﬂections

Dynkin diagrams codifying root systems

A Dynkin diagram consists of nodes and weighted (directed) edges between pairs of nodes.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Nodes of diagram ←→ fundamental roots edge weights determine angles between pairs of roots

It also codiﬁes the action of the Weyl group on a root system:

Nodes of diagram ←→ fundamental reﬂections edge weights determine orders of products of pairs of reﬂections

Dynkin diagrams codifying root systems

A Dynkin diagram consists of nodes and weighted (directed) edges between pairs of nodes.

It is a useful device for codifying a root system:

Andrew Woldar The underlying geometry of the graphs CD(k, q) It also codiﬁes the action of the Weyl group on a root system:

Nodes of diagram ←→ fundamental reﬂections edge weights determine orders of products of pairs of reﬂections

Dynkin diagrams codifying root systems

A Dynkin diagram consists of nodes and weighted (directed) edges between pairs of nodes.

It is a useful device for codifying a root system:

Nodes of diagram ←→ fundamental roots edge weights determine angles between pairs of roots

Andrew Woldar The underlying geometry of the graphs CD(k, q) Nodes of diagram ←→ fundamental reﬂections edge weights determine orders of products of pairs of reﬂections

Dynkin diagrams codifying root systems

A Dynkin diagram consists of nodes and weighted (directed) edges between pairs of nodes.

It is a useful device for codifying a root system:

Nodes of diagram ←→ fundamental roots edge weights determine angles between pairs of roots

It also codiﬁes the action of the Weyl group on a root system:

Andrew Woldar The underlying geometry of the graphs CD(k, q) Dynkin diagrams codifying root systems

A Dynkin diagram consists of nodes and weighted (directed) edges between pairs of nodes.

It is a useful device for codifying a root system:

Nodes of diagram ←→ fundamental roots edge weights determine angles between pairs of roots

It also codiﬁes the action of the Weyl group on a root system:

Nodes of diagram ←→ fundamental reﬂections edge weights determine orders of products of pairs of reﬂections

Andrew Woldar The underlying geometry of the graphs CD(k, q) Dynkin diagrams rank 2 case

A1 × A1

π/2 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Dynkin diagrams rank 2 case

A1 × A1

π/2 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Dynkin diagrams rank 2 case

A2 r2

2π/3 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Dynkin diagrams rank 2 case

A2 r2

2π/3 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Dynkin diagrams rank 2 case

r2 B2

3π/4 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Dynkin diagrams rank 2 case

r2 B2

3π/4 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Dynkin diagrams rank 2 case

5π/6 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Dynkin diagrams rank 2 case

5π/6 r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) For fundamental roots ri, rj ∈ Π, we may express this as

(ri,rj ) Aij = 2 ∈ . (ri,ri) Z

We now deﬁne the Cartan matrix for Φ by A(Φ) = (Aij).

2 0 2 −1 2 −1 2 −1 0 2 −1 2 −2 2 −3 2

A1 × A1 A2 B2 G2

The Cartan matrix another codifying device

Recall that for any pair of roots r, s ∈ Φ, one has (r,s) 2 (r,r) ∈ Z (Axiom 3 of a root system)

Andrew Woldar The underlying geometry of the graphs CD(k, q) We now deﬁne the Cartan matrix for Φ by A(Φ) = (Aij).

2 0 2 −1 2 −1 2 −1 0 2 −1 2 −2 2 −3 2

A1 × A1 A2 B2 G2

The Cartan matrix another codifying device

Recall that for any pair of roots r, s ∈ Φ, one has (r,s) 2 (r,r) ∈ Z (Axiom 3 of a root system)

For fundamental roots ri, rj ∈ Π, we may express this as

(ri,rj ) Aij = 2 ∈ . (ri,ri) Z

Andrew Woldar The underlying geometry of the graphs CD(k, q) 2 0 2 −1 2 −1 2 −1 0 2 −1 2 −2 2 −3 2

A1 × A1 A2 B2 G2

The Cartan matrix another codifying device

Recall that for any pair of roots r, s ∈ Φ, one has (r,s) 2 (r,r) ∈ Z (Axiom 3 of a root system)

For fundamental roots ri, rj ∈ Π, we may express this as

(ri,rj ) Aij = 2 ∈ . (ri,ri) Z

We now deﬁne the Cartan matrix for Φ by A(Φ) = (Aij).

Andrew Woldar The underlying geometry of the graphs CD(k, q) The Cartan matrix another codifying device

Recall that for any pair of roots r, s ∈ Φ, one has (r,s) 2 (r,r) ∈ Z (Axiom 3 of a root system)

For fundamental roots ri, rj ∈ Π, we may express this as

(ri,rj ) Aij = 2 ∈ . (ri,ri) Z

We now deﬁne the Cartan matrix for Φ by A(Φ) = (Aij).

2 0 2 −1 2 −1 2 −1 0 2 −1 2 −2 2 −3 2

A1 × A1 A2 B2 G2

Andrew Woldar The underlying geometry of the graphs CD(k, q) The Cartan matrix another codifying device

Recall that for any pair of roots r, s ∈ Φ, one has (r,s) 2 (r,r) ∈ Z (Axiom 3 of a root system)

For fundamental roots ri, rj ∈ Π, we may express this as

(ri,rj ) Aij = 2 ∈ . (ri,ri) Z

We now deﬁne the Cartan matrix for Φ by A(Φ) = (Aij).

2 0 2 −1 2 −1 2 −1 0 2 −1 2 −2 2 −3 2

A1 × A1 A2 B2 G2

Andrew Woldar The underlying geometry of the graphs CD(k, q) Lie algebras are examples of non-associative graded algebras.

Subject to a ﬁxed choice of root system and ﬁeld, one obtains a unique semisimple Lie algebra.

Lie algebras deﬁnition

A Lie algebra is a vector space L over some ﬁeld F, endowed with a binary operation [ · , · ]: L × L → L (Lie product) which is bilinear, anticommutative, and satisﬁes the Jacobi identity:

[α, [β, γ]] + [β, [γ, α]] + [γ, [α, β]] = 0, ∀α, β, γ ∈ L

Andrew Woldar The underlying geometry of the graphs CD(k, q) Subject to a ﬁxed choice of root system and ﬁeld, one obtains a unique semisimple Lie algebra.

Lie algebras deﬁnition

A Lie algebra is a vector space L over some ﬁeld F, endowed with a binary operation [ · , · ]: L × L → L (Lie product) which is bilinear, anticommutative, and satisﬁes the Jacobi identity:

[α, [β, γ]] + [β, [γ, α]] + [γ, [α, β]] = 0, ∀α, β, γ ∈ L

Lie algebras are examples of non-associative graded algebras.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Lie algebras deﬁnition

A Lie algebra is a vector space L over some ﬁeld F, endowed with a binary operation [ · , · ]: L × L → L (Lie product) which is bilinear, anticommutative, and satisﬁes the Jacobi identity:

[α, [β, γ]] + [β, [γ, α]] + [γ, [α, β]] = 0, ∀α, β, γ ∈ L

Lie algebras are examples of non-associative graded algebras.

Subject to a ﬁxed choice of root system and ﬁeld, one obtains a unique semisimple Lie algebra.

Andrew Woldar The underlying geometry of the graphs CD(k, q) L L = H Lr (Cartan decomposition) r∈Φ where each root space Lr is an H-invariant subspace of L, i.e., [H, Lr] ⊆ Lr.

If L arises from a root system, then each Lr is one-dimensional. We write Lr = heri and refer to er as a root vector.

For each r ∈ Φ, one has [h, er] = r(h)er, h ∈ H, i.e., each root r ∈ Φ is a linear functional r : H → R.

This gives Φ ,→ H∗, therefore Φ∗ ,→ H.

Lie algebras Cartan decomposition

Let H be a self-normalizing nilpotent subalgebra of L. We call H a Cartan subalgebra of L.

Andrew Woldar The underlying geometry of the graphs CD(k, q) If L arises from a root system, then each Lr is one-dimensional. We write Lr = heri and refer to er as a root vector.

For each r ∈ Φ, one has [h, er] = r(h)er, h ∈ H, i.e., each root r ∈ Φ is a linear functional r : H → R.

This gives Φ ,→ H∗, therefore Φ∗ ,→ H.

Lie algebras Cartan decomposition

Let H be a self-normalizing nilpotent subalgebra of L. We call H a Cartan subalgebra of L.

L L = H Lr (Cartan decomposition) r∈Φ where each root space Lr is an H-invariant subspace of L, i.e., [H, Lr] ⊆ Lr.

Andrew Woldar The underlying geometry of the graphs CD(k, q) For each r ∈ Φ, one has [h, er] = r(h)er, h ∈ H, i.e., each root r ∈ Φ is a linear functional r : H → R.

This gives Φ ,→ H∗, therefore Φ∗ ,→ H.

Lie algebras Cartan decomposition

Let H be a self-normalizing nilpotent subalgebra of L. We call H a Cartan subalgebra of L.

L L = H Lr (Cartan decomposition) r∈Φ where each root space Lr is an H-invariant subspace of L, i.e., [H, Lr] ⊆ Lr.

If L arises from a root system, then each Lr is one-dimensional. We write Lr = heri and refer to er as a root vector.

Andrew Woldar The underlying geometry of the graphs CD(k, q) This gives Φ ,→ H∗, therefore Φ∗ ,→ H.

Lie algebras Cartan decomposition

Let H be a self-normalizing nilpotent subalgebra of L. We call H a Cartan subalgebra of L.

L L = H Lr (Cartan decomposition) r∈Φ where each root space Lr is an H-invariant subspace of L, i.e., [H, Lr] ⊆ Lr.

If L arises from a root system, then each Lr is one-dimensional. We write Lr = heri and refer to er as a root vector.

For each r ∈ Φ, one has [h, er] = r(h)er, h ∈ H, i.e., each root r ∈ Φ is a linear functional r : H → R.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Lie algebras Cartan decomposition

Let H be a self-normalizing nilpotent subalgebra of L. We call H a Cartan subalgebra of L.

L L = H Lr (Cartan decomposition) r∈Φ where each root space Lr is an H-invariant subspace of L, i.e., [H, Lr] ⊆ Lr.

If L arises from a root system, then each Lr is one-dimensional. We write Lr = heri and refer to er as a root vector.

For each r ∈ Φ, one has [h, er] = r(h)er, h ∈ H, i.e., each root r ∈ Φ is a linear functional r : H → R.

This gives Φ ,→ H∗, therefore Φ∗ ,→ H.

Andrew Woldar The underlying geometry of the graphs CD(k, q) It follows that L + L − Lr = Lr Lr r∈Φ where + L Lr = Lr (positive root space) r∈Φ+ − L Lr = Lr (negative root space) r∈Φ− If we now append H to each of these subspaces, we obtain

U L + L = H Lr (upper Borel subalgebra) L L − L = H Lr (lower Borel subalgebra)

U L + We are interested in the upper Borel subalgebra L = H Lr .

Lie algebras Borel subalgebras

Recall that Φ = Φ+ ∪ Φ−.

Andrew Woldar The underlying geometry of the graphs CD(k, q) If we now append H to each of these subspaces, we obtain

U L + L = H Lr (upper Borel subalgebra) L L − L = H Lr (lower Borel subalgebra)

U L + We are interested in the upper Borel subalgebra L = H Lr .

Lie algebras Borel subalgebras

Recall that Φ = Φ+ ∪ Φ−. It follows that L + L − Lr = Lr Lr r∈Φ where + L Lr = Lr (positive root space) r∈Φ+ − L Lr = Lr (negative root space) r∈Φ−

Andrew Woldar The underlying geometry of the graphs CD(k, q) U L + We are interested in the upper Borel subalgebra L = H Lr .

Lie algebras Borel subalgebras

Recall that Φ = Φ+ ∪ Φ−. It follows that L + L − Lr = Lr Lr r∈Φ where + L Lr = Lr (positive root space) r∈Φ+ − L Lr = Lr (negative root space) r∈Φ− If we now append H to each of these subspaces, we obtain

U L + L = H Lr (upper Borel subalgebra) L L − L = H Lr (lower Borel subalgebra)

Andrew Woldar The underlying geometry of the graphs CD(k, q) Lie algebras Borel subalgebras

U L + L = H Lr (upper Borel subalgebra) L L − L = H Lr (lower Borel subalgebra)

U L + We are interested in the upper Borel subalgebra L = H Lr .

Andrew Woldar The underlying geometry of the graphs CD(k, q) First we determine the root system of type A2:

Φ = {r1, r2, r1 + r2, −r1, −r2, −r1 − r2}

∗ ∗ ∗ Let Π = {r1, r2} be the dual basis of the fundamental basis Π = {r1, r2}.

Then Π∗ is a basis for hΦ∗i = H, and we accordingly obtain the following canonical basis for L:

∗ ∗ {r1, r2, er1 , er2 , er1+r2 , e−r1 , e−r2 , e−r1−r2 } | {z } | {z } | {z } H L+ L−

Lie algebras type A2

EXAMPLE. We construct the complex Lie algebra L of type A2 and identify its upper Borel subalgebra LU .

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∗ ∗ ∗ Let Π = {r1, r2} be the dual basis of the fundamental basis Π = {r1, r2}.

Then Π∗ is a basis for hΦ∗i = H, and we accordingly obtain the following canonical basis for L:

∗ ∗ {r1, r2, er1 , er2 , er1+r2 , e−r1 , e−r2 , e−r1−r2 } | {z } | {z } | {z } H L+ L−

Lie algebras type A2

EXAMPLE. We construct the complex Lie algebra L of type A2 and identify its upper Borel subalgebra LU .

First we determine the root system of type A2:

Φ = {r1, r2, r1 + r2, −r1, −r2, −r1 − r2}

Andrew Woldar The underlying geometry of the graphs CD(k, q) Then Π∗ is a basis for hΦ∗i = H, and we accordingly obtain the following canonical basis for L:

∗ ∗ {r1, r2, er1 , er2 , er1+r2 , e−r1 , e−r2 , e−r1−r2 } | {z } | {z } | {z } H L+ L−

Lie algebras type A2

EXAMPLE. We construct the complex Lie algebra L of type A2 and identify its upper Borel subalgebra LU .

First we determine the root system of type A2:

Φ = {r1, r2, r1 + r2, −r1, −r2, −r1 − r2}

∗ ∗ ∗ Let Π = {r1, r2} be the dual basis of the fundamental basis Π = {r1, r2}.

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∗ ∗ {r1, r2, er1 , er2 , er1+r2 , e−r1 , e−r2 , e−r1−r2 } | {z } | {z } | {z } H L+ L−

Lie algebras type A2

EXAMPLE. We construct the complex Lie algebra L of type A2 and identify its upper Borel subalgebra LU .

First we determine the root system of type A2:

Φ = {r1, r2, r1 + r2, −r1, −r2, −r1 − r2}

∗ ∗ ∗ Let Π = {r1, r2} be the dual basis of the fundamental basis Π = {r1, r2}.

Then Π∗ is a basis for hΦ∗i = H, and we accordingly obtain the following canonical basis for L:

Andrew Woldar The underlying geometry of the graphs CD(k, q) Lie algebras type A2

EXAMPLE. We construct the complex Lie algebra L of type A2 and identify its upper Borel subalgebra LU .

First we determine the root system of type A2:

Φ = {r1, r2, r1 + r2, −r1, −r2, −r1 − r2}

∗ ∗ ∗ Let Π = {r1, r2} be the dual basis of the fundamental basis Π = {r1, r2}.

Then Π∗ is a basis for hΦ∗i = H, and we accordingly obtain the following canonical basis for L:

∗ ∗ {r1, r2, er1 , er2 , er1+r2 , e−r1 , e−r2 , e−r1−r2 } | {z } | {z } | {z } H L+ L−

Andrew Woldar The underlying geometry of the graphs CD(k, q) Similarly, we obtain matrix representations of the six root vectors:

 0 1 0   0 0 0   0 0 1        er =  0 0 0  er =  0 0 1  er +r =  0 0 0  1   2   1 2   0 0 0 0 0 0 0 0 0

 0 0 0   0 0 0   0 0 0        e−r =  1 0 0  e−r =  0 0 0  e−r −r =  0 0 0  1   2   1 2   0 0 0 0 1 0 1 0 0

Lie algebras type A2

We may choose our embedding Π∗ ,→ H as follows:

 1 0 0   0 0 0      r∗ =  0 −1 0  r∗ =  0 1 0  1   2   0 0 0 0 0 −1

Andrew Woldar The underlying geometry of the graphs CD(k, q)  0 1 0   0 0 0   0 0 1        er =  0 0 0  er =  0 0 1  er +r =  0 0 0  1   2   1 2   0 0 0 0 0 0 0 0 0

 0 0 0   0 0 0   0 0 0        e−r =  1 0 0  e−r =  0 0 0  e−r −r =  0 0 0  1   2   1 2   0 0 0 0 1 0 1 0 0

Lie algebras type A2

We may choose our embedding Π∗ ,→ H as follows:

 1 0 0   0 0 0      r∗ =  0 −1 0  r∗ =  0 1 0  1   2   0 0 0 0 0 −1

Similarly, we obtain matrix representations of the six root vectors:

Andrew Woldar The underlying geometry of the graphs CD(k, q)  0 0 0   0 0 0   0 0 0        e−r =  1 0 0  e−r =  0 0 0  e−r −r =  0 0 0  1   2   1 2   0 0 0 0 1 0 1 0 0

Lie algebras type A2

We may choose our embedding Π∗ ,→ H as follows:

 1 0 0   0 0 0      r∗ =  0 −1 0  r∗ =  0 1 0  1   2   0 0 0 0 0 −1

Similarly, we obtain matrix representations of the six root vectors:

 0 1 0   0 0 0   0 0 1        er =  0 0 0  er =  0 0 1  er +r =  0 0 0  1   2   1 2   0 0 0 0 0 0 0 0 0

Andrew Woldar The underlying geometry of the graphs CD(k, q) Lie algebras type A2

We may choose our embedding Π∗ ,→ H as follows:

 1 0 0   0 0 0      r∗ =  0 −1 0  r∗ =  0 1 0  1   2   0 0 0 0 0 −1

Similarly, we obtain matrix representations of the six root vectors:

 0 1 0   0 0 0   0 0 1        er =  0 0 0  er =  0 0 1  er +r =  0 0 0  1   2   1 2   0 0 0 0 0 0 0 0 0

 0 0 0   0 0 0   0 0 0        e−r =  1 0 0  e−r =  0 0 0  e−r −r =  0 0 0  1   2   1 2   0 0 0 0 1 0 1 0 0

Andrew Woldar The underlying geometry of the graphs CD(k, q) Subject to our matrix representation, this becomes    a c d U   L =  0 b − a e  a, b, c, d, e ∈ C

 0 0 −b 

Lie algebras type A2

A basis for LU = H ⊕ L+ is

∗ ∗ {r1, r2, er1 , er2 , er1+r2 } | {z } | {z } H L+

Andrew Woldar The underlying geometry of the graphs CD(k, q) Lie algebras type A2

A basis for LU = H ⊕ L+ is

∗ ∗ {r1, r2, er1 , er2 , er1+r2 } | {z } | {z } H L+

Subject to our matrix representation, this becomes    a c d U   L =  0 b − a e  a, b, c, d, e ∈ C

 0 0 −b 

Andrew Woldar The underlying geometry of the graphs CD(k, q) Note: exp (ad x) exp (ad y) = exp (ad [x, y]).

As ad x is a nilpotent derivation, we get that each exp (ad x) is an inner automorphism of L.

We now deﬁne the complex Lie group: G = { exp (ad x) | x ∈ L }

Finite groups of Lie type the complex Lie group

Given a complex simple Lie algebra L, for each x ∈ L we deﬁne the exponentiation map ∞ P (ad x)k exp (ad x) = k! k=0 where ad x : L → L (ad x : y 7→ [x, y])

Andrew Woldar The underlying geometry of the graphs CD(k, q) As ad x is a nilpotent derivation, we get that each exp (ad x) is an inner automorphism of L.

We now deﬁne the complex Lie group: G = { exp (ad x) | x ∈ L }

Finite groups of Lie type the complex Lie group

Given a complex simple Lie algebra L, for each x ∈ L we deﬁne the exponentiation map ∞ P (ad x)k exp (ad x) = k! k=0 where ad x : L → L (ad x : y 7→ [x, y])

Note: exp (ad x) exp (ad y) = exp (ad [x, y]).

Andrew Woldar The underlying geometry of the graphs CD(k, q) We now deﬁne the complex Lie group: G = { exp (ad x) | x ∈ L }

Finite groups of Lie type the complex Lie group

Given a complex simple Lie algebra L, for each x ∈ L we deﬁne the exponentiation map ∞ P (ad x)k exp (ad x) = k! k=0 where ad x : L → L (ad x : y 7→ [x, y])

Note: exp (ad x) exp (ad y) = exp (ad [x, y]).

As ad x is a nilpotent derivation, we get that each exp (ad x) is an inner automorphism of L.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type the complex Lie group

Given a complex simple Lie algebra L, for each x ∈ L we deﬁne the exponentiation map ∞ P (ad x)k exp (ad x) = k! k=0 where ad x : L → L (ad x : y 7→ [x, y])

Note: exp (ad x) exp (ad y) = exp (ad [x, y]).

As ad x is a nilpotent derivation, we get that each exp (ad x) is an inner automorphism of L.

We now deﬁne the complex Lie group: G = { exp (ad x) | x ∈ L }

Andrew Woldar The underlying geometry of the graphs CD(k, q) This allows the corresponding algebraic groups to be defined over Z, which enables their range of definition to be extended to finite fields.

The resulting ﬁnite simple groups are termed Chevalley groups in his honor.

Finite groups of Lie type seminal work of C. Chevalley

Chevalley constructed a basis (Chevalley basis) for the universal enveloping algebra of every complex simple Lie algebra with the property that all structure constants of the enveloping algebra are integral with respect to the basis.

Andrew Woldar The underlying geometry of the graphs CD(k, q) The resulting ﬁnite simple groups are termed Chevalley groups in his honor.

Finite groups of Lie type seminal work of C. Chevalley

This allows the corresponding algebraic groups to be defined over Z, which enables their range of definition to be extended to finite fields.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type seminal work of C. Chevalley

This allows the corresponding algebraic groups to be defined over Z, which enables their range of definition to be extended to finite fields.

The resulting ﬁnite simple groups are termed Chevalley groups in his honor.

Andrew Woldar The underlying geometry of the graphs CD(k, q) An(q) Ln+1(q) Dickson   B (q) O (q) Dickson  n 2n+1 classical∗ C (q) P Sp (q) Dickson n 2n  +  Dn(q) P Ω2n(q) Dickson

G2(q) Dickson   F4(q) Chevalley   E6(q) Dickson exceptional  E7(q) Chevalley   E8(q) Chevalley

∗ The case “q prime” was treated by C. Jordan.

Finite groups of Lie type Chevalley groups

Lie type group discoverer

Andrew Woldar The underlying geometry of the graphs CD(k, q) G2(q) Dickson   F4(q) Chevalley   E6(q) Dickson exceptional  E7(q) Chevalley   E8(q) Chevalley

∗ The case “q prime” was treated by C. Jordan.

Finite groups of Lie type Chevalley groups

Lie type group discoverer

An(q) Ln+1(q) Dickson   B (q) O (q) Dickson  n 2n+1 classical∗ C (q) P Sp (q) Dickson n 2n  +  Dn(q) P Ω2n(q) Dickson

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type Chevalley groups

Lie type group discoverer

An(q) Ln+1(q) Dickson   B (q) O (q) Dickson  n 2n+1 classical∗ C (q) P Sp (q) Dickson n 2n  +  Dn(q) P Ω2n(q) Dickson

G2(q) Dickson   F4(q) Chevalley   E6(q) Dickson exceptional  E7(q) Chevalley   E8(q) Chevalley

∗ The case “q prime” was treated by C. Jordan.

Andrew Woldar The underlying geometry of the graphs CD(k, q) 2A (q) U (q) Steinberg n n+1 classical 2 − Dn(q) P Ω2n(q) Steinberg 2 ∗ E6(q) Steinberg  3  D4(q) Steinberg   2 2m+1 B2(2 ) Suzuki exceptional 2 2m+1  G2(3 ) Ree  2 2m+1  F4(2 ) Ree

∗ 2 The family E6(q) was discovered independently by J. Tits.

Finite groups of Lie type twisted groups

Lie type group discoverer

Andrew Woldar The underlying geometry of the graphs CD(k, q) 2 ∗ E6(q) Steinberg  3  D4(q) Steinberg   2 2m+1 B2(2 ) Suzuki exceptional 2 2m+1  G2(3 ) Ree  2 2m+1  F4(2 ) Ree

∗ 2 The family E6(q) was discovered independently by J. Tits.

Finite groups of Lie type twisted groups

Lie type group discoverer 2A (q) U (q) Steinberg n n+1 classical 2 − Dn(q) P Ω2n(q) Steinberg

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type twisted groups

Lie type group discoverer 2A (q) U (q) Steinberg n n+1 classical 2 − Dn(q) P Ω2n(q) Steinberg 2 ∗ E6(q) Steinberg  3  D4(q) Steinberg   2 2m+1 B2(2 ) Suzuki exceptional 2 2m+1  G2(3 ) Ree  2 2m+1  F4(2 ) Ree

∗ 2 The family E6(q) was discovered independently by J. Tits.

Andrew Woldar The underlying geometry of the graphs CD(k, q) 3 2 2m+1 2 2m+1 2 2m+1 D4(q), B2(2 ), G2(3 ), F4(2 )

(1) The diagram D4 admits an order 3 automorphism, however 3 existence of D4(q) requires that the ﬁeld Fq be a cubic extension of a smaller ﬁeld. This precludes the existence of a complex Lie 3 group of type D4 over C.

(2) The diagrams B2, G2, F4 each admit an order 2 automorphism 2 that interchanges long and short roots. The existence of B2(q), 2 2 G2(q), F4(q) therefore requires that B2(q), G2(q), F4(q) admit graph-ﬁeld automorphisms that preserve root length. This occurs only for the ﬁelds specifed above, and certainly not for C.

Finite groups of Lie type no complex analogues

Among these 16 inﬁnite families of groups of Lie type, four such families have no complex analogues:

Andrew Woldar The underlying geometry of the graphs CD(k, q) (1) The diagram D4 admits an order 3 automorphism, however 3 existence of D4(q) requires that the ﬁeld Fq be a cubic extension of a smaller ﬁeld. This precludes the existence of a complex Lie 3 group of type D4 over C.

Finite groups of Lie type no complex analogues

Among these 16 inﬁnite families of groups of Lie type, four such families have no complex analogues:

3 2 2m+1 2 2m+1 2 2m+1 D4(q), B2(2 ), G2(3 ), F4(2 )

Andrew Woldar The underlying geometry of the graphs CD(k, q) (2) The diagrams B2, G2, F4 each admit an order 2 automorphism 2 that interchanges long and short roots. The existence of B2(q), 2 2 G2(q), F4(q) therefore requires that B2(q), G2(q), F4(q) admit graph-ﬁeld automorphisms that preserve root length. This occurs only for the ﬁelds specifed above, and certainly not for C.

Finite groups of Lie type no complex analogues

Among these 16 inﬁnite families of groups of Lie type, four such families have no complex analogues:

3 2 2m+1 2 2m+1 2 2m+1 D4(q), B2(2 ), G2(3 ), F4(2 )

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type no complex analogues

Among these 16 inﬁnite families of groups of Lie type, four such families have no complex analogues:

3 2 2m+1 2 2m+1 2 2m+1 D4(q), B2(2 ), G2(3 ), F4(2 )

Andrew Woldar The underlying geometry of the graphs CD(k, q) Fix a Sylow p-subgroup U of G (unipotent subgroup).

Let B = NG(U) (Borel subgroup).

Then the full lattice of subgroups {P | B ≤ P ≤ G} is isomorphic to the lattice of all subsets of an n-element set.

The 2n − 1 proper subgroups in this lattice are called parabolic subgroups of G. Of these, n are maximal subgroups of G. We denote these as P1,P2,...,Pn (maximal parabolics).

Finite groups of Lie type structure and partial subgroup lattice

a Let G be a ﬁnite group of Lie type of rank n over Fq, q = p .

Andrew Woldar The underlying geometry of the graphs CD(k, q) Let B = NG(U) (Borel subgroup).

Then the full lattice of subgroups {P | B ≤ P ≤ G} is isomorphic to the lattice of all subsets of an n-element set.

The 2n − 1 proper subgroups in this lattice are called parabolic subgroups of G. Of these, n are maximal subgroups of G. We denote these as P1,P2,...,Pn (maximal parabolics).

Finite groups of Lie type structure and partial subgroup lattice

a Let G be a ﬁnite group of Lie type of rank n over Fq, q = p .

Fix a Sylow p-subgroup U of G (unipotent subgroup).

Andrew Woldar The underlying geometry of the graphs CD(k, q) Then the full lattice of subgroups {P | B ≤ P ≤ G} is isomorphic to the lattice of all subsets of an n-element set.

The 2n − 1 proper subgroups in this lattice are called parabolic subgroups of G. Of these, n are maximal subgroups of G. We denote these as P1,P2,...,Pn (maximal parabolics).

Finite groups of Lie type structure and partial subgroup lattice

a Let G be a ﬁnite group of Lie type of rank n over Fq, q = p .

Fix a Sylow p-subgroup U of G (unipotent subgroup).

Let B = NG(U) (Borel subgroup).

Andrew Woldar The underlying geometry of the graphs CD(k, q) The 2n − 1 proper subgroups in this lattice are called parabolic subgroups of G. Of these, n are maximal subgroups of G. We denote these as P1,P2,...,Pn (maximal parabolics).

Finite groups of Lie type structure and partial subgroup lattice

a Let G be a ﬁnite group of Lie type of rank n over Fq, q = p .

Fix a Sylow p-subgroup U of G (unipotent subgroup).

Let B = NG(U) (Borel subgroup).

Then the full lattice of subgroups {P | B ≤ P ≤ G} is isomorphic to the lattice of all subsets of an n-element set.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type structure and partial subgroup lattice

a Let G be a ﬁnite group of Lie type of rank n over Fq, q = p .

Fix a Sylow p-subgroup U of G (unipotent subgroup).

Let B = NG(U) (Borel subgroup).

Then the full lattice of subgroups {P | B ≤ P ≤ G} is isomorphic to the lattice of all subsets of an n-element set.

The 2n − 1 proper subgroups in this lattice are called parabolic subgroups of G. Of these, n are maximal subgroups of G. We denote these as P1,P2,...,Pn (maximal parabolics).

Andrew Woldar The underlying geometry of the graphs CD(k, q) As B splits over U, there exists a subgroup T ≤ B with T =∼ B/U. We call T a maximal torus.

The Weyl group W = W (G) now appears as the quotient ∼ W = NG(T )/T

As NG(T ) need not split over T , W need not be a subgroup of G.

Finite groups of Lie type recovering the Weyl group

Let G be a ﬁnite group of Lie type, and let B = NG(U) be a ﬁxed Borel subgroup of G.

Andrew Woldar The underlying geometry of the graphs CD(k, q) The Weyl group W = W (G) now appears as the quotient ∼ W = NG(T )/T

As NG(T ) need not split over T , W need not be a subgroup of G.

Finite groups of Lie type recovering the Weyl group

Let G be a ﬁnite group of Lie type, and let B = NG(U) be a ﬁxed Borel subgroup of G.

As B splits over U, there exists a subgroup T ≤ B with T =∼ B/U. We call T a maximal torus.

Andrew Woldar The underlying geometry of the graphs CD(k, q) As NG(T ) need not split over T , W need not be a subgroup of G.

Finite groups of Lie type recovering the Weyl group

Let G be a ﬁnite group of Lie type, and let B = NG(U) be a ﬁxed Borel subgroup of G.

As B splits over U, there exists a subgroup T ≤ B with T =∼ B/U. We call T a maximal torus.

The Weyl group W = W (G) now appears as the quotient ∼ W = NG(T )/T

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type recovering the Weyl group

Let G be a ﬁnite group of Lie type, and let B = NG(U) be a ﬁxed Borel subgroup of G.

As B splits over U, there exists a subgroup T ≤ B with T =∼ B/U. We call T a maximal torus.

The Weyl group W = W (G) now appears as the quotient ∼ W = NG(T )/T

As NG(T ) need not split over T , W need not be a subgroup of G.

Andrew Woldar The underlying geometry of the graphs CD(k, q) G ∅

P ∼ {1} {2} 1 P2 =

B {1, 2}

Note: B = P1 ∩ P2. As such, we may denote B as P1,2.

Finite groups of Lie type type A2

EXAMPLE. We illustrate the case A2(q) = L3(q) in detail.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type type A2

EXAMPLE. We illustrate the case A2(q) = L3(q) in detail.

G ∅

P ∼ {1} {2} 1 P2 =

B {1, 2}

Note: B = P1 ∩ P2. As such, we may denote B as P1,2.

Andrew Woldar The underlying geometry of the graphs CD(k, q) From this, we obtain        ∗ ∗ ∗   ∗ ∗ ∗   ∗ ∗ ∗  B = 0 ∗ ∗ P1 = 0 ∗ ∗ P2 = ∗ ∗ ∗  0 0 ∗   0 ∗ ∗   0 0 ∗ 

Finite groups of Lie type type A2

We may choose our Sylow p-subgroup to be    1 ∗ ∗  U = 0 1 ∗  0 0 1 

Andrew Woldar The underlying geometry of the graphs CD(k, q)      ∗ ∗ ∗   ∗ ∗ ∗  P1 = 0 ∗ ∗ P2 = ∗ ∗ ∗  0 ∗ ∗   0 0 ∗ 

Finite groups of Lie type type A2

We may choose our Sylow p-subgroup to be    1 ∗ ∗  U = 0 1 ∗  0 0 1 

From this, we obtain    ∗ ∗ ∗  B = 0 ∗ ∗  0 0 ∗ 

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type type A2

We may choose our Sylow p-subgroup to be    1 ∗ ∗  U = 0 1 ∗  0 0 1 

From this, we obtain        ∗ ∗ ∗   ∗ ∗ ∗   ∗ ∗ ∗  B = 0 ∗ ∗ P1 = 0 ∗ ∗ P2 = ∗ ∗ ∗  0 0 ∗   0 ∗ ∗   0 0 ∗ 

Andrew Woldar The underlying geometry of the graphs CD(k, q)      1 ∗ ∗   ∗ ∗ ∗  U = 0 1 ∗ B = 0 ∗ ∗  0 0 1   0 0 ∗ 

  *   +  ∗ 0 0  0 ∗ 0 ∗ 0 0 T = 0 ∗ 0 NG(T ) = ∗ 0 0 , 0 0 ∗  0 0 ∗  0 0 ∗ 0 ∗ 0

*0 1 0 1 0 0+ ∼ W (G) = NG(T )/T = 1 0 0 , 0 0 1 = D6 0 0 1 0 1 0

Finite groups of Lie type type A2

We now recover the Weyl group.

Andrew Woldar The underlying geometry of the graphs CD(k, q)   *   +  ∗ 0 0  0 ∗ 0 ∗ 0 0 T = 0 ∗ 0 NG(T ) = ∗ 0 0 , 0 0 ∗  0 0 ∗  0 0 ∗ 0 ∗ 0

*0 1 0 1 0 0+ ∼ W (G) = NG(T )/T = 1 0 0 , 0 0 1 = D6 0 0 1 0 1 0

Finite groups of Lie type type A2

We now recover the Weyl group.

     1 ∗ ∗   ∗ ∗ ∗  U = 0 1 ∗ B = 0 ∗ ∗  0 0 1   0 0 ∗ 

Andrew Woldar The underlying geometry of the graphs CD(k, q) *0 ∗ 0 ∗ 0 0+ NG(T ) = ∗ 0 0 , 0 0 ∗ 0 0 ∗ 0 ∗ 0

*0 1 0 1 0 0+ ∼ W (G) = NG(T )/T = 1 0 0 , 0 0 1 = D6 0 0 1 0 1 0

Finite groups of Lie type type A2

We now recover the Weyl group.

     1 ∗ ∗   ∗ ∗ ∗  U = 0 1 ∗ B = 0 ∗ ∗  0 0 1   0 0 ∗     ∗ 0 0  T = 0 ∗ 0  0 0 ∗ 

Andrew Woldar The underlying geometry of the graphs CD(k, q) *0 1 0 1 0 0+ ∼ W (G) = NG(T )/T = 1 0 0 , 0 0 1 = D6 0 0 1 0 1 0

Finite groups of Lie type type A2

We now recover the Weyl group.

     1 ∗ ∗   ∗ ∗ ∗  U = 0 1 ∗ B = 0 ∗ ∗  0 0 1   0 0 ∗ 

  *   +  ∗ 0 0  0 ∗ 0 ∗ 0 0 T = 0 ∗ 0 NG(T ) = ∗ 0 0 , 0 0 ∗  0 0 ∗  0 0 ∗ 0 ∗ 0

Andrew Woldar The underlying geometry of the graphs CD(k, q) Finite groups of Lie type type A2

We now recover the Weyl group.

     1 ∗ ∗   ∗ ∗ ∗  U = 0 1 ∗ B = 0 ∗ ∗  0 0 1   0 0 ∗ 

  *   +  ∗ 0 0  0 ∗ 0 ∗ 0 0 T = 0 ∗ 0 NG(T ) = ∗ 0 0 , 0 0 ∗  0 0 ∗  0 0 ∗ 0 ∗ 0

*0 1 0 1 0 0+ ∼ W (G) = NG(T )/T = 1 0 0 , 0 0 1 = D6 0 0 1 0 1 0

Andrew Woldar The underlying geometry of the graphs CD(k, q) P = G/P1 = {gP1 | g ∈ G}

L = G/P2 = {gP2 | g ∈ G}

Now deﬁne the incidence relation I ⊂ P × L

(xP1, yP2) ∈ I ⇐⇒ xP1 ∩ yP2 6= ∅

We refer to (P, L, I) as a (thick) rank 2 building.

Rank 2 buildings deﬁnition

Let G be a rank 2 group of Lie type with maximal parabolic subgroups P1 and P2.

Andrew Woldar The underlying geometry of the graphs CD(k, q) L = G/P2 = {gP2 | g ∈ G}

Now deﬁne the incidence relation I ⊂ P × L

(xP1, yP2) ∈ I ⇐⇒ xP1 ∩ yP2 6= ∅

We refer to (P, L, I) as a (thick) rank 2 building.

Rank 2 buildings deﬁnition

Let G be a rank 2 group of Lie type with maximal parabolic subgroups P1 and P2.

P = G/P1 = {gP1 | g ∈ G}

Andrew Woldar The underlying geometry of the graphs CD(k, q) Now deﬁne the incidence relation I ⊂ P × L

(xP1, yP2) ∈ I ⇐⇒ xP1 ∩ yP2 6= ∅

We refer to (P, L, I) as a (thick) rank 2 building.

Rank 2 buildings deﬁnition

Let G be a rank 2 group of Lie type with maximal parabolic subgroups P1 and P2.

P = G/P1 = {gP1 | g ∈ G}

L = G/P2 = {gP2 | g ∈ G}

Andrew Woldar The underlying geometry of the graphs CD(k, q) We refer to (P, L, I) as a (thick) rank 2 building.

Rank 2 buildings deﬁnition

Let G be a rank 2 group of Lie type with maximal parabolic subgroups P1 and P2.

P = G/P1 = {gP1 | g ∈ G}

L = G/P2 = {gP2 | g ∈ G}

Now deﬁne the incidence relation I ⊂ P × L

(xP1, yP2) ∈ I ⇐⇒ xP1 ∩ yP2 6= ∅

Andrew Woldar The underlying geometry of the graphs CD(k, q) Rank 2 buildings deﬁnition

Let G be a rank 2 group of Lie type with maximal parabolic subgroups P1 and P2.

P = G/P1 = {gP1 | g ∈ G}

L = G/P2 = {gP2 | g ∈ G}

Now deﬁne the incidence relation I ⊂ P × L

(xP1, yP2) ∈ I ⇐⇒ xP1 ∩ yP2 6= ∅

We refer to (P, L, I) as a (thick) rank 2 building.

Andrew Woldar The underlying geometry of the graphs CD(k, q) 3 points ←→ 1-dimensional subspaces of F 3 lines ←→ 2-dimensional subspaces of F incidence ←→ containment

   ∗ ∗ ∗  P1 = 0 ∗ ∗ = stabilizer of the point he1i  0 ∗ ∗     ∗ ∗ ∗  P2 = ∗ ∗ ∗ = stabilizer of the line he1, e2i  0 0 ∗ 

Thus G/P1 and G/P2 are the point set and line set of PG(2, F). We conclude that buildings of type A2 are nothing more than Desarguesian projective planes.

Rank 2 buildings type A2

EXAMPLE. Recall the linear model for projective plane PG(2, F):

Andrew Woldar The underlying geometry of the graphs CD(k, q)    ∗ ∗ ∗  P1 = 0 ∗ ∗ = stabilizer of the point he1i  0 ∗ ∗     ∗ ∗ ∗  P2 = ∗ ∗ ∗ = stabilizer of the line he1, e2i  0 0 ∗ 

Thus G/P1 and G/P2 are the point set and line set of PG(2, F). We conclude that buildings of type A2 are nothing more than Desarguesian projective planes.

Rank 2 buildings type A2

EXAMPLE. Recall the linear model for projective plane PG(2, F): 3 points ←→ 1-dimensional subspaces of F 3 lines ←→ 2-dimensional subspaces of F incidence ←→ containment

Andrew Woldar The underlying geometry of the graphs CD(k, q) = stabilizer of the point he1i

   ∗ ∗ ∗  P2 = ∗ ∗ ∗ = stabilizer of the line he1, e2i  0 0 ∗ 

Thus G/P1 and G/P2 are the point set and line set of PG(2, F). We conclude that buildings of type A2 are nothing more than Desarguesian projective planes.

Rank 2 buildings type A2

EXAMPLE. Recall the linear model for projective plane PG(2, F): 3 points ←→ 1-dimensional subspaces of F 3 lines ←→ 2-dimensional subspaces of F incidence ←→ containment

   ∗ ∗ ∗  P1 = 0 ∗ ∗  0 ∗ ∗ 

Andrew Woldar The underlying geometry of the graphs CD(k, q)    ∗ ∗ ∗  P2 = ∗ ∗ ∗ = stabilizer of the line he1, e2i  0 0 ∗ 

Thus G/P1 and G/P2 are the point set and line set of PG(2, F). We conclude that buildings of type A2 are nothing more than Desarguesian projective planes.

Rank 2 buildings type A2

EXAMPLE. Recall the linear model for projective plane PG(2, F): 3 points ←→ 1-dimensional subspaces of F 3 lines ←→ 2-dimensional subspaces of F incidence ←→ containment

   ∗ ∗ ∗  P1 = 0 ∗ ∗ = stabilizer of the point he1i  0 ∗ ∗ 

Andrew Woldar The underlying geometry of the graphs CD(k, q) = stabilizer of the line he1, e2i

Thus G/P1 and G/P2 are the point set and line set of PG(2, F). We conclude that buildings of type A2 are nothing more than Desarguesian projective planes.

Rank 2 buildings type A2

EXAMPLE. Recall the linear model for projective plane PG(2, F): 3 points ←→ 1-dimensional subspaces of F 3 lines ←→ 2-dimensional subspaces of F incidence ←→ containment

   ∗ ∗ ∗  P1 = 0 ∗ ∗ = stabilizer of the point he1i  0 ∗ ∗     ∗ ∗ ∗  P2 = ∗ ∗ ∗  0 0 ∗ 

Andrew Woldar The underlying geometry of the graphs CD(k, q) Thus G/P1 and G/P2 are the point set and line set of PG(2, F). We conclude that buildings of type A2 are nothing more than Desarguesian projective planes.

Rank 2 buildings type A2

EXAMPLE. Recall the linear model for projective plane PG(2, F): 3 points ←→ 1-dimensional subspaces of F 3 lines ←→ 2-dimensional subspaces of F incidence ←→ containment

Andrew Woldar The underlying geometry of the graphs CD(k, q) Rank 2 buildings type A2

EXAMPLE. Recall the linear model for projective plane PG(2, F): 3 points ←→ 1-dimensional subspaces of F 3 lines ←→ 2-dimensional subspaces of F incidence ←→ containment

Thus G/P1 and G/P2 are the point set and line set of PG(2, F). We conclude that buildings of type A2 are nothing more than Desarguesian projective planes.

Andrew Woldar The underlying geometry of the graphs CD(k, q) G = BWB = ` BwB (Bruhat decomposition) w∈W

This allows the action of G on the coset spaces G/Pi to be formulated in terms of a composite action of W and B on these spaces.

Each Borel orbit on G/Pi (Schubert cell) contains a unique T -invariant coset gPi which may be identiﬁed with the coset α = wWi ∈ W/Wi where g ∈ BwB and Pi = BWiB. Moreover, every α ∈ W/Wi is so realized. We denote this orbit by Bα.

one-to-one Borel orbits Bα of G/Pi ←−−−−−−−→ cosets α ∈ W/Wi

Embedding buildings in Lie algebras general procedure

Let G be a group of Lie type with ﬁxed Borel subgroup B, ﬁxed maximal torus T < B, maximal parabolics Pi, Weyl group W , and Lie algebra L.

Andrew Woldar The underlying geometry of the graphs CD(k, q) This allows the action of G on the coset spaces G/Pi to be formulated in terms of a composite action of W and B on these spaces.

one-to-one Borel orbits Bα of G/Pi ←−−−−−−−→ cosets α ∈ W/Wi

Embedding buildings in Lie algebras general procedure

Let G be a group of Lie type with ﬁxed Borel subgroup B, ﬁxed maximal torus T < B, maximal parabolics Pi, Weyl group W , and Lie algebra L.

G = BWB = ` BwB (Bruhat decomposition) w∈W

Andrew Woldar The underlying geometry of the graphs CD(k, q) Each Borel orbit on G/Pi (Schubert cell) contains a unique T -invariant coset gPi which may be identiﬁed with the coset α = wWi ∈ W/Wi where g ∈ BwB and Pi = BWiB. Moreover, every α ∈ W/Wi is so realized. We denote this orbit by Bα.

one-to-one Borel orbits Bα of G/Pi ←−−−−−−−→ cosets α ∈ W/Wi

Embedding buildings in Lie algebras general procedure

Let G be a group of Lie type with ﬁxed Borel subgroup B, ﬁxed maximal torus T < B, maximal parabolics Pi, Weyl group W , and Lie algebra L.

G = BWB = ` BwB (Bruhat decomposition) w∈W

This allows the action of G on the coset spaces G/Pi to be formulated in terms of a composite action of W and B on these spaces.

Andrew Woldar The underlying geometry of the graphs CD(k, q) one-to-one Borel orbits Bα of G/Pi ←−−−−−−−→ cosets α ∈ W/Wi

Embedding buildings in Lie algebras general procedure

Let G be a group of Lie type with ﬁxed Borel subgroup B, ﬁxed maximal torus T < B, maximal parabolics Pi, Weyl group W , and Lie algebra L.

G = BWB = ` BwB (Bruhat decomposition) w∈W

This allows the action of G on the coset spaces G/Pi to be formulated in terms of a composite action of W and B on these spaces.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Embedding buildings in Lie algebras general procedure

Let G be a group of Lie type with ﬁxed Borel subgroup B, ﬁxed maximal torus T < B, maximal parabolics Pi, Weyl group W , and Lie algebra L.

G = BWB = ` BwB (Bruhat decomposition) w∈W

This allows the action of G on the coset spaces G/Pi to be formulated in terms of a composite action of W and B on these spaces.

one-to-one Borel orbits Bα of G/Pi ←−−−−−−−→ cosets α ∈ W/Wi

Andrew Woldar The underlying geometry of the graphs CD(k, q) U Hence we obtain embeddings W/Wi ,→ H ⊂ L .

But this means we now have an embedded transversal for the Borel orbits Bα.

+ Every object in Bα will embed in α ⊕ L , however very few of the vectors in α ⊕ L+ will correspond to such embedded objects.

Hence we need some way of identifying which vectors in α ⊕ L+ represent embedded objects from Bα.

Embedding buildings in Lie algebras general procedure

The action of W on W/Wi is equivalent to the contragredient action of W on the dual root space Φ∗ ⊂ H given by

w(r∗) = (w−1(r))∗

Andrew Woldar The underlying geometry of the graphs CD(k, q) But this means we now have an embedded transversal for the Borel orbits Bα.

+ Every object in Bα will embed in α ⊕ L , however very few of the vectors in α ⊕ L+ will correspond to such embedded objects.

Hence we need some way of identifying which vectors in α ⊕ L+ represent embedded objects from Bα.

Embedding buildings in Lie algebras general procedure

The action of W on W/Wi is equivalent to the contragredient action of W on the dual root space Φ∗ ⊂ H given by

w(r∗) = (w−1(r))∗

U Hence we obtain embeddings W/Wi ,→ H ⊂ L .

Andrew Woldar The underlying geometry of the graphs CD(k, q) + Every object in Bα will embed in α ⊕ L , however very few of the vectors in α ⊕ L+ will correspond to such embedded objects.

Hence we need some way of identifying which vectors in α ⊕ L+ represent embedded objects from Bα.

Embedding buildings in Lie algebras general procedure

The action of W on W/Wi is equivalent to the contragredient action of W on the dual root space Φ∗ ⊂ H given by

w(r∗) = (w−1(r))∗

U Hence we obtain embeddings W/Wi ,→ H ⊂ L .

But this means we now have an embedded transversal for the Borel orbits Bα.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Hence we need some way of identifying which vectors in α ⊕ L+ represent embedded objects from Bα.

Embedding buildings in Lie algebras general procedure

The action of W on W/Wi is equivalent to the contragredient action of W on the dual root space Φ∗ ⊂ H given by

w(r∗) = (w−1(r))∗

U Hence we obtain embeddings W/Wi ,→ H ⊂ L .

But this means we now have an embedded transversal for the Borel orbits Bα.

+ Every object in Bα will embed in α ⊕ L , however very few of the vectors in α ⊕ L+ will correspond to such embedded objects.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Embedding buildings in Lie algebras general procedure

The action of W on W/Wi is equivalent to the contragredient action of W on the dual root space Φ∗ ⊂ H given by

w(r∗) = (w−1(r))∗

U Hence we obtain embeddings W/Wi ,→ H ⊂ L .

But this means we now have an embedded transversal for the Borel orbits Bα.

+ Every object in Bα will embed in α ⊕ L , however very few of the vectors in α ⊕ L+ will correspond to such embedded objects.

Hence we need some way of identifying which vectors in α ⊕ L+ represent embedded objects from Bα.

Andrew Woldar The underlying geometry of the graphs CD(k, q) We now deﬁne the expanse of α to be

+ nX + nego L (α) = λrer ∈ L | r ∈ α

Then + Bα = α ⊕ L (α)

Note: This is a group-free description of the objects in each Borel orbit Bα. Thus a full determination of the objects in the embedded building depends only on the action of the Weyl group.

Embedding buildings in Lie algebras general procedure

For each α ∈ W/Wi we deﬁne the set αneg = {r ∈ Φ+ | α(r) < 0}

Andrew Woldar The underlying geometry of the graphs CD(k, q) Then + Bα = α ⊕ L (α)

Note: This is a group-free description of the objects in each Borel orbit Bα. Thus a full determination of the objects in the embedded building depends only on the action of the Weyl group.

Embedding buildings in Lie algebras general procedure

For each α ∈ W/Wi we deﬁne the set αneg = {r ∈ Φ+ | α(r) < 0}

We now deﬁne the expanse of α to be

+ nX + nego L (α) = λrer ∈ L | r ∈ α

Andrew Woldar The underlying geometry of the graphs CD(k, q) Note: This is a group-free description of the objects in each Borel orbit Bα. Thus a full determination of the objects in the embedded building depends only on the action of the Weyl group.

Embedding buildings in Lie algebras general procedure

For each α ∈ W/Wi we deﬁne the set αneg = {r ∈ Φ+ | α(r) < 0}

We now deﬁne the expanse of α to be

+ nX + nego L (α) = λrer ∈ L | r ∈ α

Then + Bα = α ⊕ L (α)

Andrew Woldar The underlying geometry of the graphs CD(k, q) Embedding buildings in Lie algebras general procedure

For each α ∈ W/Wi we deﬁne the set αneg = {r ∈ Φ+ | α(r) < 0}

We now deﬁne the expanse of α to be

+ nX + nego L (α) = λrer ∈ L | r ∈ α

Then + Bα = α ⊕ L (α)

Note: This is a group-free description of the objects in each Borel orbit Bα. Thus a full determination of the objects in the embedded building depends only on the action of the Weyl group.

Andrew Woldar The underlying geometry of the graphs CD(k, q) 1 α(r)β(r) ≥ 0 for all r ∈ Φ+ (Weyl incidence)

2 the projection of [α + a, β + b] onto L+(α) ∩ L+(β) is zero

Note: This coincides with the previously deﬁned incidence on the pre-embedded objects of the geometry (nonempty intersection of cosets).

Embedding buildings in Lie algebras incidence

Incidence: α + a ∈ Bα is incident to β + b ∈ Bβ if and only if

Andrew Woldar The underlying geometry of the graphs CD(k, q) 2 the projection of [α + a, β + b] onto L+(α) ∩ L+(β) is zero

Note: This coincides with the previously deﬁned incidence on the pre-embedded objects of the geometry (nonempty intersection of cosets).

Embedding buildings in Lie algebras incidence

Incidence: α + a ∈ Bα is incident to β + b ∈ Bβ if and only if

1 α(r)β(r) ≥ 0 for all r ∈ Φ+ (Weyl incidence)

Andrew Woldar The underlying geometry of the graphs CD(k, q) Note: This coincides with the previously deﬁned incidence on the pre-embedded objects of the geometry (nonempty intersection of cosets).

Embedding buildings in Lie algebras incidence

Incidence: α + a ∈ Bα is incident to β + b ∈ Bβ if and only if

1 α(r)β(r) ≥ 0 for all r ∈ Φ+ (Weyl incidence)

2 the projection of [α + a, β + b] onto L+(α) ∩ L+(β) is zero

Andrew Woldar The underlying geometry of the graphs CD(k, q) Embedding buildings in Lie algebras incidence

Incidence: α + a ∈ Bα is incident to β + b ∈ Bβ if and only if

1 α(r)β(r) ≥ 0 for all r ∈ Φ+ (Weyl incidence)

2 the projection of [α + a, β + b] onto L+(α) ∩ L+(β) is zero

Note: This coincides with the previously deﬁned incidence on the pre-embedded objects of the geometry (nonempty intersection of cosets).

Andrew Woldar The underlying geometry of the graphs CD(k, q) α a |Bα|  r∗ 0 1  1 ∗ ∗ P −r1 + r2 λr1 er1 q  ∗ 2 −r2 λr2 er2 + λr1+r2 er1+r2 q  r∗ 0 1  2 ∗ ∗ L r1 − r2 λr2 er2 q  ∗ 2 −r1 λr1 er1 + λr1+r2 er1+r2 q

U + Table: Objects of the embedded building of type A2 in L = H ⊕ L Each embedded object is of the form α + a for α ∈ H and a ∈ L+(α)

Embedding buildings in Lie algebras type A2 (projective plane)

EXAMPLE. We illustrate the embedding procedure for the classical projective plane PG(2, q).

Andrew Woldar The underlying geometry of the graphs CD(k, q) Embedding buildings in Lie algebras type A2 (projective plane)

EXAMPLE. We illustrate the embedding procedure for the classical projective plane PG(2, q).

α a |Bα|  r∗ 0 1  1 ∗ ∗ P −r1 + r2 λr1 er1 q  ∗ 2 −r2 λr2 er2 + λr1+r2 er1+r2 q  r∗ 0 1  2 ∗ ∗ L r1 − r2 λr2 er2 q  ∗ 2 −r1 λr1 er1 + λr1+r2 er1+r2 q

U + Table: Objects of the embedded building of type A2 in L = H ⊕ L Each embedded object is of the form α + a for α ∈ H and a ∈ L+(α)

Andrew Woldar The underlying geometry of the graphs CD(k, q) Embedding buildings in Lie algebras type A2 (projective plane)

lines z }| { ∗ ∗ ∗ ∗ r2 r1 − r2 + γr2 er2 −r1 + γr1 er1 + γr1+r2 er1+r2  r∗ 1 1 0  1 ∗ ∗ points −r1 + r2 + λr1 er1 1 0 δab  ∗ −r2 + λr2 er2 + λr1+r2 er1+r2 0 δcd δef

Table: Incidence in the embedded building of type A2.

Each of δab, δcd and δef is the kronecker delta function, where

a = 2λr1 , b = 3γr1 ; c = 2λr2 , d = 3γr2 ; and e = λr1+r2 ,

f = γr1+r2 + λr1 γr2 .

Andrew Woldar The underlying geometry of the graphs CD(k, q) Both orbits have size q2 and generate the classical biaﬃne plane, i.e., the classical aﬃne plane with one parallel class removed.

Convention: We denote the scalar multiple of the root vector

eir1+jr2 by pij for each point (p) and by `ij for each line [`].

∗ (p) = −r + p e + p e ∈ B ∗ 2 01 r2 11 r1+r2 −r2 ∗ [`] = −r + ` e + ` e ∈ B ∗ 1 10 r1 11 r1+r2 −r1

Embedding buildings in Lie algebras type A2 (projective plane)

We perform the computation for incidence between points and lines in the largest Borel orbits. These orbits are B ∗ and B ∗ −r2 −r1 respectively.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Convention: We denote the scalar multiple of the root vector

eir1+jr2 by pij for each point (p) and by `ij for each line [`].

∗ (p) = −r + p e + p e ∈ B ∗ 2 01 r2 11 r1+r2 −r2 ∗ [`] = −r + ` e + ` e ∈ B ∗ 1 10 r1 11 r1+r2 −r1

Embedding buildings in Lie algebras type A2 (projective plane)

We perform the computation for incidence between points and lines in the largest Borel orbits. These orbits are B ∗ and B ∗ −r2 −r1 respectively.

Both orbits have size q2 and generate the classical biaﬃne plane, i.e., the classical aﬃne plane with one parallel class removed.

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∗ (p) = −r + p e + p e ∈ B ∗ 2 01 r2 11 r1+r2 −r2 ∗ [`] = −r + ` e + ` e ∈ B ∗ 1 10 r1 11 r1+r2 −r1

Embedding buildings in Lie algebras type A2 (projective plane)

We perform the computation for incidence between points and lines in the largest Borel orbits. These orbits are B ∗ and B ∗ −r2 −r1 respectively.

Both orbits have size q2 and generate the classical biaﬃne plane, i.e., the classical aﬃne plane with one parallel class removed.

Convention: We denote the scalar multiple of the root vector

eir1+jr2 by pij for each point (p) and by `ij for each line [`].

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∗ [`] = −r + ` e + ` e ∈ B ∗ 1 10 r1 11 r1+r2 −r1

Embedding buildings in Lie algebras type A2 (projective plane)

We perform the computation for incidence between points and lines in the largest Borel orbits. These orbits are B ∗ and B ∗ −r2 −r1 respectively.

Both orbits have size q2 and generate the classical biaﬃne plane, i.e., the classical aﬃne plane with one parallel class removed.

Convention: We denote the scalar multiple of the root vector

eir1+jr2 by pij for each point (p) and by `ij for each line [`].

∗ (p) = −r + p e + p e ∈ B ∗ 2 01 r2 11 r1+r2 −r2

Andrew Woldar The underlying geometry of the graphs CD(k, q) Embedding buildings in Lie algebras type A2 (projective plane)

We perform the computation for incidence between points and lines in the largest Borel orbits. These orbits are B ∗ and B ∗ −r2 −r1 respectively.

Both orbits have size q2 and generate the classical biaﬃne plane, i.e., the classical aﬃne plane with one parallel class removed.

Convention: We denote the scalar multiple of the root vector

eir1+jr2 by pij for each point (p) and by `ij for each line [`].

∗ (p) = −r + p e + p e ∈ B ∗ 2 01 r2 11 r1+r2 −r2 ∗ [`] = −r + ` e + ` e ∈ B ∗ 1 10 r1 11 r1+r2 −r1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Thus point (p) is incident to line [`] precisely when the projection + ∗ + ∗ of [(p), [`]] onto L (−r2) ∩ L (−r1) is zero.

∗ ∗ ∗ ∗ ∗ [(p), [`]] = [−r2 , −r1 ] + [p01er2 , −r1 ] + [p11er1+r2 , −r1 ] + [−r2 , `10er1 ]+ ∗ [p01er2 , `10er1 ] + [p11er1+r2 , `10er1 ] + [−r2 , `11er1+r2 ]+

[p01er2 , `11er1+r2 ] + [p11er1+r2 , `11er1+r2 ]

= (p11 − p01`10 − `11)er1+r2

+ ∗ + ∗ As L (−r2) ∩ L (−r1) = her1+r2 i we conclude that (p) and [`] are incident if and only if p11 − p01`10 − `11 = 0,

i.e., p11 − `11 = p01`10

Embedding buildings in Lie algebras type A2 (projective plane)

∗ ∗ + As (−r2)(r)(−r1)(r) ≥ 0 for all r ∈ Φ we have Weyl incidence.

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∗ ∗ ∗ ∗ ∗ [(p), [`]] = [−r2 , −r1 ] + [p01er2 , −r1 ] + [p11er1+r2 , −r1 ] + [−r2 , `10er1 ]+ ∗ [p01er2 , `10er1 ] + [p11er1+r2 , `10er1 ] + [−r2 , `11er1+r2 ]+

[p01er2 , `11er1+r2 ] + [p11er1+r2 , `11er1+r2 ]

= (p11 − p01`10 − `11)er1+r2

+ ∗ + ∗ As L (−r2) ∩ L (−r1) = her1+r2 i we conclude that (p) and [`] are incident if and only if p11 − p01`10 − `11 = 0,

i.e., p11 − `11 = p01`10

Embedding buildings in Lie algebras type A2 (projective plane)

∗ ∗ + As (−r2)(r)(−r1)(r) ≥ 0 for all r ∈ Φ we have Weyl incidence.

Thus point (p) is incident to line [`] precisely when the projection + ∗ + ∗ of [(p), [`]] onto L (−r2) ∩ L (−r1) is zero.

Andrew Woldar The underlying geometry of the graphs CD(k, q) + ∗ + ∗ As L (−r2) ∩ L (−r1) = her1+r2 i we conclude that (p) and [`] are incident if and only if p11 − p01`10 − `11 = 0,

i.e., p11 − `11 = p01`10

Embedding buildings in Lie algebras type A2 (projective plane)

∗ ∗ + As (−r2)(r)(−r1)(r) ≥ 0 for all r ∈ Φ we have Weyl incidence.

Thus point (p) is incident to line [`] precisely when the projection + ∗ + ∗ of [(p), [`]] onto L (−r2) ∩ L (−r1) is zero.

∗ ∗ ∗ ∗ ∗ [(p), [`]] = [−r2 , −r1 ] + [p01er2 , −r1 ] + [p11er1+r2 , −r1 ] + [−r2 , `10er1 ]+ ∗ [p01er2 , `10er1 ] + [p11er1+r2 , `10er1 ] + [−r2 , `11er1+r2 ]+

[p01er2 , `11er1+r2 ] + [p11er1+r2 , `11er1+r2 ]

= (p11 − p01`10 − `11)er1+r2

Andrew Woldar The underlying geometry of the graphs CD(k, q) i.e., p11 − `11 = p01`10

Embedding buildings in Lie algebras type A2 (projective plane)

∗ ∗ + As (−r2)(r)(−r1)(r) ≥ 0 for all r ∈ Φ we have Weyl incidence.

Thus point (p) is incident to line [`] precisely when the projection + ∗ + ∗ of [(p), [`]] onto L (−r2) ∩ L (−r1) is zero.

∗ ∗ ∗ ∗ ∗ [(p), [`]] = [−r2 , −r1 ] + [p01er2 , −r1 ] + [p11er1+r2 , −r1 ] + [−r2 , `10er1 ]+ ∗ [p01er2 , `10er1 ] + [p11er1+r2 , `10er1 ] + [−r2 , `11er1+r2 ]+

[p01er2 , `11er1+r2 ] + [p11er1+r2 , `11er1+r2 ]

= (p11 − p01`10 − `11)er1+r2

+ ∗ + ∗ As L (−r2) ∩ L (−r1) = her1+r2 i we conclude that (p) and [`] are incident if and only if p11 − p01`10 − `11 = 0,

Andrew Woldar The underlying geometry of the graphs CD(k, q) Embedding buildings in Lie algebras type A2 (projective plane)

∗ ∗ + As (−r2)(r)(−r1)(r) ≥ 0 for all r ∈ Φ we have Weyl incidence.

Thus point (p) is incident to line [`] precisely when the projection + ∗ + ∗ of [(p), [`]] onto L (−r2) ∩ L (−r1) is zero.

∗ ∗ ∗ ∗ ∗ [(p), [`]] = [−r2 , −r1 ] + [p01er2 , −r1 ] + [p11er1+r2 , −r1 ] + [−r2 , `10er1 ]+ ∗ [p01er2 , `10er1 ] + [p11er1+r2 , `10er1 ] + [−r2 , `11er1+r2 ]+

[p01er2 , `11er1+r2 ] + [p11er1+r2 , `11er1+r2 ]

= (p11 − p01`10 − `11)er1+r2

+ ∗ + ∗ As L (−r2) ∩ L (−r1) = her1+r2 i we conclude that (p) and [`] are incident if and only if p11 − p01`10 − `11 = 0,

i.e., p11 − `11 = p01`10

Andrew Woldar The underlying geometry of the graphs CD(k, q) α a |Bα|  r∗ 0 1  1 −r∗ + 2r∗ λ e q P 1 2 r1 r1 r∗ − 2r∗ λ e + λ e q2  1 2 r2 r2 r1+r2 r1+r2  ∗ 3 −r1 λr1 er1 + λr1+r2 er1+r2 + λ2r1+r2 e2r1+r2 q  r∗ 0 1  2  r∗ − r∗ λ e q L 1 2 r2 r2 −r∗ + r∗ λ e + λ e q2  1 2 r1 r1 2r1+r2 2r1+r2  ∗ 3 −r2 λr2 er2 + λr1+r2 er1+r2 + λ2r1+r2 e2r1+r2 q

Table: Objects in the embedded building of type B2

Embedding buildings in Lie algebras type B2 (generalized quadrangle)

EXAMPLE. We provide objects of the embedded generalized quadrangle of type B2.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Embedding buildings in Lie algebras type B2 (generalized quadrangle)

EXAMPLE. We provide objects of the embedded generalized quadrangle of type B2.

α a |Bα|  r∗ 0 1  1 −r∗ + 2r∗ λ e q P 1 2 r1 r1 r∗ − 2r∗ λ e + λ e q2  1 2 r2 r2 r1+r2 r1+r2  ∗ 3 −r1 λr1 er1 + λr1+r2 er1+r2 + λ2r1+r2 e2r1+r2 q  r∗ 0 1  2  r∗ − r∗ λ e q L 1 2 r2 r2 −r∗ + r∗ λ e + λ e q2  1 2 r1 r1 2r1+r2 2r1+r2  ∗ 3 −r2 λr2 er2 + λr1+r2 er1+r2 + λ2r1+r2 e2r1+r2 q

Table: Objects in the embedded building of type B2 Andrew Woldar The underlying geometry of the graphs CD(k, q) group polygon

A2(q) triangle

B2(q),C2(q) quadrangle 2 A3(q) quadrangle 2 A4(q) quadrangle

G2(q) hexagon 3 D4(q) hexagon 2 2m+1 F4(2 ) octagon

Objects of type Ae1 the key to everything

Our interest in the rank 2 case stems from the fact that rank 2 buildings are examples of generalized polygons.

Andrew Woldar The underlying geometry of the graphs CD(k, q) A2(q) triangle

B2(q),C2(q) quadrangle 2 A3(q) quadrangle 2 A4(q) quadrangle

G2(q) hexagon 3 D4(q) hexagon 2 2m+1 F4(2 ) octagon

Objects of type Ae1 the key to everything

Our interest in the rank 2 case stems from the fact that rank 2 buildings are examples of generalized polygons.

group polygon

Andrew Woldar The underlying geometry of the graphs CD(k, q) B2(q),C2(q) quadrangle 2 A3(q) quadrangle 2 A4(q) quadrangle

G2(q) hexagon 3 D4(q) hexagon 2 2m+1 F4(2 ) octagon

Objects of type Ae1 the key to everything

Our interest in the rank 2 case stems from the fact that rank 2 buildings are examples of generalized polygons.

group polygon

A2(q) triangle

Andrew Woldar The underlying geometry of the graphs CD(k, q) G2(q) hexagon 3 D4(q) hexagon 2 2m+1 F4(2 ) octagon

Objects of type Ae1 the key to everything

Our interest in the rank 2 case stems from the fact that rank 2 buildings are examples of generalized polygons.

group polygon

A2(q) triangle

B2(q),C2(q) quadrangle 2 A3(q) quadrangle 2 A4(q) quadrangle

Andrew Woldar The underlying geometry of the graphs CD(k, q) 2 2m+1 F4(2 ) octagon

Objects of type Ae1 the key to everything

Our interest in the rank 2 case stems from the fact that rank 2 buildings are examples of generalized polygons.

group polygon

A2(q) triangle

B2(q),C2(q) quadrangle 2 A3(q) quadrangle 2 A4(q) quadrangle

G2(q) hexagon 3 D4(q) hexagon

Andrew Woldar The underlying geometry of the graphs CD(k, q) Objects of type Ae1 the key to everything

Our interest in the rank 2 case stems from the fact that rank 2 buildings are examples of generalized polygons.

group polygon

A2(q) triangle

B2(q),C2(q) quadrangle 2 A3(q) quadrangle 2 A4(q) quadrangle

G2(q) hexagon 3 D4(q) hexagon 2 2m+1 F4(2 ) octagon

Andrew Woldar The underlying geometry of the graphs CD(k, q) However, the supply is already exhausted:

A2 generalized triangle

B2 generalized quadrangle

G2 generalized hexagon

Objects of type Ae1 the key to everything

In our attempt to construct families whose behavior would resemble that of the balanced generalized polygons, we felt that Dynkin diagrams would provide the most promising pathway.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Objects of type Ae1 the key to everything

In our attempt to construct families whose behavior would resemble that of the balanced generalized polygons, we felt that Dynkin diagrams would provide the most promising pathway.

However, the supply is already exhausted:

A2 generalized triangle

B2 generalized quadrangle

G2 generalized hexagon

Andrew Woldar The underlying geometry of the graphs CD(k, q) Ae1 is an extended Dynkin diagram obtained by adjoining an imaginary root to the Dynkin diagram of type A1.

The Weyl group W (Ae1) is the inﬁnite dihedral group D∞.

The aﬃne root system Φ(Ae1) is inﬁnite.

Hence, the aﬃne Lie algebra L(Ae1) is inﬁnite-dimensional (although its Cartan subalgebra is 2-dimensional).

Objects of type Ae1 the key to everything

Or is it? ∞ Ae1

Andrew Woldar The underlying geometry of the graphs CD(k, q) The Weyl group W (Ae1) is the inﬁnite dihedral group D∞.

The aﬃne root system Φ(Ae1) is inﬁnite.

Hence, the aﬃne Lie algebra L(Ae1) is inﬁnite-dimensional (although its Cartan subalgebra is 2-dimensional).

Objects of type Ae1 the key to everything

Or is it? ∞ Ae1

Ae1 is an extended Dynkin diagram obtained by adjoining an imaginary root to the Dynkin diagram of type A1.

Andrew Woldar The underlying geometry of the graphs CD(k, q) The aﬃne root system Φ(Ae1) is inﬁnite.

Hence, the aﬃne Lie algebra L(Ae1) is inﬁnite-dimensional (although its Cartan subalgebra is 2-dimensional).

Objects of type Ae1 the key to everything

Or is it? ∞ Ae1

Ae1 is an extended Dynkin diagram obtained by adjoining an imaginary root to the Dynkin diagram of type A1.

The Weyl group W (Ae1) is the inﬁnite dihedral group D∞.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Hence, the aﬃne Lie algebra L(Ae1) is inﬁnite-dimensional (although its Cartan subalgebra is 2-dimensional).

Objects of type Ae1 the key to everything

Or is it? ∞ Ae1

Ae1 is an extended Dynkin diagram obtained by adjoining an imaginary root to the Dynkin diagram of type A1.

The Weyl group W (Ae1) is the inﬁnite dihedral group D∞.

The aﬃne root system Φ(Ae1) is inﬁnite.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Objects of type Ae1 the key to everything

Or is it? ∞ Ae1

Ae1 is an extended Dynkin diagram obtained by adjoining an imaginary root to the Dynkin diagram of type A1.

The Weyl group W (Ae1) is the inﬁnite dihedral group D∞.

The aﬃne root system Φ(Ae1) is inﬁnite.

Hence, the aﬃne Lie algebra L(Ae1) is inﬁnite-dimensional (although its Cartan subalgebra is 2-dimensional).

Andrew Woldar The underlying geometry of the graphs CD(k, q) 2 −2 2 −1 M = M = 1 −2 2 2 −4 2

We decided to work with M1.

From here, we generate the set Φ+ of positive roots:

r1, r2, r1 + r2, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,...... , ir1 + (i − 1)r2, (i − 1)r1 + ir2, ir1 + ir2,......

Objects of type Ae1 the key to everything

In fact, there are two nonisomorphic aﬃne Lie algebras of type Ae1, with respective Cartan matrices:

Andrew Woldar The underlying geometry of the graphs CD(k, q) We decided to work with M1.

From here, we generate the set Φ+ of positive roots:

r1, r2, r1 + r2, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,...... , ir1 + (i − 1)r2, (i − 1)r1 + ir2, ir1 + ir2,......

Objects of type Ae1 the key to everything

In fact, there are two nonisomorphic aﬃne Lie algebras of type Ae1, with respective Cartan matrices:

2 −2 2 −1 M = M = 1 −2 2 2 −4 2

Andrew Woldar The underlying geometry of the graphs CD(k, q) From here, we generate the set Φ+ of positive roots:

r1, r2, r1 + r2, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,...... , ir1 + (i − 1)r2, (i − 1)r1 + ir2, ir1 + ir2,......

Objects of type Ae1 the key to everything

In fact, there are two nonisomorphic aﬃne Lie algebras of type Ae1, with respective Cartan matrices:

2 −2 2 −1 M = M = 1 −2 2 2 −4 2

We decided to work with M1.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Objects of type Ae1 the key to everything

In fact, there are two nonisomorphic aﬃne Lie algebras of type Ae1, with respective Cartan matrices:

2 −2 2 −1 M = M = 1 −2 2 2 −4 2

We decided to work with M1.

From here, we generate the set Φ+ of positive roots:

r1, r2, r1 + r2, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,...... , ir1 + (i − 1)r2, (i − 1)r1 + ir2, ir1 + ir2,......

Andrew Woldar The underlying geometry of the graphs CD(k, q) In this case, we write L = he , e0 i. r ir1+ir2 ir1+ir2

This cannot occur in the (ﬁnite-dimensional) Lie algebras of groups of Lie type, where all root spaces Lr are 1-dimensional.

Personally, I believe this to be a plausible explanation as to why there do not exist generalized polygons of arbitrary even girth.

Objects of type Ae1 the key to everything

Note: For each |i| ≥ 2, the root space Lr where r = ir1 + ir2 is isotropic with respect to the Killing form, so is 2-dimensional.

Andrew Woldar The underlying geometry of the graphs CD(k, q) This cannot occur in the (ﬁnite-dimensional) Lie algebras of groups of Lie type, where all root spaces Lr are 1-dimensional.

Personally, I believe this to be a plausible explanation as to why there do not exist generalized polygons of arbitrary even girth.

Objects of type Ae1 the key to everything

Note: For each |i| ≥ 2, the root space Lr where r = ir1 + ir2 is isotropic with respect to the Killing form, so is 2-dimensional.

In this case, we write L = he , e0 i. r ir1+ir2 ir1+ir2

Andrew Woldar The underlying geometry of the graphs CD(k, q) Personally, I believe this to be a plausible explanation as to why there do not exist generalized polygons of arbitrary even girth.

Objects of type Ae1 the key to everything

Note: For each |i| ≥ 2, the root space Lr where r = ir1 + ir2 is isotropic with respect to the Killing form, so is 2-dimensional.

In this case, we write L = he , e0 i. r ir1+ir2 ir1+ir2

This cannot occur in the (ﬁnite-dimensional) Lie algebras of groups of Lie type, where all root spaces Lr are 1-dimensional.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Objects of type Ae1 the key to everything

Note: For each |i| ≥ 2, the root space Lr where r = ir1 + ir2 is isotropic with respect to the Killing form, so is 2-dimensional.

In this case, we write L = he , e0 i. r ir1+ir2 ir1+ir2

This cannot occur in the (ﬁnite-dimensional) Lie algebras of groups of Lie type, where all root spaces Lr are 1-dimensional.

Personally, I believe this to be a plausible explanation as to why there do not exist generalized polygons of arbitrary even girth.

Andrew Woldar The underlying geometry of the graphs CD(k, q) + Φ (Ae1) = {r1, r2, r1 + r2, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,...... }

+ Φ (A2) =

{r1, r2, r1 + r2}, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,......

+ Φ (B2) =

{r1, r2, r1 + r2, 2r1 + r2}, r1 + 2r2, 2r1 + 2r2,......

+ So, what happens if we truncate Φ (Ae1) at increasingly larger initial segments?

Objects of type Ae1 truncating the aﬃne root system

+ + + Observe that Φ (Ae1) contains Φ (A2) and Φ (B2) as initial segments:

Andrew Woldar The underlying geometry of the graphs CD(k, q) + Φ (A2) =

{r1, r2, r1 + r2}, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,......

+ Φ (B2) =

{r1, r2, r1 + r2, 2r1 + r2}, r1 + 2r2, 2r1 + 2r2,......

+ So, what happens if we truncate Φ (Ae1) at increasingly larger initial segments?

Objects of type Ae1 truncating the aﬃne root system

+ + + Observe that Φ (Ae1) contains Φ (A2) and Φ (B2) as initial segments:

+ Φ (Ae1) = {r1, r2, r1 + r2, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,...... }

Andrew Woldar The underlying geometry of the graphs CD(k, q) + Φ (B2) =

{r1, r2, r1 + r2, 2r1 + r2}, r1 + 2r2, 2r1 + 2r2,......

+ So, what happens if we truncate Φ (Ae1) at increasingly larger initial segments?

Objects of type Ae1 truncating the aﬃne root system

+ + + Observe that Φ (Ae1) contains Φ (A2) and Φ (B2) as initial segments:

+ Φ (Ae1) = {r1, r2, r1 + r2, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,...... }

+ Φ (A2) =

{r1, r2, r1 + r2}, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,......

Andrew Woldar The underlying geometry of the graphs CD(k, q) + So, what happens if we truncate Φ (Ae1) at increasingly larger initial segments?

Objects of type Ae1 truncating the aﬃne root system

+ + + Observe that Φ (Ae1) contains Φ (A2) and Φ (B2) as initial segments:

+ Φ (Ae1) = {r1, r2, r1 + r2, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,...... }

+ Φ (A2) =

{r1, r2, r1 + r2}, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,......

+ Φ (B2) =

{r1, r2, r1 + r2, 2r1 + r2}, r1 + 2r2, 2r1 + 2r2,......

Andrew Woldar The underlying geometry of the graphs CD(k, q) Objects of type Ae1 truncating the aﬃne root system

+ + + Observe that Φ (Ae1) contains Φ (A2) and Φ (B2) as initial segments:

+ Φ (Ae1) = {r1, r2, r1 + r2, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,...... }

+ Φ (A2) =

{r1, r2, r1 + r2}, 2r1 + r2, r1 + 2r2, 2r1 + 2r2,......

+ Φ (B2) =

{r1, r2, r1 + r2, 2r1 + r2}, r1 + 2r2, 2r1 + 2r2,......

+ So, what happens if we truncate Φ (Ae1) at increasingly larger initial segments?

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∗ + ∗ points : B ∗ = −r ⊕ L (−r ) −r2 2 2 ∗ + ∗ lines : B ∗ = −r ⊕ L (−r ) −r1 1 1

This eliminates dependence on the Weyl group.

Since the notion of expanse has already eliminated dependence on the Borel subgroup, our aﬃne truncated geometries now have a completely group-free formulation.

Objects of type Ae1 group-free formulation

Since we obtain the same asymptotics working with the largest Borel orbits of points and lines, we may restrict our truncated geometries to their aﬃne parts:

Andrew Woldar The underlying geometry of the graphs CD(k, q) ∗ + ∗ lines : B ∗ = −r ⊕ L (−r ) −r1 1 1

This eliminates dependence on the Weyl group.

Since the notion of expanse has already eliminated dependence on the Borel subgroup, our aﬃne truncated geometries now have a completely group-free formulation.

Objects of type Ae1 group-free formulation

Since we obtain the same asymptotics working with the largest Borel orbits of points and lines, we may restrict our truncated geometries to their aﬃne parts:

∗ + ∗ points : B ∗ = −r ⊕ L (−r ) −r2 2 2

Andrew Woldar The underlying geometry of the graphs CD(k, q) This eliminates dependence on the Weyl group.

Since the notion of expanse has already eliminated dependence on the Borel subgroup, our aﬃne truncated geometries now have a completely group-free formulation.

Objects of type Ae1 group-free formulation

Since we obtain the same asymptotics working with the largest Borel orbits of points and lines, we may restrict our truncated geometries to their aﬃne parts:

∗ + ∗ points : B ∗ = −r ⊕ L (−r ) −r2 2 2 ∗ + ∗ lines : B ∗ = −r ⊕ L (−r ) −r1 1 1

Andrew Woldar The underlying geometry of the graphs CD(k, q) Since the notion of expanse has already eliminated dependence on the Borel subgroup, our aﬃne truncated geometries now have a completely group-free formulation.

Objects of type Ae1 group-free formulation

Since we obtain the same asymptotics working with the largest Borel orbits of points and lines, we may restrict our truncated geometries to their aﬃne parts:

∗ + ∗ points : B ∗ = −r ⊕ L (−r ) −r2 2 2 ∗ + ∗ lines : B ∗ = −r ⊕ L (−r ) −r1 1 1

This eliminates dependence on the Weyl group.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Objects of type Ae1 group-free formulation

Since we obtain the same asymptotics working with the largest Borel orbits of points and lines, we may restrict our truncated geometries to their aﬃne parts:

∗ + ∗ points : B ∗ = −r ⊕ L (−r ) −r2 2 2 ∗ + ∗ lines : B ∗ = −r ⊕ L (−r ) −r1 1 1

This eliminates dependence on the Weyl group.

Since the notion of expanse has already eliminated dependence on the Borel subgroup, our aﬃne truncated geometries now have a completely group-free formulation.

Andrew Woldar The underlying geometry of the graphs CD(k, q) Truncating after the initial three positive root vectors gives the classical biaﬃne plane. After the initial four positive root vectors it gives the aﬃne part of the generalized quadrangle of type B2.

The aﬃne part of the generalized hexagon does not appear in our

series (perhaps due to the root space L2r1+2r2 being isotropic ??).

CD(k, q)

Graphs CD(k, q) arise as connected components of incidence graphs of aﬃne parts of truncated buildings of type Ae1.

Andrew Woldar The underlying geometry of the graphs CD(k, q) The aﬃne part of the generalized hexagon does not appear in our

series (perhaps due to the root space L2r1+2r2 being isotropic ??).

CD(k, q)

Graphs CD(k, q) arise as connected components of incidence graphs of aﬃne parts of truncated buildings of type Ae1.

Truncating after the initial three positive root vectors gives the classical biaﬃne plane. After the initial four positive root vectors it gives the aﬃne part of the generalized quadrangle of type B2.

Andrew Woldar The underlying geometry of the graphs CD(k, q) CD(k, q)

Graphs CD(k, q) arise as connected components of incidence graphs of aﬃne parts of truncated buildings of type Ae1.

The aﬃne part of the generalized hexagon does not appear in our

series (perhaps due to the root space L2r1+2r2 being isotropic ??).

Andrew Woldar The underlying geometry of the graphs CD(k, q) The End Thank you!

Vasya Ustimenko, Ivar Stakgold, Me, Felix, Joe Hemmeter (seated)

Andrew Woldar The underlying geometry of the graphs CD(k, q)