The current book contains all the material from the previous one, but we have added a lot. Most of the material in this book is due to Jacques Tits, the founder of building theory.
Introduction
- Coxeter Groups and Coxeter Complexes
- Buildings as Simplicial Complexes
- Buildings as W-Metric Spaces
- Buildings and Groups
- The Moufang Property and the Classification TheoremTheorem
- Euclidean Buildings
- Buildings as Metric Spaces
- Applications of Buildings
- A Guide for the Reader
Warning: Some mathematicians, following standard crystallographic notation, write Dm instead of D2m.] Infinite dihedral group. Each copy of the real line in the tree is an apartment, isomorphic to the Coxeter complex associated with the infinite dihedral set.
Finite Reflection Groups
Definitions
We will sometimes refer to the pair (W, V) as a finite reflection group when it is necessary to emphasize the vector space V on which W acts. For example, assume that dimV = 2 and that W is generated by two reflections :=sH and s := sH.
Examples
It is also the Weyl group in the root system Φ:={±α}, which is called the root system of type A1. And once again the W Weyl group is in a root system, called the root system of type Bn, consisting of the vectors ±ei±ej (i = j) together with the vectors ±ei.
Classification
Type G2: This is the Weyl group of the root system of the same name that we saw in Example 1.9. This is the symmetry group of a regular m-gon, but it does not correspond to any root system.
Cell Decomposition
- Cells
- Closed Cells and the Face Relation
- Panels and Walls
- Simplicial Cones
- A Condition for a Chamber to Be Simplicial
- Semigroup Structure
- Example: The Braid Arrangement
- Formal Properties of the Poset of Cells
- The Chamber Graph
We start by defining a partial order of the setΣ:=Σ(H) of cells, so thatΣ becomes an aposet (partially ordered set). If k is the length of the gallery, then exactly the k characters change, sok=l and the gallery is minimal.
The Simplicial Complex of a Reflection Group
- The Action of W on Σ(W, V )
- Examples
- The Chambers Are Simplicial
- The Coxeter Matrix
- The Coxeter Diagram
- Fundamental Domain and Stabilizers
- The Poset Σ as a Simplicial Complex
- Roots and Half-Spaces
1) The groups of fundamental reflections generate W. 2) The action of W is merely transitory in the C group of rooms. Let W be the reflection group of type H3, i.e., the group of symmetries of a regular dodecahedron in V :=R3.
Special Properties of Σ
- Σ Is a Flag Complex
- Σ Is a Colorable Chamber Complex
- Σ Is Determined by Its Chamber System
Note that this again proves that the length of a permutation is the number of inversions.] Deduce that the longest element w0 ∈ W (Section 1.5.2) has the characteristic property of taking every positive root to a negative root. It is immediately clear that the simplicial complex Σ associated with a finite reflection group is a chamber complex as defined in Section A.1.3. Recall from Section A.1.3 that the (essentially unique) type function on Σ is completely determined when we assign types to the nodes of the "fundamental" chamber C .
1.133. Using the classification of finite reflection groups (Section 1.3), find the involution σ0 for as many of them as you can.
Coxeter Groups
The Action on Roots
In the next chapter we will see that condition (A) is sufficient to allow us to construct a simple complex Σ = Σ(W, S) on which W acts, where the elements of T are reflections in a sense that will be specified. Then to each w∈W you can associate a finite subset T(w)⊆T with the following properties:. 2) For any reduced decomposition w = s1· · ·sl, the reflections ti defined in (2.1) are distinct and are precisely the elements of T(w). Parts (1) and (2) follow from (3) together with the following requirements: Given a reduced decomposition w=s1· · ·sl, the associated reflections are distinct.
This proof has the same amazing consequence that we observed for finite reflection groups in Corollary 1.70.
Examples
- Finite Reflection Groups
- The Infinite Dihedral Group
- The Group PGL 2 (Z)
The set T is now the set of reflections with respect to the walls of the chambers shown in the figure, and T×{±1} can be identified with the set of half-planes defined by these walls. The group of symmetries of the original tessellation is the group of hyperbolic isometries formed by reflections with respect to In this action, the SofW generative array is a set of reflections on three sides of one of the "chambers" C, as shown in Figure 2.5.
Note that the walls in C are precisely the fixed hyperplanes Hi of the reflections si.
Consequences of the Deletion Condition
- Equivalent Forms of (D)
- Parabolic Subgroups and Cosets
- The Word Problem
Applying (E) to s enwt, we conclude that l(swt) = d+ 2, otherwise we can replace one of the letters ins1 · · · sdt with ans in front. Sincel(t1w)< l(w), we can exchange one of the letters in this decomposition for a t1 in front of it. Note that the last statement of the theorem implies that WJ is finite if J is finite.
Therefore, in view of the folding condition (section 2.3.1), the hypothesis l(sws)< l(w) + 2 is equivalent to the equation sw=ws.
Coxeter Groups
The Canonical Linear Representation
- Construction of the Representation
- The Dual Representation
- Roots, Walls, and Chambers
- Finite Coxeter Groups
- Coxeter Groups and Geometry
- Applications of the Canonical Linear Representation
Then the elements of T on V act as orthogonal reflections with respect to the bilinear form B. From this discussion it is clear that there exists a bijection between T and the set of pairs±αof opposite roots, among which±αcorresponds to the reflectionsα∈T . We show here that if the Coxeter group W is finite, it is a finite reflection group in the sense of chapter 1.
But it turns out that the underlying field for a hyperbolic reflection array can be a more complicated polyhedron.
2.6 The Tits Cone
Cell Decomposition
Note that each cell is a polyhedral array of the type discussed above, defined by infinitely many linear equalities and inequalities. For each x, y ∈X, the line segment [x, y] traverses only finitely many walls and is contained in a finite union of cells. This is proved as in Section 1.4.6, the essential point being that a line segment in X passes only at the end of many walls.
In view of Lemma 2.85, Xf consists of the points x∈X, such that x is only contained in finitely many walls.
2.7 Infinite Hyperplane Arrangements
It is also wrong in the Euclidean case, where the closure of the Tits cone is a closed half-space. We used this second notion of "face" in the case of the breast cone. Thus, we can apply all the results we proved about Σ to convex subcomplexes.
We say disconvex if for all C, D∈ D, every minimal gallery inΣ from C to D is contained in D. Proposition 2.97. Let Σ be a subcomplex of Σ. Then the following two conditions are equivalent: ii)Σ is an intersection of subcomplexes of the form Σ±(H)(H ∈ H).
Coxeter Complexes
The Coxeter Complex
To show that Σ is simple, there are two things we need to verify (see Definition A.1):. a) All two elements A, B∈Σ have a greatest lower bound. We have already seen the canonical type function in Section 1.6.2 in the context of finite reflection groups, where we also saw examples illustrating it. Example 3.7. Let W be the group of isometries of the plane generated by the (affine) reflections with respect to the sides of an equilateral triangle.
Show that this is consistent with Definition 3.3. C, where C spans the chambers ≥A. a) Let C and D be chambers of Σ such that atd(C, D)≤d(C, D) for each chamber D adjacent to D. Show that Σ is spherical and that C and Dare opposite.
Local Properties of Coxeter Complexes
What the proposition says, then, is that lkA is the Coxeter complex associated with the matrix MJ obtained from M by choosing the rows and columns belonging to the cotype J of A. According to the above recipe, we must delete one vertex at a time from the Coxeter diagram of W. Translating this result to the connection of a simplexA of codimension 2 to an arbitrary Coxeter complex, we obtain a similar description of the minimal galleries on the substrates Σ≥A.
Since galleries correspond to words, we can use the word problem solution (Section 2.3.3) to analyze the general case.
Construction of Chamber Maps
- Generalities
- Automorphisms
- Construction of Foldings
Here AutΣ denotes the group of simplistic automorphisms ofΣ. Statement 3.32. The image of W →AutΣ is the normal subgroup Aut0Σ consisting of the type-preserving automorphisms of Σ. Statement 3.33. The homomorphism Aut(W, S) → AutΣ just defined is injective, and its image is the group Aut(Σ, C), consisting of the automorphisms of Σ that stabilize the fundamental chamber C = 1. The core of the proof is the following lemma, which constructs the φ-component of the desired φ (still assuming that C1 =C and C2=sC).
In the first case, w admits a reduced decomposition starting with s, so there exists a minimal gallery of the form C, sC,.
Roots
- Foldings
- Characterization of Coxeter Complexes
Lemma 3.45 therefore applies and provides a description of the two halvesα andαofΣdetermined byφ. Ifψ is a second fold withψ(C) =C, then we can similarly apply Lemma 3.45 to obtain the same description of the halves of Σdetermined by ψ. To this end, we assign types to the vertices of the fundamental room C by declaring that the panel fixed by the reflection s ∈ S is an s-panel.
Now we can use the same idea to prove the geometric analogue of the deletion condition.
The Weyl Distance Function
Let φ∗: S → S be the induced type change bijection (Proposition A.14), where Σ(W, S) gets its canonical type function. 3.6) To see that the right-hand side is independent of the choice of gallery, we can assume that Σ=Σ(W, S) with its canonical type function. Therefore, the right-hand side of (3.6) equals tow1−1w2, which is actually independent of the choice of gallery.
Note that pairs C, DwithC≥A,D≥B and δ(C, D) =δ(A, B) are exactly those pairs such that there is a minimal gallery from A to B of the shape.
Products and Convexity
- Sign Sequences
- Convex Sets of Chambers
- Supports
- Semigroup Structure
- Applications of Products
- Convex Subcomplexes
- The Support of a Vertex
- Links Revisited; Nested Roots
This is an immediate consequence of the fact that WA is created by the reflections that regulate xA, i.e., the reflectionsHwithA∈H; see exercise 3.62. In other words, there exists a simplex AB of cotype J1 such that CA,B = CAB, i.e., the rooms that can start a minimal gallery from A to B are precisely those that have AB as a face. In this case, as we shall see, convexity is closely related to the product in Σ. The following lemma gives the first hint of this.
This gallery is contained in Σ of convexity; therefore AC∈Σ. 2) All that needs to be proved is that Σ is a chamber subcomplex of Σ.
Buildings as Chamber Complexes
Definition and First Properties
The different τΣ thus fit together to give a type function τ defined on the union of the flats containing C . Similarly, we can talk about the Coxeter diagram of ∆; it is a graph with one vertex for each ∈ S. Strictly speaking, we should be talking about the Coxeter matrix and diagram of the pair (∆,A); but we will show in Section 4.4 that the matrix and diagram are really intrinsically associated with ∆ and do not depend on the system of apartmentsA.
The importance of the Coxeter matrix is, of course, that it completely determines the type of isomorphism of apartments.
Examples
In the language of incidence geometry, ∆ is the flag complex of a "plane" in which every point has collided with every line. The existence of many such apartments, as guaranteed by (B1), makes it plausible that ∆ is the flag complex of a projective plane. By this definition, it is indeed the case that our building ∆ is the flagship complex of a design plan.
If P is an n-dimensional polar space, then its flag complex is a rank-n construction of type Cn, i.e. with Coxeter diagram.
The Building Associated to a Vector Space
The goal of this exercise is to construct a generalized quadrilateral Q from V and −,−. The points of Q are defined as the nonzero vectors in V, and the lines of Q are the two-dimensional, totally isotropic subspaces of V. a) Show that there are 15 points, each incident on 3 lines, and 15 lines , each incident on 3 points. More generally, all four vectors inV with inner products like those of the basis vectors give rise to an octagon called ∆an apartment. Show that two vertices are opposite if and only if (i) they are non-collinear points of Q, or (ii) they are non-intersecting lines of Q. e) Show that∆5 contains vertices (but not 6) that are pairwise opposite.
