Diagram Geometry

The fundamental structure in this book is a geometry. We look at a geometry as an incidence system: abstract objects that are related by means of incidence. The concept of a point lying on a line is carried over to the more abstract notion of an element of type point being incident to an element of type line. The basic ideas and definitions are given in this chapter. The usual related concepts like homomorphisms and subgeometries, and less general concepts such as connectedness and residues, are introduced. The important notion of residual connectedness is described in different but equivalent ways (Corollary 1.6.6).

A diagram is a structure defined on a set of types I. This structure generally is close to a labelled graph and provides information on the isomorphism class of residues of rank two of geometries over I. This way diagrams lead naturally to classification questions like all residually connected geometries pertaining to a given diagram.

The study of geometries can be developed starting from a different viewpoint than the diagram geometric one of the previous chapter. It corresponds to the structure induced on the set of maximal flags, also called chambers (cf. Definition 1.2.5), of a geometry. This slightly more abstract viewpoint has advantages for the study of thin geometries as well as group-related geometries.

As stated in Definition 1.2.7, a geometry is thin if each residue of rank one has exactly two elements. These geometries are firm in the most economical way. Residually connected examples are provided by the convex polyhedra in \(\mathbb{E}^{3}\) and the tessellations of \(\mathbb{E}^{2}\) and \(\mathbb{E}^{3}\); all (residues of) faces are polygons, all (residues of) edges are digons, and all vertex residues are again polygons (see Examples 1.1.3, 1.1.4, 1.1.5). In Sect. 4.1, we study these examples in the Euclidean space. Next we look at them intrinsically, i.e., without reference to an embedding in \(\mathbb{E}^{n}\). We are most interested in the regular examples, which are geometries of Coxeter type, the subject of Sect. 4.2. It turns out that these are all quotients of geometries corresponding to Coxeter groups, one for each Coxeter type. This leads to the study of Coxeter groups, which takes up Sects. 4.3 (where many groups generated by reflections are shown to be Coxeter groups), 4.4 (where Coxeter groups are shown to have a faithful linear representation as a group generated by reflections), 4.5 (where the centers of the perspectivities induced by these reflections are studied), and 4.6 and 4.7 (where the finite Coxeter groups are determined). Having gathered enough knowledge about Coxeter groups, we return to the regular polytopes in Sect. 4.8 and classify them by Coxeter types.

In Example 1.4.9 we introduced the projective geometry PG(V) and in Example 1.4.10 the affine geometry AG(V) associated with a vector space V of finite dimension n. In Proposition 2.4.7 the geometry PG(V) was shown to have a linear Coxeter diagram A _n−1, and in Proposition 2.4.10 the geometry AG(V) was shown to belong to the linear diagram Af _n . We now turn our attention to the more general class of all geometries with a linear diagram. The shadow spaces on 1 of our motivating examples PG(V) and AG(V) are linear line spaces (in the sense that any two points are on a unique line; cf. Definition 2.5.13), and we will restrict ourselves mostly to geometries with this property. Within this class there are combinatorial structures such as matroids and Steiner systems.

In Definitions 5.2.1 and 5.1.1, projective and affine spaces were introduced by means of axioms, and in Propositions 5.2.2 and 5.1.3, the spaces ℙ(V) and \(\mathbb {A}(V)\), where V is a vector space, were shown to be examples. In this chapter we show that by and large there are no further examples.

We come to one of the central themes in the theory of geometries of spherical Coxeter type. It can be described in various intertwining ways, such as: polarities in projective spaces and their absolutes, (both algebraic and geometric) quadrics in projective spaces, geometries belonging to a diagram of type B _n , or polar spaces: line spaces satisfying the property that, for each line of the space, each point is collinear with either one or all points of that line.

In Theorems 8.3.16 and 8.4.25, we saw that the most common nondegenerate polar spaces (cf. Definition 7.4.1) of rank at least three are embeddable in a projective space. So, in the continued study of a nondegenerate polar space Z of rank at least three, it is a mild restriction to assume that Z is embedded in a projective space ℙ. Still, the methods used in this chapter only require that the rank of Z be at least two.

In this chapter, we conclude the study of polar spaces. In Chap. 7 they made their appearance as line spaces connected with diagram geometries of type B _n . In Chap. 8 they were shown to embed in projective spaces under some mild conditions, like the rank n being at least three and every line being on at least three maximal singular subspaces. Grassmannians of lines of a thick projective space over a non-commutative division ring are examples of nondegenerate polar spaces of rank three that do not satisfy these conditions. In Chap. 9, the polar spaces of rank at least two embedded in a projective space were shown to be subspaces of absolutes of quasi-polarities of the ambient projective space. In this chapter, we completely determine these polar spaces. The main result is Theorem 10.3.13 and Sect. 10.3 is devoted to its proof. Proposition 10.3.11 points out which nondegenerate polar spaces amongst those embedded in absolutes of quasi-polarities on projective spaces are proper subspaces of the absolutes. The new examples are generalizations of quadrics, called pseudo-quadrics, which are introduced in Sect. 10.2. They are characterized as the minimal polar spaces embeddable in an absolute that are invariant under perspectivities. The same property was exploited successfully in the proof of Theorem 9.5.7 (via Proposition 9.5.5), where the ambient projective space is 3-dimensional. Table 10.1 of Remark 10.3.15 surveys the relations between polar spaces (of finite rank and embeddable in projective spaces) and polar geometries, similarly to Table 6.​1 for the projective case.

The most important geometries of this book are of Coxeter type (cf. Definition 2.4.2). Their ‘building blocks’, that is, their rank two residues, are the generalized polygons (cf. Definition 2.2.7), which are precisely the rank two geometries of Coxeter type. In Chap. 4, we studied thin chamber systems of Coxeter type M and found that these are quotients of the very nice and regular universal chamber system \(\mathcal{C}(M)\) for Coxeter systems of type M. In this chapter, we study special chamber systems of Coxeter type, called buildings, in which \(\mathcal{C}(M)\) frequently occurs as a subsystem, which is called apartment.