Chamber Systems
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 geo
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Chamber Systems
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. We begin by exploring the above mentioned structure: the chamber system of a geometry. Then we study chamber systems in their own right, coming across several notions that have already been introduced for geometries, like residues, residual connectedness, and diagrams. These observations lead to the idea that geometries could be derived from chamber systems. Indeed, the main result of the present chapter is Theorem 3.4.6, which gives a correspondence between residually connected chamber systems and residually connected geometries. Throughout this chapter, I is a set of types.
3.1 From a Geometry to a Chamber System We introduce the notion of chamber system over I and show that the set of chambers of a geometry over I has such a structure. The correspondence is not bijective: Example 3.1.4 shows that not all chamber systems come from geometries and Example 3.1.8 gives two non-isomorphic geometries with isomorphic chamber systems. For |I | = 2, a criterion for a chamber system over I to be the chamber system of a geometry is given in Theorem 3.1.14. Furthermore, we introduce notions resembling those for geometries and graphs, such as chamber subsystems, (weak) homomorphisms, and quotients. Let Γ be an incidence system over I . Definition 3.1.1 A chamber system over I is a pair C = (C, {∼i | i ∈ I }) consisting of a set C, whose members are called chambers, and a collection of equivalence relations ∼i on C indexed by i ∈ I . These relations are interpreted as graph structures F. Buekenhout, A.M. Cohen, Diagram Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 57, DOI 10.1007/978-3-642-34453-4_3, © Springer-Verlag Berlin Heidelberg 2013
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3 Chamber Systems
Fig. 3.1 The chamber system of the cube drawn on the cube
on C. For each i ∈ I , the graph (C, ∼i ) is a disjoint union of cliques since ∼i is an equivalence relation. Two chambers c, d are called i-adjacent if c ∼i d. For i ∈ I , each ∼i -equivalence class is called an i-panel. The rank of C is |I |. The chamber system C over I is called firm, thick, or thin, if, for each i ∈ I , every i-panel is of size at least two, at least three, or exactly two in the respective cases. Incidence systems give rise to chamber systems in the following fashion. Lemma 3.1.2 Let C be the set of chambers of the incidence system Γ over I and, for i ∈ I , define the relation ∼i on C by c ∼i d if and only if, for each j ∈ I \ {i}, they have exactly the same j -element. The resulting pair C(Γ ) = (C, {∼i | i ∈ I }) is a chamber system over I . If Γ is a firm, thick, or thin geometry, then C(Γ ) is firm, thick, or thin, r
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