Diagram Geometry: Related to Classical Groups and Buildings by Arjeh M. Cohen, Francis Buekenhout

By Arjeh M. Cohen, Francis Buekenhout

This publication offers a self-contained advent to diagram geometry.  Tight connections with staff concept are proven. It treats skinny geometries (related to Coxeter teams) and thick structures from a diagrammatic standpoint. Projective and affine geometry are major examples.  Polar geometry is stimulated by way of polarities on diagram geometries and the entire type of these polar geometries whose projective planes are Desarguesian is given. It differs from Tits' accomplished remedy in that it makes use of Veldkamp's embeddings. The publication intends to be a uncomplicated reference in the event you examine diagram geometry.  crew theorists will locate examples of using diagram geometry.  mild on matroid thought is shed from the viewpoint of geometry with linear diagrams.  these attracted to Coxeter teams and people attracted to structures will locate short yet self-contained introductions into those subject matters from the diagrammatic perspective.  Graph theorists will locate many hugely common graphs. The textual content is written so graduate scholars can be in a position to stick to the arguments with no need recourse to extra literature. a robust element of the publication is the density of examples.  

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Extra resources for Diagram Geometry: Related to Classical Groups and Buildings (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

Example text

This geometry is also thin. The residue of a face is a triangle. 2. In each polyhedron or tessellation, the residue of an edge is a digon. Digons are very common indeed. 3 Suppose that F is a flag of Γ . (i) A subset G of XF is a flag of ΓF if and only if F ∪ G is a flag of Γ . (ii) If G is a flag of ΓF , then (ΓF )G = ΓF ∪G . (iii) If Γ is a geometry, then its residue ΓF is a geometry over I \ τ (F ). Proof (i) Let G be a subset of XF . Then G is a flag of ΓF if and only if G is a flag of Γ and G ⊆ F ∗ , which in turn is equivalent to G ∪ F being a flag of Γ .

Since obviously Gi Gj ∩ Gi Gk ⊇ Gi (Gj ∩ Gk ), we have established the ‘only if’ part of (ii). As for the converse, consider a flag X of Γ of type {i, j, k}. We will establish that X lies in the G-orbit of {Gi , Gj , Gk }, the standard flag of type {i, j, k}. In view of incidence transitivity, we may assume (without loss of generality) X = {x −1 Gi , Gj , Gk } for some x ∈ G. Thus, using incidences of the elements of this flag, we have x ∈ Gi Gj ∩ Gi Gk . So, if Gi Gj ∩ Gi Gk = Gi (Gj ∩ Gk ), we have x ∈ Gi x1 for some x1 ∈ Gj ∩ Gk , whence x −1 Gi , Gj , Gk = x1−1 Gi , x1−1 Gj , x1−1 Gk = x1−1 {Gi , Gj , Gk }.

For a, b ∈ G, we have aH ∼ bH if and only if H ∼ a −1 bH , so that it suffices to identify the neighbors of the vertex H . To this end write K = {g ∈ G | H ∼ gH }. Clearly, K = hK = Kh for every h ∈ H , so K is a union of double H -cosets. Hence a graph on which G acts transitively is determined by the knowledge of a stabilizer H and a union K of double cosets of H in G. As Δ is a graph, H ∼ rH implies rH ∼ H , whence, by application of φ(r −1 ), also H ∼ r −1 H . This shows that r −1 ∈ K, so K = K −1 .

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