Chamber systems with string diagrams (Q1059855)

There are two (canonical) constructions to get a Tits geometry \(\Gamma\) (\({\mathcal C})\) out of a chamber system \({\mathcal C}\) (both over the same set I) and, vice-versa, a chamber system \({\mathcal C}(\Gamma)\) from a Tits geometry \(\Gamma\). These two constructions are, in general, not inverse to each other, i.e. \({\mathcal C}(\Gamma ({\mathcal C}))\) and \({\mathcal C}\) need not be canonically isomorphic, nor do \(\Gamma\) (\({\mathcal C}(\Gamma))\) and \(\Gamma\). The authors give a sufficient condition in order to which the previous isomorphisms occur and, also, the diagrams of \({\mathcal C}\) and \(\Gamma\) (\({\mathcal C})\) coincide. Moreover, under this condition each flag of \(\Gamma\) (\({\mathcal C})\) is contained in a flag of type I and, if G is a group of automorphisms acting transitively on \({\mathcal C}\), then G is flag- transitive on \(\Gamma\) (\({\mathcal C}).\) At the end of the article the authors give some examples showing how the above results may be used.