On a problem on chamber systems (Q1912363)

This paper establishes the existence of irreducible simply-connected finite chamber systems with trivial diagram. This is in strong contrast with the complete reducibility result for chamber systems of geometries. The author proves that a chamber system with trivial diagram is completely reducible if and only if its cell geometry is the dual of a thin-lined polar space. He then uses this result to establish the existence of an irreducible simply-connected finite chamber system with trivial diagram. He notes that his example is not transitive; so the question of complete reducibility remains open in the transitive case.