Homogeneous coherent configurations as generalized groups and their relationship to buildings
In this remarkable paper, the author associates to each (either thick or thin) building, a certain generalized group, i.e., a partition \(G\) of \(C \times C\), where \(C\) is the set of chambers of the building, satisfying: the diagonal \(1_C\) belongs to \(G\); if \(f \in G\), then \(f^{-1} \in G\); and (the main axiom) if \(e, f, g \in G\), and \((y,z) \in g\), then the number \(a_{efg}\) of elements \(x\) of \(C\) such that \((y,x) \in e\) and \((x,z) \in f\) is independent of the choice of \(y\) and \(z\). The partition \(G\) is of course given by the Weyl distance in the building. A generalized group is thin if \(a_g g^{-1} 1_C = 1\), for all \(g \in G\). In that case, we obtain an ordinary group. If the building is thin, then by the above procedure, one obtains a Coxeter group. If the building is thin, then by the above procedure, one obtains a Coxeter group. Hence the generalized groups associated to buildings can be called generalized Coxeter groups. The procedure has also an inverse: to each generalized group and a subgroup, one can associate in two ways a chamber system. Applied to the generalized group associated to a building, and a certain subgroup generated by generalized involutions, one thus obtains the building and its Coxeter chamber system. The paper also contains a characterization of the generalized Coxeter groups associated to finite generalized polygons or to Moore geometries, amongst all generalized groups.
scientific article; zbMATH DE number 840725