Homogeneous coherent configurations as generalized groups and their relationship to buildings
type: Item
instance of: scholarly article
title: Homogeneous coherent configurations as generalized groups and their relationship to buildings
zbMATH Open document ID: 0843.51017
author: Paul-Hermann Zieschang
published in: Journal of Algebra
publication date: 1996-08-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20E42
zbMATH DE Number: 840725
zbMATH Keywords: generalized groups
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Chamber systems
zbMATH Keywords: Moore geometries
zbMATH Keywords: polygons
zbMATH Keywords: buildings
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
full work available at URL: jabr.1995.1373
OpenAlex ID: W2074709101
DOI: 10.1006/JABR.1995.1373
description: scientific article; zbMATH DE number 840725