New geometries for finite groups and polytopes
type: Item
instance of: scholarly article
title: New geometries for finite groups and polytopes
zbMATH Open document ID: 0974.51011
author: Christiane Lefèvre-Percsy
author: Nicolas Percsy
author: Dimitri Leemans
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2001-12-16T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1103055618
review text: In this rather interesting paper, the authors present two fairly general ways to create a new diagram geometry out of an old one, and they compare some connectivity properties of the old and new geometries. The first construction deals with geometries having a graph as some point-line truncation and having some additional properties such as (1) in the diagram the node corresponding to the points is only joined to the one corresponding to the lines, (2) if the two vertices (points) of an edge (line) are incident with the same element of type distinct from the type of lines, then that edge (as a line) is incident with that element. Then one replaces the set of lines (edges) by a copy of the points (vertices) and declares two points of different type to be incident when they are adjacent (hence distinct) in the graph. The rest of the new incidence relation is natural. This trick can be applied to a number of geometries with non-thick residues (ordinary \(n\)-gons and Petersen graphs, for instance). Applied to graphs themselves, one obtains self polar thick geometries, but not always partial linear spaces because the graph might contain 4-cycles. In fact, this construction goes both ways for self-polar geometries admitting a polarity without absolute points. Examples are the geometries obtained by deleting the absolute elements of a polarity of any rank 2 geometry (this way e.g. Desarguesian affine planes of even order minus a line pencil arise), or subgeometries thereof (the geometry induced by all exterior (interior) points and lines of a conic in Desarguesian projective planes of odd order; taking the interior ones in a plane of order 5, the Desargues configuration arising from the Petersen graph shows up as a special case -- it is mentioned as a sporadic example in the paper under review). The second construction is the generalization (or rather just putting in general context) of the reversed procedure related to the so-called oriflamme complex (producing a thick b…
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20D99
zbMATH DE Number: 1563228
zbMATH Keywords: incidence geometry
zbMATH Keywords: polytopes
zbMATH Keywords: thin geometries
zbMATH Keywords: diagram geometry
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
Recommended article: On geometric polygroups
Recommended article: Q3361489
Recommended article: Q3722701
Recommended article: Q4146188
Recommended article: Groups and polytopes
Recommended article: Q4704958
Recommended article: Incidence geometries for the finite symmetric groups
Recommended article: Q4867058
Recommended article: Q3824572
description: scientific article; zbMATH DE number 1563228