On the cube problem of Las Vergnas
type: Item
instance of: scholarly article
title: On the cube problem of Las Vergnas
zbMATH Open document ID: 0859.52003
DOI: 10.1007/BF00181184
author: Jürgen Bokowski
author: António Guedes de Oliveira
author: Ulrich Thiemann
author: António Veloso da Costa
published in: Geometriae Dedicata
publication date: 1997-03-25T00:00:00Z
review text: Associated with the \(d\)-dimensional (unit) cube one has its matroid \(M^d\) of affine dependencies (of rank \(d+1\), on \(2^d\) elements). As proved here, the symmetry group of \(M^d\) is the Coxeter group \(BC_{d+1}\) given by the symmetries of the \((d+1)\)-cube, modulo its 2-element center. This symmetry information is used to prove for \(d\leq 7\) a conjecture of Las Vergnas: the reorientation class of \(M_d\) is unique. The second component used for this is that the labelled contractions of rank 3 determine every oriented matroid of rank \(r\geq 3\) uniquely; thus it suffices to show that all rank 3 contractions of \(M_d\) have unique reorientation class. For \(d=7\) this is established using computer support, where the knowledge of the symmetry group is used to reduce the number of cases.
Mathematics Subject Classification ID: 52B40
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 942738
zbMATH Keywords: matroid of the cubes
zbMATH Keywords: orientations
zbMATH Keywords: reorientation class
reviewed by: Günter M. Ziegler
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181184
OpenAlex ID: W2031925074
description: scientific article; zbMATH DE number 942738