Configurations of seven lines on the real projective plane and the root system of type \(E_7\)
type: Item
instance of: scholarly article
title: Configurations of seven lines on the real projective plane and the root system of type \(E_7\)
zbMATH Open document ID: 0948.52010
author: Jiro Sekiguchi
published in: Journal of the Mathematical Society of Japan
publication date: 2000-11-12T00:00:00Z
review text: The configuration of seven lines on the projective plane are closely related with the classical topic on the twenty-eight bitangents of plane quartic curves. As it is well-known, the Weyl group of type \(E_7\) is regarded as the group of incidence-preserving automorphism of the twenty-eight bitangents. The author studies the case where the seven lines are defined over the real number field. Let \(l_1,l_2,\dots,l_7\) be mutually different seven lines on \(\mathbb P^2(\mathbb R).\) Cummings and White showed that there are eleven non-equivalent classes of systems of seven lines with condition: No three of \(l_1,l_2,\dots,l_7\) intersect at a point. The purpose of this paper is to give an interpretation of the classification of Cummings and White in terms of the root system of type \(E_7.\) To accomplish this, the following condition on these lines is added: There is no conic tangent to any six of \(l_1,l_2,\dots,l_7.\) Moreover, the notion of tetrahedral sets is necessary which consists of ten roots modulo signs in the root system of type \(E_7\) and which plays an important role in the study.
Mathematics Subject Classification ID: 52C30
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 51M30
zbMATH DE Number: 1398935
zbMATH Keywords: arrangements of lines
zbMATH Keywords: configuration space
zbMATH Keywords: real projective plane
zbMATH Keywords: root system of type \(E_7\)
zbMATH Keywords: pent-diagrams
zbMATH Keywords: tetradiagrams
zbMATH Keywords: twenty-eight bitangents of plane quartic curves
zbMATH Keywords: tetrahedral sets
zbMATH Keywords: Shrikhande graphs
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05140987
OpenAlex ID: W2080029747
DOI: 10.2969/JMSJ/05140987
description: scientific article; zbMATH DE number 1398935