## Journal of the Mathematical Society of Japan

### Configurations of seven lines on the real projective plane and the root system of type $E_{7}$

Jiro SEKIGUCHI

#### Abstract

Let $l_{1},$$l_{2},$$\ldots,$$l_{7}$ be mutually different seven lines on the real projective plane. We consider two conditions;(A) No three of $l_{1},$$l_{2},$$\ldots,$$l_{7}$ intersect at a point. (B) There is no conic tangent to any six of $l_{1},$$l_{2}$, . . . , $l_{7}$. Cummings [3] and White [16] showed that there are eleven non-equivalent classes of systems of seven lines with condition (A)(cf. [7], Chap. 18). The purposes of this article 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, it is better to add condition (B) for systems of seven lines. Moreover we need the notion of tetrahedral sets which consist of ten roots modulo slgns in the root system of type $E_{7}$ and which plays an important role in our study.

#### Article information

Source
J. Math. Soc. Japan, Volume 51, Number 4 (1999), 987-1013.

Dates
First available in Project Euclid: 10 June 2008

https://projecteuclid.org/euclid.jmsj/1213107831

Digital Object Identifier
doi:10.2969/jmsj/05140987

Mathematical Reviews number (MathSciNet)
MR1705257

Zentralblatt MATH identifier
0948.52010

#### Citation

SEKIGUCHI, Jiro. Configurations of seven lines on the real projective plane and the root system of type $E_{7}$. J. Math. Soc. Japan 51 (1999), no. 4, 987--1013. doi:10.2969/jmsj/05140987. https://projecteuclid.org/euclid.jmsj/1213107831