Journal of the Mathematical Society of Japan

Configurations of seven lines on the real projective plane and the root system of type E7

Jiro SEKIGUCHI

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Abstract

Let l1,l2,...,l7 be mutually different seven lines on the real projective plane. We consider two conditions;(A) No three of l1,l2,...,l7 intersect at a point. (B) There is no conic tangent to any six of l1,l2, . . . , l7. 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 E7. 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 E7 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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1213107831

Digital Object Identifier
doi:10.2969/jmsj/05140987

Mathematical Reviews number (MathSciNet)
MR1705257

Zentralblatt MATH identifier
0948.52010

Subjects
Primary: 52B30
Secondary: 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55] 51M30: Line geometries and their generalizations [See also 53A25]

Keywords
Arrangements of lines configuration spaces the real projective plane root systems Tetradiagrams

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


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