Eight Lines Arrangements on the Real Projective Plane and the Root System of Type $E$₈

Abstract

A classification of simple 2-arrangements of 6 lines and 7 lines is well-known (cf. B. Gr\"{u}nbaum: Convex Polytopes, Chap. 18). In the case of 6 lines, J. Sekiguchi and M. Yoshida (Kyushu J. Math. 51 (1997), 297-354) gave a parametrization of simple 2-arrangements of 6 lines in terms of the root system of type $E$₆. Furthermore, in the case of 7 lines, J. Sekiguchi showed an injective map of the totality of the so-called tetrahedral sets of the root system of type $E$₇ to certain families of simple 2-arrangements of 7 lines (cf. J. Sekiguchi: Configurations of seven lines on the real projective plane and the root system of type $E$₇, preprint).

To analyze a geometric feature of simple 2-arrangements of N lines, we propose an algorithm to count numbers of polygons included in any simple 2-arrangement. From an experimental calculation by using the algorithm, we obtain an interesting example of simple 2-arrangement of 8 lines which contains ten triangles but does not hexagons, heptagons, nor octagons.

By analyzing the W($E$₈)-orbital structure of the connected component containing this example, we discuss a relationship between simple 2-arrangements of 8 lines and the root system of type $E$₈. Partial results of our study are already published (cf. T. Fukui and J. Sekiguchi: A remark on labeled 8 lines on the real projective plane, Reports of Fac. Sci., Himeji Inst. Tech. 8 (1997), 1-11).