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A Relation between the Weyl Group W(e8) and Eight-Line Arrangements on a Real Projective Plane

The Weyl group W(E8) acts on the con guration space of systems of labelled eight lines on a real projective plane. With a system of eight lines with a certain condition, a diagram consisting of ten roots of the root system of type E8 is associated. We have already shown the existence of a W(E8)-equivariant map of the totality of such diagrams to the set of systems of labelled eight lines. The purpose of this paper is to report that the map is injective.