The action of the Weyl group on the E_8 root system

Abstract

Let Γ be the graph on the roots of the E8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let 𝛤𝑐 be the subgraph of Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of Γ that are either monochromatic, or of size at most 3, or a maximal clique in Γc for some color set c, or whose vertices are the vertices of a face of the E8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of Γ, in terms of the restrictions of f to certain special subgraphs of K of size at most 7.