The action of the Weyl group on the $E_8$ root system

Abstract

Let $\Gamma$ be the graph on the roots of the $E_8$ 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 $\Gamma_c$ be the subgraph of $\Gamma$ consisting of all the $240$ vertices, and all the edges whose color lies in $c$. We consider cliques, i.e., complete subgraphs, of $\Gamma$ that are either monochromatic, or of size at most $3$, or a maximal clique in $\Gamma_c$ for some color set $c$, or whose vertices are the vertices of a face of the $E_8$ root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $\Gamma$ 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 $\Gamma$, in terms of the restrictions of $f$ to certain special subgraphs of $K$ of size at most 7.