AbstractLet $$\varGamma $$
Γ
be the graph on the roots of the $$E_8$$
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 $$\varGamma _c$$
Γ
c
be the subgraph of $$\varGamma $$
Γ
consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$
Γ
that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$
Γ
c
for some color set c, or whose vertices are the vertices of a face of the $$E_8$$
E
8
root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$
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 $$\varGamma $$
Γ
, in terms of the restrictions of f to certain special subgraphs of K of size at most 7.
Tetrahedron Equation, Weyl Group, and Quantum Dilogarithm
