Full automorphism groups of Γ(E8, k) for 4 ≤ k ≤ 7

Determine the full automorphism group Aut(Γ(E8, k)) for k ∈ {4, 5, 6, 7}, where Γ(E8, k) has vertices that are sums of k-element strongly orthogonal subsets of E8 roots and edges when vertex differences are also such sums. Current results certify only that the Weyl group W(E8) acts by automorphisms (W(E8) ≤ Aut(Γ(E8, k))), but the complete automorphism groups have not been identified.

Background

Lemma 3.1 proves that the Weyl group W(R) acts by automorphisms on Γ(R, k). Using computational tools, the authors determined Aut(Γ(R, k)) equals W(R) (or W(R):2) in many cases.

However, for the largest E8 graphs listed in the table (k = 4, 5, 6, 7), the table marks only W(E8) ≤ Aut, indicating that while the Weyl group is contained in the automorphism group, the full automorphism group has not been determined.