Nikolaus Conference 2025

Any diagram automorphism of a finite root system \(\Phi\) gives rise to a root system \(\Phi'\) of lower rank than \(\Phi\) together with a natural embedding of Weyl\((\Phi')\) into Weyl\((\Phi)\). Further, any Chevalley group of type \(\Phi\) contains so-called twisted Chevalley groups of type \(\Phi'\). In this talk, we present a more general construction to obtain a root group grading \((U_\alpha)_{\alpha \in H_3}\) of type \(H_3\) inside any Chevalley group of type \(D_6\) by way of the folding \(D_6 \to H_3\). This folding does not correspond to a diagram automorphism of \(D_6\). Our main result is that every \(H_3\)-graded group arises as a folding of a \(D_6\)-graded group in this way. Similar results hold for the folding \(E_8 \to H_4\). This is joint work with Lennart Berg.