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arXiv · 2610.30314

Automorphisms of Twisted Chevalley Groups over Commutative Rings

Elena I. Bunina·Deep H. Makadiya
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Abstract

We classify all abstract automorphisms of the elementary and full twisted Chevalley groups of types ${}^2A_\ell$ ($\ell\geq3$), ${}^2D_\ell$ ($\ell\geq4$), ${}^2E_6$, and ${}^3D_4$ over arbitrary commutative rings, for every invariant isogeny type. The coefficient automorphism $θ$ has order dividing that of the diagram automorphism. We assume that $2$ is invertible and, for triality, that $3$ is invertible as well. Three features distinguish the answer from the usual inner--diagonal and coefficient decomposition. The elementary groups of types ${}^2A_3$ and ${}^2A_4$ admit an additive family of exceptional automorphisms supported at $θ$-fixed maximal ideals with residue field $\mathbb F_3$. In type ${}^2D_4$, diagram automorphisms may vary on idempotent components, subject to compatibility with the involution and the isogeny lattice. In triality, the coefficient automorphisms preserve the trace operator $r\mapsto r+θ(r)+θ^2(r)$ but need not commute with $θ$. We determine which elementary automorphisms extend to the full groups and describe the remaining central factors. A nonzero exceptional unitary factor extends precisely in the simply connected type ${}^2A_3$, giving a classification for the hyperbolic group $\operatorname{SU}_4(R,θ;Q_4)$. The proof glues uniquely determined normalization data for a finite root frame and then recovers the coefficient action from root-subgroup relations.

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Elena I. Bunina, Deep H. Makadiya. 2026-10-21. Automorphisms of Twisted Chevalley Groups over Commutative Rings. https://arxiv.org/abs/2610.30314

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