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arXiv · 2610.20994

Automorphisms of twisted Chevalley groups of type ${}^2E_6$ over local rings

Elena I. Bunina·Deep H. Makadiya
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Abstract

We classify abstract automorphisms of the elementary and full twisted Chevalley groups of type ${}^2E_6$ over commutative local rings with involution and $2$ invertible, for both isogeny types. Every automorphism is standard, with no invertibility assumption on $3$. In the adjoint case, it has a unique expression as a composition of conjugation by an element of the full twisted group and a ring automorphism commuting with the involution. For the simply connected form, the full group equals its elementary subgroup, and the factors are strictly inner, diagonal, and ring automorphisms. The main step replaces a linear change of basis by a group element. Invariant alternating tensors determine the transported Chevalley bracket. We correct this bracket over a faithfully flat local extension and descend the resulting conjugating element by uniqueness. Root-subgroup relations then recover the coefficient automorphism.

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Elena I. Bunina, Deep H. Makadiya. 2026-10-14. Automorphisms of twisted Chevalley groups of type ${}^2E_6$ over local rings. https://arxiv.org/abs/2610.20994

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