Triality and Algebraic Groups of Type $^3D_4$

TL;DR

This paper classifies simple algebraic groups of type $^3D_4$ over various fields based on the existence and conjugacy of outer automorphisms of order 3, using trialitarian algebra representations.

Contribution

It provides a criterion for the existence of outer automorphisms of order 3 in $^3D_4$ groups and classifies these automorphisms via symmetric compositions.

Findings

Outer automorphisms of order 3 exist iff the algebra is an endomorphism algebra of an induced cyclic composition.

Conjugacy classes of automorphisms correspond to isomorphism classes of symmetric compositions.

The classification applies over arbitrary fields of characteristic not 2.

Abstract

We determine which simple algebraic groups of type $^3D_4$ over arbitrary fields of characteristic different from 2 admit outer automorphisms of order 3, and classify these automorphisms up to conjugation. The criterion is formulated in terms of a representation of the group by automorphisms of a trialitarian algebra: outer automorphisms of order 3 exist if and only if the algebra is the endomorphism algebra of an induced cyclic composition; their conjugacy classes are in one-to-one correspondence with isomorphism classes of symmetric compositions from which the induced cyclic composition stems.