Outer automorphisms of classical algebraic groups

TL;DR

This paper investigates the existence of outer automorphisms in classical algebraic groups, revealing cases where the Tits class obstruction is not the only barrier, especially for certain outer types.

Contribution

It proves new descent theorems for specific algebras and provides explicit examples showing the Tits class obstruction's limitations in classical groups of outer type.

Findings

Outer automorphisms exist beyond Tits class obstructions in certain classical groups.

Descent theorem established for degree 6 algebras with involution.

Explicit examples where Tits class vanishes but no outer automorphism exists.

Abstract

The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one. In this paper, we prove this is not the case for classical groups of outer type, except for groups of type $^2\mathsf{A}_n$ with $n$ even, or $n=5$. More precisely, we prove a descent theorem for exponent $2$ and degree $6$ algebras with unitary involution, which shows that their automorphism groups have outer automorphisms. In all other relevant classical types, namely $^2\mathsf{A}_n$ with $n$ odd, $n\geq3$ and $^2\mathsf{D}_n$, we provide explicit examples where the Tits class obstruction vanishes, and yet the group does not have outer automorphism. As a crucial tool, we use "generic" sums of algebras with involution.