Invariants of $G_2$ and $Spin(7)$ in positive characteristic

TL;DR

This paper extends the understanding of invariants of the groups G_2 and Spin(7) acting on octonions from characteristic zero fields to arbitrary infinite fields of odd characteristic, showing the invariants are generated by the same low-degree invariants.

Contribution

It generalizes known invariant generation results from characteristic zero to fields of odd positive characteristic for G_2 and Spin(7) acting on octonions.

Findings

Invariants are generated by invariants of degree at most 4.

The same generating invariants hold over fields of odd characteristic.

Extension from characteristic zero to positive odd characteristic fields.

Abstract

Invariants of $G_2$ and $Spin(7)$, both acting on several copies of octonions, have been decribed in \cite{schw2} over a ground field of characteristic zero. In the current manuscript, we extend this result to an arbitrary infinite field of odd characteristic. More precisely, we prove that the corresponding algebras of invariants are generated by the same invariants of degree at most $4$ as in the case of a field of characteristic zero.