The Invariant Fields of the Sylow groups of Classical Groups in the natural characteristic

TL;DR

This paper determines the fields of rational invariants for Sylow p-subgroups of finite classical groups over fields of characteristic p, showing they are generated by orbit products and Steenrod-acted invariants.

Contribution

It explicitly describes the invariant fields for Sylow p-subgroups of classical groups, linking them to orbit products and Steenrod operations, a novel characterization in this context.

Findings

Invariant fields are generated by orbit products and Steenrod images.

Explicit descriptions of invariant fields for Sylow p-subgroups.

Connection between invariants and Steenrod operations in classical groups.

Abstract

Let X be any finite classical group defined over a finite field of characteristic p>0. In this paper we determine the fields of rational invariants for the Sylow p-subgroups of X, acting on the natural module. In particular we prove that these fields are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant linear forms defining X.