Invariants of Finite Orthogonal Groups of Plus Type in Odd Characteristic

TL;DR

This paper characterizes the rings of invariants for finite orthogonal groups of plus type in odd characteristic, providing explicit generators, relations, and structural properties, with potential for broader application to classical groups.

Contribution

It offers explicit descriptions of invariant rings, minimal generators, relations, and structural properties for these groups, advancing systematic invariant computation methods.

Findings

Rings of invariants are complete intersections and Cohen-Macaulay.

Constructed minimal algebra generating sets for the invariants.

Described invariants for Sylow subgroups in the defining characteristic.

Abstract

We describe the rings of invariants for the finite orthogonal groups of plus type in odd characteristic acting on the defining representations. We also describe the invariants of the corresponding Sylow subgroups in the defining characteristic. In both cases we construct minimal algebra generating sets and describe the relations among the generators. Both rings of invariants are shown to be complete intersections and thus are Cohen-Macaulay. We expect the techniques we use will generalise to give a systematic computation for rings of invariants for all of the finite classical groups in odd characteristic.