Invariants of the finite orthogonal groups in odd dimension and even characteristic

TL;DR

This paper characterizes the invariant ring of finite orthogonal groups in odd dimensions over fields of even characteristic, providing explicit generators, relations, and structural properties like being a complete intersection.

Contribution

It extends previous work by explicitly describing the invariants for these groups, including generators, relations, and Cohen-Macaulay properties, in a new characteristic setting.

Findings

The invariant ring is a complete intersection.

A minimal generating set for the invariants is constructed.

The ring is Cohen-Macaulay.

Abstract

We describe the ring of invariants for the finite orthogonal groups in odd dimension and even characteristic acting on the defining representation. We construct a minimal algebra generating set and describe the relations among the generators. This ring of invariants is shown to be a complete intersection and thus is Cohen-Macaulay. This extends the previous computation of Kropholler, Mohseni Rajaei, and Segal valid over the field of order 2.