Some four-dimensional orthogonal invariants

TL;DR

The paper computes explicit generators for the invariant rings $\mathbb{F}_p[V\oplus V^*]^G$ where $G$ is a 2-dimensional orthogonal group over the odd prime field $\mathbb{F}_p$. It treats $O_2^+(\mathbb{F}_p)$ and $O_2^-(\mathbb{F}_p)$ separately, deriving generators for $O_2^+$ via the subgroup $SO_2^+$ and a relative Reynolds operator, and, for $O_2^-$ with $p$ odd, employing Hilbert-series analysis, $s$-invariants, and the Jacobian criterion to produce a free basis and explicit generators. The $O_2^-$ case exhibits a $p$-dependent structure (notably a parameter $\lambda$ with $\lambda=-1$ when $p\equiv 3\pmod{4}$), and the Jacobian-based approach circumvents the need to compute relations. Overall, the work delivers concrete, computable descriptions of 4-dimensional modular invariant rings, advancing covariant methods in modular invariant theory and offering tools for applications in topology and commutative algebra.

Abstract

Let $p$ be an odd prime and $\mathbb{F}_p$ be the prime field of order $p$. Consider a $2$-dimensional orthogonal group $G$ over $\mathbb{F}_p$ acting on the standard representation $V$ and the dual space $V^*$. We compute the invariant ring $\mathbb{F}_p[V\oplus V^*]^G$ via explicitly exhibiting a minimal generating set. Our method finds an application of $s$-invariants appeared in covariant theory of finite groups.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Example 1.4
• proof : Proof of Theorem \ref{['thm1']}
• proof : Proof of Theorem \ref{['thm2']}
• Lemma 3.1
• proof
• Theorem 3.2
• proof