The differential invariants of $SL_2(\mathbb{F}_3)$ acting on trace-free matrices over $\mathbb{F}_3$

TL;DR

This paper computes the algebra of differential invariants under the conjugation action of SL_2(F_3) on trace-zero 2x2 matrices over F_3, using Cohen-Macaulay modules and covariant theory.

Contribution

It provides a minimal generating set for the algebra of invariants, advancing understanding of invariant theory for this specific group action.

Findings

Identified a minimal generating set for the algebra of invariants.

Applied Cohen-Macaulay module theory to invariant algebra computations.

Utilized covariant theory to facilitate the calculations.

Abstract

Let $M$ denote the vector space of $2 \times 2$ matrices with coefficients in $\mathbb{F}_3$ and trace zero. Let $G = SL_2(\mathbb{F}_3)$. Then $G$ acts on $M$ via conjugation. Let $R =(S(M^*) \otimes \Lambda(M^*))$ be the algebra of differential forms on $M$. We compute a minimal generating set for $R^G$ as a commutative-graded algebra. In doing so we utilise the theory of Cohen-Macaulay modules and results in the theory of covariants.