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

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.