On the First Fundamental Theorem for $\operatorname{GL}_2(K)$ and $\operatorname{SL}_2(K)$

Abstract

The First Fundamental Theorem of Invariant Theory describes a minimal generating set of the invariant polynomial ring under the action of some group G. In this note we give an elementary and direct proof for the GL 2 (K) and SL 2 (K) for any infinite field K. Our proof can be generalized to GL m (K) and SL m (K) for m > 2. Moreover, we present a family of counter-examples to the statements of the First Fundamental Theorems for all finite fields and m = 2.