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

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 ${\mathrm{GL}}_2(K)$ and ${\mathrm{SL}}_2(K)$ for any infinite field $K$. Our proof can be generalized to ${\mathrm{GL}}_m(K)$ and ${\mathrm{SL}}_m(K)$ for $m>2$. Moreover, we present a family of counter-examples to the statements of the First Fundamental Theorems for all finite fields and $m=2$.