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

TL;DR

This paper provides an elementary proof of the First Fundamental Theorem for the invariant rings of $ ext{GL}_2(K)$ and $ ext{SL}_2(K)$ over infinite fields, and discusses counterexamples over finite fields.

Contribution

It offers a direct proof for the theorem in the case of $m=2$ over infinite fields and extends the discussion to general $m$, including counterexamples for finite fields.

Findings

Elementary proof for $ ext{GL}_2(K)$ and $ ext{SL}_2(K)$ over infinite fields

Counterexamples for the theorem over finite fields when $m=2$

Generalization potential to $ ext{GL}_m(K)$ and $ ext{SL}_m(K)$ for $m>2$

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