论文标题
在第一个基本定理上，$ \ operatatorName {gl} _2（k）$和$ \ operatatorName {sl} _2（k）$
On the First Fundamental Theorem for $\operatorname{GL}_2(K)$ and $\operatorname{SL}_2(K)$
论文作者
论文摘要
不变理论的第一个基本定理描述了在某些$ g $的作用下，不变多项式环的生成集。在本说明中，我们为$ \ operatatorName {gl} _2（k）$和$ \ operatatorName {sl} _2（k）$提供了基本和直接的证明。我们的证明可以推广到$ \ operatatorName {gl} _m（k）$和$ \ operatatorName {sl} _m（k）$ for $ m> 2 $。此外，我们向所有有限字段的第一个基本定理和$ m = 2 $的陈述介绍了一个反规定的家族。
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 $m=2$.
