On the First Fundamental Theorem for GL₂(K) and SL₂(K)

2020/09/18 by Hana Melánová, Hana Melanova, Sergey Yurkevich +2
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Finite Group Theory Research #math.AC #math.RA #msc:13A50

paper · pdf · doi:10.48550/arxiv.2009.08904

arxiv created 2020/10/27 · arxiv updated 2020/10/28

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 GL₂(K) and SL₂(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.

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