A user friendly proof of Nagata's characterization of linearly reductive groups in positive characteristics

Abstract

Let K be an algebraically closed field of positive characteristic p, and G be a linear algebraic group over K. We give a user friendly proof of Nagata's theorem that every finite-dimensional rational representation of G is completely reducible if and only if the connected component G ⁰ is a torus and p does not divide the index (G : G ⁰).

Keywords:

• algebraic groups
• linearly reductive groups
• invariant theory

AMS Subject Classifications::

• 20G15
• 13A50