A note on the Weitzenböck problem in dimension four

ABSTRACT

Let k be an algebraically closed field of characteristic two and let 𝔾_a be the additive group of k. Given any non-trivial representation ρ:𝔾_a→GL(4,k), we let 𝔾_a act linearly, via ρ, on the polynomial ring $k[x_1,x_2,x_3,x_4]$ in four variables over k. Then we prove that the invariant ring $k{[x_1,x_2,x_3,x_4]}^{{𝔾}_a}$ is isomorphic to the polynomial ring in three variables over k.

KEYWORDS:

• 𝔾_a-invariant rings
• locally finite iterative higher derivations
• representations of 𝔾_a in positive characteristic
• the Weitzenböck problem

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 13A50
• Secondary 14R20

Acknowledgment

The author would like to express his gratitude to the referee for comments.