A new look at the classiffication of the tri-covectors of a 6-dimensional symplectic space

TL;DR

This paper classifies the polynomial invariants of tri-covectors in a 6-dimensional symplectic space, providing explicit generators and applications to normal forms in symplectic geometry.

Contribution

It explicitly determines generators for polynomial invariants under symplectic group action on tri-covectors in a 6D space, advancing understanding of symplectic invariants.

Findings

Explicit generators for polynomial invariants are provided.

Applications to normal forms of specific symplectic tensors are demonstrated.

Enhances classification methods in symplectic geometry.

Abstract

Let $\mathbb{F}$ be a field of characteristic $\neq 2$ and $3$, let $V$ be a $\mathbb{F}$-vector space of dimension $6$, and let $\Omega \in \wedge ^2V^\ast $ be a non-degenerate form. A system of generators for polynomial invariant functions under the tensorial action of the group $Sp(\Omega )$ on $\wedge ^3 V^\ast $, is given explicitly. Applications of these results to the normal forms of De Bruyn-Kwiatkowski and Popov are given.