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

ABSTRACT

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

KEYWORDS:

• Algebraic invariant function
• derivation
• linear representation
• normal forms
• symplectic group

AMS SUBJECT CLASSIFICATIONS:

• Primary: 15A21
• Secondary: 15A63
• 15A75
• 20G15

Acknowledgements

The authors wish to thank the unknown referee for his careful and thorough read of our manuscript and his valuable comments.

Notes

No potential conflict of interest was reported by the authors.