Congruence of Linear Symplectic Forms by the Symplectic Group

TL;DR

This paper classifies linear symplectic forms under symplectomorphisms in even dimensions, providing invariants and orbit descriptions, especially in dimension four, with implications for symplectic geometry and stability theorems.

Contribution

It introduces new invariants for classifying symplectic forms and describes the orbit space under symplectomorphisms, extending results to higher dimensions.

Findings

Complete description of orbit space in dimension four.

Identification of invariants like pfaffian and trace-based functions.

Extension of classification results to higher even dimensions.

Abstract

This paper concerns the action of linear symplectomorphisms on linear symplectic forms by conjugation in even dimensions. We prove that pfaffian and $-\frac{1}{2}\operatorname{tr}(JA)$ (sum function) of $A$ are invariants on the action. We use these invariants to provide a complete description of the orbit space in dimension four. In addition, we investigate the geometric shapes of the individual orbits in dimension four. In symplectic geometry, our classification result in dimension four provides a necessary condition for two symplectic forms on $\mathbb{R}^{4}$ to be intertwined by symplectomorphisms of the standard symplectic form. This stands in contrast to the lack of local invariants under diffeomorphisms. Furthermore, we determine global invariants of a class of symplectic forms, and we study an extension of a corollary of the Curry-Pelayo-Tang Stability Theorem. Lastly, we extend our results and investigate the action of linear symplectomorphisms on linear symplectic forms in dimension $2n$. We determine $n$ invariants of linear symplectic forms under this action, namely, $s_k(A)$ we defined as $\sigma_k(A)$ which is the coefficient of term $t^k$ in the polynomial expansion of pfaffian of $tJ+A$.