Congruence of Linear Symplectic Forms by the Symplectic Group

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 $tJ+A$.