$\left( 1,2\right)$-Symplectic metrics on flag manifolds and tournaments

It has been recently shown by Mo and Negreiros \cite{mn2} that a necessary condition for an invariant almost-complex structure on the complex full flag manifold $\Bbb{F}\left( n\right) $ to admit a $\left( 1,2\right) $-symplectic invariant metric is that its associated ournament is cone-free.In this paper we find a canonical stair-shaped form for such tournaments and apply it to show that the condition is also sufficient. In doing this we describe all the associated $\left(1,2\right) $-symplectic metrics, and get, in particular, a different and self-contained proof of a theorem of Gray and Wolf \cite{gw} asserting that the Cartan-Killing metric on $\Bbb{F}\left( n\right)$ is not $\left( 1,2\right) $-symplectic for $n>3$.