Tournaments and parabolic almost complex structures on flag manifolds

Using tournaments, we study parabolic almost complex structures on the classical flag manifold. The correspondence of Burstall and Salamon between tournaments and invariant almost complex structures on flag manifolds provides us a necessary condition for an almost complex structure to be parabolic. In addition, we give a new proof of the theorem, due to Mo and Negreiros, which shows that all of the parabolic almost complex structures admit (1, 2)-symplectic metrics.