f-Structures on the Classical Flag Manifold which Admit (1,2)-Symplectic Metrics

We characterize the f-structures F on the classical maximal flag manifold F(n) which admit (1,2)-symplectic metrics. This provides a sufficient condition for the existence of F-harmonic maps from any cosymplectic Riemannian manifold onto F(n). In the special case of almost-complex structures, our analysis extends and unifies two previous approaches: a paper of A.E.Brouwer 1980 on locally transitive digraphs, involving unpublished work by P.J. Cameron; and work by Mo, Paredes, Negreiros, Cohen and San Martin on cone-free digraphs. We also discuss the construction of (1,2)-symplectic metrics and calculate their dimension. Our approach is entirely graph theoretic.