$f$-Structures on the classical flag manifold which admit (1,2)-symplectic metrics

Abstract

We characterize the invariant $f$-structures $\mathcal{F}$ on the classical maximal flag manifold $\mathbb F(n)$ which admit (1,2)-symplectic metrics. This provides a sufficient condition for the existence of $\mathcal{F}$-harmonic maps from any cosymplectic Riemannian manifold onto $\mathbb F(n)$. In the special case of almost complex structures, our analysis extends and unifies two previous approaches: a paper of Brouwer in 1980 on locally transitive digraphs, involving unpublished work by Cameron; and work by Mo, Paredes, Negreiros, Cohen and San Martin on cone-free digraphs.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 graph theoretic.