Geodesic
An extension of the (1,2)-symplectic property for $f$-structures on flag manifolds
Izvestiya. Mathematics , Tome 72 (2008) no. 3, pp. 479-496

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The (1,1)-symplectic property for $f$-structures on a complex Riemannian manifold $M$ is a natural extension of the (1,2)-symplectic property for almost-complex structures on $M$, and arises in the analysis of complex harmonic maps with values in $M$. A characterization of this property in combinatorial terms is known only for almost-complex structures or when $M$ is the classical flag manifold $\mathbb{F}(n)$. In this paper, we remove these restrictions by considering an intersection graph defined in terms of the corresponding root system. We prove that the $f$-structure is (1,1)-symplectic exactly when the intersection graph is locally transitive. Our intersection graph construction may be helpful in characterizing many other Kähler-like properties on complex flag manifolds. 
@article{IM2_2008_72_3_a2,
     author = {N. Cohen and S. Pinzon},
     title = {An extension of the (1,2)-symplectic property for $f$-structures on flag manifolds},
     journal = {Izvestiya. Mathematics },
     pages = {479--496},
     publisher = {mathdoc},
     volume = {72},
     number = {3},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_3_a2/}
}
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N. Cohen; S. Pinzon. An extension of the (1,2)-symplectic property for $f$-structures on flag manifolds. Izvestiya. Mathematics , Tome 72 (2008) no. 3, pp. 479-496. http://geodesic.mathdoc.fr/item/IM2_2008_72_3_a2/