Pseudotoric structures and special Lagrangian torus fibrations on certain flag varieties

We construct pseudotoric structures (`{a} la Tyurin) on the two-step flag variety $F{\mathrm{\ell }}_{1,n-1;n}$, and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important in the study of SYZ mirror symmetry. As an application, we speculate how our constructions can explain the number of terms in the superpotential of Rietsch’s Landau-Ginzburg mirror.