Examples of Hamiltonian-minimal Lagrangian submanifolds in $\operatorname{Gr}(r, n)$

We extend the A. E. Mironov's construction of (Hamiltonian)-minimal Lagrangian submanifolds to the case of an algebraic manifold that can be equipped with a Kahler-Einstein metric symmetric under the action of the torus $T^k$. As an application, we give examples of Hamiltonian-minimal Lagrangian submanifolds of the Grassmannian $\operatorname{Gr}(r, n)$.