Examples of Mironov cycles in Grassmannians

Providing some examples of Lagrangian cycles that arise as a generalization of Mironov's construction to the case of Grassmann manifolds $\operatorname{Gr}_{{\Bbb C}}(k, n+1)$, we show that these manifolds enjoy all data necessary for this generalization, the natural real structure, and an incomplete toric action. We also provide new concrete examples.