Schubert cells of mixed type in complex Lagrangian Grassmannians

We describe CW decompositions of complex Lagrangian Grassmannians, that contain as subcomplexes, CW decompositions of real Lagrangian Grassmannians by Schubert-Arnol'd cells. The degrees of attaching maps are explicitly computed in terms of quantities that can be read off from the corresponding shifted Young diagrams of mixed type. The signs are determined by a choice of lexicographical ordering on coordinates. As an immediate consequence, we obtain the homotopy extension property for real and complex Lagrangian Grassmannians. We also show some torsion classes in the integral homology of the real Lagrangian Grassmannian are contractible inside the complex Lagrangian Grassmannian.