C-symplectic poset structure on a simply connected space

For a field k of characteristic zero, we introduce a cohomologically symplectic poset structure Pk(X) on a simply connected space X from the viewpoint of k-homotopy theory. It is given by the poset of inclusions of subgroups preserving c-symplectic structures in the group E(Xk) of k-homotopy classes of k-homotopy self-equivalences of X, which is defined by the k-Sullivan model of X. We observe that the height of the Hasse diagram of Pk(X) added by 1, denoted by c-s-depthk(X), is finite and often depends on the field k, which is a certain cyclotomic field. In this paper, we will give some examples of Pk(X).