(Co)isotropic triples and poset representations

We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; particular, we classify indecomposable structures this kind. The classification depends on the ground field, which assume only to be perfect and not characteristic 2. Our work uses theory representations partially ordered sets with (order reversing) involution; for (co)isotropic triples, relevant poset is “2 + 2 2” consisting three independent pairs, involution exchanging members each pair. A key feature that any triple either “split” “non-split.” latter case when representation underlying an itself indecomposable. Otherwise, case, decomposable necessarily direct sum a dual pair representations; “symplectification.” In course paper develop framework “symplectic representations,” can applied range problems linear algebra. Hamiltonian fields, up conjugation, example; briefly explain connection between these triples.