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