(Co)isotropic triples and poset representations

Abstract

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".If one ignores the involution, representations of the poset "2 + 2 + 2" are essentially the same as representations of a certain quiver of extended Dynkin type Ẽ6 , and our work relies on the known classification of indecomposable representations of such quivers. In particular from these results it follows that the poset representations underlying (co)isotropic triples come in two types: there are families of indecomposables which depend on a parameter taking a continuum of values ("continuous-type"), and there are those indecomposables which are characterized only by the dimensions of the spaces involved ("discrete-type"). This pattern is reflected in the classification of (co)isotropic triples; in particular, there are families of triples which depend on a parameter. Also, indecomposable triples exist in every even dimension.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.