@article{16633, abstract = {{AbstractWe obtain normal forms for infinitesimally symplectic matrices (or linear Hamiltonian vector fields) that commute with the symplectic action of a compact Lie group of symmetries. In doing so we extend Williamson's theorem on normal forms when there is no symmetry present.Using standard representation-theoretic results the symmetry can be factored out and we reduce to finding normal forms over a real division ring. There are three real division rings consisting of the real, complex and quaternionic numbers. Of these, only the real case is covered in Williamson's original work.}}, author = {{Melbourne, Ian and Dellnitz, Michael}}, issn = {{0305-0041}}, journal = {{Mathematical Proceedings of the Cambridge Philosophical Society}}, pages = {{235--268}}, title = {{{Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group}}}, doi = {{10.1017/s0305004100071577}}, year = {{1993}}, }