Conditions for symmetry reduction of polysymplectic and polycosymplectic structures

For Hamiltonian field theories on polysymplectic manifolds with a symmetry group action and a momentum map, we explore the redundancy in a set of necessary conditions that has appeared in the literature, for a generalized version of the Marsden-Weinstein symmetry reduction theorem. Next, we prove a necessary and sufficient condition for polycosymplectic reduction. We relate polycosymplectic reduction in a one-to-one way to the reduction of an associated larger polysymplectic manifold. Throughout the paper, we provide examples and discuss special cases.