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We investigate a new class of scalar multi-galileon models, which is notincluded in the commonly admitted general formulation of generalizedmulti-galileons. The Lagrangians of this class of models, some of them havingalready been introduced in previous works, are specific to multi-galileontheories, and vanish in the single galileon case. We examine them in details,discussing in particular some hidden symmetry properties which can be madeexplicit by adding total derivatives to these Lagrangians. These propertiesallow us to describe the possible dynamics for these new Lagrangians in thecase of multi-galileons in the fundamental representation of a SO(N) and SU(N)global symmetry group, as well as in the adjoint representation of a SU(N)global symmetry group. We perform in parallel an exhaustive examination of someof these models, finding a complete agreement with the dynamics obtained usingthe symmetry properties. Finally, we conclude by discussing what could be themost general multi-galileon theory, as well as the link between scalar andvector multi-galileon models.

New terms for scalar multi-Galileon models and application to SO(N) and SU(N) group representations