Symmetry is one of the most generic and useful concepts in physics and chemistry, often leading to conservation laws and selection rules. For example, symmetry considerations have been used to predict selection rules for transitions in atoms, molecules, and solids. Floquet systems also demonstrate a variety of symmetries which are spatiotemporal (i.e. dynamical symmetries (DSs)). However, the derivation of selection rules from DSs has so far been limited to several ad hoc cases. A general theory for deducing the impact of DSs in physical systems has not been formulated yet. Here we explore symmetries exhibited in Floquet systems using group theory, and discover novel DSs and selection rules. We derive the constraints on a general systems temporal evolution, and selection rules that are imposed by the DSs. As an example, we apply the theory to harmonic generation, and derive tables linking (2+1)D and (3+1)D DSs of the driving laser and medium to allowed harmonic emission and its polarization. We identify several new symmetries and selection rules, including an elliptical DS that leads to production of elliptically polarized harmonics where all the harmonics have the same ellipticity, and selection rules that have no explanation based on currently known conservation laws. We expect the theory to be useful for manipulating the harmonic spectrum, and for ultrafast spectroscopy. Furthermore, the presented Floquet group theory should be useful in various other systems, e.g., Floquet topological insulators and photonic lattices, possibly yielding formal and general classification of symmetry and topological properties.
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