  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 system's 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.
