  We point out certain symmetry induced constraints on topological order in
Mott Insulators (quantum magnets with an odd number of spin $\tfrac{1}{2}$ per
unit cell). We show, for example, that the double semion topological order is
incompatible with time reversal and translation symmetry in Mott insulators.
This sharpens the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem for 2D quantum
magnets, which guarantees that a fully symmetric gapped Mott insulator must be
topologically ordered, but is silent on which topological order is permitted.
An application of our result is the Kagome lattice quantum antiferromagnet
where recent numerical calculations of entanglement entropy indicate a ground
state compatible with either toric code or double semion topological order. Our
result rules out the latter possibility.
