  Hexagonal lattice systems (e.g. triangular, honeycomb, kagome) possess a
multidimensional irreducible representation corresponding to $d_{x^2-y^2}$ and
$d_{xy}$ symmetry. Consequently, various unconventional phases that combine
these $d$-wave representations can occur, and in so doing may break
time-reversal and spin rotation symmetries. We show that hexagonal lattice
systems with extended repulsive interactions can exhibit instabilities in the
particle-hole channel to phases with either $d_{x^2-y^2}+d_{xy}$ or $d+id$
symmetry. When lattice translational symmetry is preserved, the phase
corresponds to nematic order in the spin-channel with broken time-reversal
symmetry, known as the $\beta$ phase. On the other hand, lattice translation
symmetry can be broken, resulting in various $d_{x^2-y^2}+d_{xy}$ density wave
orders. In the weak-coupling limit, when the Fermi surface lies close to a van
Hove singularity, instabilities of both types are obtained in a controlled
fashion.
