We obtain the most general forms of rank-2 and rank-3 tensors allowed by the crystal symmetries of the honeycomb lattice of edge-sharing octahedra for crystals belonging to different crystallographic point groups, including the monoclinic point group $2/m$ and the trigonal (or rhombohedral) point group $bar{3}$. Our results are relevant for two-dimensional materials, such as $alpha$-RuCl$_3$, CrI$_3$, and the honeycomb iridates. We focus on the magnetic-field-dependent thermal conductivity tensor $kappa_{ij}(mathbf{H})$, which describes a systems longitudinal and thermal Hall responses, for the cases when the magnetic field is applied along high-symmetry directions, perpendicular to the plane and in the plane. We highlight some unexpected results, such as the equality of fully-longitudinal components to partially-transverse components in rank-3 tensors for systems with three-fold rotational symmetry, and make testable predictions for the thermal conductivity tensor.
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