  Traditional machine learning methods applied to the material sciences have
often predicted invariant, scalar properties of material systems to great
effect. Newer, coordinate equivariant models promise to provide a coordinate
system dependent output in a well defined manner, but recent applications often
neglect a direct prediction of directional (i.e. coordinate system dependent)
quantities and instead are used to predict still just invariant quantities.
This component-wise prediction of tensorial properties is achieved by
decomposing tensors into harmonic subspaces via a \textit{tensor spherical
harmonic decomposition}, by which we may also associate arbitrary tensors with
the irreducible representations of the rotation group. This essentially allows
us to read off tensors component-wise from the output representations of these
equivariant models. In this work, we present results for the prediction of
various material property tensors directly from crystalline structures. Namely,
given some material's crystalline structure, we may predict tensor components
of dielectric, piezoelectric, and elasticity tensors directly from the output
of a $SE(3)$ equivariant model.
