  Many rotational invariants for crystal structure representations have been
used to describe the structure-property relationship by machine learning. The
machine learning interatomic potential (MLIP) is one of the applications of
rotational invariants, which provides the relationship between the energy and
the crystal structure. Since the MLIP requires the highest accuracy among
machine learning estimations of the structure-property relationship, the
enumeration of rotational invariants is useful for constructing MLIPs with the
desired accuracy. In this study, we introduce high-order linearly independent
rotational invariants up to the sixth order based on spherical harmonics and
apply them to linearized MLIPs for elemental aluminum. A set of rotational
invariants is derived by the general process of reducing the Kronecker products
of irreducible representations (Irreps) for the SO(3) group using a
group-theoretical projector method. A high predictive power for a wide range of
structures is accomplished by using high-order invariants with low-order
invariants equivalent to pair and angular structural features.
