The relationship of the hyperspherical harmonics to SO(3), SO(4) and orientation distribution functions

The expansion of an orientation distribution function as a linear combination of the hyperspherical harmonics suggests that the analysis of crystallographic orientation information may be performed entirely in the axis-angle parameterization. Practical implementation of this requires an understanding of the properties of the hyperspherical harmonics. An addition theorem for the hyperspherical harmonics and an explicit formula for the relevant irreducible representatives of SO(4) are provided. The addition theorem is useful for performing convolutions of orientation distribution functions, while the irreducible representatives enable the construction of symmetric hyperspherical harmonics consistent with the crystal and sample symmetries.