Generation of basis vectors for magnetic structures and displacement modes

Increasing attention is being focussed on the use of symmetry-adapted functions to describe magnetic structures, structural distortions, and incommensurate crystallography. Though the calculation of such functions is well developed, significant difficulties can arise, such as the generation of too many or too few basis functions to minimally span the linear vector space. We present an elegant solution to these difficulties using the concept of basis sets, and discuss previous work in this area using this concept. Further, we highlight the significance of unitary irreducible representations in this method, and provide the first validation that the irreducible representations of the crystallographic space groups tabulated by Kovalev are unitary