  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
