  The paper presents the group theory of best localized and symmetry-adapted
Wannier functions in a crystal of any given space group G or magnetic group M.
Provided that the calculated band structure of the considered material is given
and that the symmetry of the Bloch functions at all the points of symmetry in
the Brillouin zone is known, the paper details whether or not the Bloch
functions of particular energy bands can be unitarily transformed into best
localized Wannier functions symmetry-adapted to the space group G, to the
magnetic group M, or to a subgroup of G or M. In this context, the paper
considers usual as well as spin-dependent Wannier functions, the latter
representing the most general definition of Wannier functions. The presented
group theory is a review of the theory published by one of the authors in
several former papers and is independent of any physical model of magnetism or
superconductivity. However, it is suggested to interpret the special symmetry
of the best localized Wannier functions in the framework of a nonadiabatic
extension of the Heisenberg model, the nonadiabatic Heisenberg model. On the
basis of the symmetry of the Wannier functions, this model of strongly
correlated localized electrons makes clear predictions whether or not the
system can possess superconducting or magnetic eigenstates.
