  The "band representation", formulated by Zak in the 80's, is widely used in
recent studies of topological phase of material. However, either the basis of
band representation contains an error in its definition or the transformation
properties employed is missing phase terms. This manuscript present a clear
definition of the basis and an alternative derivation of analytical
transformation properties of the band representation. It identifies explicitly
what occurs at high symmetry point on the surface of Brillouin zone and allows
decomposition of band representation into irreducible representations of the
space group anywhere in the representation domain of the Brillouin zone. Two
different Fourier transform conventions in defining the band representation
induced from the same localised Wannier functions are discussed and identified
as related by a simple gauge transformation. The tight binding band
representation are used as basis in deriving the tight binding Hamiltonian
using group theoretical technique. With the analytical tools afforded by the
representation matrices of the band representations, a clear methodology is
developed to obtain band connectivity in the representation domain and
establish which does not have an atomic limit. Elementary band representations
is shown to be fully connected in the atomic limit. In particular, the band
representation induced from spin-ful ${\rm p}_z$ orbital on Wyckoff position 2c
of honeycomb lattice has a connectivity with atomic limit and the topological
nature of the band in graphene can arises from such connection and spin orbit
interaction induced gap at K point. Combined with symmetry compliant tight
binding Hamiltonian, one can examine the homeomorphic transformation of bands
induced by changes in the parameters in the model and establish if band
inversion has occurred.
