We write the partition function for a lattice gauge theory exactly in terms of unconstrained variables and show that in the calculation for the effective potential, which involves uniform field configurations, only the singlet representation is relevant. We further show that this implies that the phase diagrams of the (an)isotropic SU(2) theory and the (an)isotropic U(1) theory in any dimension are identical, within this approximation, up to a re-evaluation of the numerical values of the coupling constants at the transitions. Corrections to the mean field approximation that lie within the singlet representation will not see the difference between the two either. Only nonuniform field configurations, that can be sensitive to higher dimensional representations for Yang--Mills fields, will be able to probe the difference between them.