Journal of Statistical Physics, Vol. 82, Nos. 3/4, 1996. p. 633-655.
We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube, and the Wu-Kadanoff-Wegner duality between the cube- and vertex-type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which corresponds to the three-dimensional Baxter-Bazhanov model. The vertex-type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. We write down the symmetry relations of the weight functions under the actions of the symmetry group G of the cube. The
six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected with the six spaces in which the vertex-type tetrahedron equation is defined.