  Starting from the defining transformations of complex matrices for the SO(4)
group, we construct the fundamental representation and the tensor and spinor
representations of the group SO(4). Given the commutation relations for the
corresponding algebra, the unitary representations of the group in terms of the
generalized Euler angles are constructed. These mathematical results help us to
a more complete description of the Barrett-Crane model in Quantum Gravity. In
particular a complete realization of the weight function for the partition
function is given and a new geometrical itnerpretation of the asymptotic limit
for the Regge action is presented.
