Tensor and spin representations of SO(4) and discrete quantum gravity

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.