The EPRL intertwiners and corrected partition function

Do the SU (2) intertwiners parametrize the space of the Engle, Pereira, Rovelli, Livine (EPRL) solutions to the simplicity constraint? What is the complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective in the general n-valent vertex case for the Barbero-Immirzi parameter less than 1. We find, however, that the EPRL map is not isometric. In the consequence, a partition function can be defined either using the EPRL intertwiners Hilbert product or the SU (2) intertwiners Hilbert product. We use the EPRL one and derive a new, complete formula for the partition function. Next, we view it in terms of the SU (2) intertwiners. The result, however, goes beyond the SU (2) spin-foam models' framework and the original EPRL proposal.