Imposing det E > 0 in discrete quantum gravity

We point out that the inequality det E > 0 distinguishes the kinematical phase space of canonical connection gravity from that of a gauge field theory, and characterize the eigen-vectors with positive, negative and zero-eigenvalue of the corresponding quantum operator in a lattice-discretized version of the theory. The diagonalization ofˆdet E is simplified by classifying its eigenvectors according to the irreducible representations of the octagonal group.