  We analyze a canonical construction of group-invariant CR Mappings between
hyperquadrics due to D'Angelo. Given source hyperquadric of $Q(1,1)$, we
determine the signature of the target hyperquadric for all finite subgroups of
$SU(1,1)$. We also extend combinatorial results proven by Loehr, Warrington,
and Wilf on determinants of sparse circulant determinants. We apply these
results to study CR mappings invariant under finite subgroups of $U(1,1)$.
