  We study the $SU(N)$ self-dual Chern-Simons-Higgs systems with adjoint matter
coupling, and show that the sixth order self-dual potential has $p(N)$ gauge
inequivalent degenerate minima, where $p(N)$ is the number of partitions of
$N$. We compute the masses of the gauge and scalar excitations in these
different vacua, revealing an intricate mass structure which reflects the
self-dual nature of the model.
