  A parameterization of the density operator, a coherence vector
representation, which uses a basis of orthogonal, traceless, Hermitian matrices
is discussed. Using this parameterization we find the region of permissible
vectors which represent a density operator. The inequalities which specify the
region are shown to involve the Casimir invariants of the group. In particular
cases, this allows the determination of degeneracies in the spectrum of the
operator. The identification of the Casimir invariants also provides a method
of constructing quantities which are invariant under {\it local} unitary
operations. Several examples are given which illustrate the constraints
provided by the positivity requirements and the utility of the coherence vector
parameterization.
