  We present a study of the properties of Bargmann Invariants (BI) and Null
Phase Curves (NPC) in the theory of the geometric phase for finite dimensional
systems. A recent suggestion to exploit the Majorana theorem on symmetric SU(2)
multispinors is combined with the Schwinger oscillator operator construction to
develop efficient operator based methods to handle these problems. The BI is
described using intrinsic unitary invariant angle parameters, whose algebraic
properties as functions of Hilbert space dimension are analysed using elegant
group theoretic methods. The BI-geometric phase connection, extended by the use
of NPC's, is explored in detail, and interesting new experiments in this
subject are pointed out.
