  The concept of the Schwinger Representation of a finite or compact simple Lie
group is set up as a multiplicity-free direct sum of all the unitary
irreducible representations of the group. This is abstracted from the
properties of the Schwinger oscillator construction for SU(2), and its
relevance in several quantum mechanical contexts is highlighted. The Schwinger
representations for $SU(2), SO(3)$ and SU(n) for all $n$ are constructed via
specific carrier spaces and group actions. In the SU(2) case connections to the
oscillator construction and to Majorana's theorem on pure states for any spin
are worked out. The role of the Schwinger Representation in setting up the
Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group is
brought out.
