  We show how to use quantum mechanics on the group manifold U(N) as a tool for
problems in U(N) representation theory. The quantum mechanics reduces to free
fermions on the circle, which in the large N limit become relativistic. The
theory can be bosonized giving the Das-Jevicki-Sakita collective field theory.
The formalism is particularly suited to problems involving tensor product
multiplicity (Littlewood-Richardson) coefficients. As examples, we discuss the
partition function of two-dimensional Yang-Mills theory on the sphere, and the
zero magnetic field limit of D-dimensional Eguchi-Kawai Yang-Mills theory. We
give the leading O(N^0) solution of the latter theory, using a method which
allows computing corrections. Largely (but not completely) superseded by
hep-th/9311130.
