  To an RCFT corresponds two combinatorial structures: the amplitude of a torus
(the 1-loop partition function of a closed string, sometimes called a modular
invariant), and a representation of the fusion ring (called a NIM-rep or
equivalently a fusion graph, and closely related to the 1-loop partition
function of an open string). In this paper we develop some basic theory of
NIM-reps, obtain several new NIM-rep classifications, and compare them with the
corresponding modular invariant classifications. Among other things, we make
the following fairly disturbing observation: there are infinitely many (WZW)
modular invariants which do not correspond to any NIM-rep. The resolution could
be that those modular invariants are physically sick. Is classifying modular
invariants really the right thing to do? For current algebras, the answer seems
to be: Usually but not always. For finite groups a la
Dijkgraaf-Vafa-Verlinde-Verlinde, the answer seems to be: Rarely.
