  In this paper we further analyze modular invariants for subfactors, in
particular the structure of the chiral induced systems of M-M morphisms. The
relative braiding between the chiral systems restricts to a proper braiding on
their ``ambichiral'' intersection, and we show that the ambichiral braiding is
non-degenerate if the original braiding of the N-N morphisms is. Moreover, in
this case the dimensions of the irreducible representations of the chiral
fusion rule algebras are given by the chiral branching coefficients which
describe the ambichiral contribution in the irreducible decomposition of
alpha-induced sectors. We show that modular invariants come along naturally
with several non-negative integer valued matrix representations of the original
N-N Verlinde fusion rule algebra, and we completely determine their
decomposition into its characters. Finally the theory is illustrated by various
examples, including the treatment of all SU(2)_k modular invariants, some SU(3)
conformal inclusions and the chiral conformal Ising model.
