IMPLICATIONS OF AN ARITHMETICAL SYMMETRY OF THE COMMUTANT FOR MODULARINVARIANTS

We point out the existence of an arithmetical symmetry for the commuta nt of the modular matrices S and T. This symmetry holds for all affine simple Lie algebras at all levels and implies the equality of certain coefficients in any modular invariant. Particularizing to SU(3)k, we classify the modular invariant partition functions when k + 3 is an in teger coprime with 6 and when it is a power of either 2 or 3. Our resu lts imply that no detailed knowledge of the commutant is needed to und ertake a classification of all modular invariants.