On Structure Constants of $sl(2)$ Theories

Structure constants of minimal conformal theories are reconsidered. It is shown that {\it ratios} of structure constants of spin zero fields of a non-diagonal theory over the same evaluated in the diagonal theory are given by a simple expression in terms of the components of the eigenvectors of the adjacency matrix of the corresponding Dynkin diagram. This is proved by inspection, which leads us to carefully determine the {\it signs} of the structure constants that had not all appeared in the former works on the subject. We also present a proof relying on the consideration of lattice correlation functions and speculate on the extension of these identities to more complicated theories.