  We consider sl(2) minimal conformal field theories and the dual parafermion
models. Guided by results for the critical A_L Restricted Solid-on-Solid (RSOS)
models and its Virasoro modules expressed in terms of paths, we propose a
general level-by-level algorithm to build matrix representations of the
Virasoro generators and chiral vertex operators (CVOs). We implement our scheme
for the critical Ising, tricritical Ising, 3-state Potts and Yang-Lee theories
on a cylinder and confirm that it is consistent with the known two-point
functions for the CVOs and energy-momentum tensor. Our algorithm employs a
distinguished basis which we call the L_1-basis. We relate the states of this
canonical basis level-by-level to orthonormalized Virasoro states.
