  We consider sl(2) minimal conformal field theories on a cylinder from a
lattice perspective. To each allowed one-dimensional configuration path of the
A_L Restricted Solid-on-Solid (RSOS) models we associate a physical state |h>
and a monomial in a finite fermionic algebra. The orthonormal states produced
by the action of these monomials on the primary states generate finite Virasoro
modules with dimensions given by the finitized Virasoro characters
$\chi^{(N)}_h(q)$. These finitized characters are the generating functions for
the double row transfer matrix spectra of the critical RSOS models. We argue
that a general energy-preserving bijection exists between the one-dimensional
configuration paths and the eigenstates of these transfer matrices and exhibit
this bijection for the critical and tricritical Ising models in the vacuum
sector. Our results extend to Z_{L-1} parafermion models by duality.
