  Starting from a detailed analysis of the structure of pathspaces of the
${\cal A}$-fusion graphs and the corresponding irreducible Virasoro algebra
quotients $V(c,h)$ for the ($2,q$ odd) models, we introduce the notion of an
admissible pathspace representation. The pathspaces ${\cal P_A}$ over the
${\cal A}$-Graphs are isomorphic to the pathspaces over Coxeter $A$-graphs that
appear in FB models. We give explicit construction algorithms for admissible
representations. From the finitedimensional results of these algorithms we
derive a decomposition of $V(c,h)$ into its positive and negative definite
subspaces w.r.t. the Shapovalov form and the corresponding signature
characters. Finally, we treat the Virasoro operation on the lattice induced by
admissible representations adopting a particle point of view. We use this
analysis to decompose the Virasoro algebra generators themselves. This
decomposition also takes the nonunitarity of the ($2,q$) models into account.
