Starting from a detailed analysis of the structure of path spaces of the
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 path space representation. The path spaces
over the
graphs are isomorphic to the path spaces over Coxeter A graphs that appear in FB models. We give explicit construction algorithms for admissible representations. From the finite-dimensional 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 into account the nonunitarity of the (2, q) models.
