  We investigate a specific class of CV modules for $\mathfrak{sl}_3$ and
establish an exact sequence for these modules. Utilizing dimension arguments,
we demonstrate that this module is isomorphic to the fusion product of
irreducible modules, thereby offering a new proof of the conjecture regarding
the independence of fusion products from parameters. By analyzing the
filtration of the kernel within the exact sequence, we derive the graded
character formula for fusion product modules. Moreover, we leverage the graded
character to deduce the algebraic characterization of the Littlewood-Richardson
(LR) coefficients and present an alternative proof of the saturation theorem in
type $A_2$.
