  For simply laced $SU(3)$ graphs we offer a geometric understanding of the
path creation and annihilation operators for $SU(3)$ in terms of creation and
annihilation of sequences of three vertices forming triangular cells or
collapsed triangular cells. We prove that the space of paths of a given length
can be decomposed as a direct sum of orthogonal sub-spaces constructed by
recurrent applications of the path creation operator on subspaces of essential
paths of shorter length.
