  The construction of gauge invariant states of SU(3) lattice gauge theories
has garnered new interest in recent years, but implementing them is complicated
by the need for SU(3) Clebsch-Gordon coefficients. In the loop-string-hadron
(LSH) approach to lattice gauge theories, the elementary excitations are
strictly gauge invariant, and constructing the basis requires no knowledge of
Clebsch-Gordon coefficients. Originally developed for SU(2), the LSH
formulation was recently generalized to SU(3), but limited to one spatial
dimension. In this work, we generalize the LSH approach to constructing the
basis of SU(3) gauge invariant states at a trivalent vertex - the essential
building block to multidimensional space. A direct generalization from the
SU(2) vertex yields a legitimate basis; however, in certain sectors of the
Hilbert space, the naive LSH basis vectors so defined suffer from being
nonorthogonal. The issues with orthogonality are directly related to the
`missing label' or `outer multiplicity' problem associated with SU(3) tensor
products, and may also be phrased in terms of Littlewood-Richardson
coefficients or the need for a `seventh Casimir' operator. The states that are
unaffected by the problem are orthonormalized in closed form. For the sectors
that are afflicted, we discuss the nonorthogonal bases and their
orthogonalization. A few candidates for seventh Casimir operators are readily
constructed from the suite of LSH gauge-singlet operators. The diagonalization
of a seventh Casimir represents one prescriptive solution towards obtaining a
complete orthonormal basis, but a closed-form general solution remains to be
found.
