  We utilize a diagrammatic notation for invariant tensors to construct the
Young projection operators for the irreducible representations of the unitary
group U(n), prove their uniqueness, idempotency, and orthogonality, and
rederive the formula for their dimensions. We show that all U(n) invariant
scalars (3n-j coefficients) can be constructed and evaluated diagrammatically
from these U(n) Young projection operators. We prove that the values of all
U(n) 3n-j coefficients are proportional to the dimension of the maximal
representation in the coefficient, with the proportionality factor fully
determined by its S[k] symmetric group value. We also derive a family of new
sum rules for the 3-j and 6-j coefficients, and discuss relations that follow
from the negative dimensionality theorem.
