On the structure of the canonical tensor operators in the unitary groups. I. An extension of the pattern calculus rules and the canonical splitting in U(3)

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Abstract
The structure of the totally symmetric unit tensor operators (and their conjugates) in U(n) is examined from the viewpoint of the pattern calculus and the factorization lemma. The geometrical properties of the arrow patterns of the fundamental projective (tensor) operators are demonstrated to be the origin of the existence of simple structural expressions for a class of reduced matrix elements of the totally symmetric unit projective operators. An extension of the pattern calculus rules is given whereby these matrix elements can be written out directly. This class of reduced matrix elements is sufficient to permit the construction of the general totally symmetric unit tensor operator. The canonical splitting of the multiplicity in U(3) is similarly shown to be implied uniquely by the geometrical properties of the arrow patterns of the fundamental projective operators and their conjugates. This fact is used to construct explicitly the class of U(3) unit tensor operators having maximal null space. Explicit expressions for a large class of Racah coefficients are also given, and the implications of their limit properties discussed.