On the Representations of the Semisimple Lie Groups. I. The Explicit Construction of Invariants for the Unimodular Unitary Group in N Dimensions

It is intended in the present series of papers to discuss explicit constructive determinations of the representations of the semisimple Lie groups SUn by an extension of the Racah-Wigner techniques developed for the two-dimensional unimodular unitary group (SU2). The present paper defines, and explicitly determines, a symmetric vector-coupling coefficient for the group SUn. These coefficients are utilized to construct a series of canonical invariants for SUn, of which the first I2 is the familiar Casimir invariant, and it is proved (by construction) that these invariants form a complete system of independent invariants suitable for uniquely labeling the irreducible inequivalent representations of SUn.