On the denominator function for canonical SU(3) tensor operators

Abstract

The definition of a canonical unit SU(3) tensor operator is given in terms of its characteristic null space as determined by group-theoretic properties of the intertwining number. This definition is shown to imply the canonical splitting conditions used in earlier work for the explicit and unique (up to ± phases) construction of all SU(3) WCG coefficients (Wigner–Clebsch–Gordan). Using this construction, an explicit SU(3)-invariant denominator function characterizing completely the canonically defined WCG coefficients is obtained. It is shown that this denominator function (squared) is a product of linear factors which may be obtained explicitly from the characteristic null space times a ratio of polynomials. These polynomials, denoted Gtq, are defined over three (shift) parameters and three barycentric coordinates. The properties of these polynomials (hence, of the corresponding invariant denominator function) are developed in detail: These include a derivation of their degree, symmetries, and zeros. The symmetries are those induced on the shift parameters and barycentric coordinates by the transformations of a 3×3 array under row interchange, column interchange, and transposition (the group of 72 operations leaving a 3×3 determinant invariant). Remarkably, the zeros of the general Gtq polynomial are in position and multiplicity exactly those of the SU(3) weight space associated with irreducible representation [q−1,t−1,0]. The results obtained are an essential step in the derivation of a fully explicit and comprehensible algebraic expression for all SU(3) WCG coefficients.