Denominator function for canonical SU(3) tensor operators
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 G/sup t//sub q/, 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 x 3 array under row interchange, column interchange, and transposition (the group of 72 operations leaving a 3 x 3 determinant invariant). Remarkably, the zeros of the general G/sup t//sub q/ 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.
- Research Organization:
- Duke University, Durham, North Carolina 27706 and Los Alamos National Laboratory, Theoretical Division, Los Alamos, New Mexico 87545
- DOE Contract Number:
- W-7405-ENG-36
- OSTI ID:
- 5768112
- Journal Information:
- J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 26:7; ISSN JMAPA
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
658000 -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANGULAR MOMENTUM
CALCULATION METHODS
CLEBSCH-GORDAN COEFFICIENTS
COORDINATES
DIAGRAMS
FUNCTIONS
GROUP THEORY
IRREDUCIBLE REPRESENTATIONS
LIE GROUPS
MATHEMATICAL OPERATORS
MATHEMATICS
MATRIX ELEMENTS
MULTIPLICITY
POLYNOMIALS
RACAH COEFFICIENTS
SU GROUPS
SU-3 GROUPS
SYMMETRY
SYMMETRY GROUPS
TENSORS
WEIGHTING FUNCTIONS
WIGNER COEFFICIENTS
YOUNG DIAGRAM