Generating functions for polynomial irreducible tensors

Abstract

A general method for constructing a generating function for all irreducible polynomial tensors, with respect to a given compact semisimple group, out of a given set of tensors is derived. The method is applied to the construction of polynomial bases for the IR’s of a group reduced according to a subgroup and to the finding of subgroup scalars in the enveloping algebra of a group. A number of examples are worked out.

Generator matrix elements for G2contains/implies SU(3): II. Generic representations

“…We will use "character states", for which the integrity basis is provided by the G 2 character generator (see I). It is best to start with the G 2 ⊃ SU(3) branching rules generating function (Gaskell et al 1978) G(A, B; P, Q) = [(1 − AP ) (1 − AQ) (1 − BP ) (1 − BQ)] −1 ×…”

“…We are using Dynkin's labelling of the G 2 fundamental representations: (1, 0) is the 14−plet and (0, 1) the septet. In I and in Gaskell et al (1978) that numbering is reversed.…”

Clebsch-Gordan coefficients for U(8)⊃O(8)⊃SU(3)

“…The first step consists in the construction of polynomials which are in the highest weight state of SO(3), i.e. M = L. For that, the elementary permissible diagrams (epd's) [28] have to be constructed, or in other words a complete set of basic couplings in terms of the boson creation operator b † m (m = 0, ±1) such that all states of highest weight for a fixed angular momentum can be obtained for a given total number of oscillation quanta N [29,30]. There are two epd's [29,30]…”

“…The next step is to construct the epd's (elementary couplings) of U (8). In [4] the following epd's (also denoted as integrity basis) where obtained using the method of generating functions (see Eq. (16) of Ref.…”

“…However, the construction of states is less known. The first attempts have been made in [4,5], though the SU (3) subgroup considered in [5] is not the color group and only symmetric irreps of U (8) where considered. Using a symmetric irrep in U (8) is a first valid step towards a more general construction of states and it will be followed also in this contribution.…”