In this paper basis states and generator matrix elements are given for the degenerate representations \((a,0)\) and \((0,b)\) of \(G_2 \supset SU(3)\), for which there is no missing label problem. In section 3 the generator matrix elements with respect to the \((a,0)\) basis states and the \((0,b)\) basis states and incidentally the normalization constants \(N_{p, q}\) in both cases are calculated. The authors show that for any compact Lie group the elementary unwanted states, and hence incompatibilities between fundamental basis states in the character generator, are all of degree 2. The necessary ingredients for the calculation of \(G_2 \supset SU(3)\) basis states and generator matrix elements between them are the \(G_2 \supset SU(3)\) branching rules generating function and the character generators for \(G_2\). The generating function for \(G_2 \supset SU(3)\) is (Gaskell et al. 1978): \[ G (A,B,P,Q) = {1 \over (1 - AP) (1 - AQ) (1 - BP) (1 - BQ)} \left[ {1 \over 1 - A} + {BPQ \over 1 - BPQ} \right]. \] The dummy variables \(A,B\) carry the \(G_2\) representation labels \(a,b\) as exponents and \(P, Q\) carry the \(SU(3)\) representation labels \(p,q\). If \(AP \sim \eta\), \(AQ \sim \zeta^*\), \(A \sim \theta\), \(BP \sim \lambda\), \(BQ \sim \nu^*\), \(BPQ \sim \alpha\) are interpreted as the highest states of any \(SU (3)\) representation contained in the fundamental \(G_2\) representation, then the highest states of any \(SU (3)\) representation contained in any \(G\) representation are given by the appropriate product powers of them.
The matrix elements of the eight \(SU (3)\) generators are well known (Gel’fand and Tsetlin 1950) what convinced the authors to calculate the generator matrix elements with respect to the \((a,0)\) basis states and the \((0,b)\) basis states and to determine the normalization constants \(N_{pq}\) in both cases as follows: \[ \begin{aligned} N_{pq} = & \left[ {(2a + 4)!(p + q + 2)! \over 2^{a - p - q} (a + 2)!p!q! (a - p - q)!(a + p + q + 4)!} \right]^{1/2}, \\ N_{pq} = & {1 \over (3^{2b - p - q})^{1/2} (p + 1)!} \left[ {(2b + 2)!(3b + 3)!(p + 1)!(q + 1)!(p + q + 2)! \over (b - p)!(b - q)!(b + p + 2)!(b + q + 2)!} \right]^{1/2} \cdot \\ & \cdot \left[ {1 \over (p + q - b)!(p + q + b + 3)!} \right]^{1/2}. \end{aligned} \] This paper may serve as a basis for further investigations of applying Lie groups in physics although the group-subgroup \(G_2 \supset SU (3)\) does not seem to be used directly in physics. However, the group chain \(SO (1) \supset G_2 \supset SO (3)\) was introduced by Racah (1951) for the study of atomic electrons. We quote Judd (1986) who stated: “The use of \(G_2\) is an important feature of Racah’s analysis and is largely responsible for the progress that has been made during the last decade in the analysis of actinide and rare-earth spectra.” – Final remark: This high quality paper is marred by a number of printer’s mistakes.