Sum rules for the matrices of the generators of SU(3) in an SO(3) basis

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Abstract
We consider various sum rules for the semireduced [i.e., reduced with respect to SO(3)] matrix elements of the generators of SU(3) in a basis of an irreducible representation [pq] corresponding to the group reduction SU(3) ⊆SO(3) ⊆SO(2). We use basis states which diagonalize an additional labeling operator K, but avoid their explicit construction. We build all the needed operators from the two independent SU(3) vector operators X and V, where X= (L,Q) is made of the SU(3) generators and V= (VL, VQ) is defined in terms of them. First we obtain an analytical formula for the linear sum rule satisfied by the diagonal semireduced matrix elements of Q. Then, from the set of quadratic equations fulfilled by the semireduced matrix elements of Q and VQ, we obtain explicit expressions for the quadratic sum rules satisfied by these quantities. All the above‐mentioned sum rules are independent of the selection made for K. When K is defined as the third order operator L.VL, we show that a relation between some nondiagonal matrix elements of Q and VQ exists enabling the determination of a k‐weighted quadratic sum rule for the semireduced matrix elements of Q. As by‐products of the preceding results we obtain general formulas for the eigenvalues k of K for all the values of L whose multiplicity does not exceed 3, and we show that we are able to compute analytically the individual matrix elements of Q for not too high dimensionalities by working out the case of the irreducible representation [10,5].