We display the reduction of the pseudo-orthogonal group O(2, 1) with respect to a noncompact O(1, 1) basis. After the explicit solution is obtained, we rederive the results using the method of master analytic representations.
The representation theory of the groups 80(5), SO(4, 1), SO(6) and 80(5, 1) is studied using the method of Master Analytic Representations (MAR). It is shown that a single analytic expression for the matrix elements of the generators of 8O(n + 1) and S0(n, 1) in an S0(n) basis yields all the unitary representations (for n = 4,5) and that the compact and non-compact groups have essentially the same analytic representation. Once the MAR of a group is worked out, the search for the unitary irreducible representations is reduced to a purely arithmetic operation. The utmost care has been exercised to conduct the discussions at an elementary level: knowledge of simple angular momentum theory is the only prerequisite.
An algebraic tabulation is made of the Clebsch-Gordan (CG) coefficients of SU3 which occur in the reduction into irreducible representations of the direct product (λ, μ)⊗ (1, 1) of irreducible representations of SU3. Full explanation is made of the method of handling the complications associated with the possible double occurrence of the representation (λ, μ) itself in the direct product. The phase convention employed is an explicitly stated generalization of the well-known Condon and Shortley phase convention for SU2. The relationship of the CG coefficients associated with the direct product (1, 1)⊗ (λ, μ) to those coefficients already mentioned is also exhibited.
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