Calculation Of Single Particle Isoscalar Factor For O(5)(?)O(3)

In this thesis, a recursive method for construction of symmetric irreducible representations of O(2l+1) in the O(2l+1)  O(3) basis for identical boson systems is proposed. The formalism is realized based on the group chain U(2l+1)  U(2l-1) U(2), of which the symmetric irreducible representations are simply reducible. In this paper, we focus on the l=2 case to show the basis vectors of symmetric irreducible representations of O(5) that are constructed in the O1(3)  U(1) basis. A three-term relation in determining the expansion coefficients of the O(5)  O(3) basis vectors in terms of those of the O1(3) U(1) is used, for which a Mathematica code is compiled. According to the formula for evaluating the simple isoscalar factors of O(5)  O(3), coefficients of one-particle fractional parentage of O(5)  O(3) with multiplicity are calculated by using the expansion coefficients of O(5)  O(3) in terms of O1(3)  U(1) and reduced matrix elements of O1(3)  U(1). Orthonormal relations of these coefficients are also numerically verified. The advantages of this method lies in the fact that large number of calculations due to the multiplicity can be avoid. Using the matrix representations of O(5)  O1(3)  U(1), one can then construct matrix representations of O(7)  O(3) similarly. A similar recursive procedure to construct basis vectors of Sp(2j+ 1)  O(3) in terms of those of Sp(2j+1)  Sp(2j-1)  U(1) can also be established accordingly.