Construction of Basis Vectors For Symmetric Irreducible Representations of O(5) supset O(3)

TL;DR

This paper introduces a recursive method to construct symmetric irreducible representations of the group O(5) within the O(3) basis, providing explicit formulas and matrix representations useful for boson systems.

Contribution

A novel recursive formalism for constructing symmetric irreducible representations of O(2l+1) in the O(2l-1) x U(1) basis, including explicit matrix and Wigner coefficient formulas for O(5).

Findings

Constructed basis vectors for O(5) in the O(3) basis.

Derived explicit formulas for elementary Wigner coefficients.

Presented a three-term relation for expansion coefficients.

Abstract

A recursive method for construction of symmetric irreducible representations of O(2l+1) in the O(2l + 1) supset O(3) basis for identical boson systems is proposed. The formalism is realized based on the group chain U(2l + 1) supset U(2l- 1) x U(2), of which the symmetric irreducible representations are simply reducible. The basis vectors of the O(2l+1) supset O(2l-1) x U(1) can easily be constructed from those of U(2l + 1) supset U(2l-1) x U(2) supset O(2l-1) x U(1) with no boson pairs, from which one can construct symmetric irreducible representations of O(2l+1) in the O(2l-1) x U(1) basis when all symmetric irreducible representations of O(2l-1) are known. As a starting point, basis vectors of symmetric irreducible representations of O(5) are constructed in the O1(3) x U(1) basis. Matrix representations of O(5) supset O1(3) x U(1), together with the elementary Wigner coefficients, are presented. After the angular momentum projection, a three-term relation in determining the expansion coefficients of the O(5) supset O(3) basis vectors in terms of those of the O1(3) x U(1) is derived. The eigenvectors of the projection matrix with zero eigenvalues constructed according to the three-term relation completely determine the basis vectors of O(5) supset O(3). Formulae for evaluating the elementary Wigner coefficients of O(5) supset O(3) are derived explicitly. Analytical expressions of some elementary Wigner coefficients of O(5) supset O(3) for the coupling (tau, 0) x (1,0) with resultant angular momentum quantum number L = 2 tau+ 2 - k for k = 0, 2, 3,...,6 with a multiplicity 2 case for k = 6 are presented.